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 0444509313, 9780444509314, 9780080531441

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MEASUREMENT of the THERMODYNAMIC PROPERTIES of SINGLE PHASES E X P E R I M E N T A L TH E R M O D Y N A M I C S VOLUME VI IUPAC Chemical Data Series No. 40

EXPERIMENTAL THERMODYNAMICS SERIES: Calorimetry of Non-Reacting Systems EXPERIMENTAL THERMODYNAMICS, VOLUME I Edited by J.P. McCullough and D.W. Scott Butterworths, London, 1968. Experimental Thermodynamics of Non-Reacting Fluids EXPERIMENTAL THERMODYNAMICS, VOLUME II Edited by B. Le Neindre and B. Vodar Butterworths, London, 1975 Measurement of the Transport Properties of Fluids EXPERIMENTAL THERMODYNAMICS, VOLUME III Edited by W.A. Wakeham, A. Nagashima, and J.V. Sengers Blackwell Scientific Publications, Oxford, 1991 Solution Calorimetry EXPERIMENTAL THERMODYNAMICS, VOLUME IV Edited by K.N. Marsh and P.A.G. O’Hare Blackwell Scientific Publications, Oxford, 1994 Equations of State for Fluids and Fluid Mixtures EXPERIMENTAL THERMODYNAMICS, VOLUME V Edited by J.V. Sengers, R.F. Kayser, C.J. Peters, and H.J. White, Jr. Elsevier, Amsterdam, 2000 Measurement of the Thermodynamic Properties of Single Phases EXPERIMENTAL THERMODYNAMICS, VOLUME VI Edited by A.R.H. Goodwin, K.N. Marsh, W.A. Wakeham Elsevier, Amsterdam, 2003

International Union of Pure and Applied Chemistry Physical Chemistry Division Commission on Thermodynamics

MEASUREMENT of the THERMODYNAMIC PROPERTIES of SINGLE PHASES E X P E R I M EN T A L T H E R M O D Y N A M I C S VOLUME VI

EDITED BY A.R.H. GOODWIN Schlumberger-Doll Research, Ridgefield, CT, USA

K.N. MARSH University of Canterbury, Christchurch, New Zealand

W.A. WAKEHAM University of Southampton, Southampton, UK

2003 ELSEVIER Amsterdam – Boston – London – New York – Oxford – Paris San Diego – San Francisco – Singapore – Sydney – Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands # 2003 International Union of Pure and Applied Chemistry. All rights reserved. This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK; phone: (þ44) 1865 843830, fax: (þ44) 1865 853333, email: [email protected]. You may also complete your request on-line via the Elsevier Science homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’. In the U.S.A., users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (þ1) (978) 7508400, fax: (þ1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (þ44) 207 631 5555; fax: (þ44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without written permission of the Publisher. Address permission requests to: Elsevier’s Science & Technology Rights Department, at the phone, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnosis and drug dosages should be made. First Edition 2003 Library of Congress Cataloging in Publication Data Measurement of the thermodynamic properties of single phases / edited by A.R.H. Goodwin, K.N. Marsh, W.A. Wakeham.—1st ed. p. cm. – (Experimental thermodynamics ; v. 6) At head of title: International Union of Pure and Applied Chemistry, Physical Chemistry Division, Commission on Thermodynamics. Includes bibliographical references and index. ISBN 0-444-50931-3 (acid-free paper) 1. Calorimetry. 2. Thermodynamics–Measurement. I. Goodwin, A. R. H. II. Marsh, K. N. III. Wakeham, W. A. IV. International Union of Pure and Applied Chemistry. Commission on Thermodynamics. V. Series. QC291 .M43 2002 5360 .6–dc21

2002029851

British Library Cataloguing in Publication Data Measurement of the thermodynamic properties of single phases.—(Experimental thermodynamics ; v. 6) (IUPAC chemical data series ; no. 40) 1. Thermodynamics – Measurement I. Goodwin, A. R. H. II. Marsh, K. N. III. Wakeham, W. A. IV. International Union of Pure and Applied Chemistry 541.30 690 0287 ISBN: 0-444-50931-3 ? s

The paper used in this publication meets the requirements of ANSI/NISO Z39.48–1992 (Permanence of Paper). Printed in The Netherlands.

Dedicated to the Memory of J.V. NICHOLAS who died during the preparation of this volume

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vii CONTENTS List of Contributors Foreword Acknowledgments

xi xiii xv

1

Introduction

1

2

Temperature

7

3

2.1 2.2 2.3

Thermodynamic Origin of Temperature International Temperature Scales Realising the Unit 2.3.1 Water for Triple–Point Cells 2.3.2 Use of the Water Triple–Point Cell 2.3.3 The Ice Point 2.4 Fixed Points 2.4.1 Metal Freezing Points and Melting Points 2.4.2 The Gas Triple Points 2.4.3 Other Fixed Points 2.5 Platinum Resistance Thermometry 2.5.1 ITS-90 Reference Functions 2.5.2 ITS-90 Deviation Functions 2.5.3 Resistance Measurements 2.6 Radiation Thermometry 2.6.1 Radiation Temperature Scale 2.7 Cryogenic Thermometry 2.7.1 Vapour Pressure Thermometers 2.7.2 Gas Thermometers 2.7.3 Electrical Thermometers 2.8 ITS-90 and the Thermodynamic Scale 2.9 Temperatures below 1 K 2.10 Temperatures above 2000 K 2.11 Special Thermocouples

8 10 12 13 15 16 17 17 20 22 22 26 28 29 31 33 34 34 35 36 37 38 40 41

Pressure

43

3.1

45 46 47 50 53 55 56 57 58 59 61 64 68 68 69 72

3.2

3.3 3.4

Electronic Pressure Gauges 3.1.1 Physical Principle of Resistive Strain Gauge Pressure Transducers 3.1.2 Silicon Piezoresistive Micromachined Pressure Transducers 3.1.3 Resonant Pressure Sensors Based on Mechanical Vibrations 3.1.4 Pressure Sensors Based on Capacitance 3.1.5 Silicon Capacitive Pressure Transducers 3.1.6 Performances of Piezoresistive and Capacitive Pressure Sensors 3.1.7 Scaling Limits in Pressure Transducers Piston Gauges 3.2.1 Advances in Piston Gauge Manufacturing Technology 3.2.2 Characterization Techniques 3.2.3 Practical Developments in Piston Gauges Pressure Measurements in a Diamond Anvil Cell 3.3.1 X-rays 3.3.2 Optical Techniques Low Pressure

viii

Contents 3.4.1 3.4.2 3.4.3 3.4.4

4

Mixture Preparation and Sampling Hydrocarbon Reservoir Fluids 4.1

4.2

5

Preparation of Fluid Mixtures 4.1.1 Apparatus 4.1.2 Purity Considerations 4.1.3 Preparation of Gas Mixtures 4.1.4 Preparation of Liquid Mixtures Sampling Hydrocarbon Reservoir Fluids 4.2.1 Hydrocarbon Reservoir Fluids and Formations 4.2.2 Reservoir Fluid Sampling 4.2.3 Advances in Sampling Techniques

74 80 87 88 97 98 99 100 101 104 105 106 110 118

Density

125

5.1

127 128 131 140 149 150 158 168 169 170 174 175 179 179 185 185 186 190 191 191 192 200 205 208 208 210 219

5.2 5.3 5.4

5.5

5.6

5.7

6

Interferometric Liquid-Column Manometers Piston Gauges Static Expansion Pressure Generators Pressure Transducers

Hydrostatic Balance Densimeters with Magnetic Suspension Couplings 5.1.1 Three Main Types of Buoyancy Densimeters 5.1.2 Two-Sinker Densimeter 5.1.3 Single-Sinker Densimeter Measurement of Density with Vibrating Bodies 5.2.1 Vibrating-Wire Densimeters 5.2.2 Vibrating-Tube Densimeters Bellows Volumetry 5.3.1 Theory 5.3.2 Experimental Piezometer 5.4.1 Fixed Volume Devices 5.4.2 Variable Volume Devices 5.4.3 Expansion Devices Isochoric Methods 5.5.1 Principles 5.5.2 Experimental 5.5.3 Coupled Isochoric and Burnett Methods Absolute Density Standards 5.6.1 Definition of the Absolute Density in SI Units 5.6.2 Absolute Density Measurements of Solids 5.6.3 Method of Density Comparison 5.6.4 Absolute Density Measurements of Standard Liquids In Situ Density Measurements 5.7.1 Terminology 5.7.2 In situ Densimeters 5.7.3 Applications

Speed of Sound

237

6.1

238 240 280 287 291

Measurement of the Speed of Sound 6.1.1 Gases 6.1.2 Liquids 6.1.3 Solids 6.1.4 Transducers

Contents 6.2

7

325

7.1

327 328 331 335 335 336 337 339 340 343 344 345 347 348 350 353 358 358 359 361 362 368 369 372 374

7.3

7.4

Flow Calorimetry 7.1.1 Plug-in Gas Flow Calorimeters 7.1.2 Enthalpy-Increment Calorimeters 7.1.3 Measurements in the Vicinity of the Liquid-Gas Critical Temperature 7.1.4 Enthalpy of Solution of Carbon Dioxide in Alkanolamines AC Calorimetry 7.2.1 Principle of the ac Calorimeter 7.2.2 Conditions for Heat Capacity Measurement 7.2.3 Joule-Heating ac Calorimeters 7.2.4 Light-Irradiation ac Calorimeters 7.2.5 AC Calorimetric Methods for Liquids 7.2.6 Heat Capacity Spectroscopy 7.2.7 Temperature-modulated Calorimetry Differential Scanning Calorimetry 7.3.1 Power-Compensated DSC 7.3.2 Heat Flux DSC 7.3.3 Adiabatic DSC 7.3.4 Single Cell DSC 7.3.5 Temperature Modulated DSC 7.3.6 Specialised DSC 7.3.7 Determination of Physical Properties with DSC Nano-Calorimetry 7.4.1 Micro Electro-Mechanical Systems (MEMS) 7.4.2 Nanocalorimeters as Sensors 7.4.3 Nanocalorimeters for Material Properties Determination

Properties of Mixing

387

8.1

388

8.2 8.3

9

300 302 310 311

Calorimetry

7.2

8

Thermodynamic Properties from the Speed of Sound 6.2.1 Gases 6.2.2 Liquids 6.2.3 Solids

ix

Properties of Gas Mixtures 8.1.1 Equation of State of Gases at Low and Moderate Densities. Virial Equation of State 8.1.2 Experimental Methods 8.1.3 (p, r, T) Data Reduction 8.1.4 Bibliography of Experimental Measurements on Mixtures Mixtures of Liquids Experimental Techniques for the Determination of Energetic Properties of Inorganic Solids 8.3.1 Calorimetric Methods 8.3.2 Electrochemical Methods 8.3.3 Vapour Pressure Methods 8.3.4 Some Words on Measurement Uncertainty

389 391 397 403 404 408 408 414 419 423

Relative Permittivity and Refractive Index

433

9.1

434 435

Relative Permittivity 9.1.1 Conductivity and Dielectric Losses

x

Contents

9.2 9.3

10

9.1.2 Bridge Methods and Resonance Methods. 9.1.3 Designs for Capacitors 9.1.4 Measurements 9.1.5 Concluding Remarks Refractive Index Relative Permittivities of Electrolytes 9.3.1 Fundamental Aspects of Dielectric Theory 9.3.2 Coaxial-line Techniques ( f  20 GHz) 9.3.3 Waveguide Methods (5  f/GHz  100) 9.3.4 Free-Space Methods ( f > 60 GHz) 9.3.5 Data Analysis

437 438 448 451 452 455 457 460 463 466 468

Extreme Conditions

475

10.1 Low Temperatures 10.1.1 Adiabatic Calorimeter as an Ultra-Low-Frequency Spectrometer 10.1.2 Calorimetry at Very Low Temperatures 10.2 High Temperatures 10.2.1 Resistive Pulse Heating 10.2.2 Laser Pulse Heating 10.2.3 Levitation Techniques 10.2.4 Electromagnetic Levitation 10.2.5 Future Directions 10.3 Molten Metals 10.3.1 Containers 10.3.2 Pyrometry 10.3.3 Calorimetry 10.3.4 Thermal Expansion and Density 10.3.5 Surface Tension

475 476 483 488 489 497 501 502 502 504 505 509 514 520 526

Index

537

xi LIST OF CONTRIBUTORS

. . . . . . . . . . . . . . . . . . . .

E.H. Abramson (USA) A. Aguiar-Ricardo (Portugal) J. Barthel (Germany) R. Buchner (Germany) P.M. Claudy (France) C.M.M. Duarte (Portugal) I. Egry (Germany) C.D. Ehrlich (USA) A.M. de Figureueiredo Palavra (Portugal) K. Fujii (Japan) A.R.H. Goodwin (USA) I. Hatta (Japan) A.W. van Herwaarden (Netherlands) M. Hiza (USA) J.C. Holste (USA) R. Kleinrahm (Germany) A. Kurkjian (USA) H.W. Lo¨sch (Germany) K.N. Marsh (New Zealand) V. Majer (France)

. . . . . . . . . . . . . . . . . . . . . .

M.R. Moldover (USA) J.V. Nicholas (New Zealand) J. Nighswander (UK) M. Nunes da Ponte (Portugal) J.B. Ott (USA) A.A.H. Pa´dua (France) G. Pottlacher (Austria) R. Puers (Belgium) J.W. Schmidt (USA) J.W. Stansfeld (UK) S. Stølen (Norway) H. Suga (Japan) J. Suski (France) C.M. Sutton (New Zealand) J.P.M. Trusler (UK) W. Wagner (Germany) W.A. Wakeham (UK) J.T.R. Watson (UK) D.R. White (New Zealand) L.A. Woolf (Australia) C.J. Wormald (UK) H. Yao (Japan)

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xiii FOREWORD For several decades the Commission on Thermodynamics and Thermochemistry, now named the Commission on Thermodynamics, of the International Union of Pure and Applied Chemistry (IUPAC) has pursued an active role in the definition and maintenance of standards in the fields encompassed by its title. This role includes, but is not limited to, the establishment and surveillance of international pressure and temperature scales, recommendation for calorimetric procedures, the selection and evaluation of reference standards for thermodynamic-measurement techniques of all types and the standardization of nomenclature and symbols in chemical thermodynamics. Through its subcommittees on Thermodynamic Data and on Transport Properties, the Commission has encouraged and is also responsible for the dissemination of evaluated thermodynamic data of the fluid state and representations of the transport properties of fluids. Over the years the Commission has also been responsible for the production of several texts. The most recent, titled Chemical Thermodynamics, edited by T.M. Letcher, is one of eleven monographs in the IUPAC series ‘Chemistry for the 21st Century’ that are intended to demonstrate the importance of chemistry in current areas of scientific research and industrial processes of economic importance. Under the auspices of the Commission three series of texts that summarized the state of knowledge with respect to experimental techniques in thermodynamics and thermochemistry have been produced. The first series consisted of two monographs titled Experimental Thermochemistry. Volume I, which appeared in 1956 (Interscience Publishers, Inc. New York) under the editorship of F.D. Rossini while Volume II, published in 1962 (Interscience-Wiley, New York), was edited H.A. Skinner. The text entitled Combustion Calorimetry, edited by S. Sunner and M. Ma˚nsson (Pergamon Press, Oxford) was issued in 1979 in the series ‘Experimental Chemical Thermodynamics’ and was an update of the material covered in Experimental Thermochemistry, Volumes I and II. In the third series, ‘Experimental Thermodynamics’, five volumes have already been produced. The first three books dealt with the thermodynamic measurements of non-reacting systems. The first volume, Calorimetry of Non-Reacting Systems, edited by J.P. McCullough and D.W. Scott (Butterworths, London), was published in 1968. The second volume Experimental Thermodynamics of Non-Reacting Systems, edited by B. LeNeindre and B. Vodar, published in 1975 (Butterworths, London), was concerned with the measurements of a broader class of thermodynamic properties over a wide range of temperature and pressure. The third volume, Measurement of the Transport Properties of Fluids, edited by W.A. Wakeham, N. Nagashima, and J.V. Sengers, published in 1991 (Blackwell Science Publications, Oxford), covered the measurement of properties characteristic of the relaxation of a fluid from a nonequilibrium state, the transport properties. The fourth volume, Solution Calorimetry, edited by K.N. Marsh and P.A.G. O’Hare (Blackwell Science Publications, Oxford), differed in that calorimetry of both non-reacting and reacting systems were considered. The fifth volume, Equations of State for Fluids and Fluid Mixtures, edited by J.V.

xiv

Foreword

Sengers, R.F. Kayser, C.J. Peters, and H.J. White Jr., published in 2000 (Elsevier Science, New York) presented the theoretical basis for equations of state of both fluids and fluid mixtures along with practical uses of each equation type. The concept of updating Experimental Thermodynamics of Non-Reacting Systems, Volume II in this series was conceived by the Commission on Thermodynamics in 1997 and adopted as an official IUPAC project 120/16/97 with the intention of continuing the theme of the first two books in the series by considering non-reacting systems and providing an upto-date presentation of thermodynamic measurements with a combination of strong practical bias and working equations. The Commission determined that two volumes were required: this volume and Volume VII the Measurement of the Thermodynamic Properties of Multiple Phases, edited by R.D. Weir and T.W. de Loos. Both are intended as an asset to industry as well as general academics. The editors of this volume were assigned the task of assembling an international team of distinguished experimentalists to describe the current state of development of the techniques of measurement of the thermodynamic quantities of single phases consisting of both pure fluids and compositionally complex mixtures over a wide range of conditions. Naturally, many of the author team was found among the Commission, but considerable effort was expended to locate appropriate expertise from elsewhere. The volume presented here fulfils admirably the brief given to the editors and contains a valuable summary of a large variety of experimental techniques applicable over a wide range of thermodynamic states with an emphasis on the precision and accuracy of the results obtained which is so much part of the remit of the Commission on Thermodynamics itself. Appreciation and gratitude are owed to the contributors for their willing and enthusiastic cooperation in this venture and also to the editors, faced with the task (common to all editors of co-operative efforts) of constructing a coherent whole from the independent contributions. The readers interested in the art of measurements, and in particular engaged in the measurement of thermodynamic properties, will find the material contained in this volume, which covers the literature to 2002, of considerable value as well as providing guidance for the development of new and more accurate techniques. For those who make use of literature data, yet may have little experimental expertise in the field, the volume should permit the objective judgment of quality so vital to scientific and engineering progress. At the end of 2001 IUPAC will dissolve all Commissions thus Volume VI and VII will be the last texts published by the Commission in this series. However, the work undertaken by the Commission will continue with the International Association of Chemical Thermodynamics. Ron D. Weir Chair Commission I.2, Thermodynamics International Union of Pure and Applied Chemistry

xv ACKNOWLEDGMENTS We indebted to the authors and grateful to past and present members of the IUPAC Commission I.2 on Thermodynamics for their unwaving support for this project and Christine Nichol, Ruba Vigneswaran, and Juliet Wang who faithfully produced a copy of this manuscript from the authors contributions and editors notes. Some of the illustrations that appear in this volume have been published elsewhere. The present authors, editors and publishers are grateful to all those concerned in the original publications for permission to use their Figurers again. Some of the Figurers have been edited for consistency of presentation. The following publishers and corporations have given permission for the use for the original illustrations. Academic Press for Figures 5.3 and 5.4 taken from The Journal of Chemical Thermodynamics 29, 1137, 1997; Figure 5.5 taken from The Journal of Chemical Thermodynamics 29, 1157, 1997; Figures 5.6, 5.7 and 5.9 taken from The Journal of Chemical Thermodynamics 30, 1571, 1998; Figure 5.23 taken from The Journal of Chemical Thermodynamics 25, 831, 1993; Figures 6.2 and 6.16 taken from The Journal of Chemical Thermodynamics 24, 531, 1992; Figure 6.5 taken from The Journal of Chemical Thermodynamics 17, 549, 1985; Figure 6.6 taken from The Journal of Chemical Thermodynamics. 12, 1121, 1980; Figures 7.1 and 7.2 taken from The Journal of Chemical Thermodynamics 29, 701, 1997; Figures 7.4 and 7.5 taken from The Journal of Chemical Thermodynamics 22, 269, 1990; Figure 8.1 taken from The Journal of Chemical Thermodynamics 23, 281, 1991; taken from The Journal of Chemical Thermodynamics 2, 43, 1970; Figure 8.5 from figure The Journal of Chemical Thermodynamics 6, 973, 1974; Figure 9.7 taken from The Journal of Chemical Thermodynamics 8, 709, 1976; Figure 9.8 taken from The Journal of Chemical Thermodynamics 21, 1023, 1989; Figure 10.9 taken from The Journal of Chemical Thermodynamics 19, 1275, 1987. Acoustical Society of America for Figure 6.7 taken from Journal of the Acoustical Society of America 29, 1074, 1953; Figures 6.14 and 6.15 taken from Journal of the Acoustical Society of America 93, 276, 1993. American Chemical Society for Figure 5.19 taken from Journal of Chemical Engineering Data 42, 738, 1997; Figure 7.6 taken from Industrial and Engineering Chemistry Research 1998, 37, 4137; Figure 8.3 taken from Journal of Physical Chemistry 100, 18839, 1996. American Institute of Physics for Figures 6.4 and 6.18 taken from Review of Scientific Instruments 62, 2213, 1991; Figure 6.10 taken from Review of Scientific Instruments 56, 470, 1985; Figure 9.1 taken from Review of Scientific Instruments 50, 1309, 1979; Figure 9.2 taken from Review of Scientific Instruments 41, 1087, 1970; Figure 9.3 taken from Review of Scientific Instruments 41, 1087, 1970; Figure 9.4 taken from Review of Scientific Instruments 67, 4294, 1996; Figure 9.5 taken from Review of Scientific Instruments 71, 2914, 2000; Figure 9.9 taken from Review of Scientific Instruments 62, 1411, 1991; Figures 9.16 and 9.17 taken from Journal of

xvi

Acknowledgments

Chemical Physics 107, 5319, 1997; Figure 10.5 taken from Review of Scientific Instruments 55, 1310, 1984. American Society of Mechanical Engineers for Figure 6.9 taken from Proceedings of the Fifth Symposium on Thermophysical Properties, p. 107, 1970. ASM International for Figure 10.23 taken from International Materials Review 38, 157, 1993. Austrian Fonds zur Fo¨rderung der Wissenschaftlichen Forschung Projekt P 12775 – PHY and P15055 for Figure 10.12. Bureau International des Poids et Mesures for Figure 3.17 taken from Metrologia, 36, 613, 1999; Figure 3.19 taken from Metrologia, 36, 517, 1999; Figure 3.20 taken from Metrologia 36, 617, 1999; Figures 5.28, 5.29, 5.30 and 5.31 taken from Metrologia 36, 455, 1999. Chemical Society of Japan for Figure 10.1 Bulletin of the Chemical Society of Japan 38, 1000, 1965; Figures 10.6, 10.7 and 10.8 taken from Bulletin of the Chemical Society of Japan 50, 1702, 1977. CRI, Evanston, IL, USA, for Figure 10.12. Degranges et Huot, Paris, France for Figure 3.9. DH Instruments, Incorporated, Tempe, Az, USA for Figure 3.11. Der Bunsengesellschaft for Figure 9.12 taken from Berichte der Bunsengesellschaft fu¨r Physikalische Chemie, 101, 1509, 1997; Figure 9.14 taken from Berichte der Bunsengesellschaft fu¨r Physikalische Chemie 95, 853, 1991. Elsevier Science for Figures 5.11, 5.12, and 5.13 taken from Fluid Phase Equilibria 181, 147, 2001; Figure 5.14 taken from Fluid Phase Equilibria 150–151, 815, 1998; Figure 5.24 taken from Fluid Phase Equilibria 109, 265, 1995; Figures 7.8, 7.9, and 7.10 taken from Thermochimica Acta 304/305, 157, 1997; Figure 7.23 taken from Sensors and Actuators B2, 223, 1990;. Figure 8.2 taken from Fluid Phase Equilibria 41, 141, 1988; Figure 10.3 taken from Journal of Non-Crystalline Solids 16, 171, 1974. E.I. du Pont de Nemours and Company, Wilmington, DE, USA, for Figure 7.17. European Chemical Societies for Figure 9.18 taken from Physical Chemistry Chemical Physics 1, 105, 1999. Institute for Electrical and Electronic Engineers, Inc., for Figure 5.32 taken from IEEE Transactions of Instrumentation and Measurement 50, 616, 2001. Institute of Electrical Engineers of Japan for Figure 7.25 taken from the Proceedings of the 10th International Conference on Solid-State Sensors and Actuators June 7–10 1999. Sendai, Japan, 1999. Institute of Pure and Applied Physics, Japan, for Figure 7.11 Japanese Journal of Applied Physics 11, 1995, 1981; Figure 7.12 taken from Japanese Journal of Applied Physics 38, 945, 1999. International Steel Institute of Japan for Figure 10.20 taken from Handbook of Physico-Chemical Properties at High Temperatures, Y. Kawai, Y. Shiraishi eds., 1988.

Acknowledgments xvii International Union of Pure and Applied Chemistry for Figure 9.10 taken from Pure and Applied Chemistry 62, 2287, 1990. IOP Publishing Limited for Figure 9.15 taken from Measurement Science and Technology 6, 1201, 1995; Figure 10.4 taken from Journal of Physics E: Scientific Instruments 17, 1054, 1984. Mettler Toledo Gmbh, Greifensee, Switzerland, for Figure 7.16. National Institute of Standards and Technology for Figure 5.26 taken from Journal of Research of the National Bureau of Standards C 76, 11, 1972. NMi van Swinden Laboratorium for Figures 6.3 and 6.17 taken from Proceedings of the 7th International Symposium on Temperature and Thermal measurements in Industry and Science Delft, 1999. O¨GI, Leoben, Austria, for Figure 10.10 Pion LTD for Figure 5.25 taken from High-Temperature High-Pressure 31, 91, 1999. Plenum Publishing Corporation for Figures 5.15 and 5.16 taken from International Journal of Thermophysics 18, 719, 1997; Figures 5.17 and 5.18 taken from International Journal of Thermophysics 14, 1021, 1993; Figure 5.22 taken from International Journal of Thermophysics 7, 1077, 1986; Figure 6.13 taken from International Journal of Thermophysics 22, 427, 2001; Figure 9.6 International Journal of Thermophysics in press 2001. Ruska Instruments Corporation, Houston, Tx, USA for Figure 3.10. Schlumberger Limited, New York, USA, for Figures 3.3, 4.5, 4.9, 4.12, and 4.13. Solartron Mobrey Limited, Slough, U.K. for Figure 5.43. TA Instruments, New Castle, DE, USA, for Figure 7.26. The Royal Society for Figure 6.8 taken from the Proceedings of the Royal Society A368, 125, 1979.

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1

Introduction A.R.H. GOODWIN Schlumberger-Doll Research Ridgefield, Connecticut, USA W.A. WAKEHAM University of Southampton Southampton, UK K.N. MARSH University of Canterbury Canterbury, NZ

Measurement of the Thermodynamic Properties of Single Phases A.R.H. Goodwin, K.N. Marsh, W.A. Wakeham (Editors) # 2003 International Union of Pure and Applied Chemistry. All rights reserved

1

INTRODUCTION

Commission 1.2 of the International Union of Pure and Applied Chemistry (IUPAC) has been responsible for numerous texts in fields encompassed by its title. The most recent, Chemical Thermodynamics [1], is one of eleven monographs in the IUPAC series Chemistry for the 21st Century [2] and demonstrates the importance of chemical thermodynamics in current areas of industrial significance and scientific research. Chemical Thermodynamics covered separation technology (including membrane techniques, solvent extraction and super-critical properties), colloids and microemulsions, electrolytes, adsorption, high temperature effects, nuclear applications, dielectrics, theoretical and quantum chemistry, polymer science, microgravity, new materials (including amorphous materials and glasses), enzyme catalyzed reactions, molecular and cell biology, medicine and pharmacy, food science, protein folding and design, as well as petroleum chemistry. Some experimental aspects were included in this volume. The Commission has been responsible within IUPAC for a series of monographs summarizing the state of knowledge with regard to experimental techniques in thermodynamics and thermochemistry. Two monographs [3,4] reporting methods in thermochemistry formed the first series and were updated in a text concerned with combustion calorimetry [5]. The present volume is the sixth in a series of volumes [6–10] issued by IUPAC Commission I.2 on Thermodynamics concerned with methods of measuring the thermophysical properties of substances. The first volume was concerned with calorimetry of non-reacting systems [6], while the fourth monograph the calorimetry of reacting fluids. Volume three [8] continued the theme of non-reacting systems with measurements of properties characteristic of the relaxation of a fluid from a non-equilbrium state: the transport properties. The fifth volume [10] presented the theoretical basis for equations of state of both fluids and fluid mixtures along with practical uses of each equation type. The second volume in the series, [7] focused on measurements of a broader class of thermodynamic properties and state variables over a wide range of temperature and pressure including techniques with industrial applications for chemically nonreacting systems including: pressure, p; temperature, T; (p, V, T), where V is volume; sound speed; refractive index; relative permittivity; vapor pressure; critical state; solubility; phase equilibria; molten salts; fluid metals; surface tension; adsorption; and equations of state. A very considerable effort was expended to collect a diligent set of authors who contributed to this seminal work. The entire book had a gestation period of about 10 years, drew upon 53 authors and comprised 1318 pages. It is perhaps not surprising that many of the techniques described therein remain valid today. However, there have been technical developments completely independent of thermodynamics that have profoundly affected some earlier measurement techniques and other approaches hitherto not possible have become feasible. The technical developments to which we refer include digital electronics, which has provided numerous signal analysis tools, along with materials, such as quartz and silicon. The measurement of frequency, which can now be determined with a relative precision of

Introduction

3

1011, has transformed the determination of thermophysical properties. Indeed, researchers appear to prefer methods of determining thermodynamic properties that rely on the measurement of frequency. This philosophy is reflected in the content of this volume. At the same time, the industrial demand for thermodynamic properties of fluids and solids is insatiable, requiring more properties, often of higher precision, of an ever-expanding number of materials over a wide range of temperature and pressure. However, economic reality dictates that it is not practical for any single industrial organization to maintain the breath of expertise required to perform these measurements. At the same time, in the academic research environment, the measurement of thermophysical properties has become an expensive and relatively unpopular activity. Hence, the capability to perform these measurements throughout the world has decreased, in favour of simulation techniques. IUPAC Commission I.2 therefore felt that it was essential in this environment to establish a source book of the current state-of-the-art for both present and future generations of experimentalists. A survey sponsored by IUPAC Commission I.2 of 70 individuals active in experimental thermodynamics within academia, industry, and government confirmed this preliminary view. Having had a firm endorsement of the idea from practitioners in the field the Commission formally initiated IUPAC project 120/16/ 97 in 1997. The intention was to continue the theme of non-reacting systems and to provide an up-to-date presentation of thermodynamic measurement techniques with a strong practical bias and full working equations. The success of Volume II [7] implied that a new book should also be comprehensive, covering the thermodynamic properties of solids, liquids, and gases as well as the equilibrium between them. On the other hand, many techniques described in the first book remain valid and have undergone only minor development rather than major change and in some cases have been reported in other monographs [11,12]. The Commission therefore sought to develop a text complementary to, rather than as a replacement for reference [7], concentrating on new developments and significant enhancements of earlier techniques. Even so the material that falls into these categories are more than can be accommodated in one modern monograph. Thus, the Commission determined that the total volume of material should be divided into two parts. Of course, the separation could have been performed in many ways but, after much deliberation, the Commission decided to divide the material into two volumes, one describing the properties of single phases and the other multiple phases. This volume, Volume VI, covers experimental methods primarily for single phases while Volume VII, Measurement of the Thermodynamic Properties of Multiple Phases, edited by R.D. Weir and T.W. de Loos [13], is concerned with systems containing more than one phase. The principal purpose of both volumes is to serve as a guide to the scientist or technician who are contemplating measurements of the thermodynamic properties of fluids. Emphasis is placed on those methods for which good, theoretically-based, working equations are available or with the potential for industrial application. Additionally, the volumes will also be of interest to the data evaluator who needs to make an assessment of the reliability of experimental data obtained with specific

4

Introduction

techniques. General experimental methods, not described in either of the current volumes, can be found in reference [14]. The editors of the present volume were therefore assigned the task of assembling an international team of distinguished experimentalists to describe recent developments in the techniques for measurement of thermodynamic quantities for single phases consisting of both pure fluids and compositionally complex mixtures over a wide range of conditions. Many of the authors were found among the members of Commission I.2, but considerable effort was expended to locate appropriate expertise elsewhere. This volume fulfills the brief given to the editors by the Commission and contains a summary of a large variety of experimental techniques applicable over a wide range of thermodynamic states. The precision and accuracy of the results obtained from each method, which is so much part of the remit of the Commission on Thermodynamics itself, was regarded as an essential element of the descriptions. Throughout the text we have adopted the quantities, units and symbols of physical chemistry defined by IUPAC in the text commonly known as the Green Book [15]. We have also adopted the ISO guidelines for the expression of uncertainty [16] and vocabulary in metrology [17]. Values of the fundamental constants and atomic masses of the elements have been obtained from references [18] and [19] respectively. In the remainder of the introduction an overview of the chapters is presented. Although pressure and temperature measurements have been covered elsewhere they are fundamental to thermodynamic measurements. Since the publication of reference [7] the temperature scale has been updated from the International Practical Temperature Scale of 1968 (IPTS-68) to the International Temperature Scale of 1990 (ITS-90). Pressure measurement methods have enjoyed considerable enhancement from those reported in reference [7]. The new techniques, which often rely on devices constructed with silicon technology and the measurement of frequency, provide the capability to determine pressure in extreme conditions with harsh fluids. From a measurement perspective, pure fluids are far easier to handle than are multicomponent mixtures. Although measurements on pure fluids, particularly argon, are essential for validating new techniques and working equations, they have received rhetorical comments from theorists and industrialists – why are you still measuring the properties of argon? [20]. We have therefore included a chapter which covers the important area of mixture preparation and the acquisition of samples of natural fluids that are representative of the those found in sub-surface strata under extreme conditions. The methods for the determination of volume discussed in reference [7] have changed little so that we concentrate on density metrology in Chapter 5. The notable exception to this statement is the measurement of volume of a single silicon sphere, which is both new and essential to density measurement. In this chapter a section is devoted to absolute density standards, which have achieved unforeseen precision. Measurements of the density, described in Chapter 5, of gases, liquids, and solids were not covered in the first volume. Some entirely new techniques and the use of modern electronics (in some established approaches) have brought entirely new perspective to old measurement techniques. 4

Introduction

5

Acoustic measurements can be used to probe the thermophysical properties of systems to obtain both the speed and attenuation of sound. This is an example of a technique that was possible but not routinely performed at the time of writing of reference [7]. Acoustic measurements, which rely on the determination of frequency, can now be performed with extremely high precision. Sound speed measurements can be used to determine both equation of state and heat capacity information by integration of the sound speed. Intrinsically, this approach is preferred to that of differentiation of the (p, V, T) information to obtain heat capacities and other properties. Many instruments contributing to advances in calorimetery have been described recently in reference [9]. Thus, Chapter 7 focuses on instruments that were either not described in reference [9] or are more recent developments of those that appear in reference [9]. The chapter includes a description of differential scanning calorimetry (DSC), which has become a ubiquitous instrument in material science laboratories. As was remarked earlier, in many fundamental and industrial issues, the measurements on mixtures are the most important and so we devote a special chapter to the application of the techniques presented for pure fluids to mixtures. The same chapter considers specific methods that measure excess properties of mixtures directly. In the case of liquids, we give rather little attention to routine measurements of excess properties with commercial instruments because, in our view, the fundamental understanding gained from measurements of the excess properties has been rather disappointing relative to the effort expended on it. On the other hand, measurements of the properties of mixing of solids yield a plethora of thermodynamic properties of both fundamental and industrial significance. The calorimetry of mixing for aqueous systems was covered elsewhere [9]. For gases, we have included a bibliography of studies of mixtures in the gas phase since the last volume because there have been few. The measurement of the relative permittivity was given scant attention in Volume II [7] and there was no mention of electrolytes. Relative permittivity measurements are of fundamental importance and also industrially significant for both electrical and medical applications. The relative permittivity can also be used to determine the density of non-polar fluids, as a tool to detect phase boundaries and for the determination of virial coefficients. This is another measurement where the precision of the technique has been significantly improved by the measurement of frequency. These recent methods have found several industrial applications. The chapter on extreme conditions deals with materials of industrial significance at high temperature including molten metals and alloys. It also treats calorimetric measurements at low temperatures, not covered in Chapter 9.

References 1. Chemical Thermodynamics, T.M. Letcher, ed., For IUPAC, Blackwell Scientific Publications, Oxford, 2000. 2. Watkins, C.L., J. Chem. Educ. 77, 973, 2000.

5

6

Introduction

3. Experimental Thermochemistry, F.D. Rossini, ed., For IUPAC, Interscience, New York, 1956. 4. Experimental Thermochemistry, Vol II, H.A. Skinner, ed., For IUPAC, Interscience, New York, 1962. 5. Combustion Calorimetry, S. Sunner and M. Ma˚nsson eds., For IUPAC, Pergamon, Oxford, 1979. 6. Experimental Thermodynamics, Vol I, Calorimetry of Non-Reacting Systems, J.P. McCullough and D.W. Scott eds., For IUPAC, Butterworths, London, 1968. 7. Experimental Thermodynamics, Vol II, Experimental Thermodynamics of Non-Reacting Fluids, B. Le Neindre and B. Vodar eds., For IUPAC, Butterworths, London, 1975. 8. Experimental Thermodynamics, Vol III, Measurement of the Transport Properties of Fluids, W.A. Wakeham, A. Nagashima and J.V. Sengers eds., For IUPAC, Blackwell Scientific Publications, Oxford, 1991. 9. Experimental Thermodynamics, Vol IV, Solution Calorimetry, K.N. Marsh and P.A.G. O’Hare eds., For IUPAC, Blackwell Scientific Publications, Oxford, 1994. 10. Experimental Thermodynamics, Vol V, Equations of State for Fluids and Fluid Mixtures, Parts I and II, J.V. Sengers, R.F. Kayser, C.J. Peters and H.J. White, Jr. eds., For IUPAC, Elsevier, Amsterdam, 2000. 11. Specialist Periodical Reports, Chemical Thermodynamics, Vol 1, M.L. McGlashan, Senior Reporter, Chemical Society, London, 1973. 12. Specialist Periodical Reports, Chemical Thermodynamics, Vol 2, M.L. McGlashan, Senior Reporter, Chemical Society, London, 1973. 13. Experimental Thermodynamics, Vol VII, Measurement of the Thermodynamic Properties of Multiple Phases, R.D. Weir and T.W. de Loos eds., For IUPAC, Elsevier, Amsterdam, 2002. 14. The Measurement, Instrumentation, and Sensors Handbook, J.G. Webster, ed., CRC Press, Boca Raton, Florida, 1999. 15. Mills, I., Cvitasˇ , T., Homann, K., Kalley, N. and Kuchitsu, K., Quantities, Units and Symbols in Physical Chemistry, For IUPAC, Blackwell Science, Oxford, 1993. 16. Guide to the Expression of Uncertainty in Measurement, International Standards Organization, Geneva, Switzerland, 1995. 17. International Vocabulary of Basic and General Terms in Metrology, International Standards Organization, Geneva, Switzerland, 1993. 18. Mohr, P.J. and Taylor, B.N. J. Phys. Chem. Ref. Data 28, 1713, 1999. 19. Vocke, R.D. Pure. Appl. Chem. 8, 1593, 1999. 20. Mason, E.A., Private Communication.

6

2

Temperature J.V. NICHOLAS and D.R. WHITE Measurement Standards Laboratory of New Zealand Industrial Research Lower Hutt, New Zealand 2.1 Thermodynamic Origin of Temperature 2.2 International Temperature Scales 2.3 Realising the Unit 2.3.1 Water for Triple-Point Cells 2.3.2 Use of the Water Triple-Point Cell 2.3.3 The Ice Point 2.4 Fixed Points 2.4.1 Metal Freezing and Melting Points 2.4.2 The Gas Triple Points 2.4.3 Other Fixed Points 2.5 Platinum Resistance Thermometry 2.5.1 ITS-90 Reference Functions 2.5.2 ITS-90 Deviation Functions 2.5.3 Resistance Measurements 2.6 Radiation Thermometry 2.6.1 Radiation Temperature Scale 2.7 Cryogenic Thermometry 2.7.1 Vapour Pressure Thermometers 2.7.2 Gas Thermometers 2.7.3 Electrical Thermometers 2.8 ITS-90 and the Thermodynamic Scale 2.9 Temperatures below 1 K 2.10 Temperatures above 2000 K 2.11 Special Thermocouples

Measurement of the Thermodynamic Properties of Single Phases A.R.H. Goodwin, K.N. Marsh, W.A. Wakeham (Editors) # 2003 International Union of Pure and Applied Chemistry. All rights reserved

8 10 12 13 15 16 17 17 20 22 22 26 28 29 31 33 34 34 35 36 37 38 40 41

2

TEMPERATURE

The measurement of temperature is fundamental to thermodynamic measurements and, since publication of Experimental Thermodynamics Volume II, the temperature scale has been updated from the International Practical Temperature Scale of 1968 (IPTS-68) to the International Temperature Scale of 1990 (ITS-90). This chapter covers the implementation of ITS-90 and recent determinations of the fundamental fixed points, which are essential to the practical determination of temperature using secondary thermometers. In addition, the temperature scale at T < 1 K and T > 2000 K are discussed. The use of primary acoustic thermometry for the establishment of the fixed-point temperatures is described in Chapter 6.

2.1

Thermodynamic Origin of Temperature

The scientific meaning for temperature is at the heart of thermodynamics. It arises from the Zeroth Law of thermodynamics that states that if two systems are in thermal equilibrium and one of those systems is in thermal equilibrium with a third system, then all three systems are in thermal equilibrium with each other. Thus, temperature is the property of a system that conveys information about the thermal equilibrium of the system. The Zeroth Law only establishes equality of temperatures and permits the use of any single valued function as an empirical temperature scale. In order to establish a metric scale for temperature, one that allows meaningful ratios of temperature, the Second Law of thermodynamics is used to define an absolute temperature, T, by expressing the law as dS  dQ=T

ð2:1Þ

where dS is the change in entropy and dQ is the change in heat. There are other thermodynamically equivalent ways of defining the temperature scale as described in, for example, reference [1]. Equation (2.1) gives a metric temperature scale but requires, in addition, a definition of magnitude and sign in order to define the unit. To establish the thermodynamic temperature scale the Syste`me International d’unite´s (SI), defines the kelvin, symbol K, by fixing the temperature of the triple point of water, T(H2 O, s þ l þ g) ¼ 273:16 K. Over time a variety of other temperature scales have been developed but they are no longer useful for reporting scientific data. The exception is the Celsius temperature scale. The Celsius temperature, t, is related to the absolute temperature by t= C ¼ T=K  273:15

ð2:2Þ

and the unit is the degree Celsius, symbol 8C. On this scale the ice point is 0 8C and the triple point of water is 0.01 8C. 8

Temperature

9

In principle, any suitable thermodynamic equation may be used as the basis for a thermometer. However, with the exception of the radiation thermometers used at high temperatures, thermodynamic thermometers cannot achieve the highest precision desired, and are complex and time consuming to use. To overcome these difficulties, an International Temperature Scale, ITS, is defined by the Comite´ International des Poids et Mesures (CIPM) under the Convention du Me`tre, the founding treaty for the SI, and is regularly revised with the current version agreed to in 1990 and known as ITS-90 [2,3]. The ITS are empirical temperature scales giving a close approximation to the known thermodynamic scale, but are more precise and easier to use. All temperature measurements should be traceable to the current ITS. Some earlier ITS were known as International Practical Temperature Scales (IPTS). Because of the differences between the various temperature scales and because they have the same name for their units, it is often necessary to distinguish between scale temperature and thermodynamic temperature. The symbols T90 and t90 are used for the kelvin and Celsius temperatures on the current scale, ITS-90, and previous scales are similarly denoted, for example, on the International Practical Temperature Scale of 1968 (IPTS-68), T68 and t68 are used for the kelvin and Celsius temperatures. There are three provisos concerning the scientific use of ITS. Firstly, while the scale is more precise, it does not guarantee thermodynamic accuracy; it is very dependent on the accuracy of the thermodynamic data used to establish the scale as discussed in Section 2.8. For example, recent data indicates that near 300 K the ITS90 differs from the thermodynamic scale by about 5 mK. Secondly, ITS varies with time because it is updated approximately every 20 years. This means that older thermodynamic data may not be in agreement with recent data. For example, under the IPTS-68 the normal boiling point of water, T68(H2O, 1 þ g, p ¼ 0.101 325 MPa), was 373.15 K but under ITS-90 T90(H2O, 1 þ g, p ¼ 0.101 325 MPa) ¼ 373.124 K, a difference of 26 mK. Thirdly, the ITS-90 is not strictly single valued; it exhibits nonuniqueness because of both the way it is defined and the properties of real thermometers. For example, two laboratories’ temperature measurements may differ by as much as 2 mK around 400 K, yet both comply with the ITS-90, assuming other uncertainties are negligible. Therefore, at this level of accuracy, measured thermodynamic properties may not appear to be smooth functions. This chapter introduces high precision thermometry for those requiring a close match to the thermodynamic temperature. To achieve the highest accuracies close adherence to the published guidelines [2] is necessary. Lower accuracy thermometry is covered in other publications and guidelines [4–7]. Since it is not possible to cover all thermometry applications for all possible environments; in this chapter the emphasis is on making measurements traceable to the ITS-90. In particular, the limits on accuracy and precision are examined in detail. Unless otherwise stated, all uncertainties are reported as the standard uncertainty or one standard deviation. At the extremes of temperature, the use or ITS-90 may not always be appropriate because new techniques for realising the temperature scale are constantly developed. Extensions of thermometry to very high and very low temperatures are outlined. 9

10

2.2

Temperature

International Temperature Scales

ITS-90 covers the temperature range from 0.65 K up to the highest temperature practicably measurable in terms of the Planck radiation law discussed in Section 2.6. The official text for the ITS-90 is published by the Bureau International des Poids et Mesures (BIPM), and an English version is included in the Supplementary Information for ITS-90 [2], where more detailed and practical information is given. Figure 2.1 outlines the main features of ITS-90: the fixed points, the interpolating thermometers, and the ranges for which interpolation formulae are defined. There are three basic stages in establishing the scale. First, the fixed points, that is the melting-points, freezing-points and triple-points of various substances, are constructed in accordance with the BIPM Supplementary Information. Secondly, the readings of thermometers of approved types are determined at one or more fixed points. Finally, any unknown temperature is calculated from the thermometer readings by interpolation using the readings at the fixed points and the specified interpolation equations.

Figure 2.1 An outline of the main features of ITS-90.

10

Temperature

11

Fixed points are physical systems whose temperatures are determined by a physical process and are therefore universal and repeatable. The most successful systems for temperature references have been phase transitions involving major changes of state, for example, liquid to solid or vapour to liquid. Under the proper conditions, in a fixedpoint apparatus, the phase transition will occur at a single temperature determined by the properties of the substance used and not on the apparatus. As the change involves the enthalpy of a phase transition, good temperature stability is possible. When the fixed point apparatus is properly constructed, a small amount of heat transfer between the substance and its surroundings will not cause a temperature change in the substance during the phase transition. Triple-point systems of many substances make excellent fixed points since they represent an equilibrium between the three phases of the substance: solid, liquid and vapour, which occurs at a single temperature and pressure. Freezing temperatures of pure metals are also highly repeatable but exhibit a pressure dependence, which must be understood and controlled. Normal boiling points are no longer used for defining temperatures because of their very high dependence on the pressure. Table 2.1 lists the ITS-90 fixed points with their defined values. Section 2.4 examines the main types of fixed points. Table 2.1 The defining fixed points of the ITS-90. The reference resistance ratio Wr(T90) for SPRTs is defined in Section 2.5.1. Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

T90/K 3 to 5 13.8033 &17 &20:3 24.5561 54.3584 83.8058 234.3156 273.16 302.9146 429.7485 505.078 692.677 933.473 1234.93 1337.33 1357.77

t90/ 8C  270.15 to 268.15  259.3467 &  256:15 &  252:85  248.5939  218.7916  189.3442  38.8344 0.01 29.7646 156.5985 231.928 419.527 660.323 961.78 1064.18 1084.62

Substancea

Stateb

He

V

e-H2 e-H2 (or He) e-H2 (or He) Ne O2 Ar Hg H2 O Ga In Sn Zn Al Ag Au Cu

T V (or G) V (or G) T T T T T M F F F F F F F

a

Wr(T90)

0.001 190 07

0.008 449 74 0.091 718 04 0.215 859 75 0.844 142 11 1.000 000 00 1.118 138 89 1.609 801 85 1.892 797 68 2.568 917 30 3.376 008 60 4.286 420 53

All substances except 3He are of natural isotopic composition: e-H2 is hydrogen at the equilibrium concentration of the ortho- and para-molecular forms. b The symbols have the following meanings: for substance B, V:T(B, l þ g); T:T(B, s þ l þ g); M and F:T(B, s þ l, p ¼ 0:101 325 MPa); and G is a gas-thermometry measurement.

11

12

Temperature

Four classes of thermometers are used to establish the scale: vapour-pressure thermometers, gas thermometers, platinum resistance thermometers, and radiation thermometers. The class of platinum resistance thermometers, discussed in Section 2.5, is further split into three types: capsule thermometers for (13.8 to 430) K longstem thermometers for (84 to 933) K, and high-temperature thermometers for (273.15 to 1235) K. The ITS-90 and BIPM guidelines place specific restrictions on their construction, and a thermometer that satisfies them is denoted as a standard platinum resistance thermometer or SPRT. The radiation, gas and vapour pressure thermometers must be constructed according to established physical principles, but otherwise the ITS-90 and BIPM guidelines impose no other constraints. The radiation and gas-thermometry interpolation formulae are based on thermodynamic equations but are referenced to defined temperatures on the scale and have some constants defined. For vapourpressure and resistance thermometry, the equations are empirical. The main use for the radiation, gas and vapour-pressure thermometers is to transfer the scale to more convenient reference devices, such as standard lamps or rhodium-iron resistance thermometers, with which calibrations can be made more readily. Standard platinum resistance thermometers can be used directly or to calibrate a wide range of thermometers. The thermometers constructed and calibrated to the ITS-90 requirements establish the temperature scale to which temperature measurements should be traceable. The fixed points should not be considered any more fundamental than the scale thermometers, although they do provide convenient reference points for checking the performance of any thermometer. Because real thermometers define and hold the scale, it is not easy to accurately transfer temperature readings from one ITS to another. To do so three sets of data are required: the calibration constants of the old thermometer on the old ITS scale, the calibration constants of the new thermometer on the new scale, and a measurement comparison between the old and new thermometers. If the old and new thermometers are the same device, then the comparison is obviously not necessary. If the highest accuracy is not required then the BIPM publications [2,8] give recommended values for converting between the various versions of the scale back to that adopted in 1927. These conversions are for typical thermometers, usually the ones used to develop the scale.

2.3

Realising the Unit

The water triple point, which occurs at a single temperature and pressure when ice, water and water vapour are in thermal equilibrium with each other, is used to define the kelvin, the unit of thermodynamic temperature. In order to utilise this physical system as a precision temperature reference, special cells are constructed to allow immersion of thermometers as shown in Figure 2.2. The ice point, at 0.0 8C, is very near and closely allied to the water triple point. Historically the ice point was a defining point for many temperature scales until the more precise triple-point cells were developed. The nature of most thermometers is 12

Temperature

13

Figure 2.2 A triple point of water cell showing the frozen ice mantle and thermometer-well containing some water and a small sponge to protect the cell when the thermometer is inserted.

such that for the highest accuracy and confidence, regular checks against either an ice point or water triple point are essential. Such checks build confidence in an instrument and help establish an appropriate maintenance and calibration schedule. For this reason, the ice point still has an important role in thermometry, as it has an accuracy of better than 5 mK, is very easily set up, and very inexpensive.

2.3.1

Water for Triple-Point Cells

The SI definition of the kelvin does not specify the purity of the water required to realise the triple point of water. The Supplementary Information [2] specifies that the 13

14

Temperature

water should be of high purity and have substantially the isotopic composition of ocean water. Water purity is an important consideration in the manufacture and use of a triple-point cell. Some of the observed variable properties of the triple point of water appear to arise from impurities and these effects can be minimised with good procedures for freezing and using the cells. Most of the 0.01 K difference in temperature between the ice point, which is near T(H2O, s þ l, p ¼ 0.101 325 MPa), and the water triple point T(H2O, s þ l þ g), is due to the change in pressure from standard atmospheric pressure, 0.101 325 MPa, to the triple-point pressure, 611.66 Pa, which causes a 7.5 mK change in temperature. The other main contribution comes from dissolved air, which depresses the freezing point by a further 2.5 mK. Therefore, air is the main impurity to be removed from the water. Non-volatile impurities also alter the temperature of the triple point. The freezing-point depression constant of water is 1:86 K ? mol1 of impurity in 1 kg of water. With scrupulous attention to cleanliness, the impurity level in a triple-point cell can be readily controlled to achieve an uncertainty of better than 0.1 mK. This accuracy implies 3 h and allow several pre-heated thermometers to be calibrated. After calibrating each thermometer, the existence of the plateau should be confirmed by returning the monitoring thermometer to the well. In contrast to the melting curve, the plateau for freezing, shown in Figure 2.4, should have dT/dt closer to zero because crystallisation is a purification process with only the pure metal freezing. The temperature of the freeze drops rapidly near the end of the freeze as the volume of the remaining fluid decreases and the impurity concentration increases. The temperature of the freeze plateau should also be the same as that of the melt plateau, and any difference is an indication of the level of impurities arising from loss of integrity in the cell or of a poor thermal environment for the cell and thermometer. 19

20

Temperature

Because the freezing point is pressure sensitive, the hydrostatic head of the molten metal will increase the temperature in the thermometer well. A list of recommended correction factors for both ambient pressure and thermometer immersion depth are provided in Table 2.4. Corrections may also be needed if the inert buffer gas, used to reduce oxidation of the metal and crucible, is not at 0.101 325 MPa. For a sealed cell, this pressure cannot be measured directly. Unlike the other metal fixed points, the gallium point is realised as a melting point. The melt plateau is used because the volume expansion on freezing makes it difficult to obtain a successful freeze plateau. Gallium can be obtained as an extremely high purity metal with a relative impurity 660.323 8C, with d determined from calibration point 15.

28

Temperature

29

Wr(T90), is found. This is then substituted into the appropriate reference function to give T90. While at first sight all these equations may seem overly complex, the sub-ranges make the scale more practical for a user wishing to implement the temperature range of interest. For example, a user requiring temperature measurement from (0 to 100) 8C needs only two fixed points to cover the (0 to 156) 8C range. The maximum temperature to which the thermometer would be exposed is around 156 8C. IPTS-68 would have required three fixed points and exposure of the thermometer to temperatures around 420 8C. The use of overlapping ranges leads to a difficulty in that the calculated temperature depends on the sub-range chosen. However, the non-uniqueness arising from different deviation functions is no greater than that arising from different SPRTs. Below 660 8C the non-uniqueness of ITS-90 may lead to differences of about 2 mK, but more typically 0.5 mK, which for most purposes is negligible.

2.5.3

Resistance Measurements

Because the SPRT itself is a practical thermometer, it is useful if uncertainties better than 10 mK are sought. We outline here the essential features in the use of a SPRT; further details should be found in the manufacturer’s instruction book and the BIPM guidelines. Accurate measurement of the temperature depends on accurate resistance measurement, and resistance bridges and meters designed specifically for platinum resistance thermometry are readily available. They measure resistance in the range (0.25 to 100) O, and with a 4-lead measurement technique to eliminate the effect of lead resistance. The bridges are usually automatic and operate using low-frequency ac in the range (10 to 100) Hz or dc with current reversal to eliminate thermoelectric and other dc effects. The sensing current is typically about 1 mA, but to determine the self-heating in the SPRT, a variable current source operating from (1 to 10) mA is required. Currents up to 10 mA may be required for low-temperature thermometry and for high-temperature SPRTs because of the lower resistance of the SPRTs. The SPRT should be purchased with a calibration and its accuracy determined, before and after each use, with a water triple-point cell, or for less accurate work, a well-constructed ice point. A capsule-type SPRT, shown in Figure 2.6, is used for low temperatures down to 13.8 K. The platinum resistor, R(273.16 K)&25 O, is thermally connected to a platinum sheath and the system under investigation using helium (g) at p&30 kPa and T ¼ 293 K. The capsule should be immersed, with suitable grease, in a well within a copper block whose temperature is to be measured. The four short capsule lead-wires, which pass through a platinum-glass seal, are connected to longer insulated copper leads that are thermally anchored to prevent heat transfer to the capsule. The glass seal and the gas pressure limit the maximum operating temperature of capsule thermometers. For the highest accuracy work, the capsule should not be used at temperatures above 300 K, although some capsule 29

30

Temperature

Figure 2.6 Cross-section through a typical Rð293 KÞ&25 O capsule platinum resistance thermometer. The platinum sheath is about 5 mm in diameter and about 50 mm long.

constructions can be used up to 500 K if reduced accuracy and lifetime are acceptable. Higher-temperature SPRTs have a long stem so that the glass-to-metal seal is close to room temperature. The thermal expansion of the lead wires is greater than that of the sheath material and at high and low temperature the platinum sensing element moves within the sheath; this movement must be accommodated in the construction of the thermometer. Glass or quartz sheaths are used to lower the thermal conduction along the stem; however, radiation can be piped along the transparent sheath and disturb the thermal equilibrium. For example, incandescent room lighting can raise the apparent temperature of a water triple-point cell by 0.2 mK. To avoid radiation piping, the sheath is either sandblasted above the sensor region or coated with graphite paint before each critical measurement. Chemical changes of the platinum are an important consideration in the design of the higher-temperature SPRTs. A partial pressure of 2 kPa of oxygen is used within the sheath as this concentration is low enough to prevent excessive platinum oxidation but high enough to prevent impurity oxides from breaking down to metals, which can contaminate the platinum. Contamination from, and breakdown of, the supports or sheath also limit the upper temperature. For maximum stability SPRTs with mica supports are best limited to use below 450 8C. 30

Temperature

31

ITS-90 is the first scale to use high-temperature SPRTs, consequently these thermometers are still undergoing development, and the user will need to follow the manufacturer’s recommendations closely in order to obtain the best performance. Electrical leakage becomes a problem at very high temperatures, especially for ac measurements, and is usually evident from a frequency dependent resistance. Of particular concern is the porosity of the quartz sheath to some metal vapours that can contaminate the platinum wire. A sacrificial layer of platinum foil over the sensor end of the quartz sheath will prevent the contamination. Mechanical vibration can cause strain and work hardening of the platinum wire and hence increase the triple-point resistance, R(273.16 K). For example, a large knock on a SPRT has been known to cause errors of the order of 10 mK. Annealing the SPRT above 450 8C for several hours and slowly cooling to room temperature will usually restore the original resistance. Table 2.8 lists the recommended cooling rates for SPRTs. The relative repeatability of R(273.16 K) after annealing should be  5 ? 107 , as measured by a resistance bridge; that is a higher precision than the accuracy of the standard reference resistor against which the SPRT resistance is compared. Capsule SPRTs cannot be annealed and should not be used for long periods where there is any vibration, for example within a stirred fluid bath. Mechanical strain arising from rapid changes in thermal environment can also introduce significant changes in R(273.16 K) for SPRTs. Temperature changes experienced by the thermometer should be no more than 50 K ? min1 , which is often made more manageable with a separate pre-heating furnace. The immersion depth for long stem SPRTs is large, in part because of the precision required and in part because of the length of the sensing element. Adequate immersion depths are typically 0.2 m at room temperature and up to 0.3 m at temperatures greater than 200 8C. The magnitude of any immersion error can be assessed experimentally by measuring the temperature profile versus immersion depth while keeping the temperature of interest constant.

2.6

Radiation Thermometry

All objects emit electromagnetic radiation with a spectral distribution and intensity largely determined by the temperature of the object. Thus thermal radiation can be Table 2.8 A Typical cooling schedule for a Standard Platinium Resistance Thermometer (SPRT). The SPRT may be cooled gradually or allowed to anneal at the lowest temperature of each of the three highest ranges. Temperature range/ 8C

Cooling rate/K ? h1

960 850 630 450

25 100 400 3000

to to to to

850 630 450 20

31

32

Temperature

used for thermometry (see also section 10.3.2). To a lesser extent, the emitted radiation is also determined by the optical properties of the object. To measure temperature, a radiometer, which measures the amount of radiation emitted over a known spectral band, and knowledge of the emissivity of the surface are required. Because the object is external to the radiometer, the emitted radiation is often subject to interference that introduces systematic errors into the measurement. The main sources of interference, summarised in Figure 2.7, are reflected radiation from other objects, radiation loss or gain due to scattering, and absorption or emission in the atmosphere between the target and radiometer. For an opaque body free of reflections, the radiance Lm(T) measured by the radiometer is Z? Lm ðTÞ ¼

eðlÞRðlÞLb ðl; TÞ dl;

ð2:13Þ

0

where l is the wavelength of the radiation in a vacuum, T the temperature of the object, e(l) the spectral emissivity of the object, R(l) the spectral responsivity of the radiometer, and Lb(l, T) the spectral radiance of a blackbody obtained from Planck’s radiation law Lb ðl; TÞ ¼

i1 c1 h  c2  exp 1 ; 5 lT l

ð2:14Þ

where c1 and c2 are known as the first and second radiation constants. Equation (2.13) is only useful if the spectral emissivity of the object and the spectral

Figure 2.7 Schematic showing the main sources of interference in radiation thermometry measurements.

32

33

Temperature

responsivity of the radiometer are known. The spectral responsivity of the radiometer is chosen according to the application and determined in detail by laboratory measurement. In contrast, the spectral emissivity is a property of the body under investigation. Values of e(l) for non-blackbody-like surfaces may be determined either from published data or experiment, and are often subject to uncertainties of 0.05 or more. For ITS-90, the target is a blackbody cavity so the spectral emissivity is close to unity for all wavelengths.

2.6.1

Radiation Temperature Scale

Above the freezing temperature of silver, 1234.93 K, ITS-90 uses the Planck blackbody radiation law to define the temperature in terms of the ratio of the spectral radiance at the temperature T90, Lb(l, T90), to the spectral radiance at a fixed-point temperature, Lb[l, T90(X)], where T90(X) ¼ T90(X, s þ l, p ¼ 0.101 325 MPa), is the normal freezing point of silver, gold or copper. Thus Lb ðl; T90 Þ exp½c2 =ðlT90 ðXÞ  1 ¼ Lb ðl; T90 ðXÞÞ exp½c2 =ðlT90 Þ  1

ð2:15Þ

where c2 ¼ 0:014 388 m ? K is fixed by definition. The fixed points used with radiation thermometers behave much as described in Section 2.4.1, except that the cells used to realise these points are constructed differently and the plateaux last only a few minutes. Figure 2.8 shows a diagram of a typical fixed-point blackbody cavity. In these cells, which are made from very high purity graphite, a cavity of (50 to 80) mm in length with a (1 to 6) mm aperture is used as the reference radiation source. The cavity shape and the aperture are made to ensure that the effective emissivity of the cavity is very close to unity. In use, the solid-liquid interface formed during melting or freezing should enclose the cavity. Accordingly, the furnace must have a uniform temperature close to that of the fixed point. The radiation generally propagates through an inert gas, rather than a vacuum, and a correction is required to account for this. No windows should be in the radiation path.

Figure 2.8

Metal fixed point with a blackbody cavity suitable for a radiation standard.

33

34

Temperature

Equation (2.15) defines temperature based on an idealised monochromatic radiometer. However, practical radiometers operate over finite bandwidths, typically of the order of (10 to 100) nm. One of the more difficult tasks is to solve the integral equation obtained by replacing the right hand side of Equation (2.15) with the ratio of the integrals given by Equation (2.13) with e(l) ¼ 1:0. The temperature can be found iteratively with numerical integrations of Equation (2.13). A simpler but good approximation to the solution can be found by equating each integral to a monochromatic Planck function with a temperature dependent effective wavelength. The temperature dependence of the effective wavelength is calculated from the spectral responsivity and the Planck function at several temperatures. A measured temperature is then determined with an iterative calculation by repeatedly interpolating a value for the temperature and then updating the estimate of the mean effective wavelength for the two measurements. The measurements used to characterise the radiometer are quite demanding. If the operating band is in the (600 to 900) nm range, which is typical of modern radiometers, the spectral responsivity of the radiometer must be measured with a wavelength accuracy of about 0.02 nm. Intensity ratios in Equation (2.15) of the order of 104 must be measured, and linearity corrections to the radiometer response are essential. Finally, the field of view of the radiometer is subject to size-of-source effects, which must also be corrected. Two types of reference radiometers can be distinguished. The first and most traditional is the comparator radiometer, which compares a fixed-point blackbody directly with another radiant source, and only needs a limited stability with respect to time. More recently, with the improvement in the stability of filters and detectors, transfer standard radiometers of good long-term stability have been developed. Some of the portable transfer standard radiometers may be used outside the calibration laboratory. Radiation thermometry has an uncertainty of 50 mK at the silver point where it meets the platinum resistance scale, which is itself uncertain to around 5 mK. However, the precision of the radiation measurements should be at the 10 mK level to demonstrate the quality of the fixed points and blackbody cavities.

2.7

Cryogenic Thermometry

Two types of ITS-90 interpolating thermometers are used for low temperature work: vapour-pressure thermometers and gas thermometers, as indicated in Figure 2.1. Because both of these thermometers are difficult to use at high accuracy, they are generally only used to calibrate suitable electrical thermometers. The vapourpressure relationships though can be used for less accurate measurements and are convenient because the thermometric fluid is often the refrigerant. 2.7.1

Vapour Pressure Thermometers

Unlike other parts of ITS-90, the helium vapour-pressure scale does not rely on fixed points. Instead, the ITS-90 defines the numerical relationship between the 34

Temperature

35

temperature and the vapour pressure based on experimental data: T90 =K ¼ A0 þ

9 X

Ai f½lnðp=PaÞ  B=Cgi :

ð2:16Þ

i¼1

A defined relationship is used because the uncertainties associated with the thermodynamic relation are higher than desirable. Two different substances are used: 4He and 3He; the appropriate constants and temperature ranges for Equation (2.16) are given in Table 2.9. The vapour-pressure thermometer allows the liquid and vapour phases of helium to come to thermal equilibrium. The temperature is then inferred from the absolute pressure measured at the interface between liquid and vapour. Heat losses and thermal gradients are minimised and controlled and, with care, an uncertainty of around 0.5 mK is practical.

2.7.2

Gas Thermometers

Gas thermometers with either 3He or 4He, cover the range from 3 K to the neon triple point, 24.5561 K, and require calibration at three temperatures. Gas thermometers can be used to determine the thermodynamic temperature relative to a single fixed point, usually the neon triple point. However, considerable care is required to achieve accurate results over a wide temperature range at low temperatures. Instead, ITS-90 uses three fixed points over a narrow range to realise

Table 2.9 Values of the constants of Equation (2.16) for 3He and 4He, and the temperature range for which each set of coefficients are used. Gas

3

Range A0 A1 A2 A3 A4 A5 A6 A7 A8 A9 B C

0:65  T=K  3:2 1.053 447 0.980 106 0.676 380 0.372 692 0.151 656  0.002 263 0.006 596 0.088 966  0.004 770  0.054 943 7.3 4.3

He

4

4

1:25  T=K  2:1768 1.392 408 0.527 153 0.166 756 0.050 988 0.026 514 0.001 975  0.017 976 0.005 409 0.013 259 0 5.6 2.9

2:1768  T=K  5:0 3.146 631 1.357 655 0.413 923 0.091 159 0.016 349 0.001 826  0.004 325  0.004 973 0 0 10.3 1.9

He

35

He

36

Temperature

a more reproducible scale. Accuracies of about 0.1 mK can be achieved. A simple quadratic interpolation equation is used to cover the range (4.2 to 24.6) K for 4He given by T90 ¼ a þ bp þ cp2 ;

ð2:17Þ

where p is the measured pressure and a, b and c are coefficients determined at the fixed points listed in Table 2.1. One of the chosen points should be at a temperature between (4.2 and 5) K as measured by a vapour-pressure thermometer. With 3He as the thermometer gas or with 4He over the range (3.0 to 24.6) K, an additional term is specified by ITS-90 to accommodate the temperature variation of the second virial coefficient and requires the volume of the gas thermometer. The equations are thermodynamically based with corrections to account for the known departures from ideal gas behaviour.

2.7.3

Electrical Thermometers

Gas and vapour-pressure thermometers are used to transfer the scale to more suitable temperature sensors. The most stable and reliable sensor for the low temperature range is the rhodium-iron resistance thermometer. Its construction is similar to the platinum capsule thermometer shown in Figure 2.6. The thermometer is made from rhodium wire doped with 0.5 mass per cent iron to give a resistance between (20 and 50) O at 273.16 K. The rhodium-iron thermometer is preferred to a platinum thermometer at temperatures in the range (0.5 to 30) K. It is still useful up to room temperatures, but the platinum thermometer provides superior performance. A suitable calibration equation for the rhodium-iron thermometers is

R=O ¼

n X

bi ½lnðT=K þ gÞi ;

ð2:18Þ

i¼0

where bi are adjustable parameters and g is a constant between 8 and 10. When n ¼ 6 in Equation (2.18) the interpolation uncertainties are normally within 0.3 mK. Thinfilm versions of these thermometers are available. While the rhodium-iron thermometers give very good stability, both their relatively large physical size and low sensitivities may limit their utility. Germanium resistance thermometers are smaller and provide higher sensitivities. A single crystal of germanium is used with ‘n’ or ‘p’ doping, four leads, and is mounted in a can filled with 4He or 3He gas. Individual germanium thermometers can exhibit good stability but instabilities often arise from thermal cycling. A variety of types are available and best used for narrow temperature ranges below 30 K. They are very non-linear and 36

Temperature

37

require a calibration equation of the form lnðT=KÞ ¼

n X

Ai ½ðlnðR=OÞ  M Þ=N i ;

ð2:19Þ

i¼0

where N and M are suitable constants and the Ai are determined by least-squares analysis. Calibration requires measurements at 3n points, where n&12 for a wide temperature range and n ¼ 5 for a narrow range. As with all resistance thermometers, self-heating errors occur. Because the resistance can be high compared with platinum there are also leakage resistances, and the measured ac and dc resistances may differ. Other electrical thermometers are available for low temperatures and are typically based on measurements of resistance, capacitance, or the thermoelectric effect. There are numerous articles in the literature for the interested reader to consult. In high magnetic fields at low temperatures, a systematic error may arise in the measured temperature because of the interaction between the electrical temperature sensor and the external magnetic field. However, the magnitude of the interaction varies enormously and with capacitive-based sensors is usually insignificant.

2.8

ITS-90 and the Thermodynamic Scale

Constant volume gas thermometry has been the mainstay of the previous determinations of thermodynamic temperature that formed the basis for most of the ITS scales. For ITS-90, thermodynamic determinations also came from absolute radiometry and acoustic thermometry, and at high temperatures, relative radiation thermometry. The addition of these new techniques has been particularly valuable in assessing systematic effects in the measurements. Since the publication of ITS-90 several new determinations of thermodynamic temperature have been published and a major revision of the main gas thermometry data reported. These results are summarised in Figure 2.9, which shows the measured differences between ITS-90 and the thermodynamic temperature, as determined by a variety of thermodynamic thermometers. Only absolute determinations of thermodynamic temperature are shown in Figure 2.9, and, for example, none of the relative radiation thermometry data is presented. Some of the original data given in terms of the IPTS-68 have been converted to ITS-90. The uncertainties in the measurements are not shown in Figure 2.9 but are of a similar magnitude to the measured differences, so care must be taken drawing conclusions from the results. Below 100 K the differences (T  T90) span 5 mK with most lying within 1 mK, indicating that ITS-90 is close to the thermodynamic temperature. In the range (100 to 200) K the deviations span 18 mK with the newer and revised data suggesting that the ITS-90 is in error by about 12 mK at 150 K. At temperatures between (200 to 500) K, the deviations span about 10 mK with an indication of a positive slope, d(T  T90)/dT > 0. The difference is zero at the water 37

38

Temperature

triple point, 273.16 K, because this point is defined on both scales. Above 500 K the magnitude of the deviations increases with increasing temperature and, as shown in Figure 2.9, diverge into two branches. ITS-90 was defined as an average of the two branches because there was no reason to favour one branch over the other. However, new measurements and reanalysis of some data observed prior to 1990 support the data in the upper branch. Above 500 K, new radiation-thermometry results also follow the positive branch of departures from ITS-90 with a T2 dependence that is typical of radiation-thermometry errors. These measurements suggest that ITS-90 may be in error by 85 mK at the silver point, 961.78 8C.

2.9

Temperatures below 1 K

ITS-90 extends down to 0.65 K but there is a scientific need for a temperature scale that extends to lower temperatures. A provisional low temperature scale, PLTS2000, was adopted by the CIPM in 2000 and covers the range 0.9 mK to 1 K [12]. The PLTS-2000 uses the melting pressure of 3He as the thermometric quantity and defines an equation for the melting pressure to cover the temperature range. The melting-pressure curve was established with nuclear orientation and noise thermometers. The scale achieves an accuracy of around 0.3 per cent over most of its range rising to 2 per cent at 0.9 mK. Near state-of-the-art pressure measurements are required for best accuracy. Documents reporting the methods of realisation of the scale are in preparation and are expected to be published in both a revision of the BIPM ITS-90 guidelines [2] and another archival journal. The melting pressure curve defined by PLTS-2000 is shown in Figure 2.10. Other fixed points within the temperature range are also shown on Figure 2.10 and listed in

Figure 2.9 A comparison of T90 with the thermodynamic temperature T. The filled squares are the experimental data used to determine ITS-90, with revisions excluded. The open circles are either recent measurements or prior data revised after 1990.

38

Temperature

39

Figure 2.10 The melting pressure p(3He, s þ l, T) as a function of T for 3He along with other fixed points. S, Entropic discontinuity; B, superfluid transition; A, superfluid transition; min, pressure minium; W, superconductive transition of tungsten; Be, superconductive transition of beryllium; Ir, superconductive transition of iridium; AuAl2, superconductive transition of a gold-aluminium alloy; AuIr2, superconductive transition of a gold-iridium alloy; Cd, the superconductive transition of cadmium; Zn, the superconductive transition of zinc.

Table 2.10. The melting pressure relation is non-linear and is not single-valued so a single pressure may correspond to two temperatures. It is therefore necessary to know the approximate temperature and to interpolate through the pressure minimum. The design of the melting pressure sensor allows the pressure measurement to be substituted by a capacitance measurement. At the intrinsic or other fixed points Table 2.10 Fixed points relevant to the provisional low temperature scale adopted in 2000 (PLTS-2000) that covers the temperature range (0.009 to 1) K. Fixed point

p/MPaa

T/K

Intrinsic points for solid 3He Entropy discontinuity, ‘S’ Superfluid transition, ‘B’ Superfluid transition, ‘A’ Pressure minimum, ‘min’

3.439 3.436 3.434 2.931

34 09 07 13

0.000 0.001 0.002 0.315

Superconductive transitions Tungsten, W Beryllium, Be Iridium, Ir Gold-aluminium alloy, AuAl2 Gold-indium alloy, AuIn2 Cadmium, Cd Zinc, Zn

3.381 3.353 3.120 3.025 2.971 3.050 3.622

84 72 77 39 78 53 90

0.015 3 0.022 6 0.104 0.16 0.21 0.52 0.85

a

The pressure values are derived from the temperature values through PLTS-2000.

39

902 896 444 24

40

Temperature

the capacitance can be equated to pressure. The four fixed points, cadmium, zinc, tungsten and the melting pressure minimum, p(3He, s þ l, 0.315 24 K) ¼ 2.931 13 MPa, can be used to determine coefficients of a cubic polynomial in T that is sufficiently accurate for most purposes. Of the fixed points listed in Table 2.10, the pressure minimum and the entropic discontinuity points are the most useful, while the superfluid points are more difficult to realise. The superconductive transition temperatures depend on the purity of the metals used so care is needed in the use of these points.

2.10

Temperatures above 2000 K

In principle, the ITS-90 has no upper temperature limit. However, uncertainties associated with the ITS-90 fixed points and the characterisation of radiometers propagate as T2 and become large at high temperatures. A fixed point near 2500 K could lower the uncertainty and several suitable systems have been identified for investigation [13,14]. Above 2000 K, other thermodynamic techniques, although not as precise as radiation thermometers, may be advantageous because their uncertainties propagate as T. Two thermodynamic methods are practical for high to medium accuracy. Commercial absolute cryogenic radiometers achieve radiometric accuracy better than 0.01 per cent, comparable to that of relative radiation measurements [15]. In addition, PIN diodes assembled into a trap detector can be calibrated to give 0.01 per cent uncertainties in T and are more convenient to use [16]. In either case, the main experimental difficulty is determining with sufficient accuracy the dimensions of the apertures used in both blackbody cavities and radiometers. While absolute radiometry may offer sufficient accuracy, not all systems are capable of accommodating the inclusion of a blackbody cavity. In noise thermometry [17], which uses an immersion probe, the noise voltage generated by a resistor is related to the temperature T through the Nyquist noise relationship V 2 ¼ 4kTRDf ;

ð2:20Þ

where V 2 is the mean square noise voltage, k is Boltmann’s constant, R the resistance and Df the bandwidth of the measurement. A measure of the noise voltage and the resistance can, therefore, be used to determine the thermodynamic temperature. The advantage of this method is that, since the resistance is measured each time, the sensor need not be stable, and hence a noise thermometer is suitable for use in harsh environments that otherwise degrade the temperature transducer. For example, noise thermometers are used in nuclear plants where sensor radiation damage is a severe problem. The main disadvantage of a noise thermometer is that the noise signal is small and the measurement time relatively long compared to other techniques. Noise thermometry has been used to verify the performance of very high temperature 40

Temperature

41

thermocouples and thus to give an independent measure of the thermodynamic temperature as described in Section 2.11.

2.11

Special Thermocouples

ITS-90 replaced the reference thermocouples of IPTS-68 by high-temperature SPRTs. While the SPRTs provide higher accuracy, they are relatively bulky and fragile, and for many applications, thermocouples are still preferred. A thermocouple does not detect the temperature directly but generates a voltage dependent on the temperature gradients along the whole length of the thermocouple wire from measurement junction to reference junction. Thus, the common belief that the voltage is generated at the measurement junction is a fallacy. In practice, a thermocouple installation should be designed to ensure that all junctions are isothermal and that there is no voltage generated at the junctions. It is also important that the whole length of the wire is kept homogeneous, that is, free from mechanical, chemical and metallurgical changes. Most thermocouples are constructed from alloys to optimise the output signal and resistance to chemical attack. However, alloys are susceptible to metallurgical and chemical changes with heat treatment, which make the wire inhomogeneous. In order to overcome this problem two thermocouples have been developed from high purity metals. One uses platinum and gold [18] while the other uses platinum and palladium [19]. Both thermocouples use a strain-relieving junction to overcome effects arising from the differential thermal expansion of the two metals. The output voltage for both thermocouples is lower than that of standard letter-designated basemetal thermocouples, but it is adequate given the performance of modern voltmeters. The temperature range is limited by the melting point of the metals. The goldplatinum thermocouple may be used up to 1000 8C with a precision of 3 mK over most of its range. The palladium-platinum thermocouple can be used at 1500 8C with an uncertainty of about 6 mK at 1050 8C, rising to 0.15 K at 1500 8C. At higher temperatures, there is a shortage of stable thermocouple materials. The Type B thermocouples [20] can be used at temperatures up to 1700 8C but require calibration, and the uncertainty is lower than that of the Au-Pt and Pt-Pd thermocouples. Temperatures up to 2400 8C can be reached with tungsten-rhenium thermocouples [20] but they are brittle, require frequent calibration and drift rapidly at the highest temperatures.

References 1. Pavese, F. and Molinar, G., Modern Gas-Based Temperature and Pressure Measurements, Plenum Press, New York, 1992. 2. Bureau International des Poids et Mesures, Supplementary Information for the International Temperature Scale of 1990, Se`vres, BIPM, 1990. 3. Preston-Thomas, H., Metrologia 27, 3, 1990.

41

42

Temperature

4. Nicholas, J.V. and White, D.R., Traceable Temperatures, 2nd Ed., John Wiley & Sons, Chichester, 2001. 5. Bureau International des Poids et Mesures, Techniques for Approximating the International Temperature Scale of 1990, Se`vres, BIPM, 1990. 6. Michalski, L., Eckersdorf, K. and McGhee, J., Temperature Measurement, John Wiley & Sons, Chichester, 1991. 7. McGee, T.D., Principles and Methods of Temperature Measurement, John Wiley & Sons, New York, 1988. 8. Rusby, R.L., Hudson, R.P. and Durieux, M., Metrologia 31, 149, 1994. 9. Nicholas, J.V., Dransfield, T.D. and White, D.R., Metrologia 33, 265, 1996. 10. Connolly, J.J. and McAllan, J.V., Metrologia 16, 127, 1980. 11. Bedford, R.E., Bonnier, G., Maas, H. and Pavese, F., Metrologia 33, 133, 1996. 12. Rusby, R.L., Durieux, M., Ressink, A.L., Hudson, R.P., Schuster, G., Ku¨hne, M., Fogle, W.E., Soulen, R.J. and Adams, E.D., in TEMPMEKO ’01 Proceedings of the 8th International Symposium on Temperature and Thermal Measurements in Industry and Science, B. Fellmuth, J. Seidel and G. Scholz, eds., 2001 June 19–21, VDE-Verlag, Berlin, Germany, p. 365, 2002. 13. Yamada, Y., Duan, Y., Ballico, M., Park, S.N., Sakuma, F. and Ono, A., Metrologia 38, 203, 2001. 14. Yamada, Y., Sakate, H., Sakuma, F. and Ono, A., Metrologia 38, 213, 2001. 15. Quinn, T.J. and Martin, J.E., Metrologia 33, 375, 1996. 16. Fox, N.P., Metrologia 28, 197, 1991. 17. White, D.R., Galleano, R., Actis, A., Brixy, H., De Groot, M., Dubbeldam, J., Reesink, A.L., Edler, F., Sakurai, H., Shepard, R.L. and Gallop, J.C., Metrologia 33, 325, 1996. 18. Burns, G.W., Strouse, G.F., Liu, B.M. and Mangum, B.W., In Schooley, J.F., ed., Temperature, its Measurement and Control in Science and Industry, Vol 6, American Institute of Physics, New York, p. 531, 1992. 19. Burns, G.W., Ripple, D.C. and Battuello, M., Metrologia 35, 761, 1998. 20. American Society for Testing and Materials, Manual on the Use of Thermocouples, 4th edition, ASTM, Philadelphia, 1993.

42

3

Pressure J. SUSKI Schlumberger-Riboud Product Centre Clamart, France R. PUERS Katholieke Universiteit Leuven Leuven, Belgium C.D. EHRLICH and J.W. SCHMIDT National Institute of Standards and Technology Gaithersburg, Maryland, USA E.H. ABRAMSON University of Washington Department of Chemistry Seattle, WA, USA C.M. SUTTON Measurement Standards Laboratory of New Zealand Industrial Research Limited Lower Hutt, New Zealand 3.1

3.2

3.3

Electronic Pressure Gauges 3.1.1 Physical Principle of Resistive Strain Gauge Pressure Transducers 3.1.2 Silicon Piezoresistive Micromachined Pressure Transducers 3.1.3 Resonant Pressure Sensors Based on Mechanical Vibrations 3.1.4 Pressure Sensors Based on Capacitance 3.1.5 Silicon Capacitive Pressure Transducers 3.1.6 Performances of Piezoresistive and Capacitive Pressure Sensors 3.1.7 Scaling Limits in Pressure Transducers Piston Gauges 3.2.1 Advances in Piston Gauge Manufacturing Technology 3.2.2 Characterisation Techniques 3.2.3 Practical Developments in Piston Gauges Pressure Measurements in a Diamond-Anvil Cell 3.3.1 X-rays 3.3.2 Optical Techniques

Measurement of the Thermodynamic Properties of Single Phases A.R.H. Goodwin, K.N. Marsh, W.A. Wakeham (Editors) Published by Elsevier BV on behalf of IUPAC

45 46 47 50 53 55 56 57 58 59 61 64 68 68 69

44

Pressure 3.4

Low Pressure 3.4.1 Interferometric Liquid-Column Manometers 3.4.2 Piston Gauges 3.4.3 Static Expansion Pressure Generators 3.4.4 Pressure Transducers

44

72 74 80 87 88

3

PRESSURE

The operation of pressure gauges was covered in Experimental Thermodynamics Volume II [1]. This chapter reports the advances in pressure measurements since about 1975, in particular, techniques that rely on devices constructed with Si technology, as used for the microchip, and the measurement of frequency. This combination has made a reality the measurement of pressure in extreme conditions with harsh fluids. Sections within this chapter cover electronic gauges, piston gauges, particularly for low pressure, methods that utilise X-ray and optical techniques for high pressure, and interferometric techniques.

3.1

Electronic Pressure Gauges J. SUSKI Schlumberger Industries Montrouge, France R. PUERS Katholieke Universiteit Leuven Leuven, Belgium

An electromechanical pressure transducer is a device which converts pressure into force, then mechanical strain and thence an electrical signal. There are five main types of pressure transducers suitable for high-accuracy measurement of pressure. All five types respond to the force of an applied pressure, making them independent of the fluid used. First, capacitance diaphragm gauges (CDG) where a pressure differential is applied across a tensioned diaphragm and the resulting diaphragm movement is determined capacitively [2]. Second, quartz bourdon tube gauges (QBG), in which the sensor is a fused quartz tube formed by a helix which unwinds (or winds) with any differential pressure across the tube. This pressure difference is determined from the current applied to an electromagnetic force-balance that is used to counter the tendency of the helix to unwind in response to the pressure [3]. Third, quartz resonant gauges (QRG) where an applied pressure difference loads a resonant quartz crystal, changing its oscillating frequency. Four, micro electro-mechanical systems (MEMS) resonant silicon gauges (RSG). In these miniature sensors, the pressure is measured across a silicon diaphragm by the strain-induced changes in the oscillation frequencies of two silicon resonators micro-machined on the surface of the diaphragm. Five, MEMS piezoresistive silicon (strain) gauges (PSG) where the differential pressure across a silicon diaphragm is measured by strain-induced changes in the resistance of piezoresistors doped in to the silicon diaphragm. Silicon micromachined pressure sensors can be designed for measurements in the range 10 Pa to 100 MPa. An exhaustive discussion of mechanical pressure sensors can be found in references [4,5]. 45

46

Pressure

Pressure transducers determine a pressure difference across a membrane. Three types of pressure gauges are available and these can be distinguished as follows: the measurement is relative to vacuum and referred to as absolute; differential pressure measurements referenced to an arbitrary pressure; gauge or relative pressure measurements refer to the difference between atmospheric pressure and the unknown pressure. Pressure gauge dynamic range and resolution are design criteria for electromechanical pressure transducers. The dynamic range of the device depends on both the sensor geometry (usually the ratio of membrane length to thickness) and the performance of the detection principle. Dynamic range and high resolution are required in flow rate measurements, where the pressure drop arising from fluid flow is a quadratic function of the velocity. Capacitive and resonant pressure transducers are well suited to the measurement of pressure in flowing media. The active elements of the pressure gauges described in this section are not normally exposed directly to the fluid, which is at the pressure to be measured. The pressure sensor is usually immersed in an inert fluid, for example a silicon based oil, and connected to the media under test via either a hermetically sealed bellows or a capillary tube that is also filled with silicon oil. The pressure transducer calibration must be performed with the isolating mechanism, which can be the limiting factor as far as long-term stability and accuracy are concerned. A description of bellows devices appears in Section 5.3. Bourdon tubes are also used to both isolate transducers from the media, for which the pressure measurement is required, and load the active element. Recently, it has been shown that MEMS pressure gauges constructed with a non-stoichiometric chemically vapour deposited form of Si3N4 may be directly exposed to crude oil at 428 K and 100 MPa and operate with a precision of 0.1 per cent over 0.5 a. The application of the transducers described in this section to low-pressure measurements is considered in Section 3.4.

3.1.1

Physical Principle of Resistive Strain Gauge Pressure Transducers

Silicon is a good material, because of its excellent elastic properties [6], to construct a diaphragm to detect the mechanical stress arising from an applied pressure. Silicon micromachining techniques, based on photolithography and wet chemical and dry etching processes, can be used to develop a wide range of pressure sensitive elements including membranes with resistance bridges and cantilever beams. In a strain gauge based pressure gauge, an electrical resistor is attached to the membrane and its resistance varies in proportion to the strain to which it is subjected by the applied pressure. Resistance strain gauges, initially developed to measure static and dynamic strains, can be applied to measure any quantities that can be related to strain, such as force, pressure, acceleration, and torque. The impedance of a resistor R in, for 46

Pressure

47

example, a strain gauge is given by R ¼ rL=A;

ð3:1Þ

where r is the material resistivity, L the resistor length, and A the resistors crosssectional area. The gauge factor, K, used to define the fractional change in resistance DR of a strain gauge with applied strain e, is given by K¼

DR ¼ 1 þ 2s þ Rp E; Re

ð3:2Þ

where s is Poisson’s ratio, E is Young’s modulus, and Rp is the piezoresistive coefficient. For pressure transducers constructed from silicon the piezoresistive coefficient dominates Equation (3.2) and gauge factors of order 100 can be obtained, whereas in poly-silicon material K is in the range 30 to 40 [6,7]. For strain gauges constructed from thin metal films, s varies between 0.3 and 1, the product Rp E is on the order unity, and K is greater than 2.3. Conventional thin film metal foil gauges are used where the membrane deformation is up to 0.2 times the yield strain. Metal foil strain gauges can be attached to both membranes and mechanical supports with adhesive and then used as the pressure transducers. An advantage of metal strain gauges is the linear response over a temperature range. In thick metal films the gauge factor K is usually in the range 6 to 12 [8,9], and can reach 100 for perpendicularly loaded resistors [10].

3.1.2

Silicon Piezoresistive Micromachined Pressure Transducers

Piezoresistive pressure transducers use a combination of silicon, as the construction material for the mechanical stress amplifiers (membranes, cantilever beams and bridges), and piezoresistive strain gauges. The mechanical stress amplifiers are used to transform the pressure into stress and the particular mechanical design is chosen to provide both the required sensitivity and resolution for a given application. Pressure transducers are usually formed from a membrane that is hermetically sealed to a support and also separates the reference pressure from the pressure to be measured. Silicon-to-Pyrex anodic bonding techniques [11,12] are often used for absolute pressure sensors. Pressure transducer drift may arise when either the active elements or electrical connections are constructed from materials with different thermal expansion coefficients. This is the case with silicon and Pyrex and designs, shown in Figure 3.1, have been used to minimise these effects. Silicon piezoresistive strain gauges are machined into supporting membranes with a standard integrated circuit (IC) process. Silicon based MEMS can provide pressure transducers with mm dimensions, which can withstand pressures up to 100 MPa. Mono-crystalline silicon, 47

48

Pressure

Figure 3.1 Schematic cross-section through a pressure gauge formed from an Si membrane and strain gauge doped into specific locations to detect the mechanical deformation arising from the application of a pressure difference. This Si element is sealed to the support with Pyrex, that is in turn mounted on a base to provide support for the electrical connections.

owing to the stability of the crystal, reduces the observed hysteresis. In Figure 3.1 a schematic of a typical pressure sensor is shown. The uniaxial stress placed on the resistors deposited atop the membrane, is proportional to (a=t)2 , where a and t are the membrane radius and thickness respectively. The maximum allowed uniaxial stress must be less than the rupture strength of the diaphragm, which is in the range of (1 to 10) GPa, with a margin of between (10 and 15) per cent of the rupture stress that results in an acceptable applied pressure of about 100 MPa. This upper limit defines the pressure that can be applied without changing the transducer’s performance. The piezoresistors are configured as a Wheatstone bridge and the resistance changes with mechanical deformation are determined from voltage measurements. The piezoresistive effect in silicon is anisotropic. For p-type doped Si resistors, the longitudinal K1 and transverse Kt gauge factors, corresponding to longitudinal and transverse resistors respectively, are related by K1 ¼  Kt [13]. This relationship determines the optimal layout for the resistors and, as shown in Figure 3.2, two resistors are parallel to a mechanical stress (longitudinal resistors R1 and R3) and two (transverse resistors R2 and R4) are perpendicular to the stress. In practice, the first pair of resistors are perpendicular to the edge of a square membrane, and the second pair are parallel to the membrane edge. The material resistivity r, and thus K, depend on the temperature and, if the temperature is not measured and appropriately accounted for, results in a temperature dependent response to an applied pressure. First order thermal compensation is achieved by appropriate choice of electrical carrier concentration so that dK=dT&  dR=dT, which can be obtained with a surface resistivity rs &100 O (otherwise known as sheet resistance). Each pressure transducer has to be calibrated to compensate for the effects of temperature, pressure non-linearity, and hysteresis. For silicon based sensors, a 48

Pressure

49

Figure 3.2 Schematic of a rectangular pressure transducer membrane and location of longitudinal R1 and R2 and transverse, R3 and R4, piezoresistors. TOP: Cross-section of the device, revealing the membrane obtained from anisotropic etch.

major component of the total error arises from the temperature dependent terms and are accommodated, to the required accuracy, with polynomial representation of the output as a function of T and p. The pressure transducers, described above, exhibit sensitivities to pressure equivalent to (3 to 5) per cent of the resistance obtained at maximum pressure. A comprehensive description of the sensor performance can be found in references [14] and [15]. It has been demonstrated that a pronounced piezoresistive effect in field effect transistors (FET) can be used for pressure measurements [16,17]. Metal oxide semiconductor FET (MOSFET) based pressure sensors have been developed [18], which use complementary metal oxide semiconductor (CMOS) processed ring oscillators as the sensing element. The ring oscillator has an odd number of invertors, and the mechanical stress arising from the applied pressure induces a frequency shift in each ring. The ratio of the frequency of the two oscillators, one perpendicular and one parallel to the applied stress, provides both first order temperature compensation and increased pressure sensitivity. Measurements performed with a CMOS ring oscillator pressure sensor operating at a frequency of about 10 MHz, that has a maximum determinable pressure of 1 MPa, show characteristic values of sensitivity, which equal df =dp, of about 1 Hz ? Pa1 . This CMOS based pressure gauge sensitivity is about twice that obtained for a pressure 49

50

Pressure

gauge manufactured with bipolar techniques. The temperature coefficient (df =dp)=dT is about 1:5 Hz ? kPa1 ? K1 . This CMOS device exhibited a hysteresis of about 100 Hz, which is typical for CMOS devices, and a linearity (defined as the maximum deviation of the measured output at constant temperature from a linear fit to the data obtained at p ¼ 0, p ¼ 0.5 MPa and p ¼ 1 MPa) of about 25 kHz. At a pressure difference of zero, the so called offset voltage, for this device had a temperature dependence equivalent to 100 Hz ? K1 . These specifications can be routinely obtained with pressure gauges constructed with CMOS technology. An integrated circuit can process the measured frequencies to provide the pressure.

3.1.3

Resonant Pressure Sensors Based on Mechanical Vibrations

Frequency can be measured easily with a relative precision of better than 108. Thus resonance frequency measurements of a mechanical resonator can be used as the sensing element of a pressure transducer. Resonators have been micro-machined in silicon and used by Greenwood [19,20], Greenwood and Wray [21], and more recently by Corman et al. [22] as pressure gauges. In the device reported by Corman et al. [22], an electrostatically driven resonator was encapsulated in a cavity at a pressure of 0.1 kPa. A similar device has been used to construct a vibrating object densimeter and flow meter [23,24]. High precision resonant object pressure sensors are often manufactured from mono-crystalline quartz which is stable after temperature and pressure cycling. Quartz exhibits a high material Q and a small, compared with silicon based devices, hysteresis [25,26]. Mono-crystalline quartz is approximately 24 times stronger in compression than in tension, and thus, it is desirable to design a pressure gauge such that the entire structure is largely in compression. Pressure gauges manufactured from quartz are used in the oil and gas industry where long term stability of the sensors is critical for monitoring both reservoir and borehole pressures as a function of time over at least 1 a. In the hydrocarbon recovery business, quartz pressure sensors are also used for pressure transient analysis and permanent monitoring systems for optimal reservoir management [27]. A gauge constructed for use in oil and gas reservoirs is shown in Figure 3.3. This pressure transducer consists of three cylindrical parts: a rectangular plate, which is the pressure sensitive element, that is an integral part of the cylindrical shell and two end caps have a hollow interior and are hermetically sealed to the cylinder. The temperature and pressure are determined from measurements of the resonant frequency of two shear modes of the quartz. One mode [27] is selected for the sensitivity to pressure, the other chosen for its predominant response to temperature. The mechanical strength and stability of quartz depend on surface preparation, including mechanical polishing and chemical etching [28].

50

Pressure

Figure 3.3

51

Picture of the crystal quartz gauge (CQG).

Under static conditions, both the applied pressure p and temperature T can be determined from a pair of calibration polynomials of the form p¼

n X

Aij fpi fTj ;

and

ð3:3Þ

i;j¼0



n X

Bij fpi fTj ;

ð3:4Þ

i;j¼0

where fp and fT are the frequencies of the primarily pressure sensitive (C-mode) and primarily temperature sensitive (B-mode), respectively [27]. Typically, n ¼ 4 in Equation (3.3) while n ¼ 5 in Equation (3.4) where the exact integer value is determined by statistical significance. Long term stability of the crystal quartz gauge (CQG), based on a dual-mode thickness-shear resonator design, has been demonstrated [27]. Figure 3.4 shows the fractional difference in pressure determined with four different CQG pressure gauges from that obtained with a reference piston gauge as a function of time at a temperature of 423 K and 448 K. The extraordinary stability of crystalline quartz, provides pressure transducers with a long-term stability of better than +25 ? 106 ? p (about 2.6 kPa at 103 MPa). There are two other designs of thickness-shear quartz pressure sensor available commercially from Agilent and Quartzdyne. Both use singly rotated cuts of crystalline quartz, which exhibit piezoelectric coupling to one of the two thickness-shear modes. Therefore, in these designs, a separate temperature sensing element is required. A similar device, available from Paroscientific, is described below. Further information concerning this particular device, including stability, can be found at www.paroscientific.com. 51

52

Pressure

Figure 3.4 Long-term drift of four crystal quartz gauge (CQG) gauges at p & 103 MPa. Two of the gauges were at T ¼ 423 K and two other gauges at T ¼ 448 K. On the left hand ordinate, the difference is given by Dp ¼ p(CQG)  p(ref.) in which p(ref.) is the pressure determined with a dead-weight gauge. The a indicates a pressure fluctuation solely from the automatic dead weight gauge system.

Resonant Quartz Crystal Sensor – Construction and Operation The resonant quartz crystal sensors have been designed with a resolution of better than 0.1 Pa and a precision of better than 0.01 per cent of reading, even when maintained under difficult environmental conditions. These pressure transducers usually use vibrating, single or dual beam, load-sensitive resonators. The doubleended tuning fork sensor consists of two identical beams driven piezoelectrically 1808 out of phase to minimise the energy dissipated into the mounting pads. The Q of the resonant frequency is a function of the applied pressure. The pressure sensor also includes a piezoelectrically driven oscillator whose resonant frequency is a function of temperature. The pressure sensitive structure is mounted within either a bellows or Bourdon tube, which both transmits the pressure to the gauge and separates it from the fluid of unknown p. The gauge is usually surrounded by an inert, usually silicon based, oil. This packaging procedure reduces the variations in calibration and thus performance of the device over that if it were exposed directly to the medium whose pressure was to be measured; the fluid might react with the materials used to construct the transducer. These pressure sensing mechanisms are acceleration compensated with balance masses, to reduce the effects of shock and vibration. The transducers are hermetically sealed and evacuated to eliminate air damping and maximise the Q of the resonator. High reproducibility is obtained with these devices because of both the monocrystalline quartz and the small (mm) movement. Hysteresis for a 1 MPa full scale pressure gauge is better than 8 Pa, which is equivalent to the uncertainty of +105 ? p. The resolution of the quartz pressure transducer resonance frequency is 52

Pressure

53

relatively better than 106. The signal-to-noise ratio obtained from a quartz pressure gauge is about a factor of 100 greater than that from strain gauge based pressure gauges. For measurements over a period of 1 h, the signal noise introduces a relative error of less than 106 ? p, while over a 1 d period the effect of fluctuation in, for example, thermal environment, result in a factor of 10 increase in noise. The long-term stability of the beam type quartz pressure transducers is typically better than 104 ? ps , where ps is the full scale pressure over a temperature range from (298 to 398) K. When a series of these transducers were compared with a dead-weight gauge over a period of about 10 a, the differences were within 0.01 per cent and the drift rates ranged from (  3 ? 106 ? p to  11 ? 106 ? p) a1 .

3.1.4

Pressure Sensors Based on Capacitance

For this type of pressure gauge the signal is obtained from the mechanical deformation of a membrane under the influence of an applied pressure. Capacitive type sensors differ from the piezo-sensors in that the displacement of the membrane, and not its stress, is measured directly. The basic structure of a capacitive sensor consists of two, usually, parallel plates of surface area A separated by a distance d. The zero motion capacitance is given by Cðp ¼ 0Þ ¼ eA=d;

ð3:5Þ

where e is the permittivity of the medium between the plates. Most capacitive pressure transducers rely on the change in separation d rather than a change in surface area A. The latter is intrinsically linear, however, the mechanical design is elaborate and difficult to assemble. For design purposes an idealised model of a capacitive pressure transducer is considered, where it is assumed both plates and dielectric layer deform over the entire surface area A by a distance Dd to an applied pressure p, such that the plates remain parallel, and the capacitance C is given by: CðpÞ ¼ eA=ðd þ DdÞ:

ð3:6Þ

Figure 3.5 Left: Schematic representation of a capacitor, consisting of two plates and a substance of relative permittivity er of thickness d between them. Right: Movement of the two parallel plates shown at left when a pressure is applied atop the upper plate.

53

54

Pressure

Combining Equations (3.5) and (3.6) gives CðpÞ=Cðp ¼ 0Þ ¼ 1  Dd=d;

ð3:7Þ

for small membrane displacements. The sensitivity of the capacitive pressure transducer DC/Dd can be estimated from DC=Dd ¼  eA=d 2 :

ð3:8Þ

The capacitance change will be caused by a deformation of one of the plates, as shown in Figure 3.4, and the capacitance at a pressure p is given by ZZ CðpÞ ¼

e dx dy; d w

ð3:9Þ

where w is the local deformation as a consequence of the applied pressure and is a function of Cartesian coordinates x and y. The mean membrane displacement hdi is given by: 1 hdi ¼ A

ZZ w dx dy;

ð3:10Þ

and the capacitance change by DC ¼ Cðp ¼ 0Þhdi=ðd  hdiÞ%Cðp ¼ 0Þhdi=d:

ð3:11Þ

If Dd55h, where h is the membrane thickness then hdi& 14 w at the centre of the membrane, and the pressure transducer sensitivity can be estimated with Equation (3.11). In practice, the small capacitance changes can be determined with a capacitance bridge. Finite element analysis can also be used to study the membrane motion and effect of edge clamping. In contrast with piezoresistive pressure transducers, the stress in the membranes of capacitive pressure gauges is of no direct relevance in the transduction phenomenon. From a construction viewpoint, this offers distinct advantages, particularly for MEMS, where much larger mechanical tolerances can be accepted with respect to the membrane and any mechanisms placed on it; for MEMS strain gauge based pressure transducers, the deposited structures, for example resistors, must be precisely located to obtain maximum sensitivity. Figure 3.6 shows a commercial capacitance based pressure transducer used for small differential pressure measurements. To minimise stray parasitic capacitances, the diameter of the capacitor plates is about 0.1 m. The capacitance can be determined with high accuracy (relative uncertainty of 108) from measurements with an ac capacitance bridge or with high sensitivity 54

Pressure

55

Figure 3.6 Schematic of a differential capacitive pressure transducer. The diameter of the membranes is about 0.1 m.

(relative uncertainty of 1010) from the resonance frequency of an oscillator. The latter has been applied to differential pressure measurements at radio frequencies, with an inductive-capacitive oscillator. The practical realisation of the technique is discussed in Chapter 9.

3.1.5

Silicon Capacitive Pressure Transducers

The processes required to produce Si membranes, which can be utilised as capacitive pressure gauges, were reported by Bean [29], Bohg [30], Jackson et al. [31] and Brooks et al. [32]. Figure 3.7 shows a pressure gauge based on a parallel plate capacitor that was fabricated with MEMS (or integrated circuit) methods. It has a cavity which is about (1:261:2) mm square, about 2 mm deep, and has a zero differential pressure capacitance of about 10 pF. For the parallel plate capacitor, the capacitance changes by 15 per cent for a 30 kPa differential pressure, while a 100 kPa differential pressure induces a 200 per cent change in capacitance. Capacitive pressure transducers have, when compared with strain gauge based transducers, both lower power consumption and temperature drift: the latter is about 0:25 kPa ? K1 , when the cavity is filled with gas and is almost entirely determined by the compression factor of the gas and when the cavity is evacuated, dC=dT&50 Pa ? K1 . A significant drawback of the device, shown in Figure 3.7, is that C(p ¼ 0) can be the order of the parasitic capacitances which can obscure the capacitance variation that arises from a pressure change: in this case, three terminal capacitance measurements are required. Alternatively, as Figure 3.8 shows, the capacitor can be a reference in an electronic oscillator circuit, formed from either bipolar or CMOS technology, so that its resonance frequency is proportional to the capacitance variation. The type of oscillator determines the power consumed. For a CMOS oscillator, the power consumed for the complete pressure transducer, including communication interface, is about 60 mW{I ¼ 20 mA for V ¼ 3 V} [33]. 55

56

Pressure

Figure 3.7 Cross-section through a schematic of a capacitive pressure transducer manufactured with Si processing techniques. The cavity dimensions are (1:261:260:002) mm and C(p ¼ 0) ¼ 10 pF.

3.1.6

Performances of Piezoresistive and Capacitive Pressure Sensors

Spencer et al. [34] and Chau and Wise [35] have evaluated the performance of both piezoresistive and capacitive pressure transducers. The theoretical performances of miniature capacitive and piezoresistive pressure transducers have been described [35] and a review article on micromachined pressure sensors presented by Eaton and Smith [36]. Spencer et al. [34] introduced the concept of minimum detectable signal, b, to evaluate pressure transducers. The b of a pressure transducer defines the theoretical detection limit, which is equal to the noise expressed as an equivalent pressure fluctuation. It represents the uncertainty arising from the transduction process and is a measure of transducer resolution. This definition assumes that all systematic errors cancel. The long-term drift is not considered. There are three types of noise present in electrical circuits: Schottky effect (shot noise), Johnson (thermal) noise, and 1/f noise. Schottky noise results from electric potential barriers at p-n junctions, while Johnson noise is due to energy dissipation processes. Both Schottky and Johnson noise have flat spectral densities up to GHz frequencies. 1/f noise is caused by trapping centres present near the device surface. For piezoresistive pressure transducers Johnson noise dominates. For a Wheatstone bridge based pressure transducer, the output noise is equal to the noise of one of four resistors R. The change in the resistor bridge output voltage DV

Figure 3.8 output.

Schematic of a pressure gauge, with on-board electronics and low-impedance

56

Pressure

57

arising from a pressure change Dp is given by DV ¼ aR VB Dp;

ð3:12Þ

where aR is the pressure sensitivity of the transducer and VB the applied voltage. The b is obtained by equating DV with the r.m.s. Johnson noise to give b¼

1 ð4kTRDf Þ1=2 ; aR V B

ð3:13Þ

where Df is the frequency bandwidth, R the resistance, k Boltzmans’ constant, and T temperature. For capacitive pressure sensors the noise, and therefore b, cannot be generalised and must be defined for that particular detection circuit.

3.1.7

Scaling Limits in Pressure Transducers

The pressure sensitivity of capacitive pressure transducers, assuming the circuit noise is random, is given by aC ¼

DC a4 ¼ 3; Cðp ¼ 0Þp dh

ð3:14Þ

in the small deformation limit for a square membrane free from built-in stress [35]. In Equation (3.14), DC is the capacitance change, C(p ¼ 0) the zero pressure capacitance, a the membrane length, h the membrane thickness and d the electrode separation. The pressure sensitivity for a piezoresistive transducer with a square membrane free from built-in stress is given by: aR ¼

DV a2 : Vp h2

ð3:15Þ

Comparison of Equations (3.14) and (3.15) reveals the capacitive sensor is always more sensitive to both membrane thickness and length than piezoresistive based pressure transducers. In particular, for the former decreasing the electrode separation d increases the sensitivity at the expense of the maximum measurable pressure for a membrane thickness h. The sensitivity to pressure of a piezoresistive pressure transducer is about 5 per cent of the full scale voltage reading obtained at the endpoint of the pressure range. Increasing the ratio a/h can increase this sensitivity. However, to do so reduces the membrane thickness (assuming a constant value of a) and increases the probability that pressure non-linearity will occur, complicating the calibration. 57

58

Pressure

Practical pressure transducer circuitry often introduces noise much larger than the theoretical detection limit estimated with b. For a capacitive device, if the noise determines the smallest detectable capacitance, DC, the pressure resolution for a capacitive pressure transducer dpC is then dpC ¼

DC h3 d 2 ; aC Cðp ¼ 0Þ a6

ð3:16Þ

while for a piezoresistive strain gauge based pressure transducer the resolution dpR is given by dpR ¼

DV h2 : VaR a2

ð3:17Þ

Equation (3.16) shows the resolution of the capacitive based pressure gauge is proportional to the ratio h3 d 2 =a6 , therefore, the smaller the distance d, the higher the resolution that can be achieved. Thus, capacitive based pressure gauges constructed with MEMS techniques can have an extremely high resolution by comparison with devices constructed by traditional means. For piezoresistive strain gauge based pressure transducers the performance is ultimately determined by the size and location of the strain gauge resistor; the location of maximum membrane stress resulting from an applied pressure can be estimated by finite element analysis. The minimum resistor size is determined for devices fabricated with MEMS technology, by both photolithography tracking errors and resistive heating effects, while the maximum dimensions are determined by the stress gradient, with respect to length, over the resistor [37]. In the case of both capacitive and piezoresistive based pressure transducers the pressure limit is determined by the rupture strength of the membrane material.

3.2

Piston Gauges C.D. EHRLICH and J.W. SCHMIDT National Institute of Standards and Technology Gaithersburg, Maryland, U.S.A.

There have been, since publication of Experimental Thermodynamics Volume II, significant advances in the technology and theoretical models associated with the use of piston gauges for the measurement of both pneumatic and hydraulic pressure. Most notably, the incorporation of computers and advanced electronics into commercial piston gauge systems has led to entirely new ways in which the basic piston gauge concept is used for pressure generation and measurement. Equally important, improvements in materials and manufacturing technologies have resulted in instruments of simpler design with superior performance over broader operating 58

Pressure

59

pressure ranges. For instance, the upper end of pneumatic pressure operation has been extended to pressures of about 100 MPa, and hydraulic piston gauges of simple design now operate to pressures of 0.5 GPa. The advent of analytical techniques such as finite element analysis and related computer modelling has led to improved understanding and design of piston gauges. Pneumatic piston gauges for operation at p&0:1 MPa are used as primary standards in some laboratories at uncertainty levels previously associated only with Hg manometers. This chapter will describe these and other recent advances. In their 1975 article of the same title as this chapter, Heydemann and Welch [38] described the prevailing philosophy and technology concerning the characterisation and use of piston gauges up to that time. A review of piston gauges was also presented in reference [39]. While the equations describing the basic piston gauge have not changed much, more refined models have been developed and reviewed [40– 44]. For pneumatic piston gauges these models focus primarily on the detailed flow of gas in the annular region between the piston and cylinder [45]. For hydraulic piston gauges, the focus has been on better predicting the detailed distortion of the piston and cylinder [46], both for improved design and to better understand the pressure dependence of the effective area. The Heydemann and Welch [38] treatment of controlled clearance piston gauges, while known to have its limitations, is still used effectively in many applications. The piston gauge designs described by Heydemann and Welch [38] have now been supplemented by digital piston gauges, higher pressure pneumatic piston gauges, and automated piston gauges, including a variety of sophisticated electronic sensors and data analysis hardware and software. Another significant area in which progress has taken place recently is measurement uncertainty, both in reduction of its magnitude and uniformity of its assessment. Publication of the Guide to the Expression of Uncertainty in Measurement [47], has contributed substantially to an internationally harmonised methodology for uncertainty calculations. The fabrication of pistons and cylinders with improved geometries, and the design of certain piston gauges that allow for the exchange of multiple components as part of the cross float procedure, have also led to an overall reduction of pressure measurement uncertainty. The proliferation of bilateral, multilateral and regional comparisons of piston gauges [48,49] has led to a global reduction in piston gauge uncertainties. This section is mostly concerned with the application of piston gauges to pressures above 0.1 MPa, while Section 3.4 describes the use of piston gauges at lower pressures.

3.2.1

Advances in Piston Gauge Manufacturing Technology

A piston gauge’s performance is primarily dependent on the qualities of the piston and cylinder assembly (piston-cylinder) and how the forces resulting from the applied pressure and the mass load are distributed.

59

60

Pressure Piston-Cylinder Geometry

Modelling, including finite element analysis of piston-cylinder behaviour under pressure, and manufacturing processes have improved. Previously, geometric quality was considered secondary to producing parts that behaved well under pressure, that is demonstrated high sensitivity and long spin times. Today, while good behaviour under pressure remains paramount, producing shapes that are as close as possible to ideal geometry has gained importance. Geometric defects cause piston-cylinder behaviour to deviate from models resulting in increased uncertainty. Improved piston-cylinder geometry has been achieved by the refinement of the techniques used to manufacture piston-cylinders. These include evolution in the basic manufacturing methods, in particular the lapping process by which a pistoncylinder is adjusted to its final dimensions and surface finish, as well as the techniques used to measure piston-cylinder shape and the width of the annular space. One advance in annular space measurement that has been commonly applied is the derivation of the piston-cylinder clearance from measurements of the flow of pressurised fluid through the annular space. The flow is determined from the downward displacement of the piston as its movement compensates for the loss of pressurised fluid through the annular space. Though the details of piston-cylinder manufacturing techniques are generally not published as they are considered trade secrets by the commercial piston gauge manufacturers that apply them, the results are evident in the quality of the piston-cylinders produced. High quality pistoncylinders typically achieve geometries that deviate by less than 0.5 mm from cylindrical geometry with annular separations of (0.5 to 1.5) mm and measured effective areas relatively within +40 ? 106 of nominal effective area.

Control and Reduction of Piston-Cylinder Annular Gap Emphasis has been placed on controlling and reducing the size of the annular space between the piston and the cylinder for two reasons. First, a reduction of uncertainty associated with the simplest definition of the effective area, namely, the algebraic mean of the geometric areas of the piston and cylinder, which when defined this way is necessarily located within the annular space. Apparent changes in the effective area of the piston-cylinder with the pressurised medium and operating mode have been observed. These changes, still poorly understood and difficult to quantify [50,51], increase the uncertainty in effective area for piston gauges used with different media over a wide pressure range or in different operating modes. Reducing the ratio of the annular space to the piston-cylinder radius reduces the relative influence of the annular space on the effective area. Second, reducing the size of the annular space decreases the flow of the pressurised medium through the annular space. As the flow is a cubic function of the annular gap for laminar viscous flow, even small reductions in the annular space result in significant reduction in piston fall rate and increased float time. Longer float times increase the time available to make measurements before readjusting the piston position. Currently, piston fall rates as low as 3 mm ? s1 60

Pressure

61

are available for gas operated piston gauges operated at pressures up to 350 kPa and about 1 mm ? s1 for oil operated piston gauges operated at pressures up to 200 MPa. New Piston-Cylinder Materials Although tungsten carbide remains the most frequently used material for high performance piston-cylinders, new materials have been tested. In particular, high purity ceramic has been used as the piston material in commercially available piston gauges. The low density of ceramic (about 1/5 that of tungsten carbide) allows the mass of a given piston size to be reduced, lowering the minimum mass load and thus the lowest operating pressure. When using ceramic pistons or cylinders, measures to avoid the build up of static charges need to be taken. These include grounding the piston-cylinder through the piston rotation system and the application, when possible, of metallic coatings to non-working surfaces.

3.2.2

Characterisation Techniques Effective Area from Dimensional Measurements

The pressure generated by a piston gauge can be calculated from the sum of the airbuoyancy-corrected weight of the masses and piston, divided by the piston-cylinder effective area through ( p¼

X i

)   . rair mi g 1  þ gC Aeff ðTÞ; ri

ð3:18Þ

where mi and ri are the mass and density of the weights, rair the density of the air, g the local gravitational acceleration, C the circumference of the piston, and g the surface tension of the operating fluid. Of course, when gas is used as the pressure medium g ¼ 0. The effective area, Aeff(T), might be obtained from a calibration of the gauge against either a transfer gauge whose calibration is known, or from a primary standard. A more complete discussion of the basic equations concerning piston gauges can be found in Heydemann and Welch [38] and are not repeated here. Recent advances in large diameter piston-cylinder manufacturing, combined with improved accuracy of dimensional measurements, allow the accuracy of the areas of large diameter piston gauges to compete with the accuracy of primary manometers. In particular, larger diameter gauges allow for the accurate measurement not only of the outer diameter of pistons but also of the inner diameter of cylinders. Several national laboratories can measure the diameter of pistons and cylinders with a standard uncertainty of better than 35 nm (one standard deviation) [52–55]. Thus, for an otherwise perfect cylinder with a diameter of 35 mm, a relative standard uncertainty in area of 2 ? 106 (one standard deviation) can be 61

62

Pressure

obtained. In favourable cases small shape deviations from perfect cylindricity might add relatively less than (1 or 2) ? 106 to the total uncertainty [56]. Deviations from cylindricity can be determined via straightness and roundness measurements, which can be performed with an uncertainty of 20 nm. The two types of measurements, straightness and roundness, are relative measurements in that they are referred to perfect geometry respectively but whose absolute slopes and diameters are not known. These measurements can be combined with the absolute diameter measurements to obtain a ‘bird cage’ that defines a three-dimensional object in absolute coordinates. Effective areas can be calculated based on the reconstructed bird cage by placing a ‘skin’ over the cage and integrating the forces acting over the surface with Dadson’s method [39,57,58]. Dadson and co-workers [39,57,58] separated the forces into three parts: (a) the upward force due to applied pressure on the base of the piston; (b) the upward force due to the vertical component of applied pressure on the flanks of the piston; and (c) the upward force due to fluid friction on the flanks of the piston assuming laminar viscous flow. Similarly, the effective area can be separated into components: Aeff ¼ A1 þ A2 þ A3 ;

ð3:19Þ

where A1 is the basal area of the piston, A2 is the contribution arising from the possible non-cylindrical shape of the flanks of the piston and A3 is the contribution from fluid friction on the flanks of the piston assuming laminar viscous flow. A1, A2 and A3 are defined by

1 A1 ¼ 2

Z2p

h i dy r2p ð0; yÞ ;

ð3:20Þ

0

1 A2 ¼ pð0Þ

ZL dz 0

1 A3 ¼ 2pð0Þ



Z2p dy

 drp ðz; yÞ pðzÞrp ðz; yÞ ; and dz

ð3:21Þ

0

ZL

Z2p dz

0



   dpðzÞ rp ðz; yÞ rc ðz; yÞ  rp ðz; yÞ ; dy dz

ð3:22Þ

0

where rp(z,y) and rc(z,y) represent the radii of the piston and cylinder as functions of the vertical coordinate, z, and azimuth angle, y. The local pressure p(z) within the annular space is a function of z, with z ¼ 0 defined as the bottom and z ¼ L as the top of the annular space. Inclusion of y introduces the possibility of processing complete dimensional measurements. For a gas, in which the density is proportional to 62

Pressure

63

pressure, the pressure profile p(z) is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Zz u p2 ðLÞ  p2 ð0Þ dz0 u 2 pðzÞ ¼ tp ð0Þ þ ; Iz h3 ðz0 Þ

ð3:23Þ

0

where ZL Iz ¼

dz0 ; h3 ðz0 Þ

ð3:24Þ

0

and h(z) is the mean width of the annular space at vertical position z. The areas of two large (35 mm) diameter piston gauges were determined and then compared in a round robin between four national laboratories [48,49]. The piston-cylinders were measured dimensionally by each laboratory and based on those measurement their effective areas calculated. Three participants claimed relative fractional standard uncertainties in the effective areas for these gauges of less than 5 ? 106 (one standard deviation). Although this figure is larger than the relative standard uncertainty goal of 2 ? 106 stated previously, the usefulness of these types of measurements is clear.

Modelling – Deformation Coefficient In the discussion above, the determination of piston gauge effective area from dimensional measurements applies at low-pressure, (p&0:1 MPa) because dimensional measurements are performed under isostatic conditions at this pressure. As the pressure increases, the effective areas may change depending on the design of the gauge, and the mechanical properties of the materials of construction. Estimates of the influence of the deformation on the effective area Aeff ¼ A0 ð1 þ lPÞ;

ð3:25Þ

are obtained from elasticity theory, evaluated with Young’s modulus and Poisson’s ratio for the material [59]. However, the analytical formulae are limited to gauges constructed with rather simple designs since even minor irregularities in the design can lead to complex mathematical models for the distortion of the gauge under pressure [60,61]. Equation (3.23) shows that gauges constructed from elementary designs may have non-linear pressure profiles within the crevice of the gauge, which might result in small variations in crevice width along the piston cylinder engagement length, and in the case of hydraulic gauges, introduce uncertainty from the pressure dependent 63

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viscosity of the fluid. Thus the boundary conditions may be complicated even with gauges that have simple cylindrical designs. Finite element analysis (FEA) can be used to model complex geometries. FEA essentially partitions an arbitrarily complicated solid object into small cubes, parallelepipeds or other relatively simple shapes for which the stress and displacement solutions are known exactly. The full solution for the whole object is then built up from these simple elements for which the stresses and displacements of the faces at one element are matched to the stresses and displacements on the faces of contiguous elements [46,62]. Finite element analysis techniques allow one to explore previously inaccessible lines of investigation such as: The effects of non-linear pressure profiles within the crevice; the stress distribution of more complicated gauge designs; and the effects arising from piston and cylinder barrel and/or hourglass type distortion. In addition, non axi-symmetric shapes can be analysed in more detail and with greater accuracy. Other Effects Two other effects in pneumatic piston gauges have been observed and are referred to as the gas species effect and the mode-of-operation effect [50,51]. Speculations as to their cause do not explain all of the observations, and indicate that further research is needed in this regard. Even so, well designed and made gauges (for example, those with small annular space-to-radius ratios) show only minor differences from the effective areas determined with the models described above. For example, among gauges of large diameter (50 mm) and small radial clearance (h < 0.4 mm), the differences in effective area between rival models are relatively less than 2:5 ? 106 [45,55]. Magnetostatic forces (in ferromagnetic materials) and electrostatic forces (in dielectric materials) can arise on the floating elements. If found to be present, the magnetostatic fields can be reduced by simple de-gaussing and the electrostatic fields reduced or eliminated by the deposition of conducting films onto the dielectric surfaces. As the precision and accuracy of pressures defined by piston gauges have improved, it has become possible to clearly demonstrate and quantify some effects that had previously been of an anecdotal nature. One such effect is known as the aerodynamic effect, which can occur for piston gauges operating in the gauge mode where the masses are surrounded by air or other gas. Prowse and Hatt [63] first observed this effect and its dependence on either the presence or absence of an enclosure surrounding the masses and on the diameter of the masses. Fortunately, this effect is very repeatable and the effect can be taken into account with very little increase in uncertainty [56].

3.2.3

Practical Developments in Piston Gauges

A number of practical developments have expanded the range of application of piston gauges and made them easier to use. These are described in this section. 64

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Application of Data Processing, Sensor and Automation Technologies The sensors can be add-on devices to monitor piston behaviour and ambient conditions and coupled with personal computer applications to measure and process operating parameters in real time. Piston gauges with integrated microprocessors and complete on-board measurements and data processing have been introduced that include standard communications interfaces for connection to a host computer. Piston gauge pressure generation and control accessories have also been developed that allow the piston to be floated automatically without operator intervention, with a mechanical arrangement like that shown in Figure 3.9. In some cases, masses can also be loaded and unloaded automatically resulting in a fully automated piston gauge.

Increased Range of Piston-Cylinder Sizes Improvements in piston-cylinder manufacturing and mounting techniques have increased the pressure range of piston-cylinder gauges. For gas piston gauges, pistoncylinders up to 35 mm in diameter (10 cm2 area) are common. This large diameter

Figure 3.9 Automatic mass loading pressure balance manufactured by Des Granges et Hust, Paris, France.

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allows lower pressures ( < 0.05 MPa) to be generated with sufficient mass to maintain reasonable spin times. At the other extreme, for p&0:5 GPa, smaller diameter tungsten carbide piston-cylinders reduce the mass required for high pressure oillubricated gauges. Figures 3.10 and 3.11 show two examples of piston/cylinder assemblies designed for high-pressure hydraulic operation. Simple Piston-Cylinder Mounting Systems In high pressure operation, the difficulties in predicting deformation under pressure have led to the general replacement of re-entrant piston-cylinder mounting systems with simple or free deformation mounting. In free deformation mounting, the cylinder is allowed to deform freely under the influence of the applied pressure. The deformation coefficient for a piston/cylinder assembly that uses a free deformation mounting is simpler and more reliably estimated than for a re-entrant design. Interchangeable Piston-Cylinder Modules To further simplify the interchange of piston-cylinders, piston-cylinder modules have been developed in which the critical mounting hardware required for the masses is integrated with the piston-cylinder. This arrangement allows each piston-cylinder to have its own dedicated mounting and be installed and removed from the piston gauge as a complete assembly without disturbing the mounting. Direct Operation with High Pressure Gas Piston gauges capable of operating with gas at p&100 MPa have been produced, which have reduced the uncertainty of high pressure gas measurements. Two different designs are commercially available. In one, the pressurised medium throughout the system and directly under the piston-cylinder is a gas, while a pressurised lubricant supplies oil to the piston-cylinder annulus. While this approach has the advantages of lower fall rates consistent with oil lubricated piston-cylinder assemblies, the oil vapour may migrate through the pressurised gas. In the second approach, the piston-cylinder is lubricated directly with the pressurised gas. While the risk of oil contamination is eliminated, fall rates are somewhat greater. With this second approach it appears that the thermodynamic properties (namely the heat capacity and thermal conductivity in the super critical region) of N2 gas, which is traditionally used as the pressure transmitting fluid, introduce several difficulties in the annular space between the piston and cylinder. Using He gas minimises or eliminates the difficulty. Digital Piston Gauges To combine the precision and high stability over time of the piston gauge with the operating convenience and real time reading capabilities of digital transfer standards, similar to those described in Section 3.1, piston-cylinder elements have 66

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Figure 3.10 High pressure piston and cylinder assembly manufactured by Ruska Instrument Corporation, Houston, Texas, USA.

Figure 3.11 High pressure piston and cylinder assembly manufactured by DH Instruments, Tempe, Arizona, USA.

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been combined with force measuring devices. Rather than loading mass on the piston, the mass is used to calibrate a load cell. Then, the force that results from the pressure applied to the piston-cylinder effective area is measured with the load cell, usually made from a strain gauge active element of the type used with top-loading mass balances. These ‘digital piston gauges’ allow the piston gauge principle to be used for real time measurement of an applied pressure without mass manipulation and lower the start point to zero as the mass of the piston can be tared out when zero pressure is applied. Digital piston gauges are available to cover pressure ranges up to 25 MPa.

3.3

Pressure Measurements in a Diamond-Anvil Cell E.H. ABRAMSON University of Washington Department of Chemistry Seattle, WA, USA

The pressure in a diamond-anvil cell is typically determined either through the diffraction of X-rays, by included materials for which the equations of state (EOS) are known, or through measurements of various optical transitions for which the shifts with pressure are known. Several reviews of pressure calibration at high pressure are available [64–67].

3.3.1

X-rays

Primary standards of pressure are derived from Rankine-Hugoniot curves, established in shock-wave experiments and reduced to isothermal equations of state. Use has been made of Cu [68–70], Mo [68], Pd [68], Ag [68,69], Au [70–72], W [73] and Pt [72,74]. Other X-ray standards include MgO [72,74], Re [75], and NaCl [76]. Pressures within the centre of a cell have occasionally been approximated by X-ray measurement of the surrounding Re gasket. NaCl has been used both as a pressure medium and calibrant. Equations of state for Ar [77,78], Ne [73,77] and Xe [79–82] have been published which allow these van der Waals solids to be used simultaneously as quasi-hydrostatic pressure media and pressure calibrants [83]. In the case of a non-hydrostatic load, monitoring only the strain along the lattice-vectors of the diffracting planes may cause significant error if this is considered to represent a strain due to hydrostatic stresses [84,85]. In the typical case of X-rays travelling along a direction of uniaxial loading, the average pressure will be under-estimated [68]. Meng et al. [86] have analysed this effect for cubic materials. They show that the problem is most acute for substances that are either elastically 68

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anisotropic or have large ratios of bulk to shear modulus. Their example of Au in Ne is particularly disturbing as Au is often used as a pressure calibrant. While typical measurements at higher pressures are made to precisions of 1 GPa, better can be done in the case of a hydrostatic medium. Angel et al. [87] have suggested quartz as a pressure standard for which, due to its low mosaic spread, they were able to attain a precision of 0.1 per cent at 9 GPa. 3.3.2

Optical Techniques Ruby

Observation of the pressure-induced shift of the R doublet in ruby fluorescence is by far the most common method of pressure measurement in a diamond-anvil cell. Piermarini et al. [88] calibrated the shifts of the ‘R1’ component of the doublet at a wavelength of 694.1 nm to 20 GPa, at room temperature, against the Decker [76] equation of state for NaCl. They found a linear dependence in wavelength with a slope of 2:74 ? 1018 Pa ? m1 and an uncertainty of 0.3 GPa at 20 GPa. The calibration was extended to 100 GPa [68] and 180 GPa [70] by comparison with the lattice parameters of Cu, Mo, Pd, Ag, and Au (with equations of state obtained from shock data) in an extremely non-hydrostatic environment. Measurements of ruby wave number against Cu and Ag were also taken in the quasi-hydrostatic medium of Ar up to 80 GPa [89]. The higher pressure data showed curvature in the pressurewavelength relation and were represented by the expression [68]; p=GPa ¼

o 1904 n ½1 þ ðDl=lref ÞB 1 ; B

ð3:26Þ

where lref is the wavelength measured at 0.1 MPa, and Dl ¼ l(p)  lref . For the nonhydrostatic case, B is given as 5 while for the quasi-hydrostatic case B is given as 7.665. At p&100 GPa the relative error dp=p&0:1 and 0.05 for the non-hydrostatic and quasi-hydrostatic respectively [67,68]. In both equations the slope at zero pressure was forced to fit that found by Piermarini et al. [88]. These two equations and the original calibration of Piermarini et al. [88] are all in current use with each commonly referred to as ‘the ruby pressure scale’; at room temperature and 10 GPa the differences among the three exceed the precision obtainable by almost an order of magnitude. At p < 10 GPa the principal determinant of all ruby scales is Decker’s equation of state for NaCl. This equation of state, generated solely from data at ambient pressure, departs systematically from high-pressure data by between (2 to 3) per cent and was considered ‘temporary’ when originally published [76]. Brown [90] has recently constructed another equation of state based on the more extensive thermodynamic data now available and has found it to differ by about 2.5 per cent from Decker’s at T ¼ 300 K. Not only the ruby scale, but all other scales which are based upon it, are uncertain by several per cent. 69

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The temperature dependence of the ruby fluorescence R lines has been investigated at temperatures up to 700 K, above which temperature a combination of broadening and fluorescence quenching make them less useful [91–93]. At T > 300 K the shift appears linear with temperature, with a slope of 7:7 ? 1012 m ? K1 . The mixed derivative d2l/(dp dT) is negligible [94–97]. The concentration of Cr, up to about 2 mol per cent appears to affect neither Dl due to temperature [91] nor that due to pressure shifts up to 20 GPa [88,94,96]. In a hydrostatic environment at T&293 K precisions of 0.02 GPa are obtainable [96,98], however, because a variation in temperature of 1 K will shift the wavelength by an amount equivalent to a pressure of about 0.02 GPa {(dp=dl) (dl=dT)%0:02 GPa ? K1 }, the temperature must be controlled to better than 1 K. At 473 K the precision degrades to about 0.05 GPa due to line broadening. The overlap between R1 and R2, the two components of the doublet, have been calculated as the peak intensities [91,93,96], the deconvolution of two Lorentzians, [92] and the deconvolution of mixed Gaussians and Lorentzians [99]. At elevated temperature it is common to calculate the pressure by measuring the wavelength shift from a ruby held at room temperature and compensating for dl/dT. However, this approach may confuse true line shifts with apparent shifts arising from changes in lineshape, particularly if the R1 wave number is determined as the position of maximum intensity. A ruby situated outside, but at the same temperature as the cell, may be used as the reference and the line centre wave numbers determined as described in [99] with all parameters adjusted to fit the measured data. Ruby ground to sizes suitable to the diamond anvil cell may harbour large strains which will cause a Dl equivalent to 0.2 GPa [98]; the individual grains of ruby should be examined for this effect prior to use. Inhomogeneities in the pressure medium can be manifested as broadening of the ruby lines. Non-hydrostatic, but uniform, stresses do not lead to broadening, but will cause line shifts which depend on the orientation of the ruby with respect to the stress [100,101]. Since the R2 line is less sensitive to the direction of stress it has been suggested that its use is preferable to that of the R1 line [100]. In a hydrostatic environment it is often assumed the doublet lines shift equally with pressure but differences have been predicted and measured [102,103]. Annealing by laser irradiation [104,105] can reduce non-hydrostatic stresses. Ruby fluorescence is typically excited with an Ar ion or He:Cd laser through the broad U or Y bands lying in the blue-green region of the visible spectrum [102]. As pressure increases these bands move to the blue [106] while the absorption edge in the diamond red-shifts [107]. At pressures above about 50 GPa pumping through even the lower U band becomes inefficient [108] and past 100 GPa fluorescence from the diamond can mask that of the ruby [109]. A discrimination can be made between ruby and diamond fluorescence by taking advantage of the different lifetimes [110]. At these higher pressures the R lines can also be usefully pumped through either the narrower B lines [108] or R0 lines [111]. Recently it has been claimed that two-photon absorption at a wavelength of 782 nm, obtained from a 5 mW diode laser, can stimulate R-line fluorescence at pressures up to 18 GPa [112]. 70

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Other Optical Calibrants The disadvantages of ruby fluorescence, primarily its significant shift with temperature and limited useful temperature range, but also its chemical reactivity and limited precision, have led to the development of numerous other calibrants. Before continuing, we wish to emphasise that as all these other substances have been calibrated against ruby, any adjustment to the one affects the others. Diamond is particularly useful because of its chemical compatibility with any substance held in the diamond anvil cell. The Raman line of diamond has been explored both as a function of pressure [113–115] and temperature [116,117]. The calibrations reported differ by about 1.5 GPa at a pressure of 40 GPa. Above about 13 GPa the Raman signal of the (strained) anvils is sufficiently removed from that of a grain of 12C diamond placed inside the diamond anvil cell as to make the grain useful for calibration. Shiferl et al. [118] have suggested the use of 13C diamond calibrant for lower pressures and have published data on the shift, along with a recipe for production of the material. Sm:SrB4O7 fluoresces into an isolated, narrow line at a wavelength of 685.4 nm with a pressure-induced shift comparable to that of ruby but roughly two orders of magnitude less sensitive to temperature [119–121]. It has been calibrated to 124 GPa and (separately) 900 K [122]. Although the linewidth does not increase unduly with temperature, the intensity falls slowly up to 600 K and rapidly beyond that. Synthesis of the material by solid state reaction [123,124] gives a powder; melt grown crystals of the host have been prepared [125]. Below about 2 GPa the relative precision of all pressure standards commonly used in the diamond anvil cell is quite poor. It has been demonstrated [126,127] that fluorescence from multiple-quantum-well structures in GaAlAs/GaAs offers precision in pressure an order of magnitude greater than ruby while also a factor of five less sensitive to temperature (up to 400 K) than ruby. However, the maximum measurable pressures are low, (1.8 to 3.5) GPa depending on well depth and spacing, due to a crossover from direct to indirect transitions. Quantum-wells of InAsP/InP offer similar precision and, up to room temperature, are useful to 10 GPa [128] above which pressure a phase transition occurs. These materials may be particularly useful in the study of highly compressible fluids where the largest uncertainties can be due to the limited precision of the ruby scale [98]. The Y1 fluorescence peak of Sm:YAG has a pressure induced shift comparable to that of ruby and linear to about 20 GPa, along with a negligible temperature coefficient [97,129,130]. Line intensity decreases slowly to 920 K, then more rapidly [129]. A fitting procedure has been suggested [131] which includes 10 different lines for work up to 1120 K. Beyond pressures of about 150 GPa the Y1 line becomes too weak to use, however Liu and Vohra [132] have followed the Y4 line to 300 GPa. The shift of the N2 Raman line has been measured for the fluid both in the diamond anvil cell to p ¼ 17 GPa and T ¼ 840 K and, in the shocked material, to p ¼ 41 GPa and T ¼ 5000 K. The data have been fitted as a function of pressure and temperature [133]; around room temperature the precision in pressure is 71

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about 1 GPa. The delta phase has been reported to exhibit a similar shift with pressure [134]. N2 can, thus, be used as a combined pressure medium and calibrant. Zinn et al. [134] have reported that in the presence of N2, Inconel 718 gaskets welded to the diamonds above 600 K while Re was observed to react directly with the N2. Klug and Whalley [135] have determined the pressure shifts of the antisymmetric stretches in infrared absorption spectroscopy of NaNO3 and NaNO2 in dilute solid solutions of NaBr; the spectra are measured at T ¼ 295 K to p ¼ 18.6 GPa with a scatter of < 0.2 GPa. Wong et al. [136] have calibrated the 801 cm1 line of crystalline quartz to p ¼ 5 GPa, while Grzechnik et al. [137] have suggested a stretch vibration of the carbonate ion in MgCO3 to p ¼ 29 GPa. Other means to measure pressure have been established for specialised circumstances. Determinations of the superconducting transition in Pb [138,139] and Bi [140] have been used at low temperatures. Substitution of diamonds by sintered B4C anvils in extended X-ray absorption fine structure (EXAFS) studies precludes the observation of ruby fluorescence and so the EXAFS spectrum of Cu has been used as a standard [141]. Through laser heating of samples, T > 3000 K have been attained. In such experiments, the pressures have sometimes been approximated by reference to a calibrant located within the diamond anvil cell but outside the irradiated area. The additional local pressure arising from laser heating can, however, be large [142,143]. For a sample held at p ¼ 40 GPa in an Ar pressure medium, and laser heated to T ¼ 2000 K, measurements by Fiquet et al. [144] indicate a pressure differential of as much as 9 GPa between ruby located on the periphery of the cell and a Pt X-ray standard in the centre.

3.4

Low Pressure C.M. SUTTON Measurement Standards Laboratory of New Zealand Industrial Research Limited Lower Hutt, New Zealand

A variety of pressure standards are available for the traceable measurement of low pressures in the range (0.1 to 1000) Pa but few of the higher accuracy types of instrument are well established. The uncertainties of the three most accurate types of standard are shown schematically in Figure 3.12. Interferometric liquid-column manometers are the most common type of standard, achieving standard uncertainties [47] in the range (1 to 50) mPa. Static expansion (or series expansion) pressure generators typically achieve a relative standard uncertainty of about (1 to 3) ? 103, making them more accurate than manometers for pressures below about 1 Pa. Recent developments using piston gauges (or pressure balances) have shown that they can be used with an accuracy similar to the best liquid-column manometers. Each of these three types of standard is considered in turn, with 72

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Figure 3.12 Standard uncertainties u as a function of pressure p for the low-pressure standards: 1 and 2, liquid column manometers filled with Hg; 3 and 4, liquid column manometers filled with oil; 5, single piston gauge; 6, force-balance piston gauge; 7, twin piston gauge; and 8, static expansion pressure generators.

particular emphasis on recent developments in liquid-column manometry and on emerging piston gauge techniques. Other standard instruments, such as noninterferometric and compression manometers [145], have inferior accuracy and are not discussed here. Transfer standard pressure transducers are also discussed since most pressure measurements in the (0.1 to 1000) Pa range are made indirectly. It is rare for the thermodynamic medium to be suitable for use as the operating fluid of the pressure standard. In some cases, a calibrated pressure transducer may be used to measure either the fluid or solid of interest. This is unavoidable when referencing pressure measurements to a static expansion pressure generator. Alternatively, a differential pressure transducer may be used to isolate the thermodynamic medium from the pressure standard. Care must be taken in designing and realising pressure measurements in the (0.1 to 1000) Pa range. Complexities particular to this pressure range are discussed in the text as they arise. Many of these complexities are due to the changing characteristics of both the gas used in the pressure standard and the fluid in the thermodynamic system being measured. In the (0.1 to 1000) Pa range, the nature of gas flow changes with increasing pressure from molecular through transition to viscous, making it hard to predict thermal transpiration corrections, gas interdiffusion rates and flow impedances. Gas densities change by 104 across this pressure range making it hard to predict head corrections in the presence of temperature gradients. Also, gas pressures are sufficiently low that ultra-high vacuum techniques may be required to keep impurity levels in the thermodynamic system at an insignificant level. 73

74 3.4.1

Pressure Interferometric Liquid-Column Manometers

Liquid-column manometers have been in use for measuring absolute pressures since Torricelli invented them in 1642. Yet they remain the most accurate pressure standard, achieving relative standard uncertainties of &3 ? 106 near atmospheric pressure. There are many excellent reviews of liquid-column manometers in the literature [3,146,147]. In the simple two-column manometer illustrated in Figure 3.13, the difference between pressures p1 and p2 is balanced by the difference in liquid-column heights h1 and h2 and takes the form p2  p1 ¼ r1 gh1  r2 gh2 ;

ð3:27Þ

where r1 and r2 are the average densities of the two liquid columns respectively, and g is the local gravitational acceleration. In principle, p2 is readily determined from measurements of the other parameters in Equation (3.27). In particular, heights h1 and h2, or the difference (h1 – h2), can be measured with high precision using interferometric techniques. To understand the difficulties of manometry, it is helpful to consider the properties of the manometer fluid. Hg is the most common manometer fluid because its high density allows convenient column heights (0.76 m of Hg is equivalent to &101 kPa). Hg is also easily purified by distillation and it has a highly reflective surface that can be used directly as a mirror in an optical interferometer. It also has a high acoustic impedance compared with most fluids. The most probable value for the density of Hg is 13 545:85 kg ? m3 at 293.15 K with a relative standard uncertainty of &1 ? 106 [148]. Unfortunately, Hg has four properties that complicate its use in manometry. First, Hg has a very high surface tension, which means that Hg manometers require large diameter columns to avoid significant capillary depression variations. The Hg column diameter must exceed about 45 mm for the capillary depression to be less than 10 mPa

Figure 3.13

Simple two-column manometer.

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pressure equivalent [147,149]. Second, the high density of Hg, in combination with its low viscosity, means that ripples are readily excited on the meniscus surface and are persistent. These surface disturbances are generated both by external vibrations and by capricious variations of the contact angle between the Hg surface and the tube walls as the Hg level is altered [150]. The high surface tension of Hg enhances the latter effect, making it difficult to track changes in meniscus height with traditional laser interferometers, which cannot tolerate any interruption in the reflected laser beam. As a consequence, the more accurate Hg manometers require vibration isolation and/ or some means for accommodating surface disturbances. Third, while Hg has a low thermal expansion coefficient by comparison with other liquids, at (1=r)(dr=dT) ¼ 1:81 ? 104 K1 it is sufficiently large that careful attention must be paid to both temperature measurement and temperature uniformity. For example, following Equation (3.27), if h1 is 50 mm and the uncertainty in the temperature of column 1 is 5 mK then the associated uncertainty in r1 gh1 is 6 mPa. Finally, Hg has a vapour pressure of 171 mPa at T ¼ 293.15 K. This high value has two consequences for lowpressure measurement. First, the presence of Hg vapour in the reference vacuum region (above column 1 in Figure 3.13) makes it difficult to accurately determine pressure p1 Equation (3.27). Calibrated capacitance diaphragm gauges are the preferred instruments for this measurement and are described in Sections 3.1 and 3.4.4. Second, when using an Hg manometer to measure low pressures where the gas flow is molecular or transition, the Hg vapour will diffuse through the pressure manifold at a significant rate and may contaminate the thermodynamic system under study. A differential capacitance diaphragm gauge can be used to separate the two systems but the risk of Hg contamination may be such that the capacitance diaphragm gauge cannot be properly zeroed [151]. Other issues to consider in manometry include the measurement of heights h1 and h2 (or the difference (h1  h2)), the verticality of both the columns and the height measurements, and variations in the tilt of the manometer. Height measurements are manometer specific and are discussed below. Vertical alignment of the columns and height measurements to within an angle of 1 mrad is straightforward and sufficient [152]. Tilt errors, however, can be significant because they depend on the sine of the angle and the spacing of the columns. For a manometer with two columns 0.1 m apart, a 1 mrad angular tilt along the line of the columns changes the manometer zero by 13 mPa. Variations in tilt can arise from changes in the level of the laboratory [153], from movement caused by changing the pressure which shifts the centre of gravity of the Hg, and from changes in any mounting system used to isolate the manometer from vibrations. Tilt errors can be greatly reduced with three or four column manometers [152]. Long-Range Hg Manometers High-accuracy long-range Hg manometers for measuring pressures up to at least 100 kPa continue to be actively developed because they are usually the instruments of choice for national primary pressure standards. Perkin et al. provide an excellent summary of the variety of types of state-of-the-art manometers [154]. Standard 75

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uncertainties for these manometers at 100 kPa are typically in the range (250 to 500) mPa [155–159]. Many of these Hg manometers are used to measure pressures below 1000 Pa. One type, the ultrasonic interferometric manometer, achieves a particularly high accuracy and is considered in detail in the next section. The best long-range nonultrasonic interferometric Hg manometers measure pressures below 1 kPa with a standard uncertainty of about 40 mPa. A dual cistern Schwien manometer, modified in some detail, has been shown to measure pressures below 1.5 kPa with a standard uncertainty of between (32 and 50) mPa [158]. In this manometer, capacitance sensors are used to detect the Hg surfaces and a commercial laser measurement system monitors the height of the movable cistern. Similar performance has been achieved with two other Hg manometers, both of which use laser interferometry to measure the column heights. The lowest standard uncertainty of the manometer of Alasia et al. at pressures down to 10 Pa is 42 mPa [160] while the standard uncertainty of the manometer of Harrison et al. below 20 kPa is about 40 mPa, limited by the measurement of (h1  h2) [161]. The last two manometers both use the Hg surface as the mirror in a lens/mirror cat’s-eye reflector to allow the position of the Hg surface to be measured in the presence of surface disturbances. The cat’s-eye reflector forms one arm of a Michelson interferometer. A float in the Hg supports the lens at its focal distance f from the Hg surface (Figure 3.14).

Figure 3.14 Schematic of the Cat’s-eye float reflector, which forms one part of a Michelson interferometer, to determine the position of the Hg surface in the presence of surface disturbances.

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Bennett et al. [150] consider the tolerance of the cat’s-eye reflector to variations in the angle of the Hg surface and to variations in the distance between the Hg surface and the lens. They show that the wavefront curvature w (expressed as a number of wavelengths l) resulting from an out-of-focus distance y can be calculated approximately from the relationship y&wlf 2 =r2 ;

ð3:28Þ

where r is the radius of the incident laser beam and f is the focal length. From measurements by the author, this relationship has been confirmed and the maximum value of w determined to be about 0.33. Beyond this, the interferometer fails. For example, with a 633 nm laser beam of 6 mm diameter and a 65 mm focal length lens, the allowed range of movement of the Hg surface is about +0.1 mm around the focal distance. Gas bubbles trapped to the float supporting the cat’s-eye lens can be a problem. Alasia et al. [162] have investigated float designs to minimise this effect. Harrison et al. [161] solved this problem by adding a chamber so that the Hg could be drained and then readmitted under vacuum. They also arranged the float carrying the cat’seye lens so that the laser beam reflects off a pool of Hg about 2 mm deep within the float, with the aim of suppressing vibration of the Hg surface. Short-Range Hg Manometers Some attempts have been made to develop short-range Hg manometers but few have achieved standard uncertainties below about 40 mPa. Mitsui et al. [163] constructed a 10 kPa manometer for gas thermometry, relating their measurements to a secondary Hg surface to avoid tilt errors. They used a white light interferometer with 13 mPa resolution to measure (h1  h2). Despite careful vibration isolation, the manometer was limited by vibration to an uncertainty of about 40 mPa. Alasia et al. [164] are developing a 5 kPa manometer using heterodyne laser interferometry and the cat’s-eye optics discussed earlier. Preliminary results in differential mode indicate standard uncertainties in the range (12 to 30) mPa for pressures up to 1 kPa. Ultrasonic interferometric manometers are the most accurate manometers for pressures below 1 kPa. The original Hg ultrasonic interferometric manometer was described by Heydemann et al. in 1977 [152]. It had three columns to reduce tilt errors, a range of 13 kPa, and 14 mPa resolution. After many years of development, Hg ultrasonic interferometric manometers can now achieve a standard uncertainty for pressures below 1 kPa of 3 mPa combined in quadrature with a relative standard uncertainty of 2:6 ? 106 , even with manometer ranges up to 160 kPa [165]. In the ultrasonic interferometric manometer, each column height is measured by transmitting a packet (or pulse) of sound containing (100 to 200) sine wave cycles into the Hg, from a transducer at the bottom of the column as shown in Figure 3.15. This packet is reflected by the Hg surface back to the transducer which, now acting as a receiver, measures the phase of the returned signal relative to the transmitted 77

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Figure 3.15 Schematic of a three-column Hg filled ultrasonic interferometric manometer.

signal. By measuring the phase for several different frequencies in the range (9.5 to 10.5) MHz, the column height can be determined using the method of exact fractions [156]. Length resolutions of order 0.01 mm are achieved. Ultrasonic interferometry has several advantages over other methods for measuring column height. Reflections of the ultrasound are reasonably insensitive to surface vibrations because of their relatively long 150 mm wavelength. Column heights are measured directly in contrast to optical interferometry where, except for white light interferometry, the column heights are integrated from measured changes in height. Hence, surface disturbances during pressure changes do not cause problems. In addition, there is no uncertainty due to gas refractivity, which can be a significant source of uncertainty in manometers with optical interferometry. A disadvantage of ultrasonic interferometry is the high relative temperature coefficient of the speed of sound, v, in Hg, (1=v) (dv=dT)&3:03 ? 104 K1 [166], which acts in the same direction as [r(Hg)]1 [qr(Hg)=qT]p . Overall, the relative temperature coefficient of an Hg ultrasonic interferometric manometer is (1=p) (dp=dT)&4:84 ? 104 K1 . With such a high temperature coefficient, temperature uniformity, stability and accuracy at the mK level is required to achieve mPa accuracy. A 1 mPa error will result if the temperature measurement of a 7.5 mm (1000 Pa) column of Hg is in error by 2 mK. To achieve temperature uniformity, the ultrasonic interferometric manometer is surrounded by 12 mm thick aluminium and 50 mm of expanded thermal foam [152]. At low pressures, the limiting uncertainties for the absolute Hg ultrasonic interferometric manometers are the diffraction correction and the equilibration of Hg vapour [165]. Oil Manometers Despite the obvious advantages of oil as a fluid for low-pressure manometry, there are few interferometric oil manometers in current use. The main advantages of oil are its low vapour pressure, low density, low surface tension and high viscosity. 78

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Synthetic oils for manometry are discussed by Peggs [167]. Diffusion pump oils are particularly suitable because they have vapour pressures well below 0.1 mPa. Oil densities are more than ten times lower than Hg, making it possible to measure the column heights with improved pressure-equivalent accuracy. Typical optical interferometers are limited by inherent non-linearities to an accuracy of about 2 nm [168]. With a two-column manometer, this corresponds to a pressure uncertainty of &0:4 mPa for Hg and &0:03 mPa for oil (DC704 – tetraphenyltetramethyltrisiloxane). The combination of low surface tension and high viscosity gives oil much lower capillarity and less sensitivity to vibrations. The main disadvantage of oil is its ability to dissolve gases, making it hard to establish its density with a relative uncertainty below &5 ? 105 . The low reflectivity of oil also complicates interferometry. In addition, oil has a relative temperature coefficient of density (1=r)(dr=dT)&  7:2 ? 104 K1 (for DC704) [167] which is higher than for Hg. Poulter and Nash describe an interferometric oil manometer for measuring pressures from (0.3 to 6) Pa with a standard uncertainty u(p) ¼ 0:7 mPa þ 2:5 ? 104 ? p [153]. Oil vibrations are damped over the measurement area by reducing the oil depth in each of the two columns to about 0.5 mm with a glass surface immersed in the oil. A novel polarising interferometer measures changes in the height of each column via interference between the light reflected from the oil and glass surfaces respectively. The dominant uncertainties are due to interferometer resolution and hysteresis, capillary effects and the refractive index of the oil. Unfortunately the range of this manometer cannot be increased much beyond 8 Pa because of the reduced efficacy of the ripple suppression technique [167]. Ueki and Ooiwa [169] have developed a four-column oil manometer. Heterodyne laser interferometry is used to measure the difference in column height for each pair of columns, with the laser beam reflected directly off the oil meniscus surface. A standard uncertainty of u(p) ¼ 1 mPa þ 1:6 ? 105 ? p is estimated for gauge-mode operation and pressures up to 1 kPa, dominated by the correction for refractive index and the uncertainty in the density of the oil. The manometer has not been operated in absolute mode. A four-column oil ultrasonic interferometric manometer has recently been constructed for pressures up to 140 Pa and shown to achieve a standard uncertainty u(p) ¼ 1:5 mPa þ 1:8 ? 105 ? p [151]. This manometer is only marginally more accurate than the Hg ultrasonic interferometric manometer discussed earlier because the phase non-linearities in the oil are an order of magnitude larger than with Hg and offset the benefits of the lower-density oil. The major advantage of the oil ultrasonic interferometric manometer is the absence of Hg vapour, which greatly simplifies calibration procedures and data analysis. The oil ultrasonic interferometric manometer is designed for good performance at low pressures [152]. Four 100 mm diameter columns are used for immunity to tilt errors, each 25 mm high and arranged in a 110 mm square. The columns are machined in a solid aluminium block for mechanical rigidity and thermal uniformity. Thermal shields of aluminium and expanded-foam insulation surround the manometer. A turbo-molecular pump evacuates the region above each of the two 79

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reference columns and a spinning rotor gauge is used to measure this pressure. Allmetal plumbing and manifolds are used with a minimum diameter of 18 mm. A baffle at a temperature of 223 K prevents oil migrating from the manometer to the calibration manifold. The oil ultrasonic interferometry is the same as that described earlier for the Hg ultrasonic interferometric manometer except it operates at frequencies near 5 MHz (rather than 10 MHz) because of the relatively large ultrasonic attenuation of the oil. Rather than determine the speed of sound in the oil and the oil density, the oil ultrasonic interferometric manometer is calibrated by comparison with an Hg ultrasonic interferometric manometer. A performance assessment of the oil ultrasonic interferometric manometer showed that the zero repeatability was about 0.3 mPa and the zero reading drifted by no more than 0:1 mPa ? h1 . The pressure above the reference columns was about 0.6 mPa. Three non-random effects dominate the overall uncertainty; phase nonlinearity errors (which are bounded by +3 mPa), the scale factor (determined by calibration against a Hg ultrasonic interferometric manometer with a relative standard uncertainty of 1:5 ? 105 ), and 3 mK temperature uncertainties (equivalent to a 105 relative standard uncertainty in the measured pressure). The 140 Pa maximum pressure of the oil ultrasonic interferometric manometer was dictated by the attenuation of ultrasound in the oil and the electronics available in 1977 [151]. Subsequent improvements in the electronics should make it possible to operate an oil ultrasonic interferometric manometer with a range in excess of 1 kPa.

3.4.2

Piston Gauges

The application of piston gauges to pressure measurement below 1 kPa has been limited, even though piston gauges are widely accepted fundamental standards for high pressures from 2 kPa to 0.5 GPa. We consider here the characteristics of gasoperated piston gauges that allow them to accurately determine low absolute pressures. A detailed treatment of gas- and oil-operated piston gauges (pressure balances) for higher pressures is given in Section 3.2. A piston gauge simply consists of a loaded piston in a matching coaxial cylinder. The piston is freely rotated and aerodynamic forces centre it in the cylinder so that there is no physical contact between the piston and cylinder. Gas-operated piston gauges will continue spinning in this way for several hours. Sometimes the geometry is inverted with the cylinder floating instead of the piston. Piston gauges are an alternative to liquid-column manometry for low-pressure measurement, and have two distinct advantages. First, piston gauges have a lower temperature coefficient than liquid manometers. The temperature coefficient a , of the piston gauge effective area A, is a ¼ (1=A)(dA=dT). For typical piston gauge materials a &22 ? 106 K1 for steel and a &9 ? 106 K1 for tungsten carbide. In practice, piston gauges require no special temperature environment beyond a temperature-controlled laboratory. Second, piston gauges can operate with shortterm repeatability of < 1 mPa in the presence of background vibrations and without 80

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any vibration isolation [170]. The short-term stability of a piston gauge is illustrated in Figure 3.16(a), which shows the difference in pressure Dp between two piston gauges, following a deliberate disturbance to one of them. After the disturbance, the amplitude of the oscillation in Dp decreases to about (10 to 20) mPa determined by background vibrations. The value of Dp can be accurately estimated in the presence of these vibration-induced pressure oscillations by averaging the measured values of Dp over an integral number of cycles. This is illustrated in Figure 3.16(b), which shows the four-cycle average of the data in Figure 3.16(a). When the amplitude of the variation in Dp is less than 50 mPa, the variations in the four-cycle average are predominantly random with a standard deviation of about 0.5 mPa. Several approaches have been tried to overcome the one major disadvantage of piston gauges, that is their inability to operate at pressures below a limit determined by the mass of the floating element. This limit is typically between (2 and 10) kPa for conventional gas-operated piston gauges. Inclined piston gauges have been successfully developed for pressures below 2 kPa but with uncertainties of greater than 100 mPa [171,172]. Force-balance and twin piston gauges have been more successful and these are considered in turn.

Figure 3.16 Short-term stability of a gas-operated piston gauge; (a), Intercomparison between two gas-operated piston gauges as a function of time after a deliberate disturbance in one. (b), the four-cycle average of the results shown in (a).

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Pressure Force-Balance Piston Gauges

Ultra-light pistons were used in the first approach to force-balance piston gauges, with the piston in the form of a thin disc suspended from a force-measuring device such as a microbalance. In this way, Ernsberger and Pitman [173] measured vapour pressures in the range (0.007 to 2.7) Pa with an uncertainty of better than 1 per cent. Subsequently, Warshawsky [174,175] developed several similar instruments with different pressure ranges. For example, one instrument with a 113 mm diameter piston measured pressures in the range (0.0002 to 6) Pa with an accuracy equivalent to a standard uncertainty u(p) ¼ 0:005 mPa þ 0:008 ? p. There are practical problems with these instruments, such as ensuring that the piston is not in contact with the cylinder. They also have a large leak rate and a large uncertainty in effective area associated with the large 170 mm annular gap between the piston and cylinder. More recently, Rendle and Rosenburg [176] have developed a 7 kPa forcebalance piston gauge. This instrument, which is shown in Figure 3.17, has a resolution of 2 mPa for pressures up to 2 kPa. The uncertainty of this instrument has yet to be assessed but is expected to be uðpÞ&2 mPa þ 3 ? 105 ? p for pressures up to 1 kPa, that is similar to the uncertainty in the 3.2 kPa differential and gauge pressure version [177]. In this instrument a 76 mm diameter piston is supported centrally in

Figure 3.17 Force-balance piston gauge developed by Rendle and Rosenburg [176] for operation at pressures up to 7 kPa.

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83

the cylinder by flexure hinges at each end. To cope with the relatively large, 25 mm, piston-cylinder annular gap, gas is pumped out of this annulus near the low-pressure end by a secondary pump. This reduces the gas load on the turbomolecular pump used to evacuate the low pressure reference region, which in this case is below the piston, allowing a reference pressure of 1 mPa to be maintained for measured pressures up to 2 kPa. For convenience, the secondary and turbomolecular pumps are combined into a hybrid pump. The high resolution of this force-balance piston gauge is achieved by supporting the piston on an electronic mass balance, giving a direct readout of the resultant force from the pressure to be measured acting on the piston. A counterbalance mass in the balance mechanism is adjusted to balance out the mass of the piston. A servocontrol mechanism within the electronic balance tightly constrains the vertical position of the balance pan, and hence of the piston, minimising variations in the measured pressure due to the stiffness of the flexure hinges supporting the piston. Conventional Piston Gauges Conventional piston gauges can be used to measure pressures below the limit set by the mass of the piston, either by using one piston gauge in combination with a reference pressure gauge or by using two piston gauges. In the first approach, which uses a piston gauge in combination with a reference pressure gauge, the pressure p to be measured is established in the region above the piston gauge and the pressure pr generated by the piston gauge is monitored by the reference pressure gauge as shown in Figure 3.18. Suitable gauges for measuring pr are described in both Sections 3.1 and 3.4.4.

Figure 3.18 Schematic of a piston gauge combined with a reference pressure gauge, which measures the pressure generated by the piston gauge, to determine pressure p.

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Initially, the pressure pr indicated by the reference gauge is recorded, while the piston gauge operates with the region above it evacuated to pressure pv and with a pre-determined small mass m added to it. Equation (3.18) gives pr pr ¼

ðMT þ M þ mÞg þ pv ; Aeff ðTÞ

ð3:29Þ

where MT is the piston mass and M the mass of the weight hanger. Mass m is then removed from the piston gauge, while it continues to operate under vacuum, and the pressure in the region above the piston gauge is increased to a value p to restore the reference gauge to its original reading and Equation (3.29) becomes pr ¼

ðMT þ MÞg þ p: Aeff ðTÞ

ð3:30Þ

Here it is assumed that Aeff does not vary significantly with changes in p. Hence this method relies on the modern high-quality piston-cylinder assemblies discussed in Section 3.2. From Equations (3.29) and (3.30): p¼

mg þ pv : Aeff ðTÞ

ð3:31Þ

In practice, reference readings (with the region above the piston gauge evacuated) are recorded before and after each measurement of pressure p to allow for small temporal changes in pr or in the pressure gauge zero. Using this approach, Grohmann and Lee [178] reported an accuracy equivalent to a standard uncertainty u(p) ¼ 150 mPa þ 3 ? 105 ? p for pressures from 30 Pa to about 6 kPa, with the performance limited by the reference gauge. More recently, Sutton et al. [179] have reduced the standard uncertainty of this approach to u(p) ¼ 14 mPa þ 7 ? 106 ? p by careful attention to the operating conditions of the piston gauge, described below, and operating with a higher precision reference pressure gauge. A commercial 120 kPa quartz Bourdon tube pressure gauge was used, with the temperature control of the quartz sensor improved by an order of magnitude to a standard deviation of 10 mK. By averaging the gauge reading over an integral number of cycles of the piston gauge oscillation, the piston gauge pressure pr &100 kPa was monitored with 10 mPa resolution. Pressure pr was found to change at &7 ? 105 Pa ? s1 and its average value over about 40 s was determined with a standard uncertainty of less than 10 mPa. The dominant standard uncertainties in p are due to short-term repeatability of pr (&10 mPa), interpolation of the reference value of pr at the time of measurement of p (&10 mPa), and measurement of Aeff ð&7 ? 106 ? pÞ. Sutton et al. [179] also presented a simpler version of this approach with a slightly larger standard uncertainty u(p) ¼ 14 mPa þ 9 ? 106 ? p that is much easier to 84

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implement. Instead of using added mass m, pressure p is simply determined from the change in the pressure gauge reading as the pressure in the region above the piston gauge is increased from pv to the desired value. This method relies on the accuracy and stability of the pressure gauge sensitivity, which is easily calibrated in the same configuration. In this case, the dominant standard uncertainties are due to shortterm repeatability of pr (&10 mPa), interpolation of pr (p&0) at the time of measurement of p (&10 mPa), pressure gauge calibration (&7 ? 106 ? p), and stability of pressure gauge sensitivity (&5 ? 106 ? p). In the second approach, which uses two piston gauges, the pressure to be measured is established in the region above one piston gauge in terms of a small mass m added to the other piston gauge. Sutton et al. [179] have shown that this approach is capable of realising pressures up to 10 kPa with a standard uncertainty u(p)  2 mPa þ 9 ? 106 ? p when operating the piston gauges at 100 kPa. This approach is illustrated in Figure 3.19. The two piston gauges are set up to generate nominally equal pressures p1 and p2, and a differential pressure sensor measures the residual pressure difference Dp. A pre-determined mass m is added to piston gauge 2 and the desired pressure p is established above piston gauge 1 by adjusting it to give Dp & 0. The working equation for the two piston gauge method shown in Figure 3.19 is given by p ¼ Dp þ ðp2  p1 Þ þ

mg þ pv ; Aeff;2 ðTÞ

ð3:32Þ

where Aeff,2(T) is the effective area of piston gauge 2 at its operating temperature T and pv is the pressure in the region above piston gauge 2. Quantities g, Aeff,2(T), and pv are measured, while the value for (p2  p1) is obtained from a ‘zero’ differential

Figure 3.19 Schematic of a two piston gauge configuration, for measurement of pressures up to 10 kPa [179].

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pressure measurement recorded with no added mass m and with the region above both piston gauges evacuated. For the zero measurement, p ¼ pv &0 and Equation (3.32) reduces to Dp ¼  ðp2  p1 Þ:

ð3:33Þ

In practice, (p2  p1) varies slowly with time and hence the value of (p2  p1) when p is measured is normally interpolated from prior and subsequent zero measurements. A similar approach was proposed independently by Lloyd [180] and by Dadson [181], although both suggested operating the two pressure balances with a common pressure p1 ¼ p2. Without the differential pressure sensor, this configuration would be difficult to keep in balance. The performance of this method, which depends largely on four factors, is determined by the stability of (p2  p1) and the accuracy with which it can be measured. First, vibration disturbs the pressure generated by a piston gauge but, as discussed earlier, the average value of (p2  p1) can be determined by integral cycle averaging to better than 1 mPa. Fortunately, these disturbances have no significant effect on the pressures in the regions above the piston gauges. Second, the electrical charge on the floating piston can cause (20 to 50) mPa step changes in (p2 p1), and is easily eliminated by discharging the piston [170]. Third, the pressure generated by each piston gauge depends on the height of the piston within the cylinder. For a perfect piston gauge operating with nitrogen gas at 100 kPa, this height dependence is linear &11:4 Pa ? m1 . Fourth, changes in the temperatures of the two piston gauges will change the value of (p2 p1). These changes can be minimised by linking the two piston gauges together thermally. These effects and the stability of (p2  p1) are discussed further in [170]. A twin piston gauge instrument suitable for use with this approach was developed earlier by Sutton for the accurate generation of small gauge pressures [170]. This instrument has several features aimed at minimising variations in (p2  p1) including: A high thermal conductance base to keep the temperature difference between the pressure balances small; compensating gas flows to offset the loss of gas through each piston-cylinder gap, allowing the piston gauges to operate at nearly constant height for several minutes; height sensors for the floating elements with 1 mm resolution and 10 mm repeatability; and a device for discharging the floating elements before measurements. The twin pressure balance instrument also includes a mechanical mass loader for adding or removing ring-shaped masses from one of the pressure balances while it is operating. Large pumping ports in combination with a diffusion pump allow the regions above the pressure balances to be evacuated to pressures below 10 mPa. The dominant uncertainties associated with realising pressures below 1 kPa with the twin piston gauge instrument are expected to be due to effective area Aeff;2 (&7 ? 106 ? p), measurement of Dp (&5 ? 106 ? p), and the piston gauge heights (&2:0 mPa). 86

Pressure 3.4.3

87

Static Expansion Pressure Generators

Static expansion generators, first proposed by Knudsen in 1910, are the most accurate pressure standards for pressures below &1 Pa. In these instruments, low pressures are generated in a gas by expanding a pre-determined amount of substance from a small volume into a larger evacuated volume. The generated pressure is calculated from the initial pressure and the ratio of the two volumes. Usually, the initial pressure is in the range (2 to 100) kPa, where it can be measured with a piston gauge or a manometer, and a series of expansions are used to reduce the pressure to the desired value. Pressure generation by expansion is a well-established technique, with excellent summaries in the literature [145,182] (see also section 5.4.3). Static, or series, expansion, while simple in principle, has three limitations. First, it cannot be used to directly measure pressure. Instead, static expansion is used to calibrate reference pressure transducers. Second, the performance depends upon the gas used. Inert gases such as Ar, He, Ne and N2 work well because their physical adsorption to the walls is limited. He, Ne and N2 each have the added advantage that the variation in density r with pressure p deviates from an ideal gas by only about 5 ? 104 ? r. Other more reactive gases, such as H2 and O2, are more strongly adsorbed to the walls and can be used but with reduced accuracy [145]. Recently, Jitschin et al. have shown that the effect of adsorption with O2 can be kept below 0.1 per cent for pressures above 0.1 Pa by pre-conditioning the walls of the vessel with the gas [183]. Finally, the volume of the pressure transducer under calibration affects the generated pressure. This volume is usually determined from the transducer dimensions or from an additional gas expansion. A state-of-the-art static expansion system for pressures from (105 to 103) Pa is described by Jitschin et al. [183,184]. For noble gases, this system generates pressures with a relative standard uncertainty of 103 for pressures between (0.1 and 10) Pa, and a relative standard uncertainty of 5 ? 104 for higher pressures. Starting pressures in the range (2 to 300) kPa are established with a quartz Bourdon tube pressure gauge in combination with a piston gauge. A 0.1 L volume and several 1 L and 100 L volumes are used in various combinations and sequences to give volume expansion ratios from 1:1 ? 102 to 1:6 ? 107 . Corrections are applied for temperature changes during the gas expansions. Volume ratios are determined both gravimetrically, with water, and by in situ successive gas expansions, in which the pressure is measured both before and after the expansions [185]. To minimise outgassing [182], the whole apparatus is constructed using ultra-high vacuum techniques and is routinely baked to T ¼ 673 K. Prior to use, it is evacuated with a turbomolecular pump to a base pressure below 106 Pa. The dominant uncertainty in static expansion systems is due to the volume ratios. Each volume ratio can be determined in situ with a relative standard uncertainty between (2 and 20)?104 [183,185] and when several expansions are used, these uncertainties accumulate. In practice, an in situ measurement of the volume ratio is necessary because the volumes of the connecting valves can change with use [182]. The next most significant uncertainty is due to temperature changes during gas expansion, both during the determination of the volume ratios and during 87

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normal use. Assuming isothermal conditions can result in relative uncertainties of order 103 in pressure [186]. Correcting for temperature changes to within 0.1 K contributes a relative standard uncertainty of 3 ? 104 to the pressure [183].

3.4.4

Pressure Transducers

There are five main types of pressure transducer suitable for high accuracy measurement of gas pressures from (0.1 to 1000) Pa. All have been described in the first paragraph of Section 3.1. They all respond to the force of an applied pressure difference and, provided no chemical reactions occur between the gas and the materials used to construct the pressure transducer, the readings are independent of the gas used. Temperature changes can affect the performance of all these gauges. As a consequence, capacitive diaphragm gauges and quartz Bourdon gauges are normally operated with their temperature regulated at about 318 K [45 8C]. Quartz resonance gauges have an internal quartz-crystal temperature sensor for thermal compensation. MEMS resonant Si gauges and MEMS piezoresistive Si gauges are temperature compensated. Capacitive diaphragm gauges, quartz Bourdon gauges and quartz resonance gauges are all precision instruments that perform best when they are kept free from mechanical or thermal shock, or over-pressure. Pressure transducers operated at a different temperature from the system they are measuring will be non-linear because of thermal transpiration. Unfortunately, most of this non-linearity occurs in the pressure range from (0.1 to 100) Pa and is gas

Figure 3.20 Long-term stability of low-pressure transducers over a period of about 6 years. d is the magnitude of the average relative shift in the calibration factor and pFS is the full scale range of the particular transducer. e, capacitance diaphragm gauge (CDG); r, NIST capacitance diaphragm gauge; , Quartz Bourdon tube gauge; 4, Quartz resonant gauge; &, MEMS piezoresistive silicon gauge; and 5, MEMS resonant silicon gauge.

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89

species dependent. For example, a capacitive diaphragm gauge elevated in temperature by 23 K is about 2 per cent higher in sensitivity at 1 Pa than it is at 10 Pa with Ar, but only 1 per cent higher with He [187]. Thermal transpiration, which can be approximated by the Takaishi-Sensui equation [188], is described by Poulter et al. [189]. Miiller [187] reports an extensive study of the performance of these pressure transducers. Figure 3.20, taken from this study, gives an overview of their long-term stability. The average fractional change in transducer sensitivity between successive calibrations is plotted as a function of the transducer full-scale pressure. The period between calibrations was typically one to two years. The long-term stability of capacitance diaphragm gauges is poorer than other types of pressure transducers, but capacitance diaphragm gauges have better resolution and zero stability, in part because of their availability with lower full-scale ranges [190]. The noise-limited resolution of a capacitance diaphragm gauge discussed in Section 3.1, is about 106 of the full-scale pressure pFS [187], which is about 104 Pa for a 133 Pa transducer. The zero stability is less predictable and is influenced by room temperature. Zero drifts of order &+ð3 ? 107 ? pFS Þ h1 have been observed [187]. An excellent guide to the calibration and use of capacitance diaphragm gauges is given by Hyland and Shaffer [191]. Adams [192] has reviewed the designs, methods of construction, of capacitive pressure gauges that operate at pressures between (103 to 107) Pa with relative resolution of 108 at cryogenic temperatures. The high resolution of capacitance diaphragm gauges makes them well suited for measuring the small pressure difference between a thermodynamic system under study and a reference pressure standard. In use, the pressure in the pressure standard is adjusted to give near zero capacitance diaphragm gauge reading, to avoid uncertainties due to capacitance diaphragm gauge scale errors. For the highest accuracy, a valve across the capacitance diaphragm gauge, together with isolating valves on each side, allow the capacitance diaphragm gauge zero to be checked periodically without cross-contaminating the fluid under test system and the pressure standard. After the zero check, the mixed gas in the capacitance diaphragm gauge is pumped or flushed out either via the pressure standard manifold or via a fourth valve. Waxman and Chen [193] have described a capacitive based differential pressure gauge that was used successfully in a (p, V, T) apparatus. The movement of differential pressure diaphragm in response to pressure differences can be determined from other measurements, for example, a linear variable differential transformer [194]. In this application, the capacitance diaphragm gauge may be operated at the temperature of the fluid system and a low-density gas such as He used as the pressure medium between the capacitance diaphragm gauge and the pressure standard. In this way, the fluid system can be kept at a uniform temperature and in a well-defined volume, and the head correction is reduced in magnitude. Accurately estimating the head correction can be difficult if any of the connecting manifold between the thermodynamic system and the pressure standard changes height and temperature at the same time, which is usually the case in cryogenic systems. 89

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Quartz Bourdon tube gauges and quartz resonant gauges have the best longterm stability, with relative changes is sensitivity between (104 and 105) a1. The resolution of a typical quartz Bourdon tube gauge with a 120 kPa pressure range is limited to about 300 mPa by the system used to control the temperature of the helical quartz sensor. As discussed earlier, Sutton et al. [179] have reduced this resolution to 10 mPa or &107 ? pFS by improved temperature control.

References 1. Experimental Thermodynamics Vol II, B. Le Neindre and B. Vodar, eds., For IUPAC, Butterworths, London, 1975. 2. Sullivan, J.J., J. Vac. Sci. Technol. A 3, 1721, 1985. 3. Pavese, F. and Molinar, G.F., Modern Gas-Based Temperature and Pressure Measurements, Plenum Press, New York, p. 514, 1992. 4. Goepel, W. Hesse, J. and Zamel, J.N. eds., Sensors, A Comprehensive Survey, Mechanical Sensors Vol 7, H.H. Bau, N.F. de Rooij, B. Kloeck eds., VCH, New York, 1994. 5. Proc. Trans. Eurosens. Conf. Eurosensors XIII, Netherlands, September 12–15, 1999. 6. Mosser, V., Suski, J., Goss, J. and Obermeier, E., Sensors & Actuators A 28, 113, 1991. 7. French, P.J. and Evans, A.G.R., Sensors & Actuators 7, 135, 1985. 8. Dell’Acqua, R., Dell’Orto, G. and Vincini, P., in Proc. 3rd Eur. Hybrid Microel. Conf. Avignon, p. 121, 1981. 9. Morten, B., Prudenziati, M. and Taroni, A., Sensors and Actuators 4, 237, 1983. 10. Puers, B., Paszczynski, S. and Sansen, W., Sensors and Actuators 12, 57, 1986. 11. Wallis, G. and Pomerantz, D., J. Appl. Phys. 40, 3946, 1969. 12. Cozma, A. and Puers, B., J. Micromech. Microeng. 5, 98, 1995. 13. Kanda, Y., IEEE Trans. Electr. Dev. ED-29, 64, 1982. 14. Jackobsen, E. and Baum, J., Designing Sensor Performance Specifications for MCUbased Systems, Motorola Semiconductor Application Note AN1556/D, 1996. 15. Jackobsen, E., A Cookbook Approach to Designing a Differential-Signal Amplifier for Sensor Applications, Motorola Semiconductor Application Note AN1557/D, 1996. 16. Wang, Z.Z., Suski, J. and Collard, D., Sensors & Actuators A 47, 628, 1995. 17. Dorda, G., J. Appl. Phys. 42(5), 2053, 1971. 18. Chau, M.T., Dominguez, D., Bonvalot, B. and Suski, J., Sensors & Actuators A 60, 86, 1997. 19. Greenwood, J.C., J. Phys. E. Sci. Instrum. 17, 650, 1984. 20. Greenwood, J.C., Sensors and Actuators A 31, 164, 1992. 21. Greenwood, J.C. and Wray, T., Sensors and Actuators A 37–38, 82, 1993. 22. Corman, T., Enoksson, P. and Stemme, G., Transducers ’97 1, 101, 1997. 23. Enoksson, P., Stemme, G. and Stemme, E., MEMS ’96 San Diego, USA, Feb. 11–15, 156, 1996. 24. Enoksson, P., Stemme, G. and Stemme, E., IEEE/ASME J. of MEMS 5(1), 39, 1996. 25. Karrer, H.E. and Leach, J., IEEE Trans. Ind. Electron. Contr. Instru. IECE-16, 44, 1969. 26. Ward, R.W. and EerNisse, E.P., Proc. 41st Annu. Freq. Contr. Symp. 334, 1987. 27. Matsumoto, N., Sudo, Y., Sinha, B. and Niwa, M., Proc. Joint Meeting of 13th EFTF and IEEE International Freq. Contr. Symp. 2, 1019, 1999. 28. Vig, J.R., LeBus, J.W. and Filler, R.L., Proc. 31st Annu. Freq. Contr. Symp. 31, 131, 1977.

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43. 44. 45. 46. 47.

48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.

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Mixture Preparation and Sampling Hydrocarbon Reservoir Fluids M. HIZA Delta – MH, Inc. Story, WY, USA J. NIGHSWANDER Schlumbeger Evaluation and Production Services Dyce, Scotland, UK A. KURKJIAN Schlumberger Oilfield Services Sugar Land, Texas, USA 4.1

4.2

Preparation of Fluid Mixtures 4.1.1 Apparatus 4.1.2 Purity Considerations 4.1.3 Preparation of Gas Mixtures 4.1.4 Preparation of Liquid Mixtures Sampling Hydrocarbon Reservoir Fluids 4.2.1 Hydrocarbon Reservoir Fluids and Formations 4.2.2 Reservoir Fluid Sampling 4.2.3 Advances in Sampling Techniques

Measurement of the Thermodynamic Properties of Single Phases A.R.H. Goodwin, K.N. Marsh, W.A. Wakeham (Editors) # 2003 International Union of Pure and Applied Chemistry. All rights reserved

98 99 100 101 104 105 106 110 118

4

MIXTURE PREPARATION AND SAMPLING HYDROCARBON RESERVOIR FLUIDS

It is essential to prepare mixtures of accurately known composition, from pure components, that represent mixtures often used in industry or are of theoretical interest. In particular, the preparation of mixtures with components in different phases at T ¼ 293 K and p ¼ 0.1 MPa are described in Section 4.1. Physical property measurements on such fluids are significant in developing thermodynamic models. This chapter does not directly address sampling of fluids within measurement schemes in particular those used for phase equilibrium measurements which are described in Volume VII of this series. Obtaining samples of natural fluids, that are representative of those found in sub-surface strata under extreme conditions, and of particular process streams are desirable and are described in Section 4.2. Measurements on sub samples are useful for both economic appraisal and the validation of models that require compositionally characterized fluids.

4.1

Preparation of Fluid Mixtures M. Hiza Delta-MH, Inc. Story, WY, USA

Gas mixtures, with accurately known compositions are required for a wide range of scientific and technical applications: to calibrate gas chromatographs, mass spectrometers, and other analytical instruments for measurements of mixture composition; to calibrate instruments for the measurements of thermophysical properties, such as phase equilibria, heat capacity, viscosity, thermal conductivity, relative permittivity, sound speed, (p, V, T) properties including both orthobaric and single phase fluid densities. Components can be blended into a fluid mixture by partial pressure, by volumetric measurements, or a combination of both at a known temperature. The most reliable way to produce a mixture, with accurately known composition, is to determine gravimetrically the amount of each component, regardless of how each component is added to the blend. In any case, the mixture must be prepared in a single phase and, if at all possible, maintained in that phase after preparation. The following discussion will emphasize equipment requirements, component purity considerations, preparation procedures related to gas blends of non-corrosives, and the importance of air buoyancy corrections. Applications requiring intermediate gas mixtures will be addressed along with the significance of sample homogeneity. Methods for preparing mixtures in the liquid phase will be discussed only briefly.

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Mixture Preparation and Sampling Hydrocarbon Reservoir Fluids 4.1.1

99

Apparatus

The equipment required for the preparation of fluid mixtures will vary somewhat depending on the procedure followed. For gravimetric gas mixture preparation the equipment listed here is intended to be as general as possible, but the system established by the Experimental Properties of Fluids Group at the National Institute of Standards and Technology (NIST), in Boulder, Colorado will be used as an example. {Davis and Koch [1] have reviewed methods of mass and density determination and the interested reader should consult reference [1] along with the more recent comprehensive review [2] of mass metrology and Section 5.6.}{ For gravimetric gas mixture preparation at NIST, to determine the mass of each component added to a mixture, an equal arm balance was used where each pan has a capacity of 25 kg, with a relative precision of +3:2 ? 107 (about 8 mg in 25 kg). The balance is constructed without a center post and capable of weighing objects at least 1 m high, which is sufficient for a 15.7 L aluminum gas cylinder fitted with a valve and valve protector handle. The capacity of the balance and the conservative estimate of precision is optimum for a wide range of mixture needs. The weight set provided with the balance was Class S with tolerances defined for that class [3 parts in 105 for masses of 10 g and above and 0.03 mg for masses of (1 to 10) g inclusive]. The weight set was calibrated against mass transfer standards certified by the NIST Gaithersburg laboratory responsible for maintaining those standards. The seamless aluminum compressed gas cylinders are ideal sample containers for non-corrosive gas mixtures because of both the high volume to mass ratio and lower water adsorption described in Section 4.1.2. However, the cylinder valve must be capable of sealing the cylinder connections at high vacuum in both the open and closed positions. The vacuum equipment for the sample cylinder must be capable of pumping the container to a pressure of about 105 Pa. The blending system only requires a vacuum pump capable of providing p  4 Pa for both purging and evacuation. Pressure gauges in the blending system need to cover the range from subatmospheric to the highest cylinder and sample holder pressures. A vacuum gauge, with a maximum reading of about 100 Pa is adequate to assure that the manifold is leak tight and can be evacuated and purged to maintain sample component purity. An indispensable part of the NIST blending system is a top loading electronic scale with a capacity sufficient to weigh the 15.7 L aluminum compressed gas cylinders to within +103 kg, but more important, capable of being tared with a sensitivity of 104 kg. This instrument, when calibrated with 2 kg mass, allows one to weigh each component into the sample container to some predetermined amount for a desired target composition. The sample cylinder is connected to the blending manifold with flexible capillary tubing which isolates the cylinder during weighing.

{ The text in curly brackets has been added by the editors to elucidate some points but the text in those brackets does not represent the views of the author.

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Mixture Preparation and Sampling Hydrocarbon Reservoir Fluids

An electro-chemical oxygen analyzer capable of determining O2 concentration from trace to per cent levels is located within the blending system. Baker et al. [3] described their efforts at Monsanto to develop such an instrument to measure oxygen in hydrocarbons, for process plant applications, based on a British patent in 1954 describing an invention by Hersch. The oxygen analyzer in the NIST blending system meets all of the goals described by Baker et al. {Other equipment and procedures for gas mixture preparation are described by Nelson [4].}

4.1.2

Purity Considerations

The purity of each component, always of the highest purity available, must be verified before preparing mixtures. Gas chromatographic (gc) analysis of the sample should be performed with the appropriate detectors after preliminary tests. An example of a component for which one can use different confirming purity tests is ethane which is usually supplied as a liquid with vapour. Preliminary sample evaluation can be performed with measurements of the oxygen content in the vapor space as well as cylinder pressure and temperature. The presence of moisture will be considered later. If the oxygen mole fraction is between (2 to 4) ? 106 it is reasonable to assume that the sample has not been contaminated with air during the cylinder filling operation. Measurement of the cylinder pressure and comparison with the known vapor pressure at the cylinder temperature is a simple, but less precise method of determining if the vapour space is contaminated with a non-condensable gas, such as He. Often supply cylinders are filled with low pressure He gas during storage before being put in service. As an example, of the need for these precautions, a cylinder of research grade ethane was both analyzed for oxygen and the pressure measured. There was no significant oxygen detected, but the measured cylinder pressure was 0.7 MPa higher than the vapour pressure of ethane at the same temperature. Gc analysis of the sample with Ar carrier gas confirmed a He mole fraction in the gas space consistent with the measured pressure. {Undesirable components, that are gases at, for example, either T(CO2, s þ g, p ¼ 0.1 MPa) or T(N2, 1 þ g, p ¼ 0.1 MPa), at which the required fluid is either solid or liquid, can be removed by vacuum sublimation with a cold finger as described by Bell et al. [5].} H2O(g) contamination is always a potential impurity at various levels in samples of gases removed from supply cylinders. Weaver et al. [6] determined the H2O content of a standard steel cylinder, previously dried and filled by the supplier with O2 evaporated from liquid, as a function of cylinder pressure. The H2O content was about 4 ? 105 kg ? m3 (of O2 at standard temperature and pressure) for a cylinder pressure ps &7 MPa. However, at ps &2:1 MPa the H2O content of withdrawn gas was about 1.5 times higher and increased exponentially below that pressure. This suggests that compressed gas should not be used at ps < 2.1 MPa unless an adsorption dryer is used to reduce the moisture content to an acceptable level. {H2O(g) can be removed either when the gas is passed through or resides in 0.4 nm molecular sieve (zeolite) previously baked at T > 525 K and evacuated for at least 100

Mixture Preparation and Sampling Hydrocarbon Reservoir Fluids

101

24 h [7].} Weaver et al. [6] observed that an ordinary steel cylinder that had been in service for ‘dry’ gas for some time showed about the same trend. In these tests Weaver et al. [6] used O2(g) with 105 kg ? m3 H2O at standard temperature and pressure to evaluate the performance of dried and evacuated cylinders constructed from both aluminum and stainless steel. There was a measurable but no significant increase in H2O content when O2 was removed from either of these cylinders. This tends to support the selection of Al cylinders for the preparation of mixtures with non-corrosive components. Chemical compatibility of the cylinder material and reactive components included in the mixture must be determined and alternative materials selected on a case-by-case basis; this topic will intentionally not be addressed here. The results of Weaver et al. [6] suggest that one should prepare the sample cylinder to assure that the residual moisture is removed to the lowest level possible. The procedure, which this author has found reliable in this regard for aluminum cylinders, is done in deliberate steps starting with the factory-cleaned cylinder certified to be oxygen compatible. Before the cylinder valve is attached, the cylinder is flushed with pure C2H5OH heated and flushed with dry N2(g); the valve is cleaned separately in a similar way before being installed on the cylinder. The assembled cylinder is then connected to a high vacuum system and evacuated and purged with dry nitrogen at least three times, after the final flush with N2(g) at p&0:1 MPa is left in the cylinder to facilitate heat transfer. The cylinder is then heated to approximately 325 K; and evacuated at that temperature until the ultimate vacuum obtainable by the system is reached, which takes a variable amount of time but never less than 2 d. In order to verify the vacuum attained in the cylinder, the pressure is measured before and after isolating the cylinder at 325 K for about 12 h and, provided the pressure increase over 12 h is  7 ? 104 Pa, the cylinder is considered acceptably dry. Further evacuation and heating can be performed if necessary.

4.1.3

Preparation of Gas Mixtures

The phase behavior of the required mixture must be estimated to determine the dew curve pd and thus insure the maximum gas pressure at preparation temperature is below pd. There are numerous equation of state computer programs able to provide reasonably accurate prediction of the phase behavior of a wide range of mixtures including hydrocarbons, refrigerants, noble gases and water. While it is not the intention here to provide an exhaustive list of equations of state, this author has found programs available from the NIST Standard Reference Data (http:// www.nist.gov/srd) useful including a program known under the acronym NIST14. The chosen equation of state should be used to determine the dew pressure at 298 K or the ambient temperature at which the mixture will be prepared. At a somewhat arbitrary, 0.8 times pd, the amount-of-substance density is computed so that the mass of each component required can be determined from the mixture density for the specified cylinder volume. For mixtures formed at a T > Tc, where Tc is the critical temperature, of all components the preparation pressure is determined solely by the 101

102

Mixture Preparation and Sampling Hydrocarbon Reservoir Fluids

amount of substance required for the application. For the most accurate work, the required atomic masses of the elements used in these programs should be compared with the recently accepted values [8]. Mass determinations are usually carried out by balancing the unknown mass m against standard weights of mass Ms in an air filled cabinet, not in a vacuum, with a double pan equal arm balance, which when in equilibrium the m is related to Ms through     rðair; T; p; RHÞ rðair; T; p; RHÞ m 1 ¼ Ms 1  ; rm rs

ð4:1Þ

where rm is the density of the object m, rs the density of the standard masses and r(air) the density of air. {r(air) can be determined from measurements of T, p and relative humidity (RH) with an expression reported by Davies [9].} Miller et al. [10], described the weighing procedures and buoyancy corrections for an equal arm balance with a sample cylinder on one pan and standard masses on the opposite pan. Table 4.1, which is a synopsis of Table 1 and Table 2 of reference [10], lists buoyancy corrections for a specific mass, volume and air density for some selected gases. Buoyancy corrections are insignificant when an identical reference cylinder is placed on the balance pan and the sample cylinder is filled to only a few MPa [7]. The uncertainty in the gas composition, listed in Table 4.2 for 6 components significant to natural gas mixtures, is a linear function of the amount of substance and was determined assuming an uncertainty in weighing of +5 mg. The 24.75 mol column in Table 4.2 shows the uncertainty in preparing the four mixtures listed in Table 4.3. The mixtures listed in Table 4.3 were prepared by approximately determining the mass of each component added to a cylinder on a top-loading mass balance with a resolution of 0.1 g. Prior to addition of the next component the exact mass was obtained with a double pan swing balance. The results listed in Table 4.3 demonstrate the reproducibility of the composition obtained with this gravimetric approach. For the equal arm balance gravimetric determination, the sample cylinder Table 4.1 Buoyancy correction mb for a cylinder of volume Vc and mass mc filled with substance i, on one pan of an equal arm balance with masses of volume Vs all exposed to air of density r(air) in a procedure to produce a mixture of (H2 þ CH4 þ N2 þ He). The values were obtained from reference [10], and Vc was determined experimentally as a function of pressure. i

mc/g

106 Vc/m3

106 Vs/m3

rðairÞ=kg ? m3

103 mb/kg

p¼0 H2 H2 þ CH4 H2 þ CH4 þ N2 H2 þ CH4 þ N2 þ He

4767.2032 4767.5762 4779.8321 4956.2803 4989.4624

3550.00 3550.18 3550.29 3552.21 3554.81

567.30 567.34 568.80 589.80 593.75

1.03707 1.02466 1.01404 1.02419 1.02174

3.0933 3.0564 3.0234 3.0341 3.0254

102

Mixture Preparation and Sampling Hydrocarbon Reservoir Fluids

103

Table 4.2 Uncertainty d in the molar composition of each component i in a gravimetrically prepared mixture, as a function of total amount of substance n, assuming an uncertainty in each mass determination of +5 mg. n i

CH4 N2 CO2 C2H6 C3H8 n-C4H10

1 mol

5 mol

24.75 mol

0.0312 0.0178 0.0114 0.0166 0.0113 0.0086

+ (102 d)/mol 0.0062 0.0036 0.0023 0.0033 0.0023 0.0017

0.0013 0.0007 0.0005 0.0007 0.0005 0.0003

was weighed with an identical reference cylinder on the reference pan and thus buoyancy corrections were neglected. Binary mixtures can be duplicated with significantly smaller composition differences than those listed in Table 4.3. For either compositionally complex mixtures, for example, complex hydrocarbon mixtures representative of crude oil, which combine both gas and liquid mixtures, or those with trace components (mole fractions 2 0 ¼ 2. For narrow tubes and conditions where the dt and dv are not small by comparison with tube radius these solutions are not accurate [2] and alternatives must be sought. Gillis [153] also reported an expression for the perturbation of the gas inlet tube that was located in the top plate of the cylinder. The perturbations to the resonance frequency from the tube were less than +15 ? 106 [153] and comparable with the uncertainty in determining u. A corrections for the motion of the shell was not included in the analysis of the measurements probably because there is no exact solutions to the problem or even detailed approximations to it. Based on the expressions developed for a spherical resonator, it is expected that the correction is proportional to fluid density and small provided the frequency does not coincide with the resonance of the shell. Trusler [2] has discussed the magnitude of the correction and the coupling in his text. 263

264

Speed of Sound

The cylindrical resonator described by Gillis et al. [151] and Gillis [153] is shown in Figure 6.4. The cylinder was 14 cm long and has an inner diameter of 6.5 cm. Gillis [153] used measurements with argon to determine the length L(T) from purely longitudinal modes and the radius b(T) from purely radial modes and or pure azimuthal modes. The excitation of non-axisymmetric modes in a cylindrical resonator requires the source be offset from the centre of an end plate or on one on the side walls. Gillis [153] had both a source and detector in the top end plate. These were formed from diaphragms flush with the inner surface of the resonator and connected via waveguides to transducers located outside the thermostat. The diaphragms were used to separate the gas within the waveguide, which was argon, from that in the resonator. The operational frequency range of the cavity was

Figure 6.4 Schematic cross-section through the cylindrical resonator R located in a stirred fluid thermostat [151,153]. The temperature of the cavity was determined with a platinium resistance thermometer (PRT). Waveguides connected the resonator to the source S and detector D of sound that were located outside the thermostat. Diaphragms flush with the upper inner surface of the cylindrical cavity separated the gas within the waveguide from that in the resonator.

264

Speed of Sound

265

restricted to between (1 and 8) kHz by the waveguides. This transducer arrangement will be described in Section 6.1.4. The resonance frequencies were determined using the procedure described for a spherical resonator above. This resonator has been used to determine the thermophysical properties of numerous gases [174–185] including those relevant to semiconductor processing. Measurements of the speed of sound have been proposed to determine the composition of metalorganic chemicals used in vapour deposition processes [186] and the quantity of arsine obtained from an electrochemical process [187]. Ewing et al. [149] described a cylindrical resonator, shown in Figure 6.5, that operated at frequencies about an order of magnitude higher than the resonator described by Gillis [153]; this apparatus is described here to illustrate other methods of analysis and measuring the resonance frequency. The path length of the apparatus [149] was about 100 mm and longitudinal modes of the cavity were studied to determine the sound speed at pressure below 100 kPa and temperatures between (250 and 330) K. The length of cavity was determined from measurements with argon. Each transducer was fabricated from a 50 mm by 12 mm cylinder of aluminium alloy. The active element B was a 10 mm by 0.1 mm thick piezoelectric disk attached with epoxy to the centre of the 30 mm by 2 mm diaphragm. The frequency response was flat. This arrangement provided reasonably efficient transducers at frequencies between (50 and 90) kHz that covered the first overtone at about 70 kHz where the quality factor was about 50. The acoustic cavity C was constructed from a 117 mm long 230 mm diameter 1.6 mm thick tube. It was terminated by 3.2 mm plates. The capsule platinium thermometer was located at D. Gas entered the cavity through E. The transducers were rigidly mounted in brass rings G. The lower transducer could be adjusted relative to the upper to ensure correct alignment. The transducers were connected via a fiber glass disk K and spring loaded screw J to glass-to-metal seals L and the body of the cavity which was also connected to another pin at L. Output from a synthesiser was applied to both the source, after amplification to 10 V, and an oscilloscope and the frequency measured with a counter. The output was connected to the other trace of an oscilloscope. The location of the maxima in the received signal was taken as the fl of the longitudinal mode. In their analysis [149], the resonance frequency of the longitudinal modes of the cavity, given by Equation (6.41), was recast as fl;m;n ¼

u ðl þ dÞ; 2L

ð6:48Þ

where d represents the phase shift arising from non-zero fluid velocity at the transducer faces and presence of unresolved modes with w not equal zero. The (u/L) and d were estimated from two adjacent frequencies fl and fl þ 1 and these used to locate the other frequencies in the operating range. Equation (6.48) does not formally account for the energy loss from irreversible flow of heat and momentum in the bulk of the gas and at the resonators walls. The resonance frequencies, half widths and the speed of sound are, for the longitudinal mode of a closed cavity, 265

266

Speed of Sound

Figure 6.5 The cylindrical acoustic interferometer of Ewing et al. [149]. A, transducer; B, piezoelectric disk; C, acoustic cavity; D, thermometer well; E, inlet tube; G, transducer mounting ring; H, beryllium-copper springs; I, spacer rod; J, electrical connection; K, fibreglass disk; and L, glass-to-metal seal.

strictly given by [149]   1=2 (      )  ul f 1 1 kRT 1=2 1 4ZRT 1=2 þ fl;0  gl;0 ¼ ð g  1Þ þ þ ð1  iÞ 2L p 2b L pCp;m 2b 3Mp ; u i a 2p ð6:49Þ 

266

Speed of Sound

267

where k and Z are the thermal conductivity and shear viscosity of the gas and the elastic response of the cavity has been neglected. In Equation (6.49) a is the absorption coefficient in the bulk gas and does not contribute to f for a monatomic gas. The difference between the frequencies of Equation (6.49) and the unbounded gas {Equation (6.41)} were assessed by Ewing et al. [149] including the complication of unresolved modes; there was some evidence that the longitudinal modes were coupled to the radial modes. For argon they concluded that within the experimental precision, the corrections were not justified. However, when the path length was obtained with argon and then used to calculate the perfect gas heat capacity at constant pressure of another gas the boundary layer correction is significant. This was demonstrated by Ewing et al. [188] for 2,2-dimethylpropane where uncertainties of about 1 per cent were introduced into the acoustically determined heat capacity if the calibration measurements were not analysed with Equation (6.49). A cylindrical cavity described by Younglove and Fredrick [150] is rather simple to construct and is similar to that reported by Younglove and McCarty [167] and shown in Figure 6.6. An acoustic model was used to correct the frequencies of the longitudinal modes in reference [150]. The cylindrical cavity was formed from a 67 mm long 10 mm internal diameter tube with 1 mm think walls. The electrostatic transducers formed the end plates of the cylinder. The ends of the cavity were parallel to within 0.003 mm over the cylinders diameter. The end plates also acted as solid dielectric capacitance transducers by stretching over them 0.013 mm thick polyimide, plated with gold facing into the cylinder. A disadvantage of this arrangement is that the impedance of the transducers becomes important in determining the resonance frequencies and the half widths. In this resonator, the

Figure 6.6 Cross-section through the cylindrical acoustic resonator of Younglove and McCarty [167] with schematic of the electronics used to excite and detect the resonances. Solid-dielectric capacitance transducers formed from metallised polymer film and disks were used as the cylinder end-caps.

267

268

Speed of Sound

source was set at a frequency close to a resonance in the range (40 to 70) kHz and the source modulated at about 100 Hz, which was sufficient to cause the source to sweep through the resonance. A lock-in amplifier was used to compare the detected amplitude modulated signal with the original modulation and the frequency where the phase difference was zero taken as the resonance frequency and the drive frequency determined with a counter. An alternative and simple measurement scheme, shown in Figure 6.6, with an oscilloscope and oscillator, was used by Younglove and McCarty [167] and Ewing et al. [149]. When this frequency measurement scheme is combined with either mechanical or acoustic measurements of the path length speeds of sound with a precision of between (0.1 to 0.01) per cent are obtained. The cylindrical resonators discussed have not explored the extremes of temperature and pressure or dimensions. Based on these criteria, there are two other cylindrical cavities that need be mentioned. Firstly, Carey et al. [165] have described cylindrical resonators with radii in the range (2.5 to 6.3) mm that were used to determine the sound speed and transport coefficients with, because of the small diameters, an emphasis on the transport properties. This apparatus was operated at temperature up to 1000 K. It was also operated at pressures up to 50 MPa but the upper temperature was limited to 620 K. At the extreme of length, Zuckerwar and Griffin [148] have described a cylindrical cavity of length about 17 m to measure the absorption of sound at low frequency-to-pressure ratios that cover the dispersion curve. The apparatus was operated at frequencies in the range 10 Hz, the fundamental in nitrogen, to 2500 Hz at pressures up to 10 MPa and temperatures in the range (293 to 394) K. The absorption was determined from the free decay of a standing wave after removal of the excitation. This apparatus was also used to study sound absorption in air [189]. We now consider variable path-length interferometers where the sound speed is determined from measurements of the change in length between successive half-wave lengths of longitudinal modes at a constant frequency. An example of the classical single-transducer high-frequency small-wavelength interferometer was described by Henderson and Peselnick [155] and is shown in Figure 6.7. The stainless steel piston and cylinder in this interferometer formed an acoustic cavity 25.4 mm in diameter with a length that was varied by a micrometer over a distance of about 75 mm. To over come the issue of seals, the micrometer was mounted within a pressure vessel. The combined piston and reflector assembly were ground flat to better than 1/4 wavelength of light when the piston was at the minimum displacement. The x-cut quartz transducer was optically flat and gold plated. It was mounted in the open end of the cylinder with a spring. The transducer operated at frequencies between (0.3 and 7) MHz. This wide frequency range allowed a comprehensive study of the dispersion and absorption of sound in CO2 up to liquid densities. The longitudinal modes are relatively insensitive to the parallelism of the end faces. However, at high frequencies, these effects are important because they lead to unwanted mode coupling. The variable path-length interferometer of Colclough et al. [160], shown in Figure 6.8, was operated at a frequency of about 5.6 kHz within a 30 mm diameter 268

Speed of Sound

269

Figure 6.7 A single-transducer variable path interferometer operating at a frequency of about 1 MHz of Henderson and Peselnick [155].

bore. The low frequency and narrow bore were chosen so that the apparatus operated below the first cut-off frequency and avoided the unresolved higher modes. However, this approach required rather large boundary layer corrections and placed severe demands on the accuracy with which the thermal conductivity and shear viscosity were know and the corrections to the resonance frequencies were made with the measured half-widths. The cavity was formed from brass and fitted with a nickel and tin plated brass piston. The piston was moved by means of a motorised lead screw assembly over a rather long distance of about 150 mm that was enough to cover about 5 half wavelengths. The piston displacement was measured with an optical interferometer. A moving-coil driven aluminium alloy diaphragm was used as the transducer. It was fitted with an accelerometer to determine its displacement 269

270

Speed of Sound

Figure 6.8 The low-frequency (about 5.6 kHz) variable-path interferometer of Colclough et al. [160] designed for an acoustic determination of the gas constant at T&273:16 K from sound speed measurements at pressures between (0.03 and 1.3) MPa.

amplitude. The wavelength was then determined with a relative precision of about 3 ? 105 from the analysis of admittance circles. Gammon and Douslin [34,159] described a two-transducer cylindrical variablepath length interferometer for sound speed and absorption measurements. This apparatus was used to study the speed of sound at pressure up to 27 MPa and temperatures from (98 to 420) K. The apparatus is shown in Figure 6.9 and had an inner diameter of about 51 mm and a movable piston that was keyed to prevent rotation. A quartz crystal transducer that operated at a frequency of 500 kHz was attached to the piston and at the base of the cylinder an almost identical crystal was placed. The transducer mounting surfaces were lapped in insure that they were parallel. The separation of the transducers, of between (50 and 100) mm, was measured with a micrometer that also served to move the piston. Small corrections, based on Del Grosso’s calculations [190–192] were used to account for guided-mode dispersion. The precision of the sound speed measurements was found to be about 105 ? u for helium [35,159] and it was also used to study methane [34,161] at temperatures in the range (113 to 423) K at pressures up to 27 MPa and on both sides of the gas and liquid phase boundary [34]. 270

Speed of Sound

271

Figure 6.9 The double transducer variable path interferometer of Gammon and Douslin [159]. Left: The micrometer with a precision of 0.3 mm and an accuracy of 1.3 mm is used to move the piston 10. It consists of a mechanical counter 1, a vernier scale 2, and a packing gland 3 that also holds a bellows coupling for the rod that connects to the top of the right hand figure at position A. Right: The interferometer consists of two x-cut quartz crystals 13, the upper 13 mounted on the movable piston 10 within the cylinder 11. The piston is connected to a micrometer screw 9 and through a bearing 8 and drive shaft 7 to a tempering block 5 and then to the assembly at left. Electrical feedthroughs 6 connect the crystal to the external electronics and a thermometer is located in a well 12 on the outside of the pressure vessel.

A cylindrical variable path interferometer operating at 156 kHz was reported by Zhu et al. [162] and used to measure the speed of sound at pressure up to 0.6 MPa. The piston in this apparatus could be moved over 20 wavelengths. The accuracy of the sound speed obtained from this instrument after correction for diffraction, guided mode dispersion, and the Kirchoff-Helmholtz boundary layer, was determined with argon and found to be about 3 ? 104 ? u. This result was consistent with the estimated error budget. The speed of sound in gases can also be determined with high precision by the time-of-flight of a pulse of sound over a known distance through a fluid. This 271

272

Speed of Sound

approach is best suited to dense fluids, can be used in dilute gases and have been described in references [4,6,7,17,20]. The distance the wave travels in the fluid can be determined from measurements with a fluid of known u or with a micrometer [193], which limits the precision of the measurements to about 0:002 ? u. Pulse methods work well for absorption measurements in fluids at relatively high frequencies (of about 1 MHz) where diffraction corrections are relatively small compared with the absorption of the fluid. At lower frequencies, measurements of attenuation by the pulse method are subject to large losses from beam spreading and become inaccurate. In that regime the reverberation technique offers an alternative. Many pulse methods used to measure the speed of sound in fluids have been adapted from those used to measure the elastic moduli of solids [14–16]; the significant difference is that a solid may have anisotropic elastic properties. The fluids are usually contained within almost rigid cavities of cylindrical geometry for time-of-flight measurements. Numerous instruments have been described to perform time-of-flight measurements of the speed of sound with a variable number of pulses, reflectors and transducers. A description of the working equations and methods can be found in reference [2]. Here two instruments are described that have been applied to gases: One that uses two transducers and multiple reflections between them and another that used one source transducer and two reflectors positioned at different path lengths from the source. The speed of sound determined from the observed time-of-flight of a pulse over a (total) distance will be in error due to diffraction. This error arises from the phase advance of the sound wave relative to a plane wave traversing the same distance. The measured pulse attenuation will also be error because of beam spreading. These particular techniques have also been applied to determining the speed of sound in liquids and will be discussed, along with corrections for diffraction, in Section 6.1.2. A time-of-flight instrument was developed by Greenspan and Tschiegg [194] with two matched transducers and used to determine the speed of sound in water. The method was subsequently used by Younglove [195] to measure the speed of sound in gases, including p-H2 at pressure up to 30 MPa, with a standard relative uncertainty of about 5 ? 104 . An x-cut quartz [2] crystal emitted a sound pulse and a matched x-cut crystal detected the pulse after it had travelled a known distance. A tube of fused quartz, about 25.4 mm long, separated the two crystals and determined the path length of the apparatus; quartz is advantageous because of its relatively low coefficient of thermal expansion. The end surfaces of the crystal were flat to within a few micrometers. The apparatus was mounted inside a pressure vessel similar to that shown in Figure 9.2. The emitting crystal was driven by a pulse of about 40 V ac for a duration of about 0.1 ms. This pulse excites the crystal at its resonance frequency of 10 MHz over about 40 oscillations. The acoustic pulse is reflected between the two crystals several times. The input pulse repetition frequency was then adjusted to cause constructive interference with the pulses. The time of travel between the transducers is then equal to the time between pulses. The speed of sound u was obtained from u ¼ 2df ; 272

ð6:50Þ

Speed of Sound

273

where d is the distance between the inner surfaces of the crystal and f the pulse repetition frequency. This approach has advantages over the direct measurement of time-of-flight because it avoids pulse shape distortion [2]. This apparatus was also used by Straty [196] to determine the speed of sound in saturated liquid methane from (91 to 186) K and fluid methane at temperatures between (100 and 300) K at pressures up to 35 MPa. Kortbeck et al. [38] described a dual path single transducer instrument for sound speed measurements with a precision of 0.02 per cent in gases at pressure up to 1 GPa. Ultimately, the accuracy of the method is determined by the knowledge of the path difference and accounting for errors arising from diffraction effects. This method is also known as the phase comparison pulse echo. In this apparatus, shown in Figure 6.10, an edge supported quartz crystal is located across a cylinder and used to generate pulses which travel from both transducer surfaces in opposite directions toward identical reflectors placed distances L1 (the distance between Q and the upper R in Figure 6.10) and L2 (the distance between Q and the lower R in Figure 6.10) from the source. The returning echo is also detected by the same transducer. After a time a second pulse is introduced so that the echo of the first pulse travelling over the longer path and the echo of the second pulse travelling the shorter path coincide at the transducers. The phase shift between the two echoes f is given by Df ¼ 2pft;

ð6:51Þ

Figure 6.10 The single transducer two reflector apparatus of Kortbeck et al. [38] for pulseecho measurements of the speed of sound on fluid samples at pressures up to 1 GPa: K, Kel-F polymer rod that supports the apparatus within a pressure vessel; C, conical cavity attenuator; R, reflector; Q, quartz crystal transducer clamped at its edges between conical support S and conical T; and E, a silver pin electrical connector pressed against the transducer by a phosphor bronze spring.

273

274

Speed of Sound

where t is the difference in transit time over the distance 2(L1  L2). The frequency of the transmitter is varied until the two pulses cancel each other at the phase difference Df ¼ 2ðn þ 1Þp;

ð6:52Þ

where n is an integer. Complete cancellation occurs when the amplitude of the second pulse is adjusted to be equal that of the first, that has travelled a greater distance and thus suffers greater attenuation. The corresponding null frequencies are given by ft ¼ n þ 0:5;

ð6:53Þ

where n is the order of interference and can be determined from the difference Df of successive null frequencies. The speed of sound is then given by u ¼ DL=t:

ð6:54Þ

The advantage of using two identical reflectors is that the phase shift and pulse shape distortions on reflection cancel. An alternative approach to determine the speed of sound with two reflectors is to simply determine the differences in path-length and travel-time and this will be discussed in Section 6.1.2. The transducer is clamped at the edge by a conical support and conical ring as shown in Figure 6.10. The path lengths through the fluid, in the ratio of 2 to 3, were chosen to eliminate undesirable echo overlap. The reflectors were copper, each with a plane smooth surface fixed as parallel as possible to the transducer. Copper was used for the reflectors because of its high acoustic impedance, high thermal conductivity and known thermal expansion. The length of each reflector was chosen so that unwanted reflections returning from the outer surface of the reflector reached the transducer after the arrival of the two echoes. In addition, conical cavities were placed at the ends of these reflectors to scatter and attenuate the signal transmitted in the reflector. The electronics used for this measurement must be arranged so that the method used to attenuate the second pulse does not introduce significant phase errors [39]. This apparatus was used to determine speed of sound in argon [38], nitrogen [39], neon [197], and methane [198]. Annular and Other Geometries Low frequencies are required for gases where intramolecular relaxation is slow (typically at low pressure) and for studies with strong sound attenuating gases, that occur close to the liquid-gas critical points. These low frequencies can be achieved with impractically large radius spheres, that may have a poor thermal environment, and long thin cylinders in which the ratio L/b is chosen so that the first few longitudinal modes occur at low frequencies, but with poor surface area-to-volume ratios and unresolved modes. Another possibility is a short cylinder of large radius in 274

275

Speed of Sound

Figure 6.11

A cylindrical annulus of length L with inner radius a and outer radius b.

which the first two azimuthal modes have the lowest frequencies and the ratio L/b is chosen such that the first few radial modes, azimuthal and mixed radial-azimuthal modes occur at lower frequency than the longitudinal modes. Unfortunately, overlap of key resonances precludes resolution of modes. A cylindrical annulus or square section toroid, shown in Figure 6.11, has, by comparison with a cylinder, another degree of freedom in the radius ratio a=b ¼ z (which can be varied between 0 and 1). The larger the value of this quantity the longer the path length of the azimuthal modes and the shorter the path length of the radial modes. The first few azimuthal modes occur at lower frequencies than the first radial modes. Of course, because of the larger surface area to volume ratio, the resonances are correspondingly broader. However, under some conditions the annular geometry is favourable, for example, for low-pressure measurements of the speed and adsorption of sound in thermally relaxing gases. Annular resonators have a small height, which reduces the effects of gravity, and have been used by Garland and Williams [32] to study the speed of sound near the critical point. A similar resonator was used by Jarvis et al. [199] to also measure the speed of sound near critical. Buxton [33] has reported an annular resonator capable of operating with gases that undergo molecular thermal relaxation. It is this apparatus that we will consider here. The zeroth-order working equation for an annular resonator with dimensions defined in Figure 6.11 is given by

fl;m;n

( )  u  lp2 w 2 1=2 m;n ¼ þ ; 2p L b

ð6:55Þ

the positive integer l is the order of the longitudinal mode and wm;n is an eigenvalue known exactly but dependent on the ratio z ¼ a=b. The symmetry of each mode is 275

276

Speed of Sound

determined solely by (l, m, n) so that (l, 0, 1) refers to a longitudinal mode, (0, m, 1) an azimuthal mode, that circulates about the central axis, and (0, 0, n) is for a purely radial mode. Buxton [33] applied first order perturbation theory to determine the viscous and thermal contributions for a rigid annular resonator so that Equation (6.55) can be represented by the complex frequency F ¼ f þ ig ( )  u  lp2 w 2 1=2 m;n ¼ þ þDft þ Dfv þ iðgt þ gv þ gb Þ: 2p L b

ð6:56Þ

In Equation (6.56), gb is the bulk attenuation given by the sum of Equations (6.2) and (6.5) and gv and gt are the contribution to the half-width from viscous and thermal boundary layers respectively. The frequency shifts Dfv and Dft, in Equation (6.56), that account for the viscous and thermal boundary layers, including molecular slip and temperature jump, are given by Equations (6.45) and (6.44). They depend on the thermal and viscous boundary layer contributions to the half widths gt and gv for an annular resonator given by [33] gt ¼

ðg  1Þdt pf 2VL0lmn 9 8 

bL 2 > > 2 > > > > > > e zRm ðwmn zÞ þ Rm ðwmn Þ = < l " # " # !  2  2 ; 6 > > m m 2 2 > > 2 2 > >  ðzbÞ Rm ðwmn zÞ 1  > > ; : þ b Rm ðwmn Þ 1  w wmn z mn ð6:57Þ

and dv pf i gv ¼ h 2 ðlp=LÞ þ ðwm;n =bÞ2 2VL0lmn o 9 82 0 n 13 2 2 > > ðlp=LÞ þ ½ m=ðbzÞ  zR2m ðwmn zÞ > > > > 6bL B C7 > > > > > > 4 @ A 5 n o > > > > e 2 2 2 l > > > > þ ½ m=ðbzÞ  ðw Þ þ ðlp=LÞ R mn > > m > > > > 2 " # = < 3 1 0  2 m ; 6 2 2 6 C7 > B b Rm ðwmn Þ 1  > > > w 7 6 C B   > > mn 2B > 6 w C7 > > > > þ 6 mn B > > " # C7 > >   > > 6 b B 2 C7 > > > m > > 5 4 A @ 2 2 > > > >  ðzbÞ Rm ðwmn zÞ 1  ; : wmn z 276

ð6:58Þ

Speed of Sound

277

where dt, in Equation (6.57), and dv, in Equation (6.58), are the thermal and viscous penetration lengths of Equations (6.34) and (6.35). In Equations (6.57) and (6.58) e0 ¼ 1, el > 0 ¼ 2, V ¼ pL(b2  a2 ) is the volume within the cavity, Rm (x) ¼ Jm (x) þ xNm (x) where Jm (x) is the cylindrical Bessel function and Nm(x) the cylindrical Neumann functions of order m, and L0lmn is a normalisation constant of [33] L0lmn

" " (  2 #) 1 1 m 2 ¼ Rm ðwmn Þ  1  2 el 1  z wmn " (  2 #)# z2 m  R2m ðwmn zÞ  1  : wmn z 1  z2

ð6:59Þ

Additional small contributions for openings in, and compliance of, the wall can be included in Equation (6.56). Unfortunately, an exact solution for the modes of motion of an annular shell is not possible in closed form and a detailed approximation has not been derived. General considerations may be inferred from consideration of, for example, Equation (6.20) for a sphere and the parameters adjusted to fit experiment. However, this was not done in this work. There is no a priori method of calculating the coupling constant due to the differing symmetry of the motion of the gas and shell [33]. Unfortunately, the shell resonances of the cavity used in reference [33] overlapped with the low-order azimuthal modes. Buxton [33] has also reported an expression for the perturbation of the gas inlet tube that was located in the top plate of the cylinder. Buxton [33] demonstrated that, by assuming a simple relationship between shear viscosity and thermal conductivity, analysis of the resonance line width obtained from an annular resonator can provide viscosity, thermal conductivity and relaxation times without recourse to other measurements. For example, Buxton [33] reported limiting values as r?0 of the shear viscosity Z and thermal conductivity k for SF6 that differed from literature values by up to 5 per cent for both Z and k. The precision of the product tr, where t is the relaxation time, determined by Buxton [33] varied between (1 and 15) per cent and depended on the magnitude of tr. To determine the speed of sound from the measured f with Equation (6.56), requires knowledge of the geometric parameters z, b and L as a function of T and p; Equation (6.56) suffices to determine the gas imperfections in the form of the virial equation of state but a knowledge of these dimensions allow evaluation of heat capacity. The required cavity dimensions were determined from measurements with propene for which the resonance quality factor Q { ¼ f/(2g)} was significantly greater than in argon. Buxton [33] reported the speed of sound could be determined with a relative precision of about 2 ? 105 using an annular resonator. The annular resonator, shown in Figure 6.12 with the thermostat and supports, was constructed by Buxton [33] from aluminium alloy 6082 with outer radius b ¼ 140 mm an inner to outer radius ratio z ¼ a/b ¼ 0.532 and length L ¼ 59 mm. In this resonator the first five azimuthal modes (0, m, 1) with m ¼ 1 to 5, occur at frequencies between (0.5 and 2.3) kHz when the cavity was filled with argon at a 277

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Figure 6.12 Schematic cross-section of the annular resonator, thermostat and gas handling system [33].

pressure of 100 kPa and a temperature of 300 K. At the same conditions, the Q ¼ 172 for mode (0,1,1) with argon. This frequency range allowed the study of relaxing gases at a pressure a factor of ten lower than for a practical sphere. The gas inlet tube was formed from a 2.58 mm diameter tube embedded in the top plate of the resonator that was drilled through so that the hole had a length of 67 mm. The tube was placed at r/b ¼ 0.78 where the wavefunction for the (0,0,2) radial mode was at a minimum. The tube was positioned at an angle p/2 from the two transducers where the wavefunctions for all the azimuthal modes with odd m were also a minima. The source transducer was located at r/b ¼ 0.814 in the resonator top plate. The centre of the detector was placed at r/b ¼ 0.886. The source had an active area of 20 mm2 to excite the large (3 L) volume of gas in the cavity. Thus, the diameter of the source was a significant proportion of the diameter of the resonator and it was necessary to integrate the wavefunction over the transducer area rather than approximate the source by a value at the centre of the transducer. The resonator was operated at temperatures in the range (230 to 340) K and pressures between (10 and 1000) kPa. 278

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The resonators described by Garland and Williams [32] and Jarvis et al. [199] were both used to study fluids near critical. The cavity constructed by Garland and Williams [32] had dimensions of L ¼ 3.7 mm, a ¼ 40.3 mm, and b ¼ 50.3 mm, while that of Jarvis et al. [199] had L ¼ 10 mm a ¼ 45 mm and b ¼ 56 mm. Solid dielectric capacitance transducers were used by Garland and Williams [32] and Buxton [33], while Jarvis et al. [199] used a piezo crystal source, connected to the cavity via a 1 mm diameter hole, and an electret microphone mounted flush with the inner surface. The source and detector were separated by an azimuthal angle of p. Jarvis et al. [199] corrected the resonance frequencies, in the absence of a model for viscothermal boundary layers, by adding the measured resonance line width after subtracting the known bulk attenuation. Moldover and co-workers [18,170–173] have described a dumbbell-shaped double Helmholtz resonator that also operates at a low frequency with a low Q resonance in which the gas oscillates through a duct connecting two chambers. The half-widths of the resonance have been used to determine the viscosity of gases at pressures between (0.2 and 3.2) MPa with an uncertainty of about 2 per cent. However, the frequency of the Helmholtz mode, which is on the order of 100 Hz can also be used to determine the speed of sound [18,173]. In the zeroth order approximation the Helmholtz resonance frequency of the cavity is given by fm2 &u2 r2d =ð2pLd VÞ;

ð6:60Þ

where u is the speed of sound, V the volume of both chambers and Ld and rd are the length and radius of the duct that connects the chambers. Speeds of sound determined with the Greenspan viscometer were found to lie within +2 ? 104 of literature values provided the volumes of the chambers were determined. The low frequency of the Helmholtz mode implies that this cavity may be used to determine the speed of sound in gases that undergo vibrational relaxation; in methane the relaxation contribution to Q was less than 0.3 per cent [18]. There are many other methods that have been used to determine the speed of sound in gases. One final method will be described here for measuring the speed of sound that utilises Brillouin scattering [200], which can be applied remotely to gases, liquids [201,202] and solids [203]. This approach was used by Straty [204] to measure the speed of sound in both compressed fluid and saturated methane at densities in the range (1 to 12.8) mol ? L1 . The speed of sound was obtained from the measured Brillouin frequency shift Df with u¼

Df l ; 2n sinðy=2Þ

ð6:61Þ

where n is the refractive index, which must be known at the wavelength l of the incident radiation, and y the scattering angle. In this experiment the gas within a pressure vessel was illuminated with an argon-ion laser. The frequency shifts Df were determined at a scattering angle of p/2 with a pressure scanned Fabry-Perot interferometer. The uncertainty in the measured sound-speed was about 0.5 per cent. 279

280 6.1.2

Speed of Sound Liquids

There are essentially two methods that are used to determine the speed of sound in liquids: variable path-length interferometry and time-of-flight measurements. This choice arises with liquids because the acoustic impedance of the fluid ru is no longer much less than that of the wall material as it is with gases; for gases the ratio ru/ rwuw, where the subscript w refers to the wall material, is on order 105, while liquids it is 0.1. This ratio governs sound reflection and energy transfer between the media. Thus, steady state measurements of the type described in Section 6.1.1 are inappropriate for sound speed measurements in liquids owing to coupling between the fluid and wall motion. Spherical cavities have been used to measure the sound absorption with reverberation methods [205–207] and sound speed in liquids [85,116] with resonators of internal volume between (0.5 and 100) L. An advantage of the coupling between the motion of the shell and the liquid is that the transducers can be placed on the outer surface of the cavity and do not need to come in contact with the fluid under test. Piezoelectric ceramic elements have been cemented outside the shell [205,207–209]. Of course, if the acoustic impedance of the shell could be made small then a pressure release boundary condition could be used to describe the frequencies within the cavity. This approach has an additional advantage that the viscous and thermal boundary losses vanish. However, practically it is unlikely that the impedance of the wall could be small and the cavity hold a fluid. Thus, in the remainder of this section we will focus on the time-of-flight and variable path-length interferometers. Experimental methods of measuring the speed of sound (and absorption) in liquids and the theory of operation of these techniques have been reviewed by McSkimm [3], Papadakis [4,5], Heydeman [7], and Trusler [2]. Kaatze and co-workers [210,211] have reviewed, at frequencies in the range (103 to 109) Hz, ultrasonic methods, including measurements of the speed of sound in liquids, while Leroy [20] has described the methods used specifically to determine the speed of sound in liquid water that might also be used for other liquids. However, many of the methods described for solids in reference [14] can also be used for liquids with small adaptations. The time-of-flight methods, as we did for gases, can be divided into single and multiple path length devices. Indeed the time-of-flight instrument discussed in Section 6.1.1 could equally well be used with liquids and, for example, the two matched transducer instrument of Greenspan and Tschiegg [194] was originally used for water. Here we will give several other examples of devices used recently to determine the speed of sound in liquids, including natural fluids, at elevated pressure and temperature. Wang and Nur [212] have used a time-of-flight apparatus with two transducers to study the acoustic properties of pure hydrocarbons and there mixtures including reservoir fluids [213]. Ye et al. [214] reported a single path pulse-echo apparatus that was modified by Daridon [215] and automated by Ding et al. [216] to obtain the sound speed, with an uncertainty of about 0:002 ? u [215], at high pressure in pure hydrocarbons [217,218], prepared mixtures [219,220] and natural fossil fuels at reservoir conditions [221]. This instrument has also been used to measure the speed of sound in gases [222], to detect phase boundaries [223], the subject of Volume 280

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VII in this series, including (l+l) liquid phase equilibrium [224] with a technique proposed by Williamson [225]. Zak et al. [226] have reported another pulse-echo overlap apparatus and reviewed the merits of different reflector geometries while Kozhevnikov et al. [227] reports a phase-sensitive pulse method for determining the sound speed, relative to a known value from the literature, at temperatures up to 2100 K and pressures up to 200 MPa for liquid alkali metals. This instrument operated at a frequency of about 10 MHz with samples that were about 1 mm long. Muringer et al. [228] described two methods of determining the speed of sound in liquids, both of which used time-of-flight measurements and operated at pressures up to 260 MPa and temperatures between (173 and 320) K. The first was a single pulse-echo-overlap technique that was based on the method described by Papdakis [229]. At one end of the cylindrical tube of path length 20 mm was an edge clamped x-cut quartz crystal operating at a frequency of about 2 MHz. The other end was a reflector, with a conical cavity at the end remote from the fluid. One face of the transducer was exposed to the fluid to be studied the other was in contact with an oil in a cavity 45 mm long. This arrangement was used to reduce reflections or backcoupling from the transducer. The sound speed was determined from a combination of the path length and the time required between successive pulses to create overlap of the pulses on an oscilloscope. The pulse-echo-overlap method has the disadvantage that small phase changes in the reflected pulse may result in a systematic error in the transit time. To overcome this source of error, Muringer et al. [228] adopted a phase-comparison pulse-echo method, first developed by Williams and Lamb [230], but with two reflectors made from the same material placed at unequal distances either side of the sound source. The apparatus was similar to that described by Kortbeck et al. [38] and in Section 6.1.1 for gases. As in reference [38] the time between successive pulses was adjusted so that the arrival of two echoes, one from each path, coincided at the transducer and, when the pulse amplitudes were adjusted, caused complete destructive interference. Ball and Trusler [231] have reported an apparatus that is similar in design to that of references [38] and [228] and shown in Figure 6.13. The piezo ceramic transducer 5 is clamped at its edges 6 by stainless steel rings and separated by two quartz spacers of unequal length 3 from two parallel stainless steel reflectors 2, each with conical cavities 1 in the faces, remote from the fluid, to disperse sound that passed through the reflector. These elements were held together by threaded rods, which pass through clearance holes located at a angle 2p/3 to each other in the clamping disk and the reflectors. Fused quartz was used for the spacers because it was electrically insulating and could be machined with high tolerance. The spacers were ground from solid fused quartz with diameters accurate to 10 mm and ends flat and parallel to better than +2:5 mm. The transducer was formed from lead zirconate titanate (PbZrO3/ PbTiO3) with a 10 mm diameter and 0.4 mm disc plated on both sides with nickel and operated at a frequency of 5 MHz. This transducer was supplied with single five-cycle tone burst at 5 MHz at an amplitude of 10 V peak to peak. This caused the transducer to emit pulses simultaneously in opposite directions and the two returning bursts were found to have essentially identical shape. The time at which a characteristic feature of the first returning echo on the shorter path L1 was detected at t1 and the 281

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Figure 6.13 Cross-section through the pulse-echo method of Ball and Trusler [231] that is based on a single transducer 5 clamped at its edge 6 and separated by two quartz tubes 3 of lengths L2 and L1 in the ratio 3/2 from two plane parallel reflectors 2, each with conical cavities 1 to disperse sound that passed through the reflector. The speed of sound was obtained from the difference in round trip transit times in the two paths.

time at which the second echo returned over the longer path L2 was detected at t2. The speed of sound, ignoring diffraction corrections, is then given by u¼

2ðL2  L1 Þ : t2  t1

ð6:62Þ

An advantage of determining differences in transit time is that reflection and electronic delays can be reduced. The transit times can be determined directly from an oscilloscope; however, the resolution was increased by analysis of the digitised wave-form. In this analysis, about 10 cycles of the first echo was selected and, after 282

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assuming a delay time, compared with about 10 cycles from the second echo. The analysis was repeated for a series of time differences and the optimum value determined. The path length was calibrated by measurement with water at 298 K and 0.1 MPa and comparison with the value reported by Del Grosso [232]. The variation of L with temperature and pressure was calculated from the coefficients of thermal expansion and compressibility of quartz and stainless steel. The corrections to the path length were 0.18 per cent at the maximum pressure of 200 MPa and 0.012 per cent at the maximum temperature of 473 K. Comparison of results obtained with this method for n-hexane with those reported by other workers indicated an accuracy of better than 0.5 per cent and a precision of 0.1 per cent. Diffraction of the signal is present in all pulse measurements. To ensure the cavity walls do not affect the acoustic signal as it passes through the fluid, it is important that the cavity is significantly greater in diameter than the source. Diffraction effects can be estimated from the steady-state radiation pattern of the source, and a cell diameter chosen, so that an insignificant fraction of the acoustic energy impinges on the walls. A second effect of signal diffraction is that it causes the measured transit time to be in error because of the phase advance f of the sound wave relative to a plane wave travelling the same distance [233]. To correct for this error the transit time must be increased by dt, given by dt ¼

fðLÞ ; 2pf

ð6:63Þ

where f is the frequency of the source, L the path length and f the phase advance determined either numerically or analytically from the free field diffraction equation:

4 A expðifÞ ¼ 1  p

Zp=2

  2   4b pi exp  cos2 y sin2 y dy; Ll

ð6:64Þ

0

where b represents the radius of the source. The factor Aexp(if) provides a complete description of the free-field diffraction effects for a piston like source and a parallel detector of the same radius that responds to the acoustic pressure over its surface. It is valid for a single transducer multiple reflector device provided the diameter of the reflector is equal or greater than that of the transducer. The dual transducer variable path-length interferometers of the type reported by Gammon and Douslin [34,159], which are not routinely used to determine the speed of sound in liquids, have been used to study silicate melts [234,235] and liquid metals [236,237]. Typically, single transducer variable path length devices have been used to study liquids, for example, that reported by Del Grosso and co-workers for measurement of the speed of sound in pure water [37] and sea water [36,238] and other liquids [239] with an uncertainty of 105 ? u. The work of Del Grosso has been discussed recently by Leroy [20]. For measurements at ambient pressure Del Grosso 283

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[36] used a single stationary quartz crystal transducer operating at a frequency of 5 MHz and a movable reflector that gave path-lengths between (20 and 40) mm [232]. Impedance circles were used to establish the location of each half wavelength and the position of the reflector was, in the final version of the apparatus, determined with an optical interferometer. The interferometer, for measurements at pressure, had a sliding non-rotating shaft that held the reflector. Del Grosso has examined the sources of errors between interferometers and time-of-flight measurements, and determined the effect of guided mode dispersion and free field diffraction in ultrasonic cylindrical resonators [190–192]. Del Grosso showed the effect of the side wall on the sound speed in a cylindrical cavity is negligible when the ratio of radius of the sample to that of the transducer is greater than 2. Kroebel and Mahrt [240] have also used an interferometer, albeit with greater reflector separations, to study the speed of sound in water. Both of these interferometers were operated in continuous wave mode. Heyderman [241] described a phase sensitive detection scheme for pulse measurements that was used to determine the height of the mercury column in a manometer. Of course this is essentially a variable path-length interferometer. In this approach an ultrasonic pulse is propagated through a sample and reflected back to the transducer. The phase of the echo is then compared with that of the continuous wave used to generate the pulse. The advantage of this approach is that it can be used over a wide range of length or sound speeds. Phase sensitive detection has also been used for solids, described in Section 6.1.3, and to determine the speed of sound in a mercury filled manometer [242]. Fujii and Masui [243] used these concepts to develop the apparatus shown in Figure 6.14 that combines pulse techniques, phase sensitive detection, and variable path-length interferometry to measure the speed of sound in liquids, in particular water at ambient pressure. In this apparatus, the source transducer is attached to one end of a quartz rod and a pulse of sound, which is obtained from a continuous wave source, propagates from the transducer through the quartz buffer, through the fluid and is reflected at a parallel movable plate; buffer rods have advantages including the reduction of multiple reflections from the transducer [6]. Two echoes are observed in this experiment, one that arises from a reflection at the interface between the rod and the sample and the other from part of the pulse that travels through the sample and is reflected at the moving reflector back to the transducer. The difference in phase between each echo and the continuous wave reference, used to generate the pulse, can be determined with phase sensitive detectors or from measurements as a function of varying path length [243]. The difference in distance travelled between the two echoes d is determined by the number of half wavelength constructive interference in the fluid or fringes F at a frequency given by F¼

d 1 y2  y1 ¼nþ þ ; l 2 2p

ð6:65Þ

where l is the wavelength, n is an integer number of fringes, 1/2 describes the phase 284

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Figure 6.14 Schematic of the pulse-echo variable path length interferometer with phase sensitive detection of Fujii and Masui to measure the speed of sound in liquids [243].

change on reflection at the interface between the buffer and the fluid, y the phase difference between the pulse and the continuous wave reference, and the subscripts 1 and 2 refer to the first and second echo respectively. The wavelength is then determined from the change in path length d required to observe an additional fringe. In practice, the path length is changed over about 100 fringes. The speed of sound is then determined from the wavelength and the frequency. The variable-path cavity of Fujii and Masui [243] is shown in Figure 6.15. A LiNbO3 piezo transducer, with a diameter of 10 mm and a resonant frequency of 16.5 MHz, was bonded to the lower surface of a quartz rod 25 mm in diameter and 50 mm long. The transducer was pulsed at a frequency down to about 1 kHz so that the echo from each pulse had decayed to an undetectable level before the next pulse was generated. The quartz rod, shown in Figure 6.15, formed the lower wall of the sample cavity. At the other end of the cavity was a reflector made from a stainless steel piston mounted in a cylindrical guide. The piston and cylinder assembly were machined so that the reflector was parallel to the quartz rod. The lower end of the 285

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Speed of Sound

Figure 6.15 Cross-section through the variable path-length interferometer of Fujii and Masui [243] (the lower portion of Figure 6.14) showing the transducer, quartz buffer rod, sample volume, reflector, cylindrical guide and Michelson interferometer.

reflector was immersed in the liquid and traversed about 12 mm during the experiment. The reflector movement was achieved with a stepper motor and displacement determined with a Michelson interferometer. The frequency of the laser used in the Michelson interferometer had been calibrated. Phase sensitive detection was also used to determine the reflector displacement to within 80 nm. In this experiment, the phase of the pulses and and reflector displacement were measured while the reflector was moved at a velocity of 0.1 mm ? s1. It was determined that the speed of sound obtained with a continually moving reflector introduced an insignificant error in the measurements when they were compared with those obtained from a stationary reflector moved in discrete steps between data acquisition. The apparatus was used to measure the speed of sound in water at ambient pressure at temperatures in the range (293 to 348) K with an estimated relative uncertainty of 105. Sound speed determined with this apparatus differed from literature values by about 3 ? 105 ? u. Nasch et al. [236] has reported a dual transducer variable path-length interferometer for operation with liquid metals at temperatures over 2000 K and it has been used to determine the speed of sound in liquid iron [237] at temperatures up to 1883 K. The instrument operates at frequencies in the range (5.4 to 11.8) MHz. 286

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The interferometer was constructed from polycrystalline Al2O3. To prevent heating the LiNbO3 piezoelectric crystals above the Curie temperature (about 1423 K), each one was cemented on one end of a 0.2 m long and 12.7 mm diameter Al2O3 rod and located in a water-cooled jacket. The sound from the transducers, which were situated at a lower temperature than the sample, was propagated along the rods into the liquid sample within a crucible; this approach has also been used with time-offlight measurements at high temperatures [227]. To ensure parallelism of the transducer-rod and rod-sample interfaces, important to interferometery, both ends of the rods were polished flat to within 1 mm. The lower rod was cemented into the crucible while the upper rod could be moved and the position determined with a linear variable displacement transducer. Unfortunately, the long narrow rods introduced errors from beam spreading and reflections that arose from thermally induced variations in acoustic impedance. The latter were reduced with graphite sleeves. The variable path interferometer does not require, at constant temperature, corrections for the variations in sound speed with temperature along the rod. However, the ratio of the acoustic impedance of the Al2O3 rod to that of the Fe(l) was about 1.4 and this value contributed to the larger than desired uncertainty with which the amplitude maxima could be located and thus the few per cent uncertainty in the speed of sound. A single transducer variable path-length interferometer has been described, along with other method of measuring the speed and attenuation of sound, in liquid metals by Webber and Stephens [244].

6.1.3

Solids

Measurements of the speed of sound in solids are routinely used to determine the elastic constants and thermodynamic properties as outlined in Section 6.2 and described in detail in reference [14]. The techniques used to measure the speed of sound in solids can be divided into those obtained with time-of-flight or bulk waves, guided waves that include surface acoustic waves and microscopy, continuous wave or resonance methods, and the scattering of light, neutron and X-rays. The exact choice of method depends on sample size, geometry, temperature and pressure range, and the desired precision. The sample size varies from on the order of 10 mm for time-of-flight or bulk wave methods to on order 10 mm for methods based on the interaction of X-ray scattering with acoustic waves. Smaller samples can be accommodated with higher frequency bulk wave methods at the expense of accuracy, for example, Chen et al. [245] adopted frequencies on order 1 GHz that were transmitted to the sample through a sapphire buffer rod. The uncertainties in the measured sound-speeds are on the order of 0.1 per cent for bulk wave techniques and on order 1 per cent for Brillouin, neutron and X-ray scattering methods. This variation in uncertainty can be serious, particularly when the temperature dependence of the speed of sound in solids is required because, for example, over the temperature range (4 to 300) K the sound speed may change by only a few per cent. However, neutron scattering is advantageous when the entire dispersion curve is required. A comprehensive review of the theory and all these experimental 287

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methods has been provided in reference [14]. Consequently, only a brief overview of the methods is provided here with an emphasis on both pulse and resonance techniques. Indeed, the methods described for the measurement of the speed of sound in fluids in Sections 6.1 are essentially adaptations of the methods used to determine the elastic moduli of solids [3–5,229,230,246]. The reader interested in the theory of the elastic properties of solids and the elastic properties of elements, compounds, alloys, and composite materials should consult reference [15] while those interested in the elastic properties of biological and organic material as well as those relevant to the earth and marine sciences reference [16]. For solids, it is no longer sufficient with time-of-flight measurements to simply identifying the leading edge of the echo to determine the flight time and, when combined with the dimensions of the specimen, obtain the speed of sound. Thus, other methods have been developed to determine the sound-speed including those known as phase comparison, pulse superposition, pulse-echo overlap, and sing around. Each of these methods will be described briefly. They are usually carried out with the transducer bonded to the sample and excited to provide the speed of both the longitudinal and shear waves. In the phase comparison method developed by Williams and Lamb [230], (the method reported by Muringer et al. [228] for liquids and Kortbeck et al. [38] for gases was derived from this work) a pulse is emitted from the transducer at a carrier frequency of about 10 MHz. A second pulse of the same carrier is released to interfere with the returning echo of the first pulse. This process is repeated while varying the carrier frequency until phase cancellation occurs through destructive interference. The null frequency can be obtained with sufficient accuracy to determine the speed of sound with an uncertainty of about 0.1 per cent. The pulse superposition method [3,246,247], also described in Chapter 2 of reference [14], differs in that the pulse repetition frequency of about 1 MHz is adjusted to give constructive interference between successive echoes until a maximum amplitude is observed on an oscilloscope. The pulse repetition frequency is then related to the transit time, as described in Chapter 2 of reference [14] and references [3,246–248], after accounting for the phase shift that occurs because the transducer is bonded to the sample. However, the method is limited because it cannot be used with buffer rods, individual echo pairs, broadband pulses or diffraction corrections. Anderson [249] used this method to measure the speed of sound in solids with an uncertainty of 0.1 per cent at pressures up to 0.5 GPa and temperatures between (2 and 500) K. The transducer was attached to the sample and operated at a frequency of about 10 MHz with pulse durations of between (2 and 100) ms and a repetition rate of about 1 ms. The solid sample was pressurised with either helium or nitrogen. The pulse superposition method has been used by Ledbetter and co-workers to determine the speed of sound and thus elastic constants of titanium alloys [250], aluminium alloys [251,252], austenitic (nickel þ chrochromium þ iron) (Inconel) alloys [253], and austenitic stainless steels [254] at temperatures in the range (4 to 300) K. In these measurements, the transit time was determined at room temperature and the variation of transit time as a function of temperature obtained relative to the room temperature value. The transducers were 288

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bonded to the sample with phenyl salicylate at room temperature, while at temperatures down to 70 K stop cock grease was used and silicon oil at T < 70 K. Errors arising from transducers and bonding them to the sample were reduced by performing experiments at room temperature with one and two transducers and extrapolating the results to zero transducers. The speed of sound determined with this approach has an uncertainty of about 1 per cent. In the pulse-echo overlap method two echo signals are made to overlap on an oscilloscope by driving the oscilloscopes x-axis with a continuous wave signal with a frequency equal to the reciprocal of the travel time. The phase of the pulses are synchronised with the continuous wave signal. The pulse repetition rate is determined from a signal that is phase locked to the continuous wave but at a frequency about a factor of 1000 less. Details of this method can be found in the reviews of Papadakis and Lerch, in Chapter 2 of reference [14], and Papadakis [5]. Diffraction corrections, described in [255,256], can be applied to these measurements and the method can provide sound speeds with an uncertainty of about 0.1 per cent. The pulse echo overlap technique can be used with transducers either bonded to the sample or connected to it via buffer rods. It may also be used with broad-band pulses [257] that is advantageous because overlap with the broad-band echoes can be unambiguously determined. Finally, the method can also be adapted to determine single-path travel times and are often called through transmission measurements. This method has been reviewed by Hosten in Chapter 3 of reference [14] with an emphasis on its application to anisotropic composite materials. In through transmission measurements, the transducers are placed opposite each other and are operated at frequencies between (1 and 10) MHz. The sample is placed between the transducers and the travel time determined as the orientation of the sample varied with respect to the transducers so that all the elastic constants can be extracted. The locations of the transducers are usually fixed and the sound coupled to the sample through either an air or water bath. This arrangement allows the sample orientation to be varied. In anisotropic solids, the group-speed velocity can deviate significantly in direction from the phase velocity. In the so-called sing around method, two transducers, one a transmitter, the other a receiver, are bonded to opposite faces of the specimen. The signal from the receiver is used to trigger the pulse generator to establish a pulse repetition frequency. When a steady state is achieved the pulse repetition frequency equals the reciprocal of the transit time. This method is similar to that used by Greenspan and Tschiegg [194] for water and Younglove [195] for gases. Continuous wave methods of determining the speed of sound in solids have been reviewed by Bolef and Miller [8]; these methods can also be applied to fluids. In this experiment, the transmitting and receiving transducers are plated on opposite sides of the sample and the frequency swept through resonances that occur at half wavelengths. Transient decay measurements are used to reduce the effects of cross talk. Perterson et al. [258] have used phase sensitive detection methods for this measurement. The methods of measuring the speed of sound in solids as a function of pressure, including ultrasonic methods, have been reviewed by Bassett in Chapter 20 of 289

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reference [14] and by Heydemann [7]. Ultrasonic methods operating at GHz frequencies, where the wavelength is on the order of mm, can be used with samples that are about 100 mm thick. An example has been developed by Bassett et al. [259] to determine the speed of sound from time-of-flight measurements in a diamond anvil at pressures up to 6 GPa and temperatures up to about 500 K; pressure measurements within a diamond anvil were described in Section 3.3 of this book. The apparatus described in reference [259] also permitted simultaneous inelastic Xray scattering measurements. Ultrasonic measurements of the time of flight have been obtained at higher pressures up to 10 GPa and temperature of 1300 K with a multianvil apparatus operating at frequencies between (20 and 100) MHz [260]. In both types of anvil the transducer was not exposed to stresses arising from the pressure and buffer rods were used. Bassett, in Chapter 20 of reference [14], provides descriptions of these measurements, along with the use of Brillouin scattering to determine elastic constants within a diamond anvil. The accuracy of the sound speed obtained from plane wave measurements depends on the dimension and parallelism of the sample faces. For objects that have linear dimensions on the order of 0.1 mm the flight times are on the order of 10 ns (with pulses of 1 ns duration). At higher frequencies, the uncertainties arising from transducer attachment, beam diffraction and sidewall effects increase, and the attenuation is greater. In this case, the use of either point sources or resonance methods overcomes the sample size limitation and can decrease the uncertainty of the measurement and provide redundancy. Every et al. in Chapter 4 of reference [14] described this use of point sources and point detectors as an alternative to plane wave measurement; dimensions of the transducers are less than both the wavelength of the sound and the distance between the source and receiver. The main advantages of this approach are that both longitudinal and shear waves are generated by a single pulse and the propagation occurs over a wide range of directions so that an array of detectors can be used to determine the directional dependence of the sound. Point-like sources, discussed in Chapter 4 of reference [14], may be a laser, electron beam or X-rays and as detectors small piezoelectric and capacitive devices as well as laser interferometery. The observation of the resonance frequencies within a solid of known dimensions can be used to determine the speeds of sound. For example, methods to determine the longitudinal, torsional and flexural resonances in a rod have been described by Hermann and Sockel in Chapter 13 of reference [14]. In this case, the rod, that is typically between (40 and 100) mm long and (3 and 10) mm in diameter (that is the ratio of diameter to length is less than 0.1), is suspended from carbon wires, at each end, that are connected to piezoelectric transducers. The amplitude and phase of the motion are determined at frequencies between (1 and 200) kHz. This resonance method can be used at temperatures in the range (293 to 1673) K. This approach does decouple the transduction but does not operate with small samples, however, an alternative method of parallelepiped resonance, also called resonant ultrasound spectroscopy, does. The theoretical and experimental aspects of resonant ultrasound spectroscopy have been reviewed, and compared with time-offlight measurements, by Maynard et al. [261], Migliori et al. in Chapter 10 of 290

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reference [14], and Migliori and Sarrao [262]. Migliori et al. [263] and Schwartz and Vuorinen [264], to list two, have also provided descriptions of resonant ultrasound spectroscopy. In this experiment, the frequencies of about 50 resonances are used in an inversion procedure, which has been described by Migliori et al. in Chapter 10 of reference [14], to determine the most suitable set of elastic constants simultaneously with an accuracy of about 0.05 per cent provided the sample aspect ratio is unity. The speed of sound can be determined with similar uncertainty. However, the analysis dictates that the sample geometry be rectangular parallelepiped, spherical, or cylindrical geometry. Clearly, this method of determining the speed of sound in solids offers redundancy in the resonance frequency and a method of determining the absorption. Holding a sample between two piezoelectric transducers forms a simple resonant ultrasound spectrometer, one transducer acts as a source the other a detector, and the amplitude of the response as a function of frequency is determined with phase sensitive electronics. Ledbetter et al. [265] has described an apparatus of this type that operates at frequencies between 10 kHz and 5 MHZ. However, for operation at temperatures up to 1820 K, aluminium oxide buffer rods have been used [266] while direct contact between the transducer and sample has been eliminated with magnetostrictive and electromagnetic levitation methods, similar to those described in Chapter 10 Sections 10.2.3 and 10.2.4. Non-acoustic non-contact techniques of obtaining the speed of sound include Brillouin, neutron and the X-ray methods. Brillouin and inelastic neutron scattering can provide sound speeds with an uncertainty of a few percent on samples with dimensions in the range (50 to 300) mm. With Brillouin scattering, which has been described by Grimsditch [203], the speed of sound is determined from the Brillouin frequency shift and the measurements can be performed at temperature up to 2400 K, despite the drawback of blackbody radiation. In neutron scattering, described by Stassis in Chapter 17 of reference [14] and Willis and co-workers [267,268], the sound-speeds are determined from the slopes of the measured acoustic phonon dispersion curves. Inelastic neutron scattering can be used at temperature up to 3000 K, at pressures up to 10 GPa, and the crystals can be grown in situ. Zolotoyabko in Chapter 18 of reference [14] has discussed X-ray methods.

6.1.4

Transducers

The transducers that convert mechanical work into electrical work or vice versa, and used to generate and detect sound are one of the most important components of an apparatus to measure the speed of sound. These transducers must satisfy certain criteria before they are useful for the measurements described in this chapter. All must have low output power so as not to perturb thermal equilibrium within the cavity and must operate over a wide temperature and pressure range and must be chemically inert while maintaining an acceptable signal-to-noise ratio. These requirements alone place severe limitation on the materials used to fabricate the transducers. In addition, for resonators, the transducers must have a wide frequency bandwidth that present high acoustic impedance to the fluid. The latter is required so 291

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that the transducers, which typically form part of the walls of a resonator, do not perturb the properties of the cavity. For variable path-length fixed-frequency interferometers the transducers are operated close to a mechanical resonance where the electrical impedance of the device is determined mainly by the impedance of the load. For pulsed operation, the frequency is also fixed but the bandwidth is larger to achieve useful time-domain resolution. In this section the discussion will be limited to practical transducers for the measurements that have been described in the preceding sections and include piezoelectric elements, electromagnetic devices, capacitive based methods and lasers; many of these transducers and their location have been discussed within Sections 6.1.1, 6.1.2, and 6.1.3. The theoretical and practical aspects of transducers used for the measurement of sound-speed are also described by Trusler in Chapter 5 of reference [2]. Piezoelectric elements are most often used for variable path-length interferometers and time-of-flight measurements but have also been used in resonators [102]. These applications of the piezoelectric transducer, along with the types of piezoelectric elements and their location have been discussed within Sections 6.1.1, 6.1.2, and 6.1.3 for specific apparatus. The theory of operation and development of piezoelectric transducers have been described by Mason [269], Berlincourt et al. [270], Trusler in Chapter 5 of reference [2], and, for operation at frequencies greater than 100 MHz, by Sittig [271] and will not be included in this section. Only the issues of temperature and location are considered here. When the piezoelectric element is operated below the Curie temperature, the transducer is usually exposed directly to the sample, provided no adverse reaction occurs. An advantage of the coupling between the motion of the shell and the liquid is that the piezoelectric element [205,207–209] can be placed on the outer surface of the cavity and does not need to come in contact with the fluid under test. To prevent heating the piezoelectric crystals above the Curie temperature, which for LiNbO3 is about 1423 K, and eliminate chemical compatibility issues, the transducer is usually separated from the sample. One method of separating the sample is to attach the element to one end of a rod, constructed from a material that has the appropriate acoustic and thermal properties, and the other end of the rod exposed to the sample. This buffer rod arrangement has been used successfully for measurements with liquids and solids. Bimorph discs, formed from two piezoelectric elements bonded together, have been used in bending mode, to provide lower frequencies and higher sensitivity as well as acting directly on the fluid in resonators [74,76,85,116]. Flexural bimorphs are also often coupled to a diaphragm to form a transducer. For example, when the element is bonded to the centre of an edge-clamped circular metal plate, as shown in Figure 6.5, the transducer has been used for pulsed and continuous wave methods at its resonance frequency and in broadband applications [149]. In principle, this approach could be used to eliminate direct contact of the element with the sample. For electrically conducting or magnetostrictive solids (or those coated to have at least one surface that is) longitudinal and shear waves can also be introduced into specimens with electromagnetic acoustic transducers (EMATS). These transducers, which have been used for pulse-echo and variable path-length measurements of the sound-speed in solids, do not directly contact the specimen and thus there is no phase 292

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shift corrections required for physical coupling. EMATS have been described by Dobbs [272] and there use to determine the elastic properties described by Alers and Ogi in Chapter 11 of reference [14]. Parallel plate capacitors, consisting of a rigid backplate separated from a diaphragm, that could be a tensioned metal foil, or polymer film, or edge clamped plate by a gas-filled gap have been used for gas phase sound-speed measurements. When an ac signal is applied to this type of parallel plate capacitor, electrostatic forces excite motion and it acts as a source. When sound waves deflect the diaphragm the capacitance modulates and, when the stored charge is constant, generate an alternating potential difference across the plates. Models have been reported that describe the operation of this type of microphone [273,274]. Commercially available capacitive transducers with a stiff metal foil have been used for sound-speed measurements in gases [103] including the acoustic determination of the gas constant [40]. A significant variant of this design are solid dielectric capacitive transducers, which respond to acoustic pressure or electrostatic force as described for the parallel plate capacitor. These are well suited to sound speed measurements because they are broadband, offer high sensitivity, are rugged and easy to fabricate. These transducers are typically formed from a thin (about 10 mm) dielectric membrane laid flat over a back plate surface under minimal tension. The outer surface of the film is metallized and forms the second plate of the capacitor. The stiffness of this system is dominated by a thin layer of gas between the film and the back plate. A smooth back plate provides a wide band, low sensitivity device. The sensitivity can be increased by reducing the stiffness and this can be achieved by roughening, drilling blind holes, drilling vent holes to another volume, or machining grooves in the back plate. An additional advantage of capacitive transducers is that when excited by a signal from a frequency synthesiser without a d.c. bias, sound is produced at a frequency twice that of the feed and this insures that the electrical crosstalk between the transmitter and the receiver is negligible [57]. There are several examples of this type of transducer in the literature that have been used for applications other than sound speed measurements [275–279]. For example, Kuhl et al. [275] used grooves with widths and depth in the range (0.5 to 2) mm and attributed the observed performance variation of these transducers to diaphragm tension, bending stiffness and air gap between the membrane and backplate. Anderson et al. [279] have developed a model to describe the operation of this type of device with grooves 0.2 mm wide and 3.75 mm deep that were fabricated with micromaching techniques. Trusler in Chapter 5 of reference [2] provides working equations for the design of solid dielectric capacitance transducers. Metallized polymer membranes have been used to form the dielectric layer but this limits the upper operating temperature to about 450 K [280] and routinely the upper limit is much lower [32,64,73,88] with the exact value dependent on the material. The polymer can also be an electret material, which have permanent electric charge, and when the direction of polarisation is perpendicular to the plane of the thin membrane, the materials are used in microphones because no dc bias is required [280]. Most commercial electrets are made by depositing charge in the material near one surface using corona discharge in air or an electron beam in vacuum, but films 293

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can be polarised in an electric field at elevated temperature. However, all these materials thermally depole rapidly at higher temperature, for example, polyvinylidene fluoride (the component most often used) depoles at about 370 K [105,280], and limit the upper operating temperature, while PTFE will depole at around 500 K but still have long charge retention times at 420 K [280]. Of course, metallized polymer film can be used in a microphone when a dc bias on order 100 V is applied. Unfortunately, when polyethylene terephthalate (Mylar) is used in transducers and operated with a dc bias it tends to acquire polarisation even at ambient temperatures. The use of metallized polymer has also compromised chemical compatibility [73]. Alternatively, for operation at temperatures up to 1000 K glass or mica sheets have been used as the dielectric with a thin metal membrane [165]. Glass bonded to the back plate has also been used with a metal membrane [101]. In the remainder of this section, specific transducer arrangements will be described for operation in resonators contained in a pressure vessel, so that the gas moves between the two chambers, and designs that seal the gas within the cavity from a pressure compensating fluid or ambient pressure or vacuum. When the resonator is placed within a pressure vessel, and the gas free to flow between the two volumes, many transducer types and mechanical arrangements have been used [53] including an opening or waveguide in the wall that couple sound between the transducer and the gas [54,55]. Two examples are given here, one for a cylindrical and the other for a spherical geometry cavity. Younglove and co-workers [150,167] stretched a metallized polymer film over the end plates of the cylinder (which had an inner diameter of about 10 mm) to form a solid dielectric capacitance transducer source and detector. In reference [151] a 20 mm thick Mylar with 50 nm of aluminium was used over a polished end cap, while in reference [168], to increase the sound intensity, a design similar to that reported by Kuhl et al. [275] was adopted with three concentric grooves, about 0.5 mm wide and 0.5 mm deep, cut into the back plates. The grooved back-plates were covered with 13 mm thick polyimide plated with gold facing into the cylinder and slightly tensioned. The source was driven with an ac component of 50 V and a bias of about (100 to 200) V dc with higher voltages required at higher pressures. The receiver was biased with 90 V dc. A disadvantage of using the end plates, which are a significant fraction of the surface area, as the transducers, is that the impedance of the transducers becomes important in determining the resonance frequencies and the half widths. To reduce the effects of the source transducer on the measured resonance frequency Moldover and coworkers [56,57] placed metallized Mylar over a 3.5 mm diameter rod that was inserted into the wall of a spherical resonator, of radius about 60 mm, so that the metal surface of the moving element was almost flush with the inner surface of the cavity. This design was also adopted by Ewing and co-workers [68,71,134]. More sophisticated mechanical arrangements have been developed for transducers that are used to separate the gas inside from that outside the cavity. These designs are required when the resonator is pressure compensated [40,67,83,88] or when the resonator is used as the pressure vessel [31,33,91,92,106,143]. The transducers are either fabricated as removable units that are inserted into a port through the resonator wall or form part of the wall. The electroacoustic capacitance 294

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295

Figure 6.16 Electroacoustic capacitance transducers of Ewing and Goodwin [88]. LEFT: The detector contained a commercially available electret microphone capsule. It was acoustically connected to the resonator through a 15 mm3 volume and a 0.7 mm diameter wave-guide of length 4 mm. RIGHT: The sound source was formed by clamping a 12 mm polyester membrane between the metal housing and a ceramic sleeve. A 50 nm layer of aluminium on the disk faced into the sphere and contacted the metal housing. The active area of the source was 3 mm2 and it was coupled to the resonator through a tube 0.25 mm long. The disk was tensioned with a spring through the 1.9 mm diameter back electrode which had six 0.3 mm diameter holes drilled to a depth of 0.5 mm to increase the compliance of the disk. The detector can also be formed by replacing the membrane in the right hand figure with either an electret material or applying a dc bias to the same polymer used in the source and, in both cases, locating an impedance matching JFET at the electrical lead-through.

transducers of Ewing and Goodwin [88], shown in Figure 6.16, are an example of transducers that are inserted within a port. In this case the housing was threaded and screwed into the resonator wall. The inner surfaces of the transducer housings were machined with a 40 mm radius of curvature to match that of the sphere. A nominal clearance of 10 mm between the transducer housing and the resonator reduced the area of the annular slot but allowed pressure equalisation. The sound source was formed by clamping a 12 mm polyester membrane between the metal housing and a ceramic sleeve which electrically insulated the back electrode from the housing. A 50 nm layer of aluminium on the membrane faced into the sphere and contacted the metal housing. The active area of the source was 3 mm2 and it was coupled to the resonator through a tube 0.25 mm long. The disk was tensioned with a spring through the 1.9 mm diameter back electrode which had six 0.3 mm diameter holes drilled to a depth of 0.5 mm to increase the compliance of the disk. A 60 V rms signal was superimposed on a dc bias of about 180 V and applied across the membrane. The dc bias was required because a low frequency network analyser was used to detect the amplitude and phase relative to the drive signal. The detector, shown in Figure 6.16, was based on a commercially available microphone capsule, which had a permanently polarised polyvinylidene fluoride disk as its active element. The disk had a capacitance of about 5 pF and its high 295

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Speed of Sound

impedance was matched to the external electronics with a JFET circuit mounted inside and electrical shield and close to the disk to avoid stray capacitances. The detector capsule was clamped in place within its brass housing by a PTFE sleeve and acoustically coupled to the sphere through a 15 mm3 volume and a 0.7 mm diameter waveguide of length 4 mm. This arrangement in principle forms a Helmholtz resonator but leakage conductance dominated the waveguide [134,88] so that it behaved as an open tube so that Equation (6.23) with Equation (6.29) could be used to correct the measured frequencies. Moldover and co-workers [53–55] and Ewing and co-workers [68,134] used short coupling tubes in their early investigations of the performance of spherical resonators. In the hemispherical resonator Angerstein [143] used a waveguide to reduce the perturbation on the resonance frequencies arising from the source transducer located at the position of maximum acoustic energy. Trusler and Zarari [92] used transducers that were fabricated as an integral part of the resonator wall rather than removable units. The source and detector had a 3 mm diameter active area arranged nearly flush with the inner surface of the sphere. This arrangement eliminated the slot around the removable housing and pressure equalisation was achieved with a hole through the centre of the membrane. In this case, the moving element of the detector was a 6 mm thick electret polymer film (polyvinylidene) coated with aluminium on the surface in contact with the resonator; Ewing and Trusler had used this arrangement previously [31]. The output of the detector was buffered by a JFET connected to the outside of the lead-through pin. Others have adopted this arrangement for the detector [33,83,91,91,143]. The source, which was formed from a 12 mm thick polyester film, was excited by a signal from a frequency synthesiser that was amplified to about 180 V rms without a dc bias so that the sound produced was at a frequency twice that of the feed; this reduced the electrical crosstalk between the transmitter and receiver. These polymer films limited the upper operating temperature to about 375 K. Thus to extend the temperature range up to 475 K, Estrada-Alexanders and Trusler [105] replaced the diaphragms with a 12 mm thick polyimide sheet coated with a 50 nm gold layer. Polyimide is not an electret material and the detector was operated at constant electrical charge with a dc bias of 100 V. To operate at the higher temperatures the JFET was replaced with a preamplifier, located at ambient temperature, that provided a potential-difference gain close to unity and was connected to the detector by a triaxial cable [40]. The signal generated by the detector was carried in the central conductor and the outer screen served as the ground. The intermediate coaxial conductor was connected at the output of the preamplifier but not the transducer and, because of the near unity gain of the amplifier, this arrangement effectively compensated for the cable capacitance which was large compared to that of the transducer. For operation at higher temperatures the dielectric layer was formed by melting glass powder on the back plate [101] and combined with a thin metal film to create a capacitor; pressure equalisation was achieved by a hole in the metal foil. The whole transducer, shown in Figure 6.1, consisted of a cylindrical extension, a cap, a Kovar2 plate, a ceramic insulating sleeve and a backing flange. The Kovar2 plate was one of the elements of the capacitor and was electrically isolated from the transducer body by the ceramic sleeve. The active element of the capacitive transducer was formed from an 296

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9.5 mm diameter disc of 6 mm thick aluminium foil which was electrically connected to the body of the transducer. The Kovar2 back plate and ceramic sleeve were sprayed with Corning powdered glass, suspended in acetone, and the glass melted in an argon (or nitrogen) purged furnace at a temperature of about 800 K. The cylindrical transducer housings and cap were machined from type 316 stainless steel. The cylindrical extension consisted of a circular flange and a cylindrical protrusion extending from the centre of the flange. A feedthrough was assembled in the cylindrical extension from a glass disc and Kovar2 wire, which passed through the center of the circular faces, fused to the metal flange by heating to a temperature of about 922 K. The diameter and length of the cylindrical extension were adjusted to create a sliding fit within the resonator port. The inner surface of the transducer and the resonator were flush when the transducer was fitted with the cap and glass coated Kovar2. The cap, which held the foil and glass-coated electrode, was threaded onto the cylindrical extension and the surface that formed part of the internal wall of the resonator machined to match the radius of curvature of the sphere. When the transducer was assembled, the capacitance was between (10 and 30) pF without an applied dc bias. The transducers were tested with a dc bias of up to 350 V dc without failure and they maintained satisfactory electrical characteristics when tested to temperatures up to 700 K. For operation at higher temperatures, Ripple et al. [127] have described transducers for the acoustic thermometer that can operate at temperature up to 800 K. These transducers should have noise levels no larger than the equivalent of 105 Pa? Hz1/2, a smooth frequency response from 2.5 kHz to approximately 17 kHz, and do not appreciably perturb the frequencies of the acoustic resonances. To meet these requirements Ripple et al. [127] fabricated the transducers shown in Figure 6.17. The membrane was formed from a 25 mm thick square of monocrystal-

Figure 6.17 Cross-section through the acoustic and microwave transducers of Ripple et al. [127] developed for operation at temperatures up to 800 K. LEFT: Acoustic transducers used for both excitation and detection of the sound field. The membrane is a square of 25 mm thick monocrystalline silicon that fits within a stainless steel spacer that also served to maintain a 40 mm gap between the rear face of the silicon and the back electrode. The back electrode is set within a ceramic insulator. RIGHT: The microwave probe used to connect the spherical chamber to a network analyser. The probe was inserted about 3 mm into the cavity. Three microwave transducers are used to ensure that the resonator deformations are probed in all three Cartesian co-ordinates.

297

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line silicon. This membrane fits in the well of a photoetched stainless steel disk that also serves as a spacer to maintain a 40 mm gap between a back stainless steel electrode and the silicon. The back electrode is set into a ceramic insulator fabricated from machinable alumina. The transducer, which had a capacitance of about 3 pF, was found to operate as both a source and detector over the required frequency range (0.5 to 18) kHz. Figure 6.17 also shows the 3 mm long pin transducer used to excite and detect the resonance frequencies of the transverse magnetic microwave modes that were used to determine the variation of resonator dimensions with temperature. The microwave resonances have high quality factors (approximately 104) and, thus, the impedance mismatches between the transducer, the coaxial line, and the hermetic feedthrough do not significantly affect the resonant frequencies that were measured with a network analyser. The acoustic transducer design reported by Ripple et al. [127], and shown in Figure 6.17, leads us to speculate that micro electro-mechanical system (MEMS) capacitive transducers fabricated completely with the methods used to manufacture integrated circuits (IC), and described in the literature [281–284], might be applied to techniques described in this section. In principle, MEMS could provide transducers that more nearly represent point sources and detectors that can operate at temperatures over 500 K and, when coated with appropriate layers, are chemically compatible with the fluid understudy. For example, Noble et al. [284] report a 1 mm by 1 mm square silicon nitride membrane 1 mm thick deposited over a fluid filled cavity 5 mm deep that was fabricated on a silicon substrate; transducers with 2 mm by 2 mm diaphragms were also fabricated. This 1 mm2 device, which was much larger than those of references [282] and [283], has been evaluated at ambient conditions as both sources and detectors in air at frequencies over 1 MHz (with a 6 dB bandwidth of about 0.7 MHz); the devices were also used as detectors in water at frequencies up to 10 MHz. The devices reported by Khuri-Yakub and co-workers [282,283] were fabricated from a series of cells each with diaphragms that had linear dimensions of between (20 and 40) mm and 1 mm thick. Transducers have also been fabricated from a combination of polymer film and back plates, fabricated with micromachining techniques, with surface undulations of known geometry [285–288]. It is also possible that MEMS transducers with piezoelectric zinc oxide [289] or aluminium oxide might be used. An alternative to developing transducers that can be exposed directly to high temperature and possibly a reactive sample is to use a remote transducer, at ambient temperature, coupled to a resonator by waveguides. General theoretical considerations for the use of waveguides are provided in Chapter 5 of reference [2]. Here we describe the practical implementation of a waveguide with thin metal diaphragms that also separate the fluid within the resonator from that in the waveguide [151,153]. A schematic of the housing that held both the diaphragm and waveguide is shown in Figure 6.18. The diaphragm was formed from a 10 mm diameter 25 mm thick stainless steel disk that was electron beam welded about the circumference to the housing. Two of these housings were used, one for the source and the other the detector. The waveguide used for each consisted of a horn-shaped section tapering out from the flange near the diaphragm to a cylindrical extension tube that provided spatial separation between the resonator and the transducers. The horn was 298

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Figure 6.18 Cross-section through a plug holding the diaphragm and lower portion of the waveguide including the screen, which provided a resistive acoustic termination to dampen reflections [151,153].

manufactured, by Bru¨el and Kjær for use in probe microphones, from thin-walled stainless steel and its outside diameter tapered exponentially from 33 mm at the junction with the extension tube to 12 mm over the length of 0.15 m. Standing waves in the waveguides would couple to the modes of the resonator through the diaphragm and were damped by a metal screen mounted in the 12 mm housing end. According to the manufacturer, the frequency response was flat between (0.03 and 8) kHz and then dropped by 20 dB per octave above 10 kHz. The waveguides were filled with argon and the pressure controlled to about 5 kPa above that of the gas within the resonator to minimise the stress applied to the diaphragm and maximize the acoustic transmission. Thermal insulation was placed around the waveguides to attenuate fluctuations in the phase of the acoustic wave arising from changes in room temperature. The walls of the waveguide tubes were thin and, without pressure compensation, limited the upper operating pressure to about 1 MPa. This solution is mechanically rather complex but has been used successfully for many measurements with gases [174–185]. Because of both viscous damping in the waveguide screens and acoustic impedance mismatch between the gases and the diaphragms, higher sound pressures were required to excite and detect resonance within the cavity. Thus Gillis and co-workers [151,153] used a small (about 20 mm diameter) electromagnetic (moving coil) source, taken from an earpiece, and a detector taken from a hearing aid, rather than solid dielectric capacitance transducers of the type traditionally used for measurements with gases. Non-mechanical contact excitation of the acoustic modes can also be achieved with a laser. In photoacoustic excitation, the sample is irradiated with a beam of light at a frequency of a molecular absorption and the frequency of the light modulated at 299

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the required acoustic frequency. Fiedler and Hess [290] describe photoacoustic methods with both spherical and cylindrical geometry cavities. In reference [290] the acoustic signal was detected with an electret microphone.

6.2

Thermodynamic Properties from the Speed of Sound

While the speed of sound in a phase is itself important in a number of applications, most of the interest in this quantity arises from its relation with the thermodynamic properties of the medium. We shall restrict our discussion of this relation to the cases of isotropic Newtonian fluids and isotropic elastic solids. We begin with fluid phases as these usually support only a single longitudinal sound mode with propagation speed u given by [27] u2 ¼ ðqp=qrÞS :

ð6:66Þ

Here, p is the pressure, r is the mass density and S is entropy. We shall discuss this relation, and those that may be derived from it, at some length. The case of isotropic elastic solids will be taken up later in Section 6.2.3. Equation (6.66) is strictly valid only in the limits of vanishing amplitude and vanishing frequency [2,27,291]. The situation corresponding to the first of these limits is extremely easy to approach in practice, while that corresponding to the second is usually, but not always, realised. For the purposes of this discussion, we shall assume that a practical measurement of the speed of sound in a fluid is either identical with the zero-frequency limit or that it may be corrected to that limit as discussed in Section 1. Two equivalent forms of Equation (6.66) may be derived by recalling the definition of the isentropic compressibility kS and the relation between kS and the isothermal compressibility kT : u2 ¼

1 ; rkS

ð6:67Þ

u2 ¼

g : rkT

ð6:68Þ

and

Here, g ¼ cp =cV , and cp and cV are respectively the isobaric and isochoric specific heat capacities. Equation (6.67) shows that the isentropic compressibility may be obtained from measurements of the speed of sound and the density, while Equation (6.68) shows that the isothermal compressibility may also be obtained if one also knows g. Equation (6.67) forms the basis of almost all experimental determinations of the isentropic compressibility. Since density and isothermal compressibility may be measured more-or-less directly, and often without great difficulty, Equation (6.68) provides a convenient route to g. 300

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In order to develop further the relation between u and the equation of state of the medium, it is often convenient to work in terms of molar quantities with either (T, p) or (T, rn) as independent variables, where rn is the amount-of-substance density. In terms of the former, we find 1 u2 ¼ M

"   2 #1 qrn T qrn  ; qp T r2n Cp;m qT p

ð6:69Þ

while, in terms of the latter, we have 1 u ¼ M 2

"

qp qrn

 2 # T qp þ 2 : rn CV;m qT rn T



ð6:70Þ

Here, M is the molar mass, Cp,m is the isobaric molar heat capacity and CV,m is the isochoric molar heat capacity. It is important to note that u is not a purely configurational property of the medium. Thus, in order to compute u from an equation of state in the form rn ¼ rn (T, p) or p ¼ p(T, rn ), one requires also knowledge of the heat capacity in some reference state. For example, Cp,m may be written Zp Cp;m ¼

o Cp;m ðTÞ



2 T q2 r1 n =qT dp;



ð6:71Þ

po

where Cpo, m is the isobaric heat capacity on the reference isobar p ¼ po. The analogous expression for CV,m is: Zrn CV;m ¼

o CV;m ðTÞ







T=r2n q2 p=qT 2 drn ;

ð6:72Þ

ron

where ron is the amount-of-substance density on a reference isochore. A second notable feature illustrated by Equations (6.69) and (6.70) is that neither u nor u2 enjoy a linear relationship with the equation of state. Often, a single mathematical form of the equation of state is used which is valid from the perfect-gas state up to the maximum pressure or density of interest. It is then convenient to let po ! 0 and ron ! 0, in which case Cpo, m ! Cppg, m and CVo , m ! CVpg, m , where superscript ‘pg’ denotes a property of the perfect gas. Usually, when constructing an equation of state, (T, rn) are the independent variables of choice, supplemented, in the case of a mixture, by composition variables. The thermodynamic surface in the fluid regions may then be represented by a suitable equation of state in the form p ¼ p(T, rn ) plus an expression for CVpg, m (T). 301

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Alternatively, a representation of the molar Helmholtz function Am may be adopted, in terms of which r2 u ¼ n M 2

" #  ½qðqAm =qrn ÞT =qT2rn q2 Am  : qr2n T ðq2 Am =qT 2 Þr

ð6:73Þ

n

Thermodynamic properties of fluids are generally determined from experimental speeds of sound by one of three general approaches. In the first, no restrictive assumptions are made about the functional form of the equation of state; the speeds of sound are simply combined with other measured quantities to obtain one or more additional properties through exact thermodynamic identities, such as Equations (6.67) or (6.69). This approach may be applied either to a single state point or over a region of the thermodynamic surface. In the second approach, an explicit or implicit parameterisation of the equation of state is assumed and the parameters fitted to experimental speeds of sound alone. The resulting model may then be used to predict all other observable thermodynamic properties of the phase over the region in which the assumed functional form is valid. Finally, a particular parameterisation of the equation of state may again be assumed but with the parameters fitted to the speed of sound and other thermodynamic data simultaneously. In an elaboration of this method, the functional form is itself optimised by considering a large bank of terms from which an empirical equation of state might be constructed and finding the subset of a chosen size which best represents the data according to a specified criterion [292]. In what follows, we will give emphasis to those methods in which the speed of sound is either the sole or the main experimental input. In doing so, it makes sense to treat gas and liquid phases separately. However, it is important to emphasis the obvious and often best approach which is to combine different kinds of experimental data in a multi-parameter multi-property analysis leading to a wide-ranging equation of state, usually in the form of an empirical Helmholtz function. This approach is commonly used to obtain very accurate and wide-ranging equations of state for the fluid phases of pure compounds [293–298]. It has also been applied to a few wellstudied mixtures [299]. The non-linear relation between u2 and Am complicates the analysis somewhat but speeds of sound remain extremely valuable inputs to such a procedure. Density and phase equilibrium data are also essential. The more specialised methods, to which we now turn, are advantageous when other properties are not available with commensurate accuracy or when an equation of state specialised to one particular region of the phase diagram is required.

6.2.1

Gases

We have already seen that the speed of sound in gases may be measured rapidly and accurately over a wide range of states. This has permitted many applications of such measurements and motivated the development of several different approaches to the calculation of thermodynamic properties from the experimental results. 302

Speed of Sound

303

The Gaseous Equation of State The p, rn, T relation of a real gas is given by the virial equation of state: p ¼ rn RTð1 þ Brn þ Cr2n þ   Þ:

ð6:74Þ

Here, 1, B, C, . . . are virial coefficients which depend only upon temperature and (in a mixture) composition. Combining Equations (6.70), (6.72), (6.74) with the relation Cppg, m ¼ CVpg, m þ R one obtains u2 ¼ A0 ð1 þ ba rn þ ga r2n þ   Þ;

ð6:75Þ

where A0 is given by A0 ¼

pg RT ðCV;m þ RÞ pg CV;m M

;

ð6:76Þ

and 1, ba, ga, . . . are so-called acoustic virial coefficients which also depend upon temperature and composition only. The second acoustic virial coefficient ba is given by ba ¼ 2B þ 2ðgpg  1ÞTB0 þ fðgpg  1Þ=gpg gT 2 B00 ;

ð6:77Þ

where primes denote differentiation with respect to temperature, and the third acoustic virial coefficient is given by ga ¼ fðgpg  1Þ=gpg gfB þ ð2gpg  1ÞTB0 þ ðgpg  1ÞT 2 B00 g2   : 1 þ ð1=gpg Þ ð1 þ 2gpg ÞC þ ð½gpg 2  1ÞTC 0 þ ðgpg  1Þ2 T 2 C 00 2

ð6:78Þ

The experimental quantities are almost always (u, T, p), rather than (u, T, rn), and an expansion of u2 in powers of pressure is more easily applied. Such a relation may be obtained by eliminating rn on the right hand side of Equation (6.75) with the series inversion of Equation (6.74). The result of that operation may be written u2 ¼ A 0 þ A 1 p þ A 2 p2 þ    ;

ð6:79Þ

A1 ¼ A0 ðba =RT Þ;

ð6:80Þ

where

303

304

Speed of Sound

and A2 ¼ A0 ðga  Bba Þ=ðRTÞ2 :

ð6:81Þ

For gases at low and moderate densities, Equations (6.74), for p(T, rn ), and (6.75), for u2 (T, rn ), converge rapidly. Typically, only the leading three or four terms of these equations are required to reduce the truncation error to the order of usual experimental errors [31,90,108]. Equation (6.79), for u2 (T, p), also converges rapidly [54,78,87] but there is empirical evidence that, compared with Equation (6.75), convergence is less rapid at sub-critical temperatures [90].

Determination of Perfect-Gas Properties The quantity A0 is clearly identical with the squared speed of sound in the gas in the limit of zero density or pressure – an experimentally realisable perfect-gas property. Equation (6.76), which relates A0 to other quantities, forms the basis of several of the applications of sound speed measurements. In the case of very simple gases, especially monatomic gases but also some diatomic gases such as nitrogen, the perfect-gas heat capacity may be calculated from statistical mechanics with very high accuracy, and the chemical and isotopic composition of the gas may be known in sufficient detail to permit a similarly accurate determination of the molar mass. Then, an experimental determination of A0 at the temperature of the triple point of water, where T is known by definition, leads to the value of the gas constant. Indeed, the presently accepted value of R is based on just such measurements [40]. Alternatively, knowing R, an experimental determination of A0(y) at a fixed empirical temperature y permits the evaluation of the corresponding thermodynamic temperature T(y). Acoustic thermometry in this form has been used to advantage at temperatures below 20 K [300]. Better, in many situations, is an experimental determination of the ratio A0 (y1 )=A0 (y2 ) which permits evaluation of the ratio T(y1 )=T(y2 ) unburdened by the uncertainty of R. If one of the temperatures is made identical with the temperature of the triple point of water then an absolute determination of thermodynamic temperature is possible by this method. Acoustic thermometry of this kind has been reported recently for temperature between (90 and 323) K [45,46], including determinations of the triple-point temperatures of mercury and gallium [45]. Apart from its applications in basic thermal metrology, Equation (6.76) also provides one of the best available means of evaluating the perfect-gas heat capacities of those pure compounds that are sufficiently volatile and stable at the temperatures of interest to be studied in the gas phase [31,54,63,64,68,71,73,78– 80,87,89,90,92,96,97,99,102,111,112,174–178]. Colgate and co-workers [65,69,74,76,77,100] determined the perfect-gas heat capacities of volatile and nonvolatile hydrocarbons that were a dilute gaseous solute in an argon gas solvent. 304

Speed of Sound

305

From Equation (6.76) for CVpg, m we obtain pg ¼ R=fðMA0 =RTÞ  1g: CV;m

ð6:82Þ

Given the small uncertainty of R and the possibility of measuring both T and A0 with high accuracy, the uncertainty dCVpg, m of CVpg, m is usually dominated by the uncertainty dM in the molar mass and, in that case, we have 

 pg pg pg pg dCV; m =CV; m &fg =ðg  1ÞgðdM=M Þ:

ð6:83Þ

Since gpg =(gpg  1) is at least 3.5 for a diatomic gas, and often much greater for polyatomic gases, the relative uncertainty of M is substantially amplified in determining the corresponding relative uncertainty of CVpg, m . A further very useful application of Equation (6.76) is in the determination of composition in a binary gaseous mixture of components A and B. In this case, A0 is given by A0 ¼

pg RTðCV; A þ RÞ

!

pg CV; A MA

 1 þ a1 x ; 1 þ a2 x þ a3 x2

ð6:84Þ

where x is the mole fraction of B and the coefficients are: pg pg pg  CV;A Þ=ðCV;A þ RÞ; a1 ¼ ðCV;B pg pg =CV;A Þ  2; a2 ¼ ðMB =MA Þ þ ðCV;B

a3 ¼ ½1 

pg pg ðCV;B =CV;A Þ½1

ð6:85Þ

 ðMB =MA Þ:

From Equation (6.84), a quadratic equation may be obtained and solved for x. Usually, A0(x) is a single-valued function and just one root will exit in the domain [0,1]. The precision with which A0 may be determined in practise usually leads to a sensitive measure of the composition [114,128]. Of course, the perfect-gas heat capacity and the molar mass of each component must be known with appropriate accuracy. The system {(1x)O2 þ xAr} is one of the rare examples of a mixture for which A0(x) is not a single valued function. Determination of Virial Coefficients and Intermolecular Potential Parameters Given the relations presented above between the speed of sound and the virial coefficients of the gas, it is clear that results at a single temperature are insufficient to deduce any meaningful information about the latter. However, results obtained on a 305

306

Speed of Sound

sequence of isotherms covering a range of temperatures may be used to obtain at least the second and third virial coefficients reliably. The simplest approach is to determine the leading coefficients of Equation (6.79) by fitting them directly to results along each isotherm. It is then necessary to adopt a model for each of the corresponding ordinary virial coefficients as functions of T. In the case of an analytical model, one can obtain expressions for the temperature derivatives of the virial coefficients and thence relations, containing the model parameter, between A1, A2, A3, . . . and the corresponding virial coefficients B, C, D, . . . . Then, having obtained gpg from A0, it is a simple matter of fitting the parameters of the models to the experimental values of A1, A2, A3, . . . : A1(T) is used to determine B(T); then A2(T) and B(T) are used to determine C(T), and so on. This procedure is of course subject to errors associated not only with the experimental quantities but possibly also with the assumption implied by the models adopted for B, C, D, . . . . The second virial coefficient is particularly insensitive to such errors and analyses using, for example, polynomials in inverse temperature give almost exactly the same results as more sophisticated models based on intermolecular potential-energy functions [68]. The situation for the higher virial coefficients is less clear but there is strong evidence that third virial coefficients may be obtained with good accuracy when a theoretically-based model is used. Much work has been carried out with the hardcore-square-well intermolecular potential model [64,68,71,78,87]. This model contains three parameters: the diameter s of the hard cores, the depth e of the attractive well, and the range l of the attractive well. Analytical expressions for B and C follow from this model and there is an analytical expression for D for the specific case of l ¼ 2. In order to fit the acoustic data it is necessary to obtain a different set of parameter values for each virial coefficient in turn but good results are then obtained. For a number of systems investigated, the truncated virial equations obtained gave results which compared favourably with independent (p, rn, T) data [301]. Equation (6.75) can easily be used to analyse the acoustic isotherms instead of Equation (6.79). The amount-of-substance densities required in this expansion at each state point are computed from the virial equation of state with trial values of the virial coefficient. The analysis proceeds with determination of the acoustic virial coefficients and fitting of parameters in models for the ordinary virial coefficients. Improved values of rn may then be obtained and the analysis iterated to convergence. In practice this is easy and rapid, requiring at most three cycles. It has already been remarked that Equation (6.75) may converge more rapidly than Equation (6.79) and the slight additional complication of this approach is often justified [31,90,92]. It is interesting to note that the second and third virial coefficients obtained by these procedures are remarkably insensitive to the details of the intermolecular potential model chosen. Experimental second acoustic virial coefficients generally require a three-parameter intermolecular potential model. These model often extrapolate with remarkable accuracy to temperatures far beyond those studied acoustically [31,68]. When one considers the third and higher virial coefficients then non-additivity of the intermolecular potential energy is an issue. In the approach 306

Speed of Sound

307

described above in connection with the square-well model, a different effective pair potential, incorporating non-additive effects, is obtained for each virial coefficient [301]. Alternatively, one can use the same intermolecular pair potential for both B and C and account for non-additive effects explicitly with a three-body term. This approach has been used to advantage with a three-parameter model pair potential and a one-parameter model three-body term, thereby permitting an excellent representation of B and C using just four parameters [302]. The analysis described so far is sub-optimal in one respect because it involves sequential fitting of parameters: first one finds the acoustic virial coefficients; and then one fits model parameters to the acoustic virial coefficients. An improved procedure has been described in which the intermolecular potential parameters are fitted simultaneously to the whole sound-speed surface [73,303]. When applied to propane [303], the results were found to agree with experimental (p, rn, T) and heat capacity data over a large region of temperature and pressure, extending far beyond the domain in which the speed of sound itself had been measured. Empirical Equation of State for the Gas Phase The success of the methods described in the preceding section is attributed largely to the facts that the virial equation of state is rigorously based in theory and that even crude intermolecular potential models can represent accurately the temperature dependences of the virial coefficients. Of course there are many other equations of state which may be applied to the gas phase and it is possible, in principle, to fit the parameters of any of them to speeds of sound alone. For example, a surface-fitting procedure has been described in which an empirical Helmholtz function was employed and, in this case, a comparison with directly measured (p, rn, T) data was favourable [122]. This success was perhaps to be expected in view of the fact that the functional form used for Am was in fact equivalent to a truncated and depleted (i.e. one with some lower-order terms omitted) virial equation with empirical functions for the virial coefficients. Generally, however, the results obtained with equations of state not of virial form are disappointing unless (p, rn, T) data are included in the fit. Of course, multi-property fitting is the method of choice when different kinds of data are available and a wide-ranging equation of state is desired. Numerical Integration Methods For Gases The procedures described above involve assumptions about the functional form of the equation of state. The most successful of them make those assumptions at the molecular level and the results are very good in the region within which a virial equation of state converges rapidly. However, beyond that domain, other methods are needed. These methods are based on numerical integration of the differential equations which link the speed of sound with other thermodynamic properties. This approach is not entirely independent of non-acoustic data (initial conditions are required) but it is based on rigorous thermodynamics without restrictive assumptions about the equation of state. We have already seen that the perfect-gas heat 307

308

Speed of Sound

capacities may be obtained from the zero-density speed of sound. Thus, if the (p, rn, T) relation is also obtained, we will have sufficient information to compute all of the observable thermodynamic properties of the gas. In order to obtain the (p, rn, T) relation for the gas, it is convenient to introduce the dimensionless compression factor, Z ¼ p=rn RT, in terms of which

u

2

2 3       )2   ( M qZ R qZ 4 Zp 5;  ZþT RTZ 2 qp T Cp;m qT p

 ¼

ð6:86Þ

and 

qCp;m qp



 (    2  ) R qZ 2 q Z ¼ þT 2T : p qT p qT 2 p T

ð6:87Þ

Equations (6.86) and (6.87) comprise a non-linear second-order system of partial differential equations which can be solved numerically for Z(T, p) subject to specified initial conditions [109]. The required data are the speed of sound as a function of temperature and pressure over the whole domain in which we wish to develop the solution, plus appropriate initial conditions from which to start the numerical integration. Typically, the speeds of sound will be measured on closely-spaced isotherms which each extend from the same upper bound pmax down to a sufficiently low pressure for A0 to be obtained by extrapolation. The solution will then be developed within a rectangular domain in (T, p) co-ordinates, extending down to zero pressure where Z ¼ 1 and u2 ¼ A0. An interpolation procedure is required from which a speed of sound may be obtained anywhere within this region and it is quite possible to achieve this by means of numerical interpolation within a table of experimental data [92]. This has the advantage of avoiding fitting functions entirely but it is necessary to ensure that the experimental data are sufficiently closely spaced. In considering along which edge or edges of the domain to impose initial conditions, it is instructive to refer back to Equation (6.66). If the path of an isentrope is known in advance then, given u(T, p), Equation (6.66) can be integrated for r along that path starting from one initial value. The same procedure can of course be applied on a sequence of different isentropes and the (T, p) co-ordinates of the initial values will then define a path which cuts all isentropes. Since any rigorous procedure used to obtain r (or Z) from u must be equivalent to a solution of Equation (6.66), we can deduce immediately that initial values of r (or, equivalently, of Z) will be required along a path which cuts across the isentropes [109]. Further since, as illustrated in Figure 6.19, the slope (qp=qT)S of an isentrope is everywhere positive, the only edge of the solution domain which cuts all isentropes is the isotherm at the lowest temperature. Since the present system of equation is of the second order, two sets of initial values are required. Thus, in addition to Z, we require also (qZ=qT)p along the same path. This additional information is in fact 308

Speed of Sound

309

thermodynamically equivalent to establishing the initial path of the isentropes and so the analogy with a solution of Equation (6.1) is complete. A simple Euler algorithm for integrating Equations (6.86) and (6.87) begins at the lowest temperature T0. From the initial conditions, it is straightforward to obtain (qZ=qp)T and thence to obtain Cp,m from Equation (6.86). Then, differentiating Cp,m with respect to pressure and applying Equation (6.87), one obtains (q2 Z=qT 2 )p at T0. This provides sufficient information to estimate Z and (qZ=qT)p on a new isotherm at T1 ¼ T0 þ dT by means of Taylor expansions about T ¼ T0 correct to the second derivative; further iterations of the whole procedure allow the solution to be developed isotherm-by-isotherm until the upper limit in temperature is reached. The method is stable in the sense that the effects of small errors in the initial conditions decay with increasing temperature along isobars. Although the first-order Euler method described here is very simple to implement, it is rather inefficient and very small temperature steps are required to maintain sufficiently small truncation errors in the Taylor expansions. A second-order algorithm, such as the predictor-corrector method, is much more efficient and only slightly more difficult to implement [92]. The whole procedure can be carried out without the use of fitting functions; even the differentiation of Z and Cp,m with respect to pressure can be accomplished accurately by numerical means [92,109,113,304]. Equations (6.86) and (6.87) are written with (T, p) as the independent variables but an entirely equivalent system exists with (T, rn) as the independent variables. In this case, the most convenient domain in which to develop the solution is one bounded from above by an isochore. The initial conditions required are Z and (qZ=qT)rn on the lowest isotherm. It is still possible to work with u(T, p) as the inputs because whenever a speed of sound at specified (T, rn) is called for, the corresponding compression factor and pressure have already been evaluated [109,113].

Figure 6.19 Temperature-pressure diagram for methane showing isentropes and phase boundaries: -----, isentropes; ——, phase boundaries.

309

310

Speed of Sound

Probably the greatest limitation of the numerical integration methods is the requirement for initial values of (qZ=qT)p or (qZ=qT)rn . Inspection of Equation (6.86) shows that, because of the pre-factor (R=Cp, m ), the procedure can tolerate a larger uncertainty in fZ þ T(qZ=qT)p g than would be acceptable in fZ  T(qZ=qp)T g. However, the requirement for the temperature derivative is demanding and in practice it will usually be necessary to have (p, rn, T) data on several isotherms spanning a range of temperatures near T0 in order to obtain this quantity with sufficient accuracy [113]. Nevertheless, these methods have been applied successfully to methane [92], ethane [117] and argon [113] and they would no doubt be useful in the investigation of new substances where it was impractical or uneconomic to make (p, rn, T) measurements over the whole surface. The propagation of errors has been studied in some detail [109,304].

6.2.2

Liquids

In the case of liquids, it is often relatively easy to make accurate (p, rn, T) measurements at moderate pressures. For example, vibrating-tube densimeters suitable for pressures up to 40 MPa are available and, with proper calibration, they can be quite accurate. One use of sound speed measurements in the regime where (p, rn, T) data are available is as an alternative to calorimetry in the determination of heat capacity from Equation (6.69). The speed of sound, in combination with (p, rn, T) data, is also the main source of experimental values of the heat-capacity ratio g and the isentropic compressibility kS of pure liquids and mixtures. At higher pressures, (p, rn, T) measurements are much more difficult and it is in this region that sound speed measurements in liquids are probably of the greatest value. The method of analysis is then one of numerical integration. Numerical Integration Methods for Liquids Although the procedure for obtaining thermodynamic properties of liquids from the speed of sound is somewhat similar to that described for gases, there are sufficient differences to warrant a separate discussion. The working equations with (T, p) as the independent variables are the mass-basis equivalent of Equation (6.69), " u ¼ 2

 2 #1 T qr  2 ; r qT c p p T



qr qp

ð6:88Þ

and 

qcp qp





q2 v ¼ T qT 2 T 310

 ; p

ð6:89Þ

Speed of Sound

311

where v ¼ 1/r is the specific volume. The method has generally been applied in a rectangular (T, p) space with initial values of r and cp specified along the isobar at the lowest pressure p0 (often 0.1 MPa) [305–309]. The initial values of density must be sufficiently accurate to determine both the first and second temperature derivatives; alternatively, a separate determination of thermal expansion can be made. From the speed of sound and the initial values on the isobar at p ¼ p0, it is possible to determine both (qr=qp)T and (qcp =qp)T and hence to estimate r and cp on a new isobar at p1 ¼ p0 þ dp by means of first-order Taylor expansions. Repetition of the procedure allows development of the solution isobar-by-isobar until the upper limit in pressure is reached. A second-order integration method would greatly increase the efficiency of the procedure. The issues associated with where to impose the initial conditions are identical in principle with those discussed in connection with gases. However, (qp=qT)S in the liquid is much greater and, as illustrated in Figure 6.19, the isentropes rise steeply from the vapour pressure curve on a (p, T) diagram. Consequently, the isobar at the lowest pressure comes close to meeting the criterion of cutting all isentropes which pass through the solution region. However, it does not fully satisfy that criterion because some additional isentropes flow in through the isotherm defining the lowtemperature boundary of the integration domain. This fact may be the cause of error in previous applications of the method. Most workers have used polynomial function of temperature to represent r and cp on each isobar and the usually slow variation of those quantities may help to stabilise the method despite the incomplete imposition of boundary conditions.

6.2.3

Solids Elastic Constants

The elastic properties of an isotropic solid may be specified by a pair of quantities such as the bulk modulus K and the shear modulus G. Other commonly used parameters are Young’s modulus E, Poisson’s ratio s, and the Lame´ constants l and m. The shear modulus G is actually identical with the second Lame´ constant m and the other parameters are interrelated as follows [310]: E ¼ 9GK=ð3K þ GÞ ¼ 3ð1  2sÞK

9 > =

s ¼ ð3K  2GÞ=ð6K þ 2GÞ ¼ ðE=2GÞ  1 > ; l ¼ K  2G=3

ð6:90Þ

These elastic constants relate various components of stress and strain under isothermal conditions. In the case of pure shear stress, the resulting strain takes place without change of volume and so an isothermal and reversible shear process is also isentropic. Consequently, the shear modulus is the same for both static and dynamic processes in an elastic body. However, compressive stress gives rise to a change in 311

312

Speed of Sound

volume so that an isothermal compression is not generally isentropic. The isothermal bulk modulus K ¼ 1/kT therefore differs from the isentropic bulk modulus, which is KS ¼ 1=kS , and the two are related by the thermodynamic relationship: 1=KS ¼ 1=K  Ta2 =ðrcp Þ;

ð6:91Þ

where a is the coefficient of thermal expansivity. It follows that the isentropic analogue ES of Young’s modulus is given by ES ¼ E=f1  ETa2 =ð9rcp Þg:

ð6:92Þ

Longitudinal and Shear Waves Unlike fluid phases, solids generally support both a longitudinal or compressive sound mode, in which the direction of stress and strain is parallel to the direction of propagation, and two orthogonal shear or transverse wave modes in each of which the direction of shear stress is perpendicular to the direction of propagation. In an isotropic solid, the two shear modes are degenerate, each propagating with speed uS given by u2S ¼ G=r

ð6:93Þ

The speed uL of longitudinal sound waves in a bulk specimen is given by u2L ¼ ðKS þ 4G=3Þ=r:

ð6:94Þ

However, it should be noted that, for example, the actual phase speed u of a compression wave propagating along the axis of a solid bar generally depends upon the lateral dimension of the bar. In fact, u approaches uL only when the lateral extent of the bar is much greater than the wavelength. For bars of smaller cross section, the phase speed u is generally smaller than uL and, when the lateral dimensions are much smaller than the wavelength, it reaches a limit uE given by u2E ¼ ES =r;

ð6:95Þ

in an isotropic elastic solid uL > uE > uS. Determination of Elastic Constants From Acoustic Measurements Acoustic, especially ultrasonic, methods are the most common means of determining the elastic constants of solids. High frequency (e.g. 10 MHz) ultrasonic measurements typically provide directly values of uS and uL [311]. When combined with a 312

Speed of Sound

313

measurement of the density, G and KS may then be determined. The difference between kS and kT in a solid material is typically very small (1 per cent or less [312]) and so the isothermal bulk modulus K is easily obtained from KS by means of a calculated correction according to Equation (6.91). Very approximate values of a and cp will suffice for that purpose. Once G and K have been obtained, s follows from Equations (6.90). Low frequency (e.g. 100 kHz) ultrasonic resonance experiments typically provide directly values of uS and uE from which G and ES are readily obtained [313]. A calculated correction then permits determination of the isothermal Young’s modulus E. Measurements at both low and high ultrasonic frequencies have been made for many materials over wide ranges of temperature [312]. Since the typical experimental uncertainties in the sound speeds are between (0.1 and 1) per cent, the differences between the dynamic (isentropic) and static (isothermal) elastic constants are on the margins of significance. Few measurements have been reported at pressures above ambient. However, given measurements of the two sound speeds along an isotherm, rough values of a and cp, and a measurement of the density at ambient pressure, it would be easy to determine the elastic constants as a function of pressure by means of a numerical integration.

References 1. Van Dael, W., in Experimental Thermodynamics, Vol II, Experimental Thermodynamics of Non-Reacting Fluids, Ch. 11., B. Le Neindre and B. Vodar eds., Butterworths, London, 1975. 2. Trusler, J.P.M., Physical Acoustics and Metrology of Fluids, Adam-Hilger, Bristol, 1991. 3. McSkimin, H.J., in Physical Acoustics, Vol I, Part A, Ch. 4., W.P. Mason, ed., Academic Press, New York, 1964. 4. Papadakis, E.P., in Physical Acoustics: Principles and Methods, Vol XII, Ch. 5., W.P. Mason and R.N. Thurston eds., Academic Press, New York, 1976. 5. Papadakis, E.P., in Physical Acoustics, Vol XIX, Ch. 2., R.N. Thurston and A.D. Pierce eds., Academic Press, New York, 1990. 6. Papadakis, E.P., in Physical Acoustics, Vol XIX, Ch. 3., R.N. Thurston and A.D. Pierce eds., Academic Press, New York, 1990. 7. Heydeman, P., in Physical Acoustics: Principles and Methods, Vol VIII, Ch. 4., W.P. Mason and R.N. Thurston eds., Academic Press, New York, 1971. 8. Bolef, D.I. and Miller, J.G., in Physical Acoustics: Principles and Methods, Vol VIII, Ch. 3., W.P. Mason and R.N. Thurston eds., Academic Press, New York, 1971. 9. Rayne, J.A. and Jones, C.K., in Physical Acoustics: Principles and Methods, Vol VII, Ch. 3., W.P. Mason and R.N. Thurston eds., Academic Press, New York, 1970. 10. Garland, C.W., in Physical Acoustics: Principles and Methods, Vol VII, Ch. 2., W.P. Mason and R.N. Thurston eds., Academic Press, New York, 1970. 11. Webber, G.M.B. and Stephens, R.W.B., in Physical Acoustics: Principles and Methods, Vol IV, Part B, Ch. 11., W.P. Mason, ed., Academic Press, New York, 1968. 12. Papadakis, E.P., in Physical Acoustics: Principles and Methods, Vol IV, Part B, Ch. 15., W.P. Mason, ed., Academic Press, New York, 1968.

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201. Hirshberg, J.G., Byrne, J.D., Woutes, A.W. and Boynton, G.C., Appl. Opt. 23, 2624, 1984. 202. Cummins, H.Z. and Gammon, R.W. J. Chem. Phys. 44, 2787, 1966. 203. Grimsditch, M. in Handbook of Elastic Properties of Solids, Liquids and Gases, Vol I, Dynamic Methods for Measuring the Elastic Properties of Solids, Ch. 14., M. Levy, H.E. Bass, and R.R. Stern eds.-in-chief, A.G. Every and W. Sachse, vol. eds., V. Keppens, supervising ed., Academic Press, New York, 2001. 204. Straty, G.C., Cryogenics 15, 729, 1975. 205. Wilson, D.A. and Liebermann, L.N., J. Acoust. Soc. Am. 19, 286, 1947. 206. Leonard, R.W., J. Acoust. Soc. Am. 20, 244, 1948. 207. Moen, C.J., J. Acoust. Soc. Am. 23, 62, 1951. 208. Kurtze, G. and Tamm, K., Acustica 3, 33, 1953. 209. Ohsawa, T. and Wade, Y.,Jpn. J. Appl. Phys. 6, 1351, 1967. 210. Eggers, F. and Kaatze, U., Meas. Sci. Tech. 7, 1, 1996. 211. Kaatze, U., Hushcha, T. and Eggers, F., J. Soln. Chem. 29, 299, 2000. 212. Wang, Z.J., and Nur, A., J. Acoust. Soc. Am. 89, 2725, 1991. 213. Wang, Z.J., Nur, A.M. and Batzle, M.L., J. Pet. Tech. 42, 192, 1990. 214. Ye, S., Alliez, J., Lagourette, B., Saint-Guirons, H., Arman, J. and Xans, P., Revue Phys. Appl. 25, 555, 1990. 215. Daridon, J.L., Acustica 80, 416, 1994. 216. Ding, Z.S., Alliez, J., Boned, C. and Xans, P., Meas. Sci. Technol. 8, 154, 1997. 217. Daridon, J.L. and Lagourette, B., High Temp. High Press. 32, 83, 2000. 218. Dutour, S., Daridon, J.L. and Lagourette, B., Int. J. Thermophys. 21, 173, 2000. 219. Ye, S., Lagourette, B., Alliez, J., Saint-Guirons, H., Xans, P. and Montel, F., Fluid Phase Equilib. 74, 177, 1992. 220. Daridon, J.L., Lagourette, B. and Xans, P., Fluid Phase Equilib. 100, 269, 1994. 221. Daridon, J.L., Lagrabette, A. and Lagourette, B., J. Chem. Thermodyn. 30, 607, 1998. 222. Lagourette, B., Daridon, J.L., Gaubert, J.F. and Saint-Guirons, H., J. Chem. Thermodyn. 27, 259, 1995. 223. Ye, S., Daridon, J.L., Lagourette, B., Alliez, J., Saint-Guirons, H. and Xans, P., High Press. Res. 8, 423, 1991. 224. Lu, Z., Daridon, J.L., Lagourette, B., and Ye, S. Rev. Sci. Instrum. 70, 2065, 1999. 225. Williamson, R.C., J. Acoust. Soc. Am. 45, 1251, 1969. 226. Zak, A., Dzida, M., Zorebski, M. and Ernst, A., Rev. Sci. Instrum. 71, 1756, 2000. 227. Kozhevnikov, V.F., Arnold, D.I., Briggs, M.E., Naurzakov, S.P., Viner, J.M. and Taylor, P.C., J. Acoust. Soc. Am. 106, 3424, 1999. 228. Muringer, M.J.P., Trappeniers, N.J. and Biswas, S.N., Phys. Chem. Liq. 14, 273, 1985. 229. Papdakis, E.P., J. Acoust. Soc. Am. 42, 1045, 1967. 230. Williams, J. and Lamb, J., J. Acoust. Soc. Am. 30, 308, 1958. 231. Ball, S.J. and Trusler, J.P.M., Int. J. Thermophys. 22, 427, 2001. 232. Del Grosso, V.A., J. Acoust. Soc. Am. 47, 947, 1970. 233. Williams, A.O., J. Acoust. Soc. Am. 23, 1, 1951. 234. Katahara, K.W., Rai, C.S., Manghnani, M.H. and Balogh, J., J. Geophys. Res. 86, 11779, 1981. 235. Secco, R.A., Manghnani, M.H. and Liu, T.-C., Geophys. Res. Lett. 18, 1397, 1991. 236. Nasch, Ph.M., Manghnani, M.H. and Secco, R.A., J. Acoust. Soc. Am. 65, 682, 1994. 237. Nasch, Ph.M., Manghnani, M.H. and Secco, R.A. J. Geophys. Res. 99, 4285, 1994.

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238. Del Grosso, V.A., The Velocity of Sound in Sea Water at Zero Depth, NRL Report 4002, 1952. 239. Del Grosso, V.A., Materials Suitable for Sound Applications, Part 1, Ultrasonic Velocities and Impedances of Selected Liquids, NRL Report 4191, 1953. 240. Kroebel, W. and Mahrt, K.-H., Acustica 35, 154, 1976. 241. Heydemann, P.L.M., Rev. Sci. Instrum. 42, 983, 1971. 242. Tilford, C.R., Metrologia 24, 121, 1987. 243. Fujii, K. and Masui, R., J. Acoust. Soc. 93, 276, 1993. 244. Webber, G.M.B. and Stephens, R.W.B., in Physical Acoustics, Vol IV, Part B, Ch. 11., W.P. Mason, ed., Academic Press, New York, 1968. 245. Chen, G., Yoneda, A., Getting, I.C. and Spetzler, H.A., J. Geophys. Res. 101, 25161, 1996. 246. McSkimm, H.J., J. Acoust. Soc. Am. 33, 12, 1961. 247. McSkimm H.J. and Andreatch, P., J. Acoust. Soc. Am. 34, 609, 1962. 248. Papdakis, E.P., J. Appl. Phys. 35, 1474, 1964. 249. Anderson, O.L., in Progress in Very High Pressure Research, F.P. Bundy, W.R. Hibbard and H.M. Strong eds., Wiley, New York, pp. 225–255, 1961. 250. Naimon, E.R., Weston, W.F. and Ledbetter, H.M., Cryogenics 14, 24, 1974. 251. Naimon, E.R., Ledbetter, H.M. and Weston, W.F., J. Mat. Sci. 10, 1309, 1975. 252. Read, D.T. and Ledbetter, H.M., J. Eng. Mat. Technol. 99, 181, 1977. 253. Weston, W.F., Ledbetter, H.M. and Naimon, E.R., Mat. Sci. Eng. 20, 185, 1975. 254. Ledbetter, H.M., Weston, W.F. and Naimon, E.R., J. App. Phys. 46, 3855, 1975. 255. Papadakis, E.P., J. Acoust. Soc. Am. 40, 863, 1966. 256. Papadakis, E.P., in Physical Acoustics: Principles and Methods, Vol XI, Ch. 5., W.P. Mason and R.N. Thurston eds., Academic Press, New York, 1975. 257. Papadakis, E.P., J. Acoust. Soc. Am. 45, 1547, 1969. 258. Peterson, G., Chcik, B.B., Fortunko, C.M., and Hirao, M., Rev. Sci. Instrum. 65, 192, 1972. 259. Bassett, W.A., Reichmann, H.-J., Angel, R.J., Spetzler, H. and Smyth, J.R., Am. Mineral. 85, 283, 2000. 260. Lieberman, R.C. and Li, B., Rev. Mineralogy 37, 459, 1998. 261. Maynard, J.D., McKenna, M.J. Migliori, A. and Visscher, W.M., in Physical Acoustics, Vol XX, Ch. 8., R.N. Thurston and A.D. Pierce eds., Academic Press, New York, 1992. 262. Migliori, A. and Sarrao, J.L. Resonant Ultrasound Spectroscopy, Wiley, New York, 1997. 263. Migliori, A., Sarrao, J.L., Visscher, W.M., Bell, T.M., Lei, M., Fisk, Z., and Leisure, R.G., Physica B183, 1, 1993. 264. Schwaz, R.B. and Vuorinen, J.F. J. Alloys Comp. 310, 243, 2000. 265. Ledbetter, H., Fortunko, C. and Heyliger, P., J. Appl. Phys. 78, 1542, 1995. 266. Goto, T. and Anderson, O.L., Rev. Sci. Instrum. 59, 1405, 1988. 267. Willis, B.T.M., Acta Crystallogr. A 42, 514, 1986. 268. Willis, B.T.M., Carlile, C.J., Ward, R.C., David, W.I.F. and Johnson, M.W., Europhys. Lett. 2, 767, 1986. 269. Mason, W.P., Piezoelectric Crystals and Their Applications to Ultrasonics, Van Nostrand, New York, 1950. 270. Berlincourt, D.A., Curran, D.R. and Jaffe, H., in Physical Acoustics, Vol I, Part A, Ch. 3., W.P. Mason, ed., Academic Press, New York, 1964. 271. Sittig, E.K., in Physical Acoustics: Principles and Methods, Vol IX, Ch. 5., W.P. Mason and R.N. Thurston eds., Academic Press, New York, 1972.

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272. Dobbs, E.R., in Physical Acoustics: Principles and Methods, Vol X, Ch. 3., W.P. Mason and R.N. Thurston eds., Academic Press, New York, 1973. 273. Zuckerwar, A.J., J. Acoust. Soc. Am. 64, 1278, 1978. 274. Zahn, R., J. Acoust. Soc. Am. 69, 1200, 1981. 275. Kuhl, W., Schodder, G.R. and Schroder, F.-K., Acustica 4, 519, 1954. 276. Matsuzawa, K., J. Phys. Soc. Japan 13, 1533, 1958. 277. Matsuzawa, K., Japan J. Appl. Phys. 15, 167, 1960. 278. Sessler, G.M. and West, J.E., J. Acoust. Soc. Am. 34, 1787, 1962. 279. Anderson, M.J., Hill, J.A., Fortunko, C.M., Dogan, N.S. and Moore, R.D., J. Acoust. Soc. Am. 97, 262, 1995. 280. Van Turnhout, J., in Electrets, Ch. 3., G.M. Sessler, ed., Springer, New York, 1980. 281. Holm, D. and Hess, G., J. Acoust. Soc. Am. 85, 476, 1989. 282. Haller, M.I. and Khuri-Yakub, B.T., IEEE T. Ultrason. Ferr. 43, 1, 1996. 283. Jin, X., Ladabaum, I., Degertekin, F.L., Calmes, S. and Khuri-Yakub, B.T., J. Microelectromech. Sys. 8, 100, 1999. 284. Noble, R.A., Jones, A.D.R., Robertson, T.J., Hutchins, D.A. and Billson, D.R., IEEE T. Ultrason. Ferr. 48, 1495, 2001. 285. Hohm, D. and Gerhard-Multhaupt, R., J. Acoust. Soc. Am. 75, 1297, 1984. 286. Sprenkels, A.J., Groothengel, R.A., Verloop, A.J. and Bergveld, P., Sens. Act. 17, 509, 1989. 287. Suzuki, K., Higuchi, K. and Tanigawa, H., IEEE T. Ultrason. Ferr. 36, 620, 1989. 288. Scheeper, P.R., Van Der Donk, A.G.H., Olthuis, W. and Bergveld, P., Sens. Act. A, 44, 1, 1994. 289. Percin, G., Atalar, A., Levent Degertekin, F. and Khuri-Yakub, B.T., App. Phys. Lett. 72, 1397, 1998. 290. Fiedler, M. and Hess, P., in Topics in Current Physics Vol 46, Ch. 4., P. Hess, ed., Springer-Verlag: Berlin, 1989. 291. Morse, P.M. and Ingard, K.U., Theoretical Acoustics, McGraw-Hill, New York, p. 233, 1968. 292. Span, R., Collmann, H.J. and Wagner, W., Int. J. Thermophys. 19, 491, 1998. 293. Setzman, U. and Wagner, W., J. Phys. Chem. Ref. Data 20, 1061, 1991. 294. Span, R. and Wagner, W., J. Phys. Chem. Ref. Data 25, 1509, 1996. 295. Tegeler, C., Span, R. and Wagner, W., J. Phys. Chem. Ref. Data 28, 779, 1999. 296. Span, R., Lemmon, E.W., Jacobsen, R.T., Wagner, W. and Yokozeki, A., J. Phys. Chem. Ref. Data 29, 1361, 2000. 297. de Reuck, K.M. and Craven, R.J.B., International Thermodynamic Tables of the Fluid State, Vol 12: Methanol, Blackwell Scientific, Oxford, 1993. 298. Lemmon, E.W., Jacobsen, R.T., Penoncello, S.G. and Friend, D.G., J. Phys. Chem. Ref. Data 29, 331, 2000. 299. Lemmon, E.W. and Jacobsen, R.T., Int. J. Thermophys. 20, 1629, 1999. 300. Colclough, A.R., Proc. Roy. Soc. Lond. A 365, 349, 1979. 301. Gillis, K.A. and Moldover, M.R., Int. J. Thermophys. 17, 35, 1996. 302. Trusler, J.P.M., Wakeham, W.A. and Zarari, M.P., Mol. Phys. 90, 695, 1997. 303. Trusler, J.P.M., Int. J. Thermophys. 18, 635, 1997. 304. Dayton, T., Beyerlein, S.W. and Goodwin, A.R.H., J. Chem. Thermodyn. 31, 847, 1999. 305. Davis, L.A. and Gordon, R.B., J. Chem. Phys. 46, 2650, 1967. 306. Sun, T.F., Kortbeek, P.J., Trappeniers, N.J. and Biswas, S.N., Phys. Chem. Liq. 16, 163, 1987. 307. Sun, T.F., Schouten, J.A. and Biswas, S.N., Int. J. Thermophys. 12, 381, 1991.

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7

Calorimetry K.N. MARSH Department of Chemical and Process Engineering University of Canterbury Christchurch, New Zealand J.B. OTT Department of Chemistry Brigham Young University Provo, UT, USA C.J. WORMALD Department of Chemistry University of Bristol Bristol, UK H. YAO Department of Condensed Matter Physics Tokyo Institute of Technology Tokyo, Japan I. HATTA Department of Applied Physics Nagoya University Nagoya, Japan P.M. CLAUDY Laboratoire des Mate´riaux Organiques a` Proprie´te´s Spe´cifiques Centre National de la Recherche Scientifique Vernaison, France S. VAN HERWAARDEN Xensor Intergration Delft, Netherlands 7.1

Flow 7.1.1 7.1.2 7.1.3

Calorimetry Plug-in Gas Flow Calorimeters Enthalpy-Increment Calorimeters Measurements in the Vicinity of the Liquid-Gas Critical Temperature 7.1.4 Enthalpy of Solution of Carbon Dioxide in Alkanolamines

Measurement of the Thermodynamic Properties of Single Phases A.R.H. Goodwin, K.N. Marsh, W.A. Wakeham (Editors) # 2003 International Union of Pure and Applied Chemistry. All rights reserved

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7.3

7.4

AC Calorimetry 7.2.1 Principle of the ac Calorimeter 7.2.2 Conditions for Heat Capacity Measurement 7.2.3 Joule-Heating ac Calorimeters 7.2.4 Light-Irradiation ac Calorimeters 7.2.5 AC Calorimetric Methods for Liquids 7.2.6 Heat Capacity Spectroscopy 7.2.7 Temperature-modulated Calorimetry Differential Scanning Calorimetry 7.3.1 Power-Compensated DSC 7.3.2 Heat Flux DSC 7.3.3 Adiabatic DSC 7.3.4 Single Cell DSC 7.3.5 Temperature Modulated DSC 7.3.6 Specialised DSC 7.3.7 Determination of Physical Properties with DSC Nano-Calorimetry 7.4.1 Micro Electro-Mechanical Systems (MEMS) 7.4.2 Nanocalorimeters as Sensors 7.4.3 Nanocalorimeters for Material Properties Determination

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336 337 339 340 343 344 345 347 348 350 353 358 358 359 361 362 368 369 372 374

7

CALORIMETRY

Calorimetry is a fundamental thermodynamic measurement required for the design of thermal processes and the understanding of molecular interactions with a statistical mechanical model. The major advances in calorimetry since Experimental Thermodynamics, Volume II have been described recently in Experimental Thermodynamics, Volume IV. This chapter focuses on instruments that were not previously described; these include various flow calorimetric techniques for measuring enthalpy differences of fluids, ac calorimetry, and differential scanning calorimetry.

7.1

Flow Calorimetry K.N. MARSH Department of Chemical and Process Engineering University of Canterbury Christchurch, New Zealand J.B. OTT Department of Chemistry Brigham Young University Provo, UT, USA C.J. WORMALD Department of Chemistry University of Bristol Bristol, UK

In recent years flow calorimetry has become the method of choice for the measurement of heat effects occurring in mixing processes, or for the determination of the heat capacity of fluids and fluid mixtures. Experimental Thermodynamics, Volume IV, Solution Calorimetry, edited by Marsh and O’Hare, was published in 1994 [1] and documented the advances in experimental techniques used for solution calorimetry since the 1962 publication of Experimental Thermochemistry, Volume 2, edited by Skinner [2]. In Chapter 4 of Solution Calorimetry, Grolier documents the use of flow calorimetry to measure the heat capacity of both pure fluids and fluid mixtures. Detailed descriptions of the Picker et al. flow microcalorimeter [3] and the Ernst et al. high-pressure calorimeter [4] are given in reference [1]. Chapter 5 of reference [1], by Albert and Archer, describes various flow calorimeters developed by Wood and coworkers [5], that were designed primarily to measure the enthalpies of dilution and heat capacities of aqueous electrolyte solutions at temperatures from ambient to 600 K and pressures up to 60 MPa. In Chapter 11 of reference [1], Simonson and Mesmer describe the Oak Ridge National Laboratory heat flux flow calorimeter [6] used for electrolyte solutions. Ott and Wormald in Chapter 8 of reference [1] discuss a variety of flow techniques developed to measure the excess 327

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enthalpies of liquid and gas mixtures. These included high-temperature, highpressure calorimeters developed by Christensen and Izatt [7], for the measurement of the enthalpy of mixing of organic liquids, and the calorimeters developed by Wormald and co-workers [8,9], for measurements on supercritical fluid mixtures, particularly mixtures containing steam as a component. Stokes, in Chapter 7 of reference [1] described the isothermal dilution calorimeter. The advances, albeit few, in both apparatus design and experimental methods since the publication of reference [1] are described herein. In particular, that not described in Solution Calorimetry [1] was a flow calorimeter developed by CastroGomez et al. [10] for the measurement of enthalpy increments at temperatures from (200 to 500) K and pressures up to 20 MPa. Wormald and co-workers have also described two enthalpy increment calorimeters. One was a water cooled countercurrent heat exchange calorimeter [11], which operated at pressures up to 15 MPa and temperatures below 698 K, and the other a flash vapourising calorimeter [12], which operates over the temperature range (240 to 523) K at pressures up to 10 MPa. In addition, Wormald has detailed a new flow calorimetric apparatus [13] having a series of plug-in calorimeters which can be used to measure gas phase excess enthalpies, heat capacities and Joule-Thomson coefficients. Fuangswasdi et al. [14] have described a modification of the high pressure calorimeter discussed in reference 7 where the thermostat is formed from a eutectic molten salt bath and the heat leak, made from nickel bolts in the work of Christensen and Izatt [7], has been replaced by a thermoelectric cooling unit consisting of an 80 junction chromel-alumel thermopile. Finally, flow calorimetry has been applied to measurements in the near critical region and to enthalpies of solution of carbon dioxide or hydrogen sulphide in reactive solvents.

7.1.1

Plug-in Gas Flow Calorimeters

In a series of papers, Wormald described calorimeters for the measurement of endothermic and exothermic enthalpies of mixing of gases with either a differential [15] arrangement or a single-stage mixing calorimeter [13]. He reported that development of these calorimeters was slow, involving much experimentation, with many unsuccessful designs. In particular, the process of removing a prototype calorimeter from the oil-filled thermostat for modifications was found very time consuming. This problem has been largely solved by the design of an apparatus, shown in Figure 7.1, which features plug-in calorimeter modules that can be easily and quickly removed. Three plug-in calorimeters have been developed. These are a single-stage gas mixing calorimeter, a heat capacity calorimeter, and a mixing calorimeter. The third was designed with an adjustable throttle to allow isothermal measurements on gas mixtures when the mixing process is exothermic. The throttle can be adjusted so that the Joule-Thomson cooling effect exactly offsets the exothermic mixing process. The apparatus can be used on gases or condensable liquids, which are vaporised in flash boilers. The vapours pass through heat exchange coils, D, over platinum resistance thermometers sleeves, E, and into a glass socket consisting of two concentric tubes, F, the outer tube terminating in a hemispherical 328

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329

Figure 7.1 Schematic of the flow calorimetric apparatus [13]: A, Thermostatted bath; B, flash boiler (one of two shown for clarity); C, entry port for gaseous components; D, heat exchanger coil; E, platinum resistance thermometer sleeve; F, hemispherical ball and socket joint; G, mixing calorimeter; H, glass tube; I, ball and socket joint; J, mercury manometer with which the pressure drop across the calorimeter is determined; K, water cooled double surface condenser; L, liquid collection bulb; M, trap, which can be filled with ice or solid carbon dioxide; N, condensate receiver; O, tap for the admission of air; and P, mercury manometer for the determination of the system pressure.

socket joint, the inner tube projecting 20 mm above the level of the joint. The calorimeter, G, was mounted in a glass tube, H, fitted at the bottom with a ball joint which plugs into the hemispherical socket and was sealed by an O-ring. The gases flow up the calorimeter where they are mixed over a heater. The mixture exits through a second ball and joint socket, I. When the clip is removed from this joint the calorimeter can be readily removed. The pressure drop across the calorimeter was measured on a mercury manometer, J. The vapour mixture was condensed at K and collected in bulb, L. Details of the plug-in gas mixing calorimeter are shown Figure 7.2. The gases flowing through tubes A and B mix at I and flow over a heater, J. The direction of gas flow was reversed by baffle K and the gas then flowed over the outside of the heating chamber. Mixing was promoted by eight turns of nickel wire, L, and a series of stainless steel gauze baffles, M. The gases then flowed over two platinum resistance thermometers, P. The calorimeter was tested on (nitrogen þ cyclohexane) and three strategies for correcting the Joule-Thomson effect have been described [13]. The plug-in heat capacity calorimeter is shown in Figure 7.3(c). The heat capacity calorimeter was designed to prevent radiation from the heater reaching the resistance thermometer. A tapered cylinder of fine stainless steel gauze was used to absorb radiant energy from the heater and transfer it to the gas stream. Ten steel gauze discs were used to ensure that gas, which had come into contact with the heater, was well mixed with the rest of the gas stream. To minimise heat loss through the walls, the distance from the heater to the thermometer was short. The plug-in mixing calorimeter fitted with a throttle is of similar design to that shown in Figure 7.3(b). The difference is that a Kel-F cone, attached by a rod to a bellows assembly at the top of the calorimeter, was positioned just above the heater. For mixing processes which are exothermic, the cone can be moved towards the top 329

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Figure 7.2 The ‘plug in’ calorimeter module and its mounting assembly [13]: A and B, Gas inlet tubes; C and D, platinum resistance thermometers; E, two concentric tubes, with outer diameters of 12 mm and 5 mm; F, hemispherical glass socket which terminates the outer tube while the inner tube extends 20 mm above the socket; G, connects the annulus between the concentric tubes to a mercury manometer, 10, shown in Figure 7.1; H, a ball joint fitted to a 12 mm diameter tube on the base of the calorimeter; I, gas mixing region; J, heater; K, baffle to reverse the direction of fluid flow over the outside of the heating section; L, an eight turn coil of soft nickel wire wound into the shape of a spring that fits loosely over the outside of the central tube; M, stainless steel gauze to fit the lower part of the baffle; N, gauze disk which holds the upper part of the baffle; O, copper disc which returns heat that has travelled along the heater leads back to the gas; P, two platinum resistance thermometers; and Q, ground glass joint, the socket of which fits a standard B24 cone that attaches the calorimeter module to the supporting tube.

of the tube containing the heater, and the throttling effect can be adjusted to exactly offset the enthalpy of mixing. Only the pressure drop across the calorimeter needs to be measured. Figure 7.3(a) shows a plug-in Joule Thomson calorimeter. All the calorimeters were mounted in cylindrical evacuated and silvered jackets, and all contain a piece of thin PTFE sheet rolled into a cylinder located between the 330

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331

Figure 7.3 Plug in calorimeters. (a) Joule-Thomson calorimeter; (b) mixing calorimeter; (c) heat capacity calorimeter.

calorimeter assembly and the glass tube which contains it. This simple measure prevents direct contact of the gas stream with the glass wall, and reduces the heat leak by a factor of about five. With all the calorimeters, measurements were made over a range of flow rate to characterise the heat leaks and make appropriate corrections.

7.1.2

Enthalpy-Increment Calorimeters

Enthalpy increment flow calorimeters have been described by Wormald and coworkers [11,12] and by Castro-Gomez et al. [10]. The Castro-Gomez et al. [10] 331

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Figure 7.4 Schematic drawing of the enthalpy-increment flow calorimeter [10]. A, Thermoelectric cooling device; B, syringe pump; C, calorimeter heater; D, calorimetric vessel; E, conditioning block; F, connecting tube; G, back-pressure regulator; H, collection tank; I, calorimeter thermostat; J, vacuum connection.

thermoelectric flow calorimeter can be used for the measurement of the enthalpy of fluids at temperatures from (200 to 500) K and pressures up to 20 MPa. A schematic of the apparatus is shown in Figure 7.4. Energy was removed from the calorimeter vessel, D, at a constant rate by a current regulated thermoelectric cooling device, A. Before pumping fluid from the syringe pump B, the power removed from the calorimeter by the thermoelectric cooling module was matched by the addition of electric power from the heater C to maintain the temperature in the calorimeter vessel constant to within +0.002 K of the chosen bath temperature. When fluid flowed through the calorimeter at a known rate and inlet temperature the power supplied to the heater was changed to maintain the calorimeter temperature constant. The enthalpy increment was determined from the flow rate and the change in power supplied to the heater. The fluid was contained in the high-pressure syringe pump whose flow rate, accurate to +0.3 per cent, could be adjusted over a wide range. The fluid was pumped through the conditioning block E, the connecting tube F, the calorimeter vessel, and the back-pressure regulator G, to the collecting tank, H. The flow rate could also be determined to +0.3 per cent from the mass of fluid discharged into the container placed on a force balance. Further details of the enthalpy increment calorimeter are shown in Figure 7.5. 332

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Figure 7.5 Schematic of the enthalpy increment flow calorimeter [10]. A, Inlet-temperature probe; B, outer aluminium can; C, copper conditioning block; D, connecting tube; E, outlettemperature probe; F, vacuum connection; H, cooling module; I, aluminium shield with heater and cooling coil; J, connection to cooling unit; K, fluid entrance from pump; L, heater connections; M, connection to high vacuum; N, thermostat heater; O, outer aluminium can; P, matching heater; Q, bath oil; R, thermoelectric cooling device; S, calorimetric vessel; T, stirrer; U, thermostat cooler; V, fluid outlet to back-pressure regulator; W, trim heater; X, fins for enhanced heat exchange.

The conditioning block, located above the calorimeter vessel, consisted of a solid copper block, C, with a spiral groove into which the flow tube was silver soldered. The temperature of this block was controlled to +0.1 K. A small cooling chamber situated atop the block could be used for low-temperature operation. The aluminium shield surrounding the copper block contained heating and cooling coils. A vacuum-tight aluminium can, B, surrounded this shield and the space between the shield and the outer-can contained radiation shields. The connecting tube D passed from the base of the conditioning block to the bottom of the calorimeter vessel. A controlled trim heater, W, was fitted to the top of the inlet 333

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tube to allow fine adjustments to the inlet temperature. A heat exchanger X was attached to the section of the connecting tube inside the calorimeter to reduce energy losses by conduction along the tube. The flow tube (1 mm i.d.), centered in a stainless steel connecting tube, was surrounded by several alternative layers of glass fabric insulating material and 0.05 mm thick stainless steel radiation shields. The connecting tube could be evacuated. A thermistor, sealed in a stainless steel probe 0.66 mm o.d. and 0.65 m long, was inserted in the flow tube through a seal at the top of the conditioning block. This probe could be moved a total of 120 mm from the bottom of the connecting tube. This allowed a detailed study of the temperature profile in the connecting tube as a function of flow rate and inlet temperature. From this study the range of conditions that would give enthalpy increments independent of the flow rate was established. The calorimeter itself was inside an air bath that was in turn immersed in an oil bath. The calorimeter was filled with a heat transfer fluid that was stirred rapidly by a stirrer with two impellers. The sample fluid flowed through a 0.2 mm internal diameter coiled tube. A 50 O heater dissipated most of its power near the lower end of the connecting tube. At the same time, power was removed from the calorimeter by a thermoelectric cooling module. In operation, the power to the thermoelectric cooling module was set and the heater power adjusted to maintain the calorimeter isothermal to within +0.005 K of the initial bath temperature. The power change could be determined to +0.05 per cent. The calorimeter was controlled by a computer with the various sensor devices connected through a low-thermal scanner to a digital voltmeter with 0.1 mV resolution. Measurements on water, p-xylene, and heptane indicated that the apparatus was capable of measurements with an uncertainty of +0.5 per cent. Measurements on the enthalpy increments for mixtures of f(1  x)CO2 þ xC2 H6 g at temperatures from (230 to 350) K [16] and enthalpy increments of f(1  x)H2 O þ xCH3 OHg from (180 to 320) K [17] have been reported. The flash vaporising flow calorimeter described by Wormald et al. [12] was designed to operate from (240 to 523) K at pressures to 10 MPa. When the initially cold liquid was pumped into the calorimeter it was vaporised and the electrical energy supplied to the oil bath surrounding the calorimeter adjusted to exactly offset the incremental enthalpy change. Measurements on hexane indicated that the apparatus was capable of providing results with an uncertainty of +2 per cent. Recently, Wormald has reported another enthalpy increment calorimeter [18] specifically designed for operation in the vicinity of the critical region. The novel feature is that the near critical fluid flows horizontally from the hot zone to the cold zone so that gravitational effects are zero. The pressure transducer was level with the calorimeter so that hydrostatic head did not affect the pressure measurement. Preheated fluid was pumped through a heat exchange coil immersed in a bath of silicone oil maintained at a temperature close to the critical temperature of the fluid. The hot fluid then flows through a second heat exchange coil immersed in a bath of ethanol maintained at a temperature of 298.15 K by an array of Peltier cooling elements. A resistance thermometer in the bath is the sensing element of a feed-back control loop used to adjust the power to the cooling elements so that the heat transferred to the ethanol was continuously removed and the temperature remained close to 298.15 K. 334

Calorimetry

335

Enthalpy increments have been measured for near-critical benzene, and these were found accurate to between (0.5 and 1) per cent. 7.1.3

Measurements in the Vicinity of the Liquid-Gas Critical Temperature

Flow calorimeters allow one to make excess enthalpy measurements at high temperatures and pressures, including pressure and temperature conditions near the critical locus of a mixture. Care must be taken, however, if the measurements are made too close to the locus. Pressure gradients are always present when fluids are E flowing in the calorimeter and, even if the gradient is small, the effect on Hm can be E E E significant since Vm and (qVm =qT)p can be large, and hence, (qHm =qp)T , given by  E  E qHm qVm ¼ VmE  T ; ð7:1Þ qp T qT p is also large. Furthermore, the correlation length becomes long near the critical locus, and the time required to effect complete mixing in the calorimeter may become larger than the residence time, resulting in incomplete mixing. Ott and Wormald [19] suggest that to obtain accurate results, measurements should not be made at temperatures within 1 K and pressures within 10 kPa of the critical locus. Keeping in E mind these restrictions, accurate Hm measurements can be made in a flow calorimeter in the vicinity of the locus. 7.1.4

Enthalpy of Solution of Carbon Dioxide in Alkanolamines

Mathonat and co-workers [20,21] describe a modification of the flow mixing unit of a commercially available (SETARAM C-80) calorimeter to measure the enthalpy of absorption and the solubility of CO2 in aqueous solutions of alkanolamines. Special care had to be taken to stabilise the base line of this instrument. The aqueous solution of alkanolamine gave a steady base line and was well-behaved when the carbon dioxide was introduced. When carbon dioxide was used to establish the base line, the introduction of the aqueous alkanolamine gave a large perturbation and it took an excessive time to obtain a stable calorimeter signal. As a consequence, the accuracy was less than normally achieved by flow calorimetry. Further, for systems where the kinetics of the reaction is not fast, for example the adsorption of carbon dioxide in methyldiethanolamine at low temperature, the measured heat can become flow dependent, not due to incomplete mixing but because of incomplete reaction. The problem can be overcome by increasing the length of the mixing tube. An advantage of a flow calorimeter operating under pressure is its ability to measure the solubility of a gas from the break in the curve of enthalpy of adsorption as a function of gas loading, as shown in Figure 7.6. Carson et al. [22] have described the use of the isothermal dilution calorimeter to make similar enthalpy of adsorption measurements of CO2 in alkanolamines. 335

336

Calorimetry

Figure 7.6 Measured enthalpies DHabs (kJ per mole of monoethanolamine) versus CO2 loading a (moles of CO2 per mole of monoethanolamine) for absorption of CO2 in an aqueous solution 30 mass per cent monoethanolamine at a temperature of 313.15 K and pressure of 2.0 MPa [21].

7.2

AC Calorimetry H. YAO Department of Condensed Matter Physics Tokyo Institute of Technology Tokyo, Japan I. HATTA Department of Applied Physics Nagoya University Nagoya, Japan

AC calorimetry was first proposed by Corbino [23] and more recently by Kraftmakher [24] to measure the heat capacity of metals at high temperature. High-resolution ac calorimetry has been made possible by the advent of the lock-in amplifier that was first applied to the measurement of heat capacities of metals in the late 1960s [25,26]. Subsequently, ac calorimetry has been applied to the study of dielectrics [27,28], liquid crystals [29–32], biological substances [33–35], free standing films [36], and adsorbed gases [37,38]. AC calorimetry has several advantages over other calorimetric techniques. Some of these advantages may be directly attributed to the use of lock-in amplifiers, for 336

Calorimetry

337

example, it is possible to measure the heat capacity of a sample with a temperature resolution of about 1 mK and precision of about +0.1 per cent because small temperature oscillations can be detected with a lock-in amplifier with high sensitivity. Thus the method is suitable for measuring the heat capacity anomaly near a phase transition precisely. Further, the amount of sample required is about 1 mg and so the technique can be used to measure the heat capacity of expensive or rare samples. AC calorimetry has been used over a wide range of temperatures from 1 K [25] to 3600 K [39] and it is relatively easy to construct an apparatus to measure the heat capacity in the presence of a magnetic field [25], at high pressure [40,41], and simultaneously with other measurements [42]. On the other hand, the disadvantages of ac calorimetry are: the measurement of the enthalpy of a phase transition at a firstorder transition is difficult, so that the uncertainty of the measurement is reduced to about +5 per cent. However, this can be compensated for, in part, by calorimeter design in which the relaxation method [43] or non-adiabatic scanning methods [44] are combined with ac calorimetry. It is possible to improve on the uncertainty in phase transition measurements [45,46].

7.2.1

Principle of the ac Calorimeter

AC calorimetry is a method of measuring the heat capacity in which an oscillatory heat flux is applied to a sample and its heat capacity is determined from the resultant temperature oscillations. The simplest model for an ac calorimeter is shown schematically in Figure 7.7(a). The heat capacity at constant pressure Cp (shown in Figure 7.7(a) as C) of calorimeter þ contents is loosely coupled to the surrounding heat bath at temperature Tb by a massless thermal resistance R. It is assumed that the thermal diffusivity of the sample is high enough to ignore the effect of

Figure 7.7 Thermal models of an ac calorimeter. A sample with heat capacity C(:Cp ) is coupled to a thermal bath with a thermal resistance R. Tb is the bath temperature. (a) lumpedconstant model when the thermal diffusivity of a sample is high. (b) distributed-constant model when the thermal diffusivity of a sample is low. A plate sample of specific heat c(:cp ), density r and thermal conductivity k has a thickness l, and area A.

337

338

Calorimetry

temperature gradients inside the sample. An ac heat flux Pac ¼ P0 (eiot þ 1) is continuously supplied, at an angular frequency o, to the sample; in Peltier ac calorimetry Pac ¼ P0 eiot [47]. Conservation of energy yields the relationship: Cp dT=dt ¼ P0 ðeiot þ 1Þ  ðT  Tb Þ=R;

ð7:2Þ

where T is the sample temperature. The sample temperature T in the steady state is determined by T ¼ jTac jeiðotyÞ þ P0 R þ Tb ;

ð7:3Þ

h i1=2 jTac j ¼ ðP0 =oCÞ 1 þ 1=ðote Þ2 ;

ð7:4Þ

y ¼ p=2  arctanð1=ote Þ;

ð7:5Þ

te ¼ RCp :

ð7:6Þ

where jTac j is given by

where y and te are

and

In Equations (7.3) through (7.6) Tac is the ac temperature, y the phase difference between the ac temperature and the ac heat flux, and te the external relaxation time for the sample temperature to relax to the steady-state value. AC calorimetric measurements are usually performed under the condition that the heat leak is negligibly small, so that ote 441;

ð7:7Þ

and it is then adiabatic. The lower limit of o that can be employed is determined by Equation (7.7). Under the condition that the heat capacity Cp is inversely proportional to the ac temperature jTac j the heat capacity is given by Cp ¼ P0 =½ojTac j:

ð7:8Þ

If the adiabatic condition (7.7) is not fully satisfied, the heat capacity can be derived from the correction for heat leakage with the method described in Section 7.2.5, or by measuring the thermal resistance R directly [48]. 338

Calorimetry 7.2.2

339

Conditions for Heat Capacity Measurement

Consider the case where the thermal diffusivity of a sample is poor and the temperature gradient inside the sample cannot be ignored [25]. A one-dimensional distributed-constant model of an ac calorimeter is illustrated in Figure 7.7(b). In this model, the sample has a slab-like shape with thickness L and an area A. An ac heat flux Pac ¼ P0 (eiot þ 1)=A is supplied to the surface at x ¼ 0. If the surface dimensions are much larger than L and the surface is heated homogeneously, the central part of the sample can be treated as one-dimensional thermal diffusion normal to the surface. In the system, the ac temperature at x ¼ L is described by Tac ðLÞ ¼

P0 R n o n o; 1=2 cosh ½o=ð2aÞ Lð1 þ iÞ þ ARk½o=ð2aÞ1=2 ð1 þ iÞ sinh ½o=ð2aÞ1=2 Lð1 þ iÞ ð7:9Þ

where k is the thermal conductivity, r density and a the thermal diffusivity, which is defined by a ¼ k=rcp , where cp is the specific heat capacity at constant pressure. In Equation (7.9), the term [o=(2a)]1=2 is the inverse of the thermal diffusion length, which defines when the amplitude of the ac temperature wave in the sample has decayed by e1. When the thickness L is less than the thermal diffusion length fL[o=(2a)]1=2  1g, Equation (7.9) can be expanded in a series in L[o=(2a)]1=2 and, when both the higher order terms are omitted and the real and imaginary parts separated, gives: h i1=2 pffiffiffiffiffi jTac ðLÞj ¼ ðP0 R=oC Þ 1 þ 1=ðote Þ2 þ ðoti Þ2 þ 40ti =tc ;

ð7:10Þ

and   pffiffiffiffiffi y ¼ p=2  arctan 1=ote  10oti =2 ;

ð7:11Þ

where ti ¼ L2 =

pffiffiffiffiffi  90a :

ð7:12Þ

In Equations (7.10), (7.11), and (7.12) ti is the internal relaxation time that is required for the temperatures inside the sample to be uniform. Since it is necessary that the temperatures inside the sample be uniformly modulated, ac calorimetric heat-capacity measurements must be performed under 339

340

Calorimetry

the isothermal condition oti 551:

ð7:13Þ

Therefore, the measurement angular frequency o must be low if the thermal diffusivity of the sample a is low. If the sample thickness L is 0.1 mm, o is roughly 10 Hz for metals, 1 Hz for dielectric crystals, and 0.1 Hz for polymers and nonmetallic liquids. If the adiabatic and isothermal conditions are satisfied, y is about p=2. In the discussion above, it is assumed that the ac temperature responds linearly to the ac heat flux. This assumption is invalid at a first-order transition. However, if the enthalpy change of the transition is small, it is observed as a distortion of the ac temperature waveform and additional analysis of the waveform is required. This analysis has been performed on ac calorimetric measurements of antiferromagnet chromium [49].

7.2.3

Joule-Heating ac Calorimeters

Joule-heating calorimeters have the advantage that the ac heat flux is stable and its amplitude can be determined precisely from measurements of both the current and voltage. At present, the Joule-heating type calorimeters allow heat capacity measurements to be made with a precision of +0:01 per cent. The drawback is that an electric heater is required for ac heating. It increases the heat capacity of the equipment and, therefore, a relatively large amount of sample is necessary in order to maintain the precision. It also results in an increase of ti . Figure 7.8 shows an example of a Joule-heating ac calorimeter [49]. A sample of mass (30 to 50) mg was loaded into a gold cup (13 mm in diameter, 1 mm deep and 0.25 mm thick), and hermetically closed with a gold lid (0.2 mm thick) by cold pressure welding under a helium gas atmosphere to avoid the deterioration of a sample during a measurement. A strain gauge (3.7 mm wide and 10 mm long),

Figure 7.8 Schematic view of a sample cell in a Joule-heating ac calorimeter [49]. S, sample; C, gold cup; L, gold lid; W, gold wire; H, heater; T, thermistor.

340

Calorimetry

341

Figure 7.9 Schematic diagram of a Joule-heating ac calorimeter [49]. SC, Sample cell; B, thermal bath (copper); S1, inner shield (brass, heater is wound around it); S2, outer shield (stainless steel); C1, inner chamber; C2, outer chamber; ST, stainless steel stem; TP, Teflon plate.

attached to the bottom of the cup, using a diluted GE7031 varnish, served as a thin plate heater. The temperature of the cell was detected with a 0.3 mm diameter smallbead thermistor attached to the centre of the lid with GE varnish. When the amount of liquid sample was large, a helical coil of fine gold wire was wound in a spiral shape and immersed in the sample to reduce the effective internal relaxation time ti . Figure 7.9 shows a schematic diagram of the apparatus [49]. The thermal bath B was a cylindrical copper block (60 mm diameter and 55 mm height). It had an inner space 30 mm in diameter and 25 mm high in which the sample cell SC was placed. Two polyimide-insulated copper wires (0.35 mm in diameter and 30 mm in length) were used to support the cell. The thermal link between the cell and the bath was provided in part by these wires and in part by the nitrogen gas of p&100 kPa surrounding the cell. The temperature of the bath could be controlled by the heater wound around the inner shield S1. The inner chamber C1 and the outer chamber C2 can be evacuated separately. Figure 7.10 shows the block diagram of the calorimeter [49]. A 12-bit arbitrary waveform generator, which had a maximum relative variation of output amplitude with respect to temperature of +50 ? 106 K1 , was used as the signal source oscillator for ac heating. Usually, the heating frequency was 0.031 25 Hz, and the heating power was 0.5 mW. This ac heating caused an ac temperature response of typically 7 mK (rms), which determined the temperature resolution of the measurement. The thermistor for monitoring the sample temperature comprised 341

342

Calorimetry

Figure 7.10 Block diagram of a Joule-heating ac calorimeter [49].

one arm of a Wheatstone bridge, which balanced when the resistance of the thermistor was equal to that of a programmable 6-digit variable resistor. The variable resistor was composed of metal-foil precision resistors with temperature coefficient of +3 ? 106 K1 max. The resistors r1, shown in Figure 7.10, in the bridge were of the same type and nominal value (r1 ¼ 1 kO in the present apparatus). The bridge was driven at 11.5 Hz using a function generator as the signal source. The imbalance signal of the bridge was fed into a lock-in amplifier. The output of the lock-in amplifier was read with a 5.5-digit digital multimeter, synchronised with the ac heating over 6 heating cycles. Usually, 768 readings were collected in this process, and the data were Fourier-transformed with a computer. To obtain high stability over long time periods, it was necessary to monitor the overall sensitivity of the bridge, which was affected by the drift of the oscillator output, and that of the sensitivity of the lock-in amplifier. To do this, the setting of the variable resistor was changed by a certain amount, and the change in the output of the lock-in amplifier was measured. This procedure was carried out after each reading process described above. The amplitude of the signal source for the bridge was adjusted so that the power dissipation in the thermistor did not exceed 30 pW. To obtain the highest signal-to-noise ratio, the thermistor resistance was between (2 and 20) kO. The temperature of the bath was measured with a platinum resistance thermometer and an automatic resistance bridge. Since it is not necessary to scan the bath temperature in ac calorimetry, the bath temperature can be changed stepwise so that the heat capacity can be measured with high temperature resolution. 342

Calorimetry 7.2.4

343

Light-Irradiation ac Calorimeters

The main advantage of light-irradiation type ac calorimeters is that extremely small samples, for example, with linear dimensions of (16160:1) mm, can be studied because it is not necessary to attach a heater to the sample. The drawback is that the absolute value of the heat capacity is difficult to obtain with high accuracy, because the amount of ac heat flux absorbed by a sample is difficult to estimate. Another drawback is that the amount of ac heat flux absorbed from irradiating light is not as constant as that from Joule heating. A schematic of a light-irradiation-type ac calorimeter [28] is shown in Figure 7.11. In this example, the sample is assumed to be a thin plate-like solid. If the sample is transparent, an opaque film (e.g. carbon or metal) is deposited on the surface of the sample. The temperature was measured with fine thermocouple wires (about 25 mm in diameter) spot-welded crosswise with the crossed point attached to the sample with a thermally conductive adhesive such as silver paste or GE 7031 varnish. In this configuration it serves as two pairs of thermocouples. In this application one pair was used for ac temperature detection and the other was used for measuring the temperature difference between the sample and the bath. An ac heat flux was applied to the sample by chopping the light beam from a halogen lamp with a mechanical chopper; diode lasers have also been used as modulated sources [50]. Since the ac signal was extremely small (a few nV) and the resistance of the thermocouple low (in this case 10 O), the voltage was amplified with a high-quality signal transformer and fed to a lock-in amplifier. The thermal bath was made of copper and its temperature was maintained constant with a temperature controller. The absolute value of the ac heat flux can be estimated, for example, by replacing a sample with a reference sample whose surface is coated with the same opaque film as the sample. With this method one may obtain the absolute value of heat capacity with an uncertainty of about +10 per cent.

Figure 7.11

Block diagram of a light-irradiation ac calorimeter [28].

343

344 7.2.5

Calorimetry AC Calorimetric Methods for Liquids

To measure the heat capacity of a small amount of liquid with high accuracy, which is required, for example, in the study of the thermal denaturation of proteins, there are two main problems. One is the stability of the applied ac energy. The other is the heat leak from the cell to the bath. To solve the former, the cell can be directly Joule heated with ac microcalorimetry [46]. The cross sectional view of a sample cell is shown in Figure 7.12. The cell was constructed from a metal tube with 290 mm inner diameter, 20 mm wall thickness and 5 cm long. The internal cell volume was about 3.3 mL. Both ends of the tube were connected to a signal source for ac heating. An ac electric current was applied through the tube to generate a very stable ac heat flux in the cell without adding a heater. The frequency of the ac temperature was 0.5 Hz. The tip of a chromel-constantan thermocouple, with a wire diameter of 13 mm, was attached in the middle of the tube with GE703l varnish. Both ends of the tube were extended out of the bath and connected to the outside of the calorimeter with Teflon tubes through which the liquid samples were introduced. The space between the tube and the bath was evacuated. One of the thermocouples was connected to a digital lock-in amplifier through a transformer. A synchronous output signal of the oscillator was supplied to the lock-in amplifier as a reference phase signal. The output of the lock-in amplifier was fed to a computer. The leakage of heat is generally unavoidable with small samples. AC calorimetry has the advantage that a correction for the leakage can be made by measuring both the amplitude and the phase of the ac temperature [46]. In principle, the heat leakage can be avoided by making te longer than the ac heating period. Thus the heat capacity can be measured without correction when the adiabatic condition, (7.7), is satisfied. Since the thermal diffusivity of liquids such as water and organic solvents is small, it is necessary to carry out the measurements at low frequency. Therefore, the effects of the leakage become less avoidable and correction necessary. In one method of correction both the ac temperature and its phase are measured three times while

Figure 7.12 Block diagram of the ac microcalorimeter for liquid samples [46]. A, thermocouple; B, Teflon tube; C, lead wire.

344

Calorimetry

345

an oscillating heat flux P0 is supplied per unit length of the cell and three separate measurements are required. First, when the cell is empty, the ac temperature of the cell is given by DTc eifc ¼ P0 =ðKc þ ioCp;c Þ;

ð7:14Þ

where DTc is the amplitude of ac temperature, fc is the phase difference between the ac temperature and the ac heat flux, Cp;c is the heat capacity of the tube-like cell per unit length and Kc is the leakage conduction at the surface of the sample per unit length. Subscripts s, r and c denote sample, reference and cell, respectively. Second, when the cell is filled with a liquid sample having a specific heat capacity cp,s and density rs, the ac temperature of the sample cell is given by DTs eifs ¼

P0 ; Ks þ ioðCp;c þ Vrs cs Þ

ð7:15Þ

where V is the inner volume of the tube-like cell per unit length. Third, when the cell is filled with a reference liquid having known specific heat capacity cp,r and mass density rr, we obtain DTr eifr ¼

P0 : Kr þ ioðCp;c þ Vrr cp;r Þ

ð7:16Þ

From Equations (7.14) to (7.16), we obtain the absolute value of the heat capacity per unit volume of the liquid sample cp,srs as cp;s rs ¼

sin fs =DTs  sin fc =DTc cp;r rr : sin fr =DTr  sin fc =DTc

ð7:17Þ

This relation is useful because, even if the leakage conductance is large, the heat capacity can be obtained by measuring both the amplitude and the phase of the ac temperature. Using this technique the heat capacity of a liquid sample can be measured with a repeatability of +0.01 per cent and an uncertainty of +0.5 per cent.

7.2.6

Heat Capacity Spectroscopy

There are several techniques for measuring the frequency or time dependence of the heat capacity [48,51,52]. The heat capacity of a material has been traditionally thought of as a static quantity. However, we can define the dynamic heat capacity or the complex heat capacity in the same way as we define dynamic dielectric constant or the complex dielectric constant. The dynamic dielectric constant is given in terms of a time-dependent dipole-dipole correlation function and the dynamic heat 345

346

Calorimetry

capacity can be expressed in terms of a time-dependent energy-density correlation function. There are two methods for heating ac calorimeters. One method uses Joule heating and the other uses irradiation by light. Dynamic thermal effusivity measurements can be made on a bulk sample with a flat surface up to high frequencies [52]. The thermal effusivity, given by rcp k where k is the thermal conductivity, can be obtained by detecting the ac temperature at the surface of the sample. Since the thermal conductivity does not exhibit a sharp anomaly at a phase transition, the thermal effusivity can be used to obtain information on the dynamic heat capacity. For a solid sample, a metal film, used as a heater and a temperature sensor, is placed on the surface. In a liquid sample, a metal film is used for a heater and a temperature sensor is placed at the surface of a substrate made from an insulating solid material which is immersed in the liquid sample. For both cases, the sample system is shown in Figure 7.13. The density, specific heat capacity, and thermal conductivity of the sample are given by r, cp and k, respectively, and the same properties for the substrate, which for a solid sample is air, are annotated by the subscript s. By solving the heat diffusion equation under the boundary condition shown in Figure 7.13, we can obtain the ac temperature at the surface of the sample from: qeip=4 T ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; orcp k þ ors cp;s ks

ð7:18Þ

where q is heat flux with an angular frequency o caused by the heater. In Equation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (7.18), q and ors cp;s ks are known. Therefore, from the measurement of the ac pffiffiffiffiffiffiffiffiffiffi temperature T we can obtain the value of rcp k at an angular frequency o. When pffiffiffiffiffiffiffiffiffiffi there is a frequency dispersion in the heat capacity, rcp k becomes complex, that is, rcp k ¼ Re(rcp k)  iIm(rcp k) where Re is the real and Im is the imaginary part.

Figure 7.13 Schematic of a measurement of the complex thermal effusivity rcp k where r is mass density, cp is the specific heat capacity at constant pressure and k the thermal conductivity. A metal film, used as both a heater and thermometer, is placed at the surface between a bulk sample and a substrate denoted with the subscript s.

346

Calorimetry

347

From the measurement of the amplitude and the phase of the ac temperature T, we can obtain the complex heat capacity. When an ac electric current I(t) ¼ I exp (iot=2) is applied to the metal film with electrical resistance R, a heat flux with the angular frequency o arises. As a result, the ac temperature of the heater oscillates at o. Since the electrical resistance of the metal film heater is proportional to the temperature in a small temperature range, the resistance R(t) oscillates at o. To measure the ac temperature at the metal film, the ac voltage across the resistance given by I(t)R(t) is measured. The ac voltage has a small signal at 3o=2 as well as at o=2. Only the signal at 3o=2 is proportional to the ac temperature at o which is related to the thermal effusivity. To detect such a small signal a bridge is used. This approach is called a three-omega (or 3o) method [52].

7.2.7

Temperature-modulated Calorimetry

Temperature-modulated calorimetry, in which a conventional differential scanning calorimeter is used, has a number of advantages [53,54]. It is superior to lightirradiation ac calorimetry, as the absolute value of the heat capacity can be determined easily. In addition, it is superior to Joule-heating ac calorimetry as the heat capacities of both the temperature sensor and heater do not affect the measurement. In comparison with conventional differential scanning calorimetry, the effects of external noises and drift can be easily excluded because the ac temperature response to periodic heating is used in analysing the data. In addition, the heat capacity can be measured as a function of frequency. The details of temperature-modulated calorimetry are described in Section 7.3.5. In this section, the discussion concerning temperature-modulated calorimetry is limited to the issues relating to ac calorimetry. In this case, we are interested in the temperature dependence of heat capacity so the temperature of the heater is changed by stepwise superposition on the ac heating. A naive model for both heat-flux differential scanning calorimetry and temperature-modulated calorimetry is shown in Figure 7.14 where C is the heat capacity, T the ac temperature and the subscripts s, r, and 0 refer to the sample, reference and base plate respectively. In Figure 7.14, K0 is the heat transfer coefficients between the base plate and the cell and K the heat transfer coefficient between the heater and the base plate. The reference cell contains a pan and reference material, which in some experiments may be omitted. In heatflux differential scanning calorimetry, the contribution of C0 can be compensated and it is usual to ignore the contribution of K0 . In this case, the heat capacity, C, of a sample is given by Cs  Cr and the relationship: C¼

KDT ; dT=dt

ð7:19Þ

where DT is the temperature difference between the reference and the sample sides and dT/dt is the heating rate. In temperature-modulated calorimetry, C may be 347

348

Calorimetry

Figure 7.14 Schematic representation of the model for temperature-modulated calorimetry where C(:Cp ) is the molar heat capacity, T the ac temperature, K0 the heat transfer coefficients between the base plate and the cell, and K the heat transfer coefficient between the heater and the base plate. The subscripts s, r, and 0 refer to the sample, reference and base plate respectively.

expressed by: C¼

KDTac eip=2 ; oTs

ð7:20Þ

where DTac is the ac temperature difference between the reference and the sample sides, o is an angular frequency and Ts is the ac temperature of the sample side. Ideally, the phase difference between DTac and Ts is p/2, and then C is real, and Equations (7.19) and (7.20) are essentially the same. In heat-flux differential scanning calorimetry, the principle of operation has been established within the framework of the naive model described by Equation (7.19), while for temperature-modulated calorimetry, in which there are rapid temperature changes, the consequence of omitting K0 must be considered. When K0 is sufficiently large the ac temperature of both cells follow that of the base plate, without appreciable delay, and K is simply modified by the heat capacities of the base plates and the cells. However, when K0 is small its contribution must be included and other approaches are needed to solve the problem [54].

7.3

Differential Scanning Calorimetry P.M. CLAUDY Laboratoire des Mate´riaux Organiques a` Proprie´te´s Spe´cifiques Centre National de la Recherche Scientifique Vernaison, France

Differential scanning calorimetry (DSC) was the name given by Perkin-Elmer to a commercially available power-compensated calorimeter. DSC may be considered a 348

Calorimetry

349

development of differential thermal analysis (DTA), and heat flux scanning calorimeters have been referred to as quantitative DTA. IUPAC has used the term DSC for any apparatus capable of measuring thermal power in a temperature scan irrespective of the mode of operation. In a DSC experiment, the reference and sample are heated in a furnace, whose temperature is varied linearly with time. The difference in temperature between the sample and reference is determined as a function of the furnace temperature. Initially, the temperature difference is zero. When there is a physical change in the sample, for example, a phase transition, the heat exchanged with the surrounding environment changes, and a temperature difference is observed between the sample and the reference. The temperature difference vanishes at the end of the transition in an ideal DSC. The variation in temperature difference as a function of furnace temperature can provide information about the physical or chemical change that occurs in the sample. Unfortunately, the area under the temperature variation vs. furnace temperature peak is not proportional to the heat and it is not reproducible. Boersma [55] described an apparatus where the thermal paths were reproducible, that changes DTA into DSC. It is advantageous for the DSC to have a small time constant of response, typically below 60 s. The sample under study, normally with a mass in the range (1 to 30) mg, and the reference material are sealed into metallic crucibles, each with a volume on the order of 10 mL. The metal used to construct the crucible should not react with the sample under study or with the purge gas flushing the DSC apparatus. The crucible material should also have high thermal conductivity and low heat capacity. Aluminium is most often used to construct the crucible, but other materials such as gold, copper, sapphire and glass have been utilised, with the choice depending on the specific experimental conditions. A differential heat flow calorimeter, such as the Calvet type microcalorimeter, provides a quantitative [56–58] measure of heat when used in an isothermal mode. However, its time constant of response is about a factor of 10 greater than that of a normal DSC. Consequently, the Calvet type calorimeter will not be referred to as a DSC, even if operated in scanning mode [59–60], in this section. DSC uses concepts from three scientific disciplines: thermophysics, which takes into account the heat transfer taking place in the apparatus; chemical thermodynamics, which describes the equilibrium state of the system; and kinetics, which gives the rate at which a change is taking place in a sample as it proceeds towards equilibrium when subjected to a change in thermodynamic variables. Each type of DSC (or particular mode of operation) that will be described in the remainder of Section 7.3 requires a working equation to relate the quantity q of heat exchanged between the sample and furnace to either the sample temperature T or time t: q ¼ f fT or tg:

ð7:21Þ

Typically, models derived from electrical equivalent circuit are used to describe heat transfer within a DSC apparatus. 349

350

Calorimetry

The electrical equivalents, listed in Table 7.1, provide linear relationships between heat flow rate and the temperature difference. This implies that the heat transfer occurs by thermal conduction and localised thermal time constants, rather than distributed time constants that are typical between discrete components. In DSC it is assumed that the crucible and contents have a single thermal time constant. This assumption is valid when the DSC is used at temperatures in the range (100 to 800) K with heating rates, dT/dt, are between (0 to 1800) K ? s1 .

7.3.1

Power-Compensated DSC

Power-compensated DSC [61–65] consists of two independent furnaces placed in the same isothermal surrounding. One furnace contains the sample under study and the other contains a reference sample that has no physical or chemical transformation in the temperature range of interest. The sample and reference material are usually sealed into separate metallic crucibles and then placed within separate furnaces that are heated at the same rate. Any physical or chemical transformation of the sample will, because of the adsorption or production of heat, manifest itself as a change in the heating rate. The calorimetric signal, DP, is then the difference between the power supplied to the reference crucible and power applied to the sample crucible to maintain both at the same constant heating rate. The temperature of the furnace at time t is given by, Tf ðtf Þ ¼ Ti ðt ¼ 0Þ þ tðdT=dtÞ;

ð7:22Þ

where Ti(t ¼ 0) is the initial furnace temperature. It is assumed that the sample has a transformation at a temperature between Ti and the final furnace temperature Tf, that absorbs heat qstr . The qsfu needed to heat the furnace with the sample from the Table 7.1 Analogy between thermal and electrical properties along with interrelating equations. Thermal Temperature Heat Heat flow rate Thermal conductance Heat capacity at constant pressure

Electrical Voltage Quantity of electricity Current Resistance Capacitance

T q F G Cp Equations:

F ¼ dq=dt F ¼ GDT dq ¼ Cp dT F ¼ Cp dT=dt

I ¼ dQ=dt l ¼ ðE2  E1 Þ=R dQ ¼ CdE I ¼ CdE=dt

350

E Q I R C

Calorimetry

351

initial temperature Ti (t ¼ 0) to the final temperature Tf is given by

¼

qsfu

qstr

ZTf  Ztf  s s s þ Cp;fu þ Cp;cr þ Cp dT þ Psfu dt; Ti

ð7:23Þ

0

where the subscripts fu and cr refer to furnace and crucible respectively and the superscript s defines the sample so that Cps is the heat capacity of the sample. It has s s been assumed that Cps ¼ Cp;pr where Cp;pr is the heat capacity of the product of the sample transformation. The external environment is maintained at constant temperature Text and there may also be an exchange of power Pext with the surroundings. To obtain qstr , a second furnace, as identical to the first one as possible, is heated in the same surroundings with a crucible that contains a reference material. The heat r capacity of the crucible Cp;cr is similar to that of the crucible containing the unknown sample. For the reference furnace, qrfu is given by

qrfu ¼

ZTf 

Ztf  r r Cp;fu þ Cp;cr þ Cpr dT þ Prfu dt;

Ti

ð7:24Þ

0

where Cpr is the heat capacity of the reference material and the superscript r defines the reference calorimeter. Subtracting Equation (7.23) from (7.24), gives qsfu  qrfu ¼ qstr þ

ZTf 

s Cp;fu



r Cp;fu

þ

s Cp;cr



r Cp;cr

Ti

þ

Cps



Cpr



Ztf dT þ



Psfu  Prfu dt:

0

ð7:25Þ Rearranging (7.25) gives

d qsfu  qrfu DP ¼ dt 

dqstr  s r s r þ Cp;fu  Cp;fu þ Cp;cr  Cp;cr þ Cps  Cpr ðdT=dtÞ þ Psfu  Prfu : ð7:26Þ ¼ dt The applied differential power DP is proportional [65] to the sum of the power absorbed by the transformation when it is corrected for the differences between the heat capacities of the crucibles and furnace used for both the reference and sample 351

352

Calorimetry

and the different power exchanged between each and the surroundings. In this case, the energy of a transformation within the sample, between an initial and final state, is determined directly. If, as is not unreasonable to do, we assume that the heat capacities and heat losses of the sample and the reference chambers are equal then Equations (7.25) and (7.26) simplify to: qsfu  qrfu ¼ q;

ð7:27Þ



d qsfu  qrfu dq : ¼ DP ¼ dt dt

ð7:28Þ

and

A typical furnace, shown in Figure 7.15(a) with the sample holder, contains a heating element and a platinum resistance thermometer which is located between the heating element and the sample container wall to minimise the thermal paths. The temperature control algorithm, shown in Figure 7.15(b), is intended to maintain the temperature of each furnace equal. As an alternative to the arrangement shown in Figure 7.15, Dosch [66] has proposed that a transistor act as both furnace and temperature sensing element. When a transistor is fed with a constant collector current, the thermal power developed depends mostly on the base intensity. The temperature is given by the forward voltage of the emitter base diode. In principle, the results obtained from a power-compensated DSC apparatus should not need correction. However, the temperature is measured underneath the crucible and this value, which depends on the thermal resistance between the crucible

Figure 7.15 (a) Power compensated DSC furnace layout. (b) Schematic of the power compensated DSC temperature control system.

352

Calorimetry

353

and the thermometer [67–70], differs from the true temperature. For the same reason, the energy absorbed by the sample transformation requires compensation [71]. When these potential systematic errors are correctly accounted for, with appropriate calibration materials, one can obtain precise measurements with a DSC [72,73].

7.3.2

Heat Flux DSC

If the base area of the reference and sample crucibles are identical, then heat flow DSC may be called heat flux DSC. In most calorimeters this condition is met, so for the remainder of this text heat flow DSC will be called heat flux DSC. A heat flux DSC apparatus [74] consists of a furnace containing a thermoelectric device to measure heat fluxes and two crucibles, one containing the sample under study and the other the reference material. In this type of DSC either the heat exchanged between the crucibles and furnace or the temperature difference between the sample and reference are measured. The former is essentially a miniaturised Calvet type calorimeter [56,57] while the latter is similar to the calorimeter used for DTA [74,75]. There are also modifications of this DSC that can provide the temperature difference in proportion to the power exchanged between each crucible and the furnace. The heat flux exchanged between the furnace and each crucible is measured with a heat meter that converts the heat flux into an electrical signal. The energy exchanged between the furnace and the sample and the reference crucibles may be written as: 8   dT s s > < Ps ¼ Cps þ Cp;cr þ Ftr dt ;   dT r > : Pr ¼ C r þ C r p p;cr dt

ð7:29Þ

where Ftr is the heat flow rate as a result of the transformation. Equation (7.29) s r assume that Cp;fu ¼ Cp;fu . If the heating rates of the sample and the reference are the same then dT s dT r dT ; & ¼ dt dt dt

ð7:30Þ

and the calorimetric signal DT is given by   dT s r þ Ftr : DT ¼ kðPs  Pr Þ ¼ Cps þ Cp;cr  Cpr  Cp;cr dt In Equation (7.31) k is a constant determined by calibration. 353

ð7:31Þ

354

Calorimetry

Figure 7.16 (a) Cross-section through the Mettler TA 2000B DSC. (b) Cross-section through the furnace in which the thermoelectric disk resides.

Two types of DSC are commercially available. The first has a three-dimensional heat flux meter based on a set of thermopiles and is essentially a miniaturised Calvet type microcalorimeter that is used in scanning mode [76]. This apparatus is useful when sensitivity is required and low heating rates are acceptable. The volume of the crucible is about 200 mL. Higher sensitivity devices have been constructed using thermopiles made of semi-conducting material [77]. The second type uses a thermoelectric disk with the thermopiles in a plane. A DSC apparatus of this type [78,79] is shown in Figure 7.16(a). The furnace is cooled by evaporation of liquid nitrogen, where its flow is controlled by a valve, and heated by a current flowing through a resistor. A temperature controller adjusts the quantity of heat added and the flow of cold gas so that the temperature of the furnace follows the programmed values as close as possible. The thermoelectric disk, shown in Figure 7.16(b), is made of glass and has 5 differential Au/Ni thermocouples [80] deposited on it. One junction is below the reference crucible and the other below the sample crucible so that the output is proportional to the temperature difference between the crucibles. The signal is converted to a temperature difference DT from a knowledge of the variation of the thermocouple emf with temperature. A thermoelectric disk fitted with greater than 5 thermocouples has been developed to improve the sensitivity [81]. Temperature differences can also be measured using platinum resistors vacuum deposited on either a glass or ceramic substrate [82]. The thermal analysis apparatus, formerly known as DuPont 910 DSC and shown in Figure 7.17, uses a constantan

Figure 7.17 DuPont 910 DSC apparatus.

354

Calorimetry

355

disk [83] with spaces for two (or more) crucibles. Alumel wires are attached to the constantan disk below each crucible forming a differential constantan-alumel thermocouple. Another thermocouple is placed under the reference crucible to determine the temperature of the thermoelectric disk. When a heat flux DSC is integrated with a thermal gravimetric analysis (TGA), the thermal effects can be isolated from those arising from mass changes that occur during a temperature scan [84].

Model of a Heat Flux DSC In a simple electrical equivalent of a heat flux DSC apparatus, shown in Figure 7.18, the electrical voltage E determines the furnace temperature U, which feeds the sample capacitor Cs through a resistor R that represents the sum of all the thermal conductors in the furnace. The thermal capacitance of the sample Cs is the sum of the capacitance of the crucible, the sample and the measuring device. When the sum of the currents in both the reference and the sample are zero then 8 dU s E  U s > ¼ < Cs þ Ftr dt Rs ; r r > : C r dU ¼ E  U dt Rr

ð7:32Þ

where C r is the reference capacitance, and Rs and Rr are resistances and Us and Ur the temperatures of the sample and reference respectively.

Figure 7.18 Equivalent circuit of a heat flux DSC apparatus. Rs, the resistance between the furnace and sample crucible; Cs , the heat capacity of the crucible and the sample; Us, the temperature of the sample; Rr, the resistance between the furnace and reference crucible; Cr , heat capacity of the crucible and the reference; Ur, temperature of the reference; E, the furnace temperature.

355

356

Calorimetry The calorimetric signal DT is given by: DT ¼ U r  U s ;

ð7:33Þ

and may be determined by an apparatus constructed so that the resistors and capacitors are approximately equal so that Rs &Rr ¼ R; C s &Cr ¼ C:

and

ð7:34Þ

Equation (7.34) describes a symmetric DSC. Combining Equation (7.32) with Equations (7.33) and (7.34) gives

Ftr ¼

DT dDT þC ; R dt

ð7:35Þ

which demonstrates the advantage of a differential arrangement by eliminating the dependence on E. An additional benefit of this arrangement is that the contribution of the furnace noise to the calorimetric signal is minimised. This simple model neglects the heat exchange that occurs between the furnace and the crucibles and also between the both crucibles [82,85–88]. Most of the heat transfer occurs through the thermoelectric disk but some heat flows through the gas flushing the DSC apparatus. As long as heat transfer occurs through the gas by conduction it can be accounted for by including a resistance in parallel with the resistors on the disk. The electrical schematic of this model is shown in Figure 7.19 and the working equations, including the resistance between sample and reference crucibles, Rcc, are given by

s Ccr þ Cdk

dU s E  U s Ur  Us ¼ þ F þ ; tr dt Rs Rcc

ð7:36Þ

dU r E  U r U s  U r ¼ þ ; dt Rr Rcc

ð7:37Þ

and

r Cr þ Ccr þ Cdk

r s where Ccr and Ccr are the capacitance of the reference and sample crucibles respectively. In Equation (7.37), Cdk is the capacitance of the thermoelectric disk.

356

Calorimetry

357

Figure 7.19 Equivalent circuit of a heat flux DSC with coupled cells. Rs, the resistance between the furnace and sample crucible; Rr, the resistance between the furnace and reference crucible; Us, the temperature of the sample; Rcc, the resistance between the sample and reference crucibles; Cdk , the heat capacity of the disk; Cs , the heat capacity of the sample; Ccr , the heat capacity of the reference crucible; Ccs , the heat capacity of the sample crucible; C r , the heat capacity of the reference material, Ur, the temperature of the reference; fs , the heat flux in the sample; and E, the furnace temperature.

Solving Equations (7.36) and (7.37) [91] gives:  Ftr ¼ DT

1 2 þ R Rcc

 C

dDT : dt

ð7:38Þ

Equation (7.38) was obtained assuming a symmetric calorimeter and applying the same assumptions as those used for Equation (7.35). For this model only the first derivative is taken into account. For the real apparatus, [89,90], the higher order derivatives may be significant and the relationship

Ftr ¼ k1 DT þ k2

dDT d2 DT þ k3 þ ; dt dt2

ð7:39Þ

needs to be considered. In Equation (7.39) ki , i ¼ 1, 2, 3, are parameters determined by calibration. Not surprisingly, this approach gives a better representation of the behaviour of the DSC in either a linear or modulated scanning mode. 357

358 7.3.3

Calorimetry Adiabatic DSC

In this type of DSC no heat flows from the calorimeter to the surroundings. This is achieved with an actively controlled thermal shield so that the temperature of the shield follows that of the calorimeter [91, 92]. For the adiabatic DSC the working equations are similar to Equations (7.25) and (7.26) for the power compensated DSC with Pfu ¼ 0:

q ¼ qstr þ

ZTf 

r Cp;fu þ Cps

Ti



9 > > > > dT > =

  dT dq r ¼ þ Cp;fu þ Cps dt dt dt dqstr

> > > > > ;

;

ð7:40Þ

where q is defined by Equation (7.27). When this apparatus is operated as a scanning calorimeter and the heat computed from measurements with the sample furnace containing either no material or a material of known heat capacity then it may be classed as a DSC. Adiabatic calorimeters are usually used to study dilute solutions of biochemical compounds in water. Usually, this type of transformation has a small enthalpy change, the reaction is slow, and occur in a narrow temperature range of (273 to 373) K. Privalov et al. [93] reported a DSC for this kind of study. This calorimeter was also described in Chapter 4 of reference [1].

7.3.4

Single Cell DSC

In a single cell DSC apparatus two measurements are performed in series: one with either the reference material or an empty crucible, the other with the sample. The mechanical simplicity of this arrangement, requiring only one furnace and crucible [94–96], places rather more severe, but still attainable, demands on the temperature control than for the DSC described previously. When the DSC contains a sample the heat flow in the single cell is given by:  dT dqs dqstr  s ; ¼ þ Cp;fu þ Cps dt dt dt

ð7:41Þ

and when the cell contains a reference the heat flow is given by: dqr dT r : ¼ Cp;fu dt dt 358

ð7:42Þ

Calorimetry

359

Subtracting (7.42) from (7.41) gives the working relationship for this DSC: dðqs  qr Þ dqstr dT ¼ : þ Cps dt dt dt

7.3.5

ð7:43Þ

Temperature Modulated DSC

Temperature modulation has been used in both ac calorimetry, which was described in Section 2 of this chapter, and DTA [97,98]. More recently temperature modulation has been combined with DSC [99]. The temperature of a DSC furnace modulated at an angular frequency o is given by Tfu ðtÞ ¼ Ti þ ðdT=dtÞt þ A sinðotÞ;

ð7:44Þ

where Tfu(t) is the temperature of the furnace at time t and Ti the initial temperature at t ¼ 0. Thus the calorimetric signal contains an additional modulated component superimposed on that arising from the continuous linear temperature increment [100–105]. This periodic change in the calorimetric signal can be used to assist with the interpretation of the DSC results. Model of a Temperature Modulated DSC The electrical analogue circuit for the calorimeter, shown in Figure 7.20, includes complex impedances for the reference and sample. The oscillating part of the temperature difference U r  U s , which is equal to the modulated component of the calorimetric signal DTo, has been obtained from a lumped equivalent circuit analysis

Figure 7.20 Equivalent circuit of a modulated DSC. Z s and Z r are complex impedances that describe the sample and reference respectively; Rs is the resistance between the furnace and sample crucible; Rr, the resistance between the furnace and reference crucible; Us, the temperature of the sample; Rcc, the resistance between the sample and reference crucibles; Ur, the temperature of the reference; and Eo , the furnace temperature.

359

360

Calorimetry

[106–109] with the result: DTo ¼

Eo Rcc RðZ r  Zs Þ ; Rcc ðR þ Zr ÞðR þ Z s Þ þ RZ r ðR þ Z s Þ þ RZ s ðR þ Z r Þ

ð7:45Þ

where Eo is the oscillatory temperature of the furnace, Rcc the resistance between the sample and the reference and the complex impedances Zr and Zs are of the reference and sample, respectively. In the derivation of Equation (7.45) it was assumed that the resistance of the sample and reference were equivalent, that is Rs ¼ Rr. For purely capacitive impedances, Equation (7.45) can be recast in complex notation as: DTo iRcc RoðC r  C s Þ ; ¼ 2 Rcc þ 2R  Rcc R Cr Cs o2 þ iRoðRcc þ RÞðC r þ C s Þ Eo

ð7:46Þ

where the real part (Re) of Equation (7.46) is given by   DTo  Rcc RoðC s  C r Þ   ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi; Re Eo  ðRcc þ 2R  Rcc R2 C s C r o2 Þ2 þ ½RoðRcc þ RÞðC s þ C r Þ2

ð7:47Þ

and imaginary (Im) component by tan f ¼

Rcc þ 2R  Rcc R2 o2 C s C r : RoðCs þ C r ÞðRcc þ RÞ

ð7:48Þ

An alternative model, based on the electrical equivalent circuit shown in Figure 7.18, has been developed for temperature modulated DSC by Wunderlich et al. [110] and Hatta and Muramatsu [111]. The calorimetric signal DT, obtained from the power difference between sample and reference, can also be separated with a Fourier transform into two parts, one for the continuous component that should represent the signal obtained without modulation, and the other for the periodic signal. The periodic part has an in-phase and an out-of-phase component, which provides a real and complex heat capacity [112–115]. The real part is called the ‘reversible heat capacity’ and the imaginary part the ‘irreversible heat capacity’. Alternatively, the measured calorimetric signal DT may be represented by DT ¼ a1 sinðotÞ þ a2 cosðotÞ þ bt þ c;

ð7:49Þ

where a1, a2, b and c are adjustable parameters adjusted to represent measurements of the calorimetric signal over a period [116]. The amplitude of the modulation, A, 360

Calorimetry

361

the phase, f, and the calorimetric signal without modulation, DTo¼0 , are obtained from: 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > A ¼ a21 þ a22 ; > > = a1 ð7:50Þ tan f ¼ ; and > a2 > > > DTo¼0 ¼ bt þ c: ; Schawe [117] and Reading [118] have compared the different computational methods used in modulated DSC. The influences of both the experimental parameters and thermophysical properties of the sample on modulated DSC have been studied [119,120] including dissymmetry of the apparatus [121,122], phase shift [123,124], harmonics in the modulating temperature [125], and the thermal conductivity of the sample [126,127]. Other workers [128,129] have attempted to attribute a physical significance to the complex heat capacity. Ozawa and Kanari [130] have suggested that there are unaccounted for non-linearities in modulated DSC. Thus, it is fair to say that the interpretation and analyses of measurements obtained with a modulated DSC is, at the present time, the subject of debate. Numerical simulations of the heat transfer within a modulated DSC experiment have been reported by Kanari and Ozawa [131,132] and Wang and Harrison [133]. It is plausible that numerical methods, based on specified thermodynamic and kinetic models, will provide an estimated calorimetric signal that can be compared with that observed experimentally. Some of the developments in modulated DSC can be found in references [134– 136]. A large number of applications of modulated DSC have been published, including the determination of heat capacity [137–142], glass transitions [143–151] and other phase transitions [152–158]. Analytical uses of modulated DSC have also been reported [159–164], for example, the characterisation of polymers [159,164] and petroleum products [165]. Measurements of thermal conductivity have also been reported [166,167] with DSC as have thermometer calibration [168].

7.3.6

Specialised DSC

DSCs have been constructed for specific applications, including the study of gas reactions at high pressure [169,170], high temperature heat capacity measurements, and rapid reactions with short time constants. Each of these will be considered here. For gas reactions at high pressure the main requirement is a means of containing the gas within the DSC. Two methods of pressurisation have been reported. The first involves the construction of a cell specifically designed to seal the gas within a pressure vessel. The pressure is generated by either decomposition of the sample or a known quantity of another material, of known heat capacity, that decomposes at a lower temperature than the sample under study. The second method is to install the DSC in a high-pressure vessel and calibrate the instrument at the same pressure and 361

362

Calorimetry

with the same gas used for the experiment. Obviously, the heat capacity of the crucibles is large which increases the time constant of the DSC and, because the heat transfer occurs through a thick metallic wall, the sensitivity of the DSC decreases. At high temperatures, typically between (800 to 1900) K, DSC is usually used to determine heat capacities [171,172]. At these temperatures, heat transfer occurs by both conduction and radiation [173], and the thermocouple location requires considerable detailed study [174]. A high temperature DSC heated with modulated light has been described in reference [175]. In DSC, decreasing the size of both the sample and the measuring device can decrease the time constants. To achieve this, devices based on silicon integrated heatflow transducers have been constructed [176,177]. Brill and Gmelin [178] described a silicon based DSC that can be programmed with non-linear heating rates.

7.3.7

Determination of Physical Properties with DSC

DSC scans are usually performed at a constant heating rate and the results are often found to depend on heating rate because the calorimetric signal contains both kinetic and thermal effects. To separate the thermal and kinetic effects two approaches are used. One is to maintain the system as close as possible to equilibrium by reducing the temperature scan rate. The other is to determine the time-temperature response of the system and correct the results for the kinetic effect. In any case, the interpretation of DSC results may require additional measurements obtained with ancillary techniques. For example, the gas evolved from thermal decomposition of the sample can be analysed with thermal gravimetric techniques [179,180], mass spectrometry, Fourier transform infrared spectroscopy, or gas-liquid chromatography [181,182]. Gas evolved during a scan may also effect the heat measurement and change the sensitivity of the apparatus because of changes in the thermal conductivity of the gas surrounding the sample. This can be overcome by purging the DSC with the same gas as that evolved during the decomposition. X-ray diffractions studies can be performed during DSC to assist in determining the relation between structural and thermal effects [183–186]. Methods to observe the sample optically with, for example, thermomicroscopy, have also been developed [187]. A comparison of the performance of several DSC’s, based on experimental information, has been reported in reference [188]. Typically, the thermal noise for the furnace in scanning, heating, cooling or isothermal mode is 102 K [189]. The thermopile sensitivity to heat flow ranges between (2 to 2000) mV ? mW1 . For power compensated DSC the heat is usually accounted for with an uncertainty of about 1 mW. Reproducibility and uncertainty of the temperature measurements is usually about +0.1 K. The time constant of response of a DSC can be between (0.2 and 30) s, with power-compensated DSC tending toward the short time constants while heat flux DSC are usually at the upper end of the range. Shorter time constants provide higher resolution for consecutive thermal effects. The lower bound of DSC sensitivity is, typically, less than about 1 mW [190]. 362

Calorimetry

363

The uncertainties in the thermodynamic properties obtained with a DSC also depend on the calibration of the apparatus for temperature [191–201], heat [202– 206], and electrical power [207–210]. The International Confederation of Thermal Analysis (ICTA), has recommended calibration procedures [193,194,210], and these have been reported as international standards ISO/CD 11357–1. Usually, polynomial relationships are used to represent the error between the observed and the real temperature. For a heat flux DSC these relationships are used to convert the electrical signal into a temperature difference DT, and thus into a power difference DP. For power-compensated DSC these relationships are used to obtain a correction factor for the experimentally determined power. In the remainder of this section the thermodynamic properties obtained with DSC and important experimental aspects are discussed.

Phase Transitions and the Base Line For a first order transition, such as melting, where the phases are in equilibrium and the temperatures and heating rates are constant, a quantity qsþl of heat is absorbed [211–213]. According to the equivalent circuit, shown in Figure 7.19, the time constant of response t of the sample and reference are given by the electrical analogues: tr ¼ Rr C r

and

ts ¼ R s C s :

ð7:51Þ

The temperature of the reference crucible is given by U r ¼ Ei þ

dT ft  tr ½1  expðt=tr Þg; dt

ð7:52Þ

where Ei is the initial furnace temperature and the heating rate given by dU r =dt ¼

dT ½1  expðt=tr Þ; dt

ð7:53Þ

does not depend on what happens in the crucible. Equation (7.53) is not valid for a coupled cell calorimeter [83,87,88]. The temperature Us of the sample crucible is given by Equation (7.52) with the superscript r, for the reference, replaced by s. For t44tr (or t44ts on the sample side), the heating rate of each crucible is the same as that of the furnace. The temperature of the reference (or of the sample) is lower than the temperature of the furnace by (dT=dt)tr (or (dT=dT)ts on the sample side). This difference in temperature between the crucible and furnace, called the temperature lag, depends on the time constant of the DSC. 363

364

Calorimetry

At the phase transition, U s ¼ T sþl , and the time from the start of the experiment to the time of the transition t1 is given by: t1 ¼

T sþl  Ti þ ts : ðdT=dtÞ

ð7:54Þ

The duration of time tl þ s at the transition is given by:

sþl

t

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qsþl Rs ¼ 2 with ðdT=dtÞ

tsþl Z þt1

qlþs ¼

ðdT=dtÞt dt: Rs

ð7:55Þ

t1

It is assumed here that t ¼ 0 at the start of melting. The temperature of the furnace at the end of the transformation E 0 is given by E 0 ¼ Ti þ ðdT=dtÞðtsþl þ t1 Þ:

ð7:56Þ

When the transformation is complete, the sample responds only to its heat capacity, and the sample temperature at time t after the transition is given by:

U s ¼ Ei þ ðdT=dtÞðt þ tsþl þ t1 Þ þ E 0 þ T sþl  ðdT=dtÞts 6½1  expðt=ts Þ:

ð7:57Þ

The calorimetric signal DT is given by Equation (7.33) namely: DT ¼ ðE  U s Þ  ðE  U r Þ;

ð7:58Þ

and if the DSC is symmetric and Rr ¼ Rs ¼ R, then Equation (7.58) becomes DP ¼

DT ¼ Fs  Fr : R

ð7:59Þ

The calorimetric signal is then proportional to the difference in power exchanged between the furnace and each crucible. During the heating stage the calorimetric signal is given by: DT ¼ U r  U s ¼ ðdT=dtÞðtr  ts Þ þ ts expðt=ts Þ  tr expðt=tr Þ:

ð7:60Þ

For most calorimeters tr =ts and the calorimetric signal DP=0 even if the heating rate of the crucibles of the furnace are identical. Indeed for t44tr &ts the 364

365

Calorimetry calorimetric signal DT is given by DT&ðdT=dtÞðtr  ts Þ;

ð7:61Þ

while for a symmetric apparatus the electrical analogue is DT=R&ðdT=dtÞðC r  C s Þ:

ð7:62Þ

Thus, for the symmetric apparatus, the calorimetric signal is proportional to the difference between the heat capacities of the reference and the sample. During the transition, the calorimetric signal is the difference between the constant temperature T sþl and the temperature of the reference side. Since this temperature is a linear function of time, the slope of the calorimetric signal (expressed as a temperature) is constant and equal to the heating rate of the furnace and thus the reference: DT ¼ Ti þ ðdT=dtÞðt  ts Þ  T sþl :

ð7:63Þ

At the end of the transition, the product crucible is heated again at a rate dT=dt, and the calorimetric signal is given by DT&(dT=dt)(tr  ts ), as it was before the transition. Numerical evaluation of (the equivalent electrical circuit) Equation (7.60), for U r :T r and U s :T s , and Equation (7.62) over the temperature range (0 to 200) K, with realistic values of capacitance, resistance, and dT=dt listed in Table 7.2, are shown in Figure 7.21. The results obtained from this simulation are consistent with experiment in two respects. First, as Figure 7.21 shows, the temperature of the furnace is greater than the T r and T s. Second, the phase transition temperature for the sample T s remains constant for the period of the transition while the furnace and reference temperature continue increasing at the same rate dT=dt giving rise to the variation of DT. After the transition, dT s/dt eventually returns to the same value as that of the furnace and reference. The observed enthalpy change depends on the extent to which any transformation (which may be a phase transition or chemical reaction) has occurred [213]. For a Table 7.2 The resistances Rr and Rs and heat capacities Cr and Cs (defined in Figure 7.19) for both the reference, with superscript r, and sample, with superscript s, crucibles. The simulation was performed with the sample phase transition (denoted by subscript trs) data over a temperature range with a constant heating rate. Calorimeter

Sample

Simulation

Rr ¼ 0:1 K ? mW1 C r ¼ 0:6 J ? K1 Rs ¼ 0:1 K ? mW1 C s ¼ 0:3 J ? K1

qtrs ¼ 1:5 J Ttrs ¼ 50 K

0  T=K  110 b ¼ 0:1 K ? s1

365

366

Calorimetry

transformation, measurements of Dtrs H(T, p) can be used to determine the apparent heat capacity Cp at constant pressure from:         qH qH qH qx ¼ þ Cp ¼ ; qT p qT p;x qx p;T qT

ð7:64Þ

where x is the extent of transformation. The apparent heat capacity is the sum of the true heat capacity Cp, x at constant extent of transformation and the configurational heat capacity Cp, conf :  Cp ¼ Cp;x þ Cp;conf ¼

qH qT



 þDtrs Hðp; TÞ

p;x

 dx ; dT

ð7:65Þ

where Dtrs H(p, T) is the enthalpy of the transformation (phase transition). The power change DP for the system is given by DP ¼ ðdT=dtÞðCpr  Cps Þ þ Dtrs Hðp; TÞ

dx : dt

ð7:66Þ

The rate of transformation is given by:   dx Ea ¼ f ðxÞAa exp  ; dt RT

ð7:67Þ

where f (x) is an unspecified function of the extent of transformation, Ea an activation energy, and Aa is the frequency factor. This relationship, although widely used, has been criticised [214,215]. The difference between the two heat capacities in Equation (7.66), Cpr  Cps , obtained when there is no thermal event (or transition) is the so-called base line shown in Figure 7.21. The enthalpy change of the transition is determined, by integration, with respect to time over the transition, of the calorimetric signal minus the base line. The determination of the DSC base line, which is difficult, has been discussed in the literature [216–219]. Usually, the base line signal is represented by an arbitrary polynomial function. For a transformation A ! B, the calorimetric signal given by Equation (7.62) can be used to determine the apparent heat capacity of the system before and after the transformation. This apparent heat capacity before the transformation is the heat capacity of A while that after is the heat capacity of B. If the extent of transformation with respect to time is known, then the true heat capacity of the system is given by Cp ¼ ð1  xÞCp ðAÞ þ xCp ðBÞ: 366

ð7:68Þ

Calorimetry

367

Figure 7.21 Simulation of a first order transition. Calorimetric signal DT and temperatures Tfu, T s, T r of the furnace, sample and reference respectively. The DSC base line is shown also.

Equation (7.68) is solved iteratively until the difference between two computed base lines is less than some chosen value [220].

Heat Capacity at Constant Pressure Measurement of heat capacity [221–226] requires three temperature scans, each with the same reference crucible and the same heating rate. The first scan is performed with an empty crucible to obtain DT1. The second scan, with the same crucible r containing a known mass, mr, of a reference material whose heat capacity Cp;s is also well known provides DT2. Finally, the third scan, which provides DT3, is obtained with a sample of known mass ms, but unknown heat capacity Cps , which is then determined from the combined measurements of the calorimetric signal with:

r Cps ¼ Cp;s

mr DT3  DT1 : ms DT2  TD1

ð7:69Þ

A single temperature scan can be performed provided the apparatus has been calibrated previously [227]. Sample purity can be determined from measurements of dT sþl =dt along the melting line with the method reported by Skau [228], and utilised by others [229– 231], where it is assumed that the sample does not decompose during the course of the temperature scan. Impurities are assumed to be soluble in only the liquid phase and in sufficiently dilute amounts for the mixture to be considered an ideal solution. Assuming thermodynamic equilibrium is achieved, the relationship between the impurity mole fraction xi, the temperature of the sample, T, and the melted fraction, 367

368

Calorimetry

x, is given by: T ¼ Tfus 

RT 2 xi 1 ; Dfus H x

ð7:70Þ

where Dfus H is the enthalpy of fusion and is determined experimentally. The (dTfus =dt) obtained must be corrected using Sondak’s method [232] or other computational means [233–235]. The DSC can be programmed to perform a series of discrete temperature increments [236–239], called step heating, where the calorimeter is maintained at each temperature until equilibrium is achieved. This method has been used for purity determination, using Skau’s method [240–243], and to obtain heat capacities with higher accuracy than obtained with method described above [244]. Step heating has been also used for the study of kinetics [245–249], where the analysis of the DSC signal provides the rate constant. This method is time consuming and, consequently, little work has been devoted to its development. Porosity and Kinetics The temperature and pressure at which a phase transition occurs may depend on the curvature of the phases [250]. If a porous solid is saturated with an excess of a liquid it is possible to determine, at a temperature, the radius of the pores within the solid along with the volume of the freezing liquid [251–255] from a DSC operated with temperature decrements. Variants of this technique have been used to study other systems such as emulsions [256–258] and coatings or powders [259]. The kinetics of chemical reactions have been studied with DSC [260–265] even though the interpretation of the results is somewhat difficult [266]. Examples of the chemical kinetics studied include potentially dangerous reactions [267] and the glass transition [268–272]. DSC can be used to quantitatively determine the amount of one component in a mixture if that species exhibits a heat effect over the DSC temperature range [273–275].

7.4

Nano-Calorimetry S. VAN HERWAARDEN Xensor Intergration Delft, Netherlands

Around 1780 Lavoisier and Laplace developed the so called ice calorimeter which determined the heat evolved by a living organism within it from measurements of the quantity of ice the organ melted. In particular, it was used to determine the metabolism of entire living animals, about (1012 to 1015) cells, placed within the 368

Calorimetry

369

calorimeter [276] and a variant of this type of calorimeter is still in use [277]. Bataillard [278] reports that a nanocalorimeter was used to study the metabolism of less than 106 bacteria which is a reduction in cell sample size over the animal within the ice calorimeter by a factor of between (106 to 109). The advent of micro electromechanical systems (MEMS), also known as micro-system technology (MST), that has evolved from the methods used to manufacture integrated circuits (IC) has permitted the development of calorimeters that can operate with significantly smaller sample sizes. MEMS can be manufactured with dimensions from (103 to 109) m with high precision and reproducibility. MEMS calorimeters have been fabricated with dimensions on the order of 103 m that incorporate heating elements with well defined thermal properties, thermometers and mechanical structures that can hold the samples. The term ‘nanocalorimeters’, consistent with the current and accepted definition of microcalorimeter, is used to describe these calorimeters because of the power detected by the instrument rather than the dimensions of the object formed with integrated circuit technology. Nanocalorimeters have been used to measure thermodynamic properties of fluids. The main application of these devices is outside those traditional measurements and differs from those of differential scanning calorimeters (DSC). The devices reported at the time of this writing operate over a rather limited range of temperature but the small size provides both rapid response times and high sensitivity.

7.4.1

Micro Electro-Mechanical Systems (MEMS)

MEMS is a technology used to manufacture three-dimensional silicon based structures with specific geometrical, mechanical and electrical properties to execute certain tasks [279]. In a MEMS nanocalorimeter the microstructure includes on one substrate all the required components, including a heating resistor, a temperature sensor, a sensor to determine temperature differences, a well-defined thermal conductance and a container [280–282]. In a calorimeter constructed with traditional methods, for example the DSC described in Section 7.3, the assembly is formed from several individual components that are mechanically joined to form one calorimeter. In the case of a DSC it consists of the following individual components: an oven surrounding the instrument; a thermocouple to measure the temperature of the crucible; and a sample holder that is usually a metal foil (that acts as a thermal conductor) with two cup-like indentations to hold the sample. Silicon-Based MEMS With MEMS technology, the nanocalorimeter is formed as a single integrated structure including heater, sensor, and sample holder on a Si base. Typically, MEMS uses circular silicon wafers about 100 mm in diameter and about 0.5 mm thick as the substrate. An individual device has dimensions on the order of 103 m and up to 105 devices may be processed simultaneously with the number depending on both device 369

370

Calorimetry

size and wafer quantity. Resistors and thermopiles are formed atop the silicon wafer by depositing specific substances that are then subsequently shaped with a combination of photolithography and either wet or dry etching. The fabrication steps for a membrane type calorimeter are shown, not to scale, in Figure 7.22. In this nanocalorimeter, the membrane with a heater at the centre is the sample holder. The heat flows laterally through the membrane to the support rim, which is the heat sink, and creates a temperature difference across the membrane that is determined with the thermopile. The membrane is formed by removing the silicon beneath the thermopile with a process called back etching. The membrane so formed is about 5 mm thick, and comparable with the thickness of the thermopile. The sensitivity of this calorimeter increases by about a factor of 100 compared with what it would have had with a 500 mm thick membrane. The silicon membrane can be between (5 to

Figure 7.22 Schematic of the sequence used to process a membrane-type nanocalorimeter. (a) Starting material silicon wafer 100 mm diameter, 0.5 mm thick; (b) lithography: after spinning photo-sensitive resist on the wafer, the resist is exposed, with UV-radiation through a glass mask, and subsequently developed; (c) silicon thermopile and heating resistor formed after ion implantation again using photolithography with a photoresistant material and mask; (d) fabrication of SiO2 dielectric isolation layer and aluminium interconnection layer {with lithographic methods as shown in (b)}; and (e) a completed nanocalorimeter after etching away silicon underneath the thermal path, and application of an enzyme layer.

370

Calorimetry

371

50) mm thick with a cross-sectional area between (1 and 100) mm2, and because the thermal conductivity of silicon is about 150 W ? K1 ? m1 , the thermal resistance is of the order (10 to 100) K ? W1 . The thermopile integrated into the membrane typically has a sensitivity between (10 to 100) mV ? K1 . The heating resistor, which is usually integrated into the middle of the membrane, has a resistance of about 1 kO and, when supplied with voltages from (1 to 10) mV, can dissipate between (1 to 100) nW. This particular design is appropriate for liquids. Many nanocalorimeters can be fabricated on one wafer and the individual devices are separated from each other with a diamond cutting machine. Electrical connections to the MEMS are achieved by ultrasonically welding wires between contact pads and an external mount. MEMS usually require packaging that is specific to their end use and standard plastic IC packages are not usually used. Further details on MEMS and their processing may be found in reference [283]. For gases, where the fluid thermal conductivity k is on the order of 10 times less than the corresponding liquid, for example, k(H2O, 1,373 K)/k(H2O, g, 373 K) &30, the heat generated by the resistor tends to flow into the silicon support rather than the fluid. Increasing the thermal resistance of the membrane increases the sensitivity of the device. Typically, this is achieved by replacing the 5 mm thick silicon membrane with a 0.5 mm thick non-stoichiometric form of Si3N4, which has a thermal resistance 500 times that of silicon. A thermopile of similar thickness can also be fabricated from poly-silicon. These thin membranes are, necessarily, more fragile and sensitive than the thicker counterpart. A device with the non-stoichiometric silicon nitride membrane, with an area of 1 mm2, has an overall sensitivity of 50 V ? W1 at a thermopile sensitivity of 10 mV ? K1 and a membrane thermal resistance of 5000 K ? W1 .

Micro-Devices Based on Flexible Film The membrane used in the nanocalorimeter can also be formed from thin plastic membranes with low thermal conductivity, such as Mylar. This approach eliminates the need for micro-machining to create the membrane. The nanocalorimeters formed from polymer films can be mechanically more robust then the MEMS fabricated device. Muehlbauer et al. [284,285] developed a bio-nanocalorimeter from a thermopile formed from 50 bismuth-antimony junctions, each 1 mm thick, evaporated onto a 40 mm thick Mylar film. This sheet was bent into a 3 mm diameter cylinder, as shown in Figure 7.23, with the thermopile on the inside. A glucose sensitive enzyme was placed on the outside of the Mylar tube. This arrangement is a calorimetric-based biosensor. A thermoelectric generator has been constructed from bismuth-telluride thermopiles sputtered onto a Kapton foil [286]. When the Kapton foil is folded into a block with linear dimensions of (9.5 by 7.5 by 2.8) mm and aluminium-oxide deposited on both the top and bottom of the block, a generator was obtained. In this case the internal resistance was 1 MO and an output of approximately 4 V and 16 mW obtained when a 20 K temperature difference was applied between the top and bottom of the block. This approach could be used in nanocalorimeters. 371

372

Calorimetry

Figure 7.23 TOP: Schematic of the Mylar film based calorimeter for the determination of glucose concentration. BOTTOM: Cross section of the active element of the calorimeter. Mylar-film based thermopile-enzyme glucose sensor fabricated by Muehlbauer et al. [284, 285].

7.4.2

Nanocalorimeters as Sensors

Calorimeters can be used to determine physical properties, including the enthalpy of chemical reaction, biochemical transformation, and the heat capacity of a material, from measurements of a temperature difference across a thermal resistance through which heat flows. The temperature difference is usually determined from measurements of the electrical characteristics of, as examples, thermocouples, resistors, or transistors. Nanocalorimeters are, necessarily, physically small with low internal heat capacities so that they can be used with small sample volumes maintaining a necessary sensitivity without the calorimeter dominating the thermal process. Nanocalorimeters are usually operated isothermally and have been used for the study of biological systems.

Nanocalorimetry in Biology Bataillard [278] has reviewed the use of nanocalorimeters in biomedical and biochemical research. One application reported in reference [278] is the detection of the quantity of both glucose and urea in blood from measurements of the enthalpy of an enzymatic reaction. Another important application is the use of nanocalorimeters with living cells, where either the sensitivity of cells to medication or toxins can be determined from measurement of the heat, which can be up to 300 pW per cell, produced during metabolism. In addition, the concentration of nutrition in the fluid surrounding the cells can be determined from the metabolic heat output. The measurement of cell metabolism is the nanoscale alternative of the whole-body calorimeter of two centuries ago. 372

Calorimetry

373

Muehlbauer et al. [285,286] has described a nanocalorimeter suitable for such measurements that is based on a flexible (Mylar) foil and was shown in Figure 7.23 above. Bataillard [278] used a MEMS, with a silicon membrane, integrated silicon thermopiles, and an enzyme coating that was placed on the opposite side of the membrane to that of the thermopiles, to detect constituents in blood and other liquids. Both of these devices could be used for medical diagnostics techniques including, as examples, the determination in blood of the concentration of glucose, which is significant for diabetes, urea, penicillin and creatinine, which is an indicator of kidney malfunction. To detect glucose concentration the reaction GOD C6 H12 O6 þ O2   ? C6 H12 O7 þ H2 O2 ;

ð7:71Þ

can be used, for which the enthalpy of reaction Dr H ¼ þ79 kJ ? mol1 . In Equation (7.71), GOD is an acronym for glucose-oxidase (b-D-Glucopyranose aerodehydrogenase), an enzyme that catalyses the glucose-specific reaction and, thus, makes the sensor selective. The sensitivity of the sensor can be enhanced by adding a second enzyme, catalase, which decomposes the hydrogen peroxide according to the reaction H2 O2 ?H2 O þ 12O2 ;

ð7:72Þ

for which Df H 0 ¼ þ 100 kJ ? mol1 . Thus the total enthalpy of reaction is about 180 kJ ? mol1 . The nanocalorimeter described in reference [278] has a sensitivity of about 50 mV ? mol1 ? L1 , which gave a minimum glucose concentration detection threshold of 20 mmol ? L1 . However, H2O2 can be formed from reactions other than Equation (7.70) and a systematic error is introduced when the sensor is coated with both enzymes. When the calorimeter membrane is coated with the enzyme urease, the concentration of urea can be determined from the enthalpy of reaction of Dr H ¼ þ 61 kJ ? mol1 , while the concentration of Penicillin G can be obtained when the membrane is coated with the reagent enzyme b-lactamase for which Dr H ¼ þ 67 kJ ? mol1 . Other species can be detected with other membrane coatings [278]. Kidney disease can lead to an increase in creatinine concentration in blood, from 30 mmol ? L1 to 1000 mmol ? L1 , and in urine where the normal range is (4 to 18) mmol ? L1 and the elevated concentration between (40 to 50) mmol ? L1 . These variations of creatinine can be determined with the enzyme creatinine deiminase coated onto a nanocalorimeter membrane. The enthalpy of reaction gives a concentration resolution of between (5 to 10) mmol ? L1 and a sensitivity of about 1 mV ? mmol1 ? L1 . The advantage of using a calorimetric technique is that no energy is released in the sensor other than by the enzymatic oxidation. Verhaegen et al. [287] have developed a nanocalorimeter matrix of 96, essentially identical, calorimeters on a single 150 mm diameter silicon wafer. In this calorimeter, living cells are kept in a fraction of the elements while the remainder act as a 373

374

Calorimetry

reference. The temperature difference between the living cell filled and reference cavities is determined with a thermopile. The calorimeter is based on a large 0.6 mm thick dielectric membrane formed from silicon oxide and silicon nitride which is coated in a 20 mm thick layer of rubber to support the otherwise very fragile dielectric membrane. The thermopile consists of 666 p-type polysilicon-aluminium thermocouples with a combined sensitivity of 130 mV ? K1 . The calorimeter has a sensitivity of about 20 V ? W1 . With &107 cells in the cavity, and cell metabolism of between (100 to 300) pW, an output on the order of 10 mV has been obtained for a single-element nanocalorimeter [287]. In this apparatus, the effect of medication or other alterations in the cell’s environment on the cells metabolism can be observed from voltage measurements with mV resolution. This sensor may be used for rapid drug screening. Nanocalorimeters as Gas Detector Calorimeters can be used for real-time in situ detection of specific gaseous species within a gas mixture, such as detecting potential illness based on a chemicals concentration within exhaled breath. Mitrovics et al. has described a modular based sensor system for the detection of gases in gas mixtures [288]. In this so called electronic nose, there are elements based on the following methods: quartz crystal microbalance and surface acoustic wave device, for the determination of variations in mass; metal oxide sensors, for which specific gases alter the electrical resistance; and nanocalorimeters [289]. The nanocalorimeters included in this instrument are shown schematically in Figure 7.22. The membrane in these calorimeters is formed from a 6 mm thick, 3.5 mm wide and 3.5 mm deep silicon membrane. Four of these calorimeters are mounted in a single ceramic pin grid array (PGA) of a standard IC housing. Each nanocalorimeter in the matrix has a different catalytic or enzymatic coating so that each gives a unique response to specific gases. This method gives a multi-dimensional signal allowing the determination of type and concentration of components in a gas mixture.

7.4.3

Nanocalorimeters for Material Properties Determination

The methods used to fabricate MEMS are particularly useful for the measurement of the thermal properties of compounds that may be vapour deposited atop, an otherwise essentially identical, nanocalorimeter membranes. The variation of the calorimeters thermal characteristics can be used to evaluate the thermal properties of additional layers. The thermal properties of monocrystalline silicon, polycrystalline silicon as well as the dielectric layers Si3N4 and SiO2, and conductive layers of Al have been determined with nanocalorimeters [290]. Literature in this field has been reviewed by von Arx who also reported measurements of the Seebeck coefficient [290]. 374

Calorimetry

375

Table 7.3 Seebeck coefficient, S, thermal conductivity, k, and heat capacity Cp of microsensor materials at room temperature [289] and [280]. Material

k W  K71  m71

Cp MJ71  K71  m73

S mV  K71

mono silicon poly silicon aluminium silicon nitride silicon oxide

150 20 to 30 200 3 1

1.6 1.8 2.5 2.2 1.6

+250 to 750 +100 to 300 2

Figure 7.24 shows the MEMS device used by von Arx [290] to determine the Seebeck coefficient of polysilicon. It contains a single polysilicon beam, 1, a polysilicon heater 3, and a polysilicon resistor, 4, which is used to calibrate the temperature difference across, 1. Methods used to fabricate this device are similar to

Figure 7.24 TOP: Photograph obtained with a scanning electron microscope (SEM) of the device used by von Arx [290] to determine the Seebeck coefficient of a complimentary metal oxide semiconductor (CMOS) polysilicon layer. The cavity is formed by front etching through windows in the CMOS dielectrics. BOTTOM: (a) schematic of the layout observed from above; (b) cross-section through the device. 1, beam; 2, etched cavity; 3, polysilicon heater; 4, polysilicon resistor for temperature sensing; 5, metal connections; 6, metal cover for temperature homogenisation; 7, silicon substrate; 8, one of the windows in CMOS dielectrics; 9, polysilicon sample being characterised; 10, hot contact to 9; 11, cold contact to 9.

375

376

Calorimetry

those shown in Figure 7.22 except that a cavity was etched below the polysilicon beam from atop the wafer rather than below. This fabrication technique is know as surface micromachining, while that used to form the cavity shown in Figure 7.22 is called bulk micromachining. The former can provide smaller cavities within the substrate and, because the silicon substrate is not completely removed, the device is more robust. However, the fabrication process is somewhat more complicated because of the requirements placed on the photolithography alignment and the compatibility of the etching processes with the existing layers. Measurements with devices have provided the thermal properties, listed in Table 7.3, of materials and layers commonly used in MEMS and nanocalorimeters.

Portable Nanocalorimeters Setaram has developed a portable calorimeter, similar in cross-section to that shown in Figure 7.22. It has a square 10 mm by 10 mm silicon substrate in which was fabricated 8.5 mm by 8.5 mm square membrane 50 mm thick [282,291]. This calorimeter has a sensitivity of &1 V ? W1 [292]. A monocrystalline siliconaluminium thermopile with 164 junctions, with an estimated sensitivity of about

Figure 7.25 TOP: Schematic of a MEMS nanocalorimeter where the sample holder is thermally isolated from its surroundings. A 300 mm by 300 mm square membrane 3 mm thick, is suspended by a 100 mm wide and 3 mm thick wire above an etched square cavity 2.5 mm by 2.5 mm. A heater and gold-nickel thin-film thermocouple are integrated into the membrane [293]. BOTTOM: Cross-section along X-X0 showing the pyramidal shape of the cavity over which the membrane is suspended [293].

376

Calorimetry

377

100 mV ? K1 , and a heater are integrated into the membrane. The nanocalorimeter is mounted in standard ceramic PGA housing. Kimura et al. [293] have developed a MEMS nanocalorimeter in which the sample holder (membrane) is almost thermally isolated from its surroundings as shown in Figure 7.25. A 300 mm by 300 mm square membrane 3 mm thick, is suspended by a 100 mm wide and 3 mm thick wire above an etched square cavity 2.5 mm by 2.5 mm. A heater and gold-nickel thin-film thermocouple, which is used to measure the temperature increase, are integrated into the membrane. This device has a low mass, and is thermally disconnected from the surroundings. Therefore it is very sensitive with a very short time constant of response. Temperature-controlled gas sensors have also been constructed [294].

Nanocalorimeter Probe for Thermal Analysis An apparatus based on atomic force microscopy methods, shown in Figure 7.26, has been developed to determine the thermal conductivity, thermal diffusivity and heat capacity of a solid by TA Instruments (mTA 2990 Micro-Thermal Analyser) [295]. In this apparatus a probe, with a tip of radius 50 nm, is lowered close to the sample surface and moved in the x, y plane while maintaining a fixed distance z from the surface. These measurements provide the topography of the surface which must be flat to better than 10 mm. In this particular device, a laser is used to determine the vertical displacement. The tip is a Wollaston wire, which is coated platinum that acts as both the resistive heater and thermometer. This probe can be used to determine the thermal properties of the sample in 30 mm by 30 mm square blocks with a spatial resolution 0.99 for calcium activities below 6 ? 106 at 1113 K [180]

Figure 8.6 Conductivity and pressure range of pure (> 99 per cent) ionic conductivity for some of the more usual solid electrolytes.

416

Properties of Mixing

417

and fluorine partial pressures down to 1043 Pa at 1073 K and to 1065 Pa at 767 K [181]. CaF2 was initially used to study Ca-silicates [182] and carbides [183] but has later been applied to binary alloys like Th [184] and U alloys [185]. The use of CaF2 is limited by its high reactivity as discussed by Levitskii [186] and Kleykamp [187]. Other fluorine conducting electrolytes are PbF2, CeF3 and SrF2 [188]. The latter and CaF2 have recently been applied to study the activity of SrO in La1xSrxMnO3d [189], whereas CeF3 has been used to determine the Ce activity in Pd-Ce alloys [190]. A third group of much used solid electrolytes are the b-aluminas. In Na-balumina [191] Naþ is the ion-conducting species situated in oxygen-deficient layers separating the four oxide layers thick ‘spinel blocks’ of the structure. Na-b-alumina has been used extensively to investigate binary Na-metal systems [192,193]. By proper ion exchange the Naþ ion conducting species can be exchanged by other ions. Zn, Co, Ni, Cu, Ca and Mn conducting b-alumina have been used for studies of Zn, Co and Ni silicates and for studies of mixing properties of CaO and MnO, NiO and CoO [194]. Cell Arrangements and Materials Compatibility Although the experimental setup is simple in principle, several factors make construction of electrochemical cells difficult: sealing of the cell, oxygen permeability, and materials stability and compatibility in general. Experimental cell designs have been discussed by Kleykamp [187] and by Pratt [171]. In single compartment cells, a cell stack is kept in a common protective atmosphere. Transport of the ionic conducting species via the gas phase should be insignificant. A solution to this may be to choose a reference electrode with potential similar to that of the working electrode. An electrolyte may also be used in form of a crucible in order to increase the gas-phase path distance between the electrodes. However, in cases where the difference in equilibrium pressure is large, sealing of the cell is necessary. This can be done by sealing one electrode with an alumina-based cement or with a fused gasket of a high melting glass, or by using O-rings outside the furnace. Commonly used cells uses Pt/O2 or Pt/air as reference electrodes. At very low partial pressures of oxygen, care must be taken to avoid direct permeation of oxygen through stabilised zirconia from the air (or reference electrode) [195,196]. The effect may be avoided by use of reference electrodes, with activity near that observed at the working electrode. A well-defined buffer system like a metal–metal oxide or a metalmetal fluoride mixture is one solution to the problem. The thermodynamic properties of these buffer systems are well known over extended temperature ranges. Although, well-defined equilibria involving phases with fixed stoichiometry are preferred, nonstoichiometric phases have also been used, e.g. the Fe/Fe1yO buffer [197]. The variation of the composition with temperature and the corresponding changes in thermodynamic properties must be taken into consideration. Another way of avoiding direct permeation of oxygen through the electrolyte is to have a controlled continuous adjustment of activity at the reference electrodes. By containing the sample in the inner of two concentric zirconia tubes, the electrodes on the outer tube may be used to measure and control the oxygen partial pressure between the two tubes, and the reference electrode potential [198,199]. 417

418

Properties of Mixing

Materials stability and compatibility also must be taken into consideration. Often it will be necessary to avoid direct contact between the electrolyte and the electrode, which may cause interfacial reactions. In certain cases it is advantageous to separate the electrodes from the electrolyte by a compatible intermediate material. Liquid silver has been used to separate a Fe-O-SiO2 electrode from the ZrO2 electrolyte [200]. Correspondingly, the choice of the reference and working electrode is important and knowledge of the phase equilibria in the system of interest is essential for a valid interpretation of electrochemical cell measurement [187]. The species present at the working electrode should be co-existing phases in the system of interest. Care must be taken since the co-existing phases may change with temperature. In the Fe-Mo-O system Fe and Mo2Fe3 are in equilibrium with Fe2Mo3O8 below 1189 K, and with Fe2MoO4 above 1189 K, as shown in Figure 8.7 [201]. In general the number of different types of co-existing phase working electrodes needed in a study is determined by the number of phases which are to be thermodynamically characterised. Many materials have been used as electrical connectors [187]. Most common is Pt. However, Pt easily forms very stable intermediate phases with actinides and lanthanides and in these cases W or Mo have been used. Correspondingly, W may react with refractory oxides like Rh2O3 forming WO2 and an intermediate layer of Rh has then been proposed to prevent this [187]. Special Methods Since the electrolyte is an ion conductor, the activity of that species in a closed electrolyte cell can be controlled [202,203]. Ions can be pumped in or out of a closed compartment and the change in composition determined accurately. Hence, the composition of a non-stoichiometric compound can be changed in small steps and both the composition and the activity of the species are simultaneously determined. Detailed information of the compositional dependence of the partial Gibbs energy under isothermal conditions can thus be obtained by this technique, termed coulometric titration. The technique is most commonly used to study phases with properties that are highly dependent on small changes in stoichiometry. Early examples are studies of sulfides, selenides and tellurides [203–205], while more recent

Figure 8.7 Co-existing phases in the Mo-Fe-O system below and above 1190 K.

418

419

Properties of Mixing

ones include careful studies of the partial Gibbs energy of oxygen in UO2 þ x [206] and U1yGdyO2x [207]. In order to reach equilibrium in a reasonable time the material to be studied must have a high diffusion coefficient for the conducting ion of the electrolyte. An additional advantage of the technique is that phase boundaries can be determined accurately and this has been used for simultaneous mapping of phase equilibria and thermodynamic properties both in intermetallic e.g. Pd-Y [208] and oxide systems, e.g. Cu-Ge-O [209]. A method closely related to coulometric titration, the pin-point method [210], has similarly been used to map the phase relations in the Au-Ca system [211]. While electrochemical cells usually yield activities and partial Gibbs energies, the partial molar entropy of oxygen in single-phase or multiphase mixtures can also be determined. The difference between the Seebeck coefficients of two concentric thermocells kept at different temperatures has been used [212]. 8.3.3

Vapour Pressure Methods

A chemical compound may vaporise congruently or non-congruently, so that AB2 ðsÞ ¼ AB2 ðgÞ;

ð8:34Þ

AB2 ðsÞ ¼ ABðsÞ þ BðgÞ:

ð8:35Þ

or

Vapour pressure methods are used to determine the pressure pi of the volatile species i in equilibrium with a solid compound with a well-defined composition. The activity can then be deduced through ai ¼

fi pi & 0: 0 fi pi

ð8:36Þ

Here fi and pi, are the fugacity and the partial pressure of the species i, and fi0 and p0i are the fugacity and activity of the species i in its standard state. The activity is determined directly by measurement of the vapour pressure of an element or a compound at a certain temperature (static or effusion methods) or indirectly, through equilibration of the sample with a well defined gas phase. The techniques are here treated under two main headings: effusion and equilibration methods. Knudsen Effusion Methods The most usual effusion methods are based on equilibration of a substance in a Knudsen cell. A small fraction of the vapour molecules effuse through a small effusion orifice in the lid of the cell (diameter (0.1 to 1) mm) ideally without 419

420

Properties of Mixing

disturbing the equilibrium in the cell. The equilibrium partial vapour pressure of species i is given by the steady-state evaporation rate [213]

dmi 1 pi ¼ dt Af

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pkB T ; Mi

ð8:37Þ

where (dmi =dt) and Mi are the mass rate of effusion and the molar mass of the effusing species, A is the area of Knudsen cell orifice and f is a correction factor, the Clausing factor [213]. The methods are used with good results in the vapour pressure range between (107 and 10) Pa for temperatures up to 2800 K, see [214]. Various methods are used for the analysis of the effused vapour. If the mass of the molecules in the vapour is known, the equilibrium vapour pressure may be determined directly through determination of the mass rate of effusion by measurement of the mass loss [213] or through the use of impact or recoil momentum sensors [215]. However, the most usual and versatile method for analysis of the gas is mass spectrometry [216]. The mass loss technique [213] gives the total vapour pressure only and is suitable mainly for samples that vaporises congruently, or compounds that vaporises incongruently but with one dominating vapour species. The evaporation rate is deduced from the mass loss or through collecting and analysing the vapour effusing from the cell. Some recent techniques are based on the latter method where the vapour is condensed on a collector plate. The activity of Au in Au-alloys has been studied through the use of a twin Knudsen cell with the sample in one cell and the reference, pure Au, in the other cell [217]. The activity is obtained through analyses of the decay rate of 2 radioisotopes of Au by a Ge-detector. Other analyses of the collected vapour are based on neutron activation or microprobe techniques. The momentum sensor techniques [215] are based on the force transferred from a gas to a surface on impact or recoil. Impact momentum sensors [218] are generally not very sensitive partly because molecules simultaneously condense and re-vaporise from the target. Recoil based techniques are, hence, preferred. In one version, the vapour pressure is deduced from the change in mass of a Knudsen cell that is observed on opening/shutting the orifice at the measuring temperature [219]. In the torsion recoil method [220] the Knudsen cell is suspended on a fibre. Two orifices are made in the cell perpendicular to the fibre and in opposite directions. The vapour pressure is deduced from the torsion force that results from the vapour effusing through the two orifices. The recoil of the anti-parallel effusing vapour twists the supporting torsion fibre to a degree determined by the elastic torsion momentum of the fibre. The deflection angle is the measurand. A third variant is based on measurement of the recoil momentum of a linear pendulum [221]. Momentum sensor techniques have recently been used to study alloys, for example, the Mn activity of Fe-Co-Mn alloys [222] and the Pb activity of Al-Pb alloys [223] but also for fluorides and chalcogenides [224,225]. 420

Properties of Mixing

421

In mass spectrometry techniques the effusing vapour is ionised by an ionisation source and the product analysed with a mass spectrometer. The different vapour species are identified and the partial pressures of all species determined. The partial pressure of species i, of a compound or a solution with a specific composition, is at a specific temperature: pi ¼ kIi T=si ;

ð8:38Þ

where k is a pressure calibration factor, si ionisation cross section of species i and Ii the intensity of species i. The pressure calibration factor may be determined ‘in situ’ by use of a twin-type Knudsen cell with sample in one cell and the reference materials in the other or through separate experiments on the reference material [226]. In a study of a binary alloy, the obvious choice of reference material is the pure metal whose activity is measured. Some important precautions must be taken. Since high temperatures may be used, the choice of the Knudsen cell material is important and reactions between sample and the cell must be avoided. Care must also be taken to avoid fragmentation of the gas molecules on ionisation [216]. The mass spectrometric analysis allows detailed thermodynamic studies of compounds where the vapour consists of more than one species e.g. for NaDyI4 (s) where the main gaseous species are NaI (g) and DyI3 (g) [227]. Knudsen effusion mass spectrometry have also been used extensively in studies of mixing properties of different types of solid solutions, recent examples being Zr1-xTe2 [228], Na2O-SiO2 [229] and Cr-Fe-Ni [230]. This is by far the most usual and versatile vapour pressure technique and a large number of examples of uses are given in earlier reviews [216,231]. Equilibration Methods A number of techniques are based on direct measurement of the total vapour pressure in equilibrium with a compound at a given temperature. The most usual methods are based on the use of pressure gauges covering pressures from (107 to 100) kPa [232]. Methods based on thermogravimetric determination of the mass of the vapour [233], and on atomic absorption spectroscopy have also been reported [234]. Total pressure measurements has recently been used to study the vaporisation of lathanum trihalides [235] and the Te activity over Ge-Te alloys [236]. The use of this methodology to inorganic solids has been discussed [237]. The equilibrium vapour pressures may also be determined indirectly, for example through measurement of the exact composition of a non-stoichiometric compound in equilibrium with a gas with a well-defined activity of the volatile species. While Knudsen effusion studies by mass spectrometry depends on complex and expensive instrumentation, some equilibration studies are readily performed in rather simple experimental set-ups. By use of thermogravimetry, the composition of a non-stoichiometric compound can be determined as a function of the vapour pressure of a volatile species such as oxygen in the case of La2xSrxCuO4d [238] 421

422

Properties of Mixing

and the technique is complementary to coulometric titration. While only certain discrete partial pressures of oxygen in practice are feasible by thermogravimetry, oxygen permeability and materials compatibility problems in general are less of a problem. The temperature of decomposition of carbonates of the YBCO high-temperature superconductor to oxides [239]:

2 YBa2 Cu3 O6:3 ðCO2 Þ0:19 ðsÞ þ 2:62 CO2 ðgÞ þ 0:2 O2 ðgÞ ¼ 5 CuOðsÞ þ 3BaCO3 ðsÞ þ Y2 BaCuO5 ðsÞ;

ð8:39Þ

at different well defined partial pressures of CO2(g) also facilitates determination of thermodynamic properties through second or third law treatments of the equilibrium pressure data. The main systematic error is often related to inadequate equilibration and it is important that the equilibrium pressure is obtained both on decomposition (i.e. on heating) and on carbonatisation (i.e. on cooling). It is often advantageous to start out with a partly decomposed sample in order to reduce nucleation problems. A range of different methods measures the solubility of hydrogen in metals and alloys. Manometric methods [240] and gas volumetric methods [241] have been used to determine pressure-composition isotherms at selected temperatures for a range of alloys [242–245]. In the isopiestic method two condensed phases are equilibrated via the gas phase [246,247]. The composition and pressure of the gas phase is determined by use of a reference compound for which the partial pressure of a volatile component is known as a function of temperature and composition. Experiments can be performed isothermally by equilibration of one sample with the reference sample [248]. The sample is taken out and its composition determined analytically. Alternatively, several samples are equilibrated at the same time in a temperature gradient. This method is well suited for studies of non-stoichiometric compounds and alloys. Various binary and ternary systems with Zn, Cd, As, Sb or Te as volatile components have been studied [249–251]. The technique is also applicable to oxides such as the Hf-O system [252] and metals with a low vapour pressure, i.e. for Mn [253] or Mg [254]. In the dew point method, the sample is kept in an evacuated silica glass tube which is placed in a temperature gradient [232]. The sample is contained in the hot end and the temperature of the cold end controlled to the temperature where the vapour of the volatile component just starts to condense. The activity of the volatile species in the compound is given from the dew point temperature. This method has been used to study the Zn activity of ternary alloys [255]. In the transpiration method the vapour is transported by an inert carrier gas into a condenser [232]. If the inert gas is saturated with the vapour, the vapour pressure can be determined from the mass of the sample or from the mass of the condensate. As an example, the Pb activity over Cu-Fe-Pb has been determined at 1773 K [256]. 422

Properties of Mixing 8.3.4

423

Some Words on Measurement Uncertainty

It is evident that the accuracy of an enthalpy determined by direct-reaction calorimetry will depend largely on the completeness of the reaction and on the corrections made in order to take this source of systematic error into consideration. Local saturation and precipitation are similarly possible sources for systematic uncertainties in solution calorimetry. Correspondingly, obvious and less obvious sources of systematic errors may be found for all experimental techniques. Hence, it is difficult to give a certain common uncertainty to a particular measurement technique and the uncertainty is to a large extent determined not only by the technique itself but also by the temperature of the reaction, the type of compound studied and so on. It is, hence, difficult to estimate the uncertainty of an experiment and results obtained by different methods often do not agree within the stated uncertainties/reproducibilities. Calorimetrically determined enthalpies of formation for three inorganic compounds LaNi5, GeSe2 and Si3N4 obtained by leading scientists using combustion calorimetry, solution calorimetry and direct reaction calorimetry are given in Table 8.2. Four of the five determinations for LaNi5 are equal within the stated estimate of the uncertainty. For the two other compounds, larger systematic errors in some of the determinations are inferred. The Fcombustion mean value for GeSe2 is 18:7 kJ ? mol1 , 22 per cent more negative than the value obtained by direction reaction calorimetry. The combined uncertainty of the F-combustion and direct reaction values is 4:8 kJ ? mol1 . For b-Si3N4, the Fcombustion values are 24:2 kJ ? mol1 , 3 per cent more positive than those obtained by solution calorimetry and again larger than the combined estimated uncertainties. Entropies obtained from heat capacity calorimetry are more accurate. The entropies of a- and b-quartz and of CuS, as determined by different calorimetric techniques, Table 8.2 Selected values of the standard enthalpy of formation Df Hm for LaNi5 , GeSe2 and Si3 N4 obtained with different methods at a temperature of 298.15 K. Compound

DfHm/kJ ? mol1

LaNi5

GeSe2

Si3N4

a a b a b

 126.3 + 7.5  159.1 + 8.3  165.6 + 10.2  161.4 + 10.8  157.8 + 18.1  102.2 + 2.6  104.0 + 3.0  84.4 + 1.8  787.8 + 3  828.9 + 3.4  827.8 + 2.5  850.9 + 22.4  852.0 + 8.7

423

Method

Reference

HCl solution HCl solution Al solution Al solution liquid reaction F-combustion F-combustion Direct reaction F-combustion F-combustion F-combustion Solution Solution

Semenko [257] O’Hare [258] Colinet [259] Colinet [260] Kleppa [261] O’Hare [262] O’Hare [263] Kleppa [264] Margrave [265] O’Hare [266] O’Hare [266] Navrotsky [267] Navrotsky [267]

424

Properties of Mixing

Table 8.3 Selected values of the entropy increment DTT21 Sm obtained with various instruments for CuS and a-SiO2. Compound

DTT21 Sm =J ? K1 ? mol1

T1/K

T2/K

Method

Reference

CuS CuS CuS CuS a-SiO2 a-SiO2 ab-SiO2 ab-SiO2 ab-SiO2

67.36 67.15 50.84 50.50 41.34 41.43 74.92 74.79 74.82

0 0 298.15 298.15 0 0 298.15 298.15 298.15

298.15 298.15 780.5 780.5 298.15 298.15 1000 1000 1000

Adiabatic Adiabatic Adiabatic Drop Adiabatic Adiabatic Adiabatic DSC Drop

Westrum [268] Ferrante [269] Westrum [268] Ferrante [269] Westrum [270] Gurevich [271] Grønvold [272] Hemmingway [273] Richet [274]

are compared in Table 8.3. In these particular cases the agreements between the different studies are excellent. Still, in other cases larger scatter must be expected. The quality of DSC determinations of heat capacity and implicitly of entropy does in particular depend on the procedures used during experiments [257–274]. The next question is: how does the results of calorimetry, electrochemical and vapour pressure methods compare? The formation properties of BaZrO3 have been extensively studied. The directly measured Gibbs energy of the reaction BaOðsÞ þ ZrO2 ðsÞ ¼ BaZrO3 ðsÞ;

ð8:40Þ

obtained by electrochemical measurements [275–277] and by Knudsen effusion mass spectrometry [278–280] gives enthalpies of formation of the ternary oxides from the binary oxides, ZrO2 and BaO at 298.15 K range from ( 45.2 [278] to  123.4 [280]) kJ ? mol1 . Calorimetrically determined enthalpies of formation are (  110:2+3:7) kJ ? mol1 [281] and 109:2 [282] kJ ? mol1 . Also the entropy derived from the Gibbs energies varies considerably; from (67.5 [276] to 129.4 [277]) J ? K1 ? mol1 . Although more consistent results are obtained in other cases, deconvolution of directly measured Gibbs energies to enthalpic and entropic contributions is in general difficult.

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428 107. 108. 109. 110. 111. 112. 113. 114. 115. 116.

117. 118. 119. 120. 121. 122. 123.

124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141.

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9

Relative Permittivity and Refractive Index M.R. MOLDOVER Process Measurements Division National Institute of Standards and Technology Gaithersburg, MD, USA K.N. MARSH Department of Chemical and Process Engineering University of Canterbury Christchurch, NZ J. BARTHEL and R. BUCHNER Institut fu¨r Physikalische und Theoretische Chemie Universita¨t Regensburg Regensburg, Germany 9.1

9.2 9.3

Relative Permittivity 9.1.1 Conductivity and Dielectric Losses 9.1.2 Bridge methods and resonance methods 9.1.3 Designs for Capacitors 9.1.4 Measurements 9.1.5 Concluding Remarks Refractive Index Relative Permittivities of Electrolytes 9.3.1 Fundamental Aspects of Dielectric Theory 9.3.2 Coaxial-line Techniques (f  20 GHz) 9.3.3 Waveguide Methods (5  f =GHz  100) 9.3.4 Free-Space Methods (f > 60 GHz) 9.3.5 Data Analysis

Measurement of the Thermodynamic Properties of Single Phases A.R.H. Goodwin, K.N. Marsh, W.A. Wakeham (Editors) # 2003 International Union of Pure and Applied Chemistry. All rights reserved

434 435 437 438 448 451 452 455 457 460 463 466 468

9

RELATIVE PERMITTIVITY AND REFRACTIVE INDEX

Measurements of the relative electric permittivity (dielectric constant or relative permittivity) e(p, T) and refractive index of fluids n(p, T) as a function of the pressure and the temperature can be used to determine a wide range of thermodynamic properties. The physics that is used to relate the measurements of e(p, T) and measurements of n(p, T) to thermodynamic properties is essentially the same; however, the experimental techniques differ greatly. The measurement of e(p, T) of non-electrolytes is described in Section 1. Here, the electrical conductivity of the fluid is a key factor in determining the frequency of the measurements, which in turn determines the geometry of the sample and the instrumentation. Experimental techniques for measuring n(p, T) are given in Section 2 while Section 3 describes the measurement of both e(p, T) and n(p, T) of electrolytes.

9.1

Relative Permittivity M.R. MOLDOVER Process Measurements Division National Institute of Standards and Technology Gaithersburg, MD, USA

Measurements of the relative electric permittivity of fluids as a function of the pressure and the temperature have extraordinarily diverse applications which include: (1) Precise detection of the onset of phase separation [1] in fluid mixtures at high pressures; (2) inferring the heating value of natural gas [2] from on-line measurements in pipelines; (3) precise measurements of the density anomaly of liquid sulfur near sulfur’s ring-to-polymer phase transition at 432 K [3], and (4) studies of evaporation and condensation of 3He near its liquid-vapor critical point at 3.3 K [4]. These applications do not require difficult (and expensive) mechanical or dimensional measurements. Instead, they exploit the ease in measuring e(p, T) accurately, non-invasively, automatically, and often, using commercially available electronic instruments. These applications also exploit the Clausius-Mossotti relation [5] that connects dielectric-constant data to the molar density r and the molar polarizability p(r, T) of a nonpolar fluid: pðr; TÞ:

e1 1 : eþ2 r

ð9:1Þ

The Clausius-Mossotti relation is useful whenever p(r, T) has simple dependencies on density, temperature, and frequency. For insulating fluids, p(r, T) is essentially independent of frequency over many decades of frequency; 434

Relative Permittivity and Refractive Index

435

for poor conductors (such as polar liquids with ionic impurities), there is a narrower range of useful frequencies. For small, non-polar molecules, p(r, T) is nearly independent of temperature and density. As an example, p(r, T) of methane changes by less than 0.5 per cent at densities ranging from that of a dilute gas to that of a compressed liquid temperatures in the range (91 to 300) K, and pressure p < 34 MPa [6]. For polar molecules, p(r, T) has significant density and temperature dependencies that are often approximated at low densities by the Debye equation: p ¼ 4 p NA ½a þ 2 =ð3kB TÞ=3 where a and  are the molecular polarizability and dipole moment respectively and are approximated by constants [7]. At high densities, this approximation fails. For example, in water at 700 K, a [and p(r, 700 K)] changes by a factor of 2 as the density is increased from 50 kg ? m3 to 1000 kg ? m3 [8]. In liquid mixtures of polar and nonpolar components, p(r, T) may deviate by 50 per cent from a linear composition dependence [9]. With the applications enumerated above in mind, we consider how electrical conductivity, measurement frequency, electrical instrumentation, mechanical stability, and geometry led to the four different designs for capacitors used to determine e(p, T). In concluding, several physical phenomena that might interfere with measurements of e(p, T) are described and reference data for verifying the performance of systems designed to measure e(p, T) are recommended.

9.1.1

Conductivity and Dielectric Losses

At frequencies below several hundred megahertz, the dielectric constant e(p, T) can be deduced from the ratio of two electrical impedances: the impedance of a capacitor filled with the fluid under study divided by the impedance of the same capacitor when it is evacuated. In general, this impedance ratio is a complex number er called the relative electrical permittivity. The notation er :e0  ie00 is used with the understanding that e0 and e00 may be frequency-dependent. At any measurement frequency, the real part of the impedance ratio is the dielectric constant e0 . The imaginary part of the impedance ratio, e00 ¼ s=(oe0 ) accounts for electrical dissipation within the dielectric fluid where s is the conductivity. (Here o ¼ 2pf is the angular frequency and e0 &8:854187 ? 1012 F ? m1 is the electric constant or permittivity of free space.) The ratio e0 e0 =s:td is the characteristic time required for charges placed within a dielectric sample to reach the surface of the sample. The ratio e0 =e00 ¼ Qd may be considered the ‘quality factor’ of the dielectric fluid. When Qd 441, or, equivalently when otd 441, the dielectric loss is small and the equivalent circuit of the fluid-filled capacitor is that of a capacitor in parallel with a large resistor. Whenever otd 551, the equivalent circuit is that of a capacitor with a very small resistor in parallel with it; the capacitor is nearly short circuited. Resonance methods are useful for determining e0 only when Qd 441 and this inequality sets an approximate lower bound to usable resonance frequencies. For pure liquid water at 298 K the bound is f 441:3 kHz, where we have used the values s ¼ 5:5 S ? m1 [10] and e0 ¼ 78.44 [11]. Thus, for pure water, resonance methods cannot be used below 105 Hz. If the water contains ionic impurities, higher 435

436

Relative Permittivity and Refractive Index

frequencies will be required because ionic impurities can increase the number of charge carriers and the conductivity by factors of 10 without producing a detectable change in e0 . Many useful models separate e00 into two terms [12]. The first term is a dielectric relaxation e00relax and the second term is the static, or zero-frequency, electrical conductivity s: e00 ¼ e00relax þ s=(oe0 ). This decomposition of e00 is not unique; it depends upon the choice of e00relax . The phenomenological Debye-Drude model and its elaborations are used to describe the frequency-dependence (or dispersion) of the complex dielectric constant of small polar molecules, such as water, over a wide range of conditions [11]. The model introduces a relaxation time t that accounts for the time required for the dipole moment to reorient in response to an applied electric field. In this model, the frequency dependencies of e0 and e00relax are e0 ¼ e? þ

e00relax ¼ 

es  e? and; 1 þ o2 t2

ðes  e? Þot : 1 þ o2 t2

ð9:2Þ

ð9:3Þ

For liquid water near 298 K, e? &4, es & 78, and t & 1011 s [11]. Upon inserting these parameters in Equation (9.2), one concludes that if e(p, T) were measured at 100 MHz, the fractional correction of e0 to obtain its zero-frequency value would be only (ot)2 &4 ? 105 and Qd & e?(ot) & 630. This example shows that resonance measurements near 100 MHz can determine the zero-frequency value of e0 quite accurately. At higher frequencies, Qd decreases and resonance measurements become increasingly difficult. In the vicinity of consolute points [13], the Maxwell-Wagner effect adds another frequency dependence to the otherwise simple relationship between the dielectric constant and the density. Maxwell and Wagner calculated the impedance of model dielectrics composed of inhomogeneous mixtures of two components, each having different values of e0 and s (with e00relax ¼ 0). Remarkably, these models lead to frequency dependent values of e0 and e00 that are qualitatively similar to those given by Equation (9.2) with a relaxation time on the order of e0eB/sB. (Here the subscript B denotes the properties of the component of the mixture with the larger volume fraction [13].) Typically, the Maxwell-Wagner relaxation time is several orders of magnitude longer than the relaxation time associated with the reorientation of dipoles and it too is very sensitive to ionic impurities. Near consolute points, concentration fluctuations in mixtures become large and both e0 and e00 show dispersion that is similar to the models of Maxwell and Wagner. When measuring er in conducting solutions at very low frequencies, the electrical double-layers that form at electrode-solution interfaces must be considered [14–16]. Crudely, double-layers act as capacitors Cdl with values on the order of (10 to 100) F ? cm2 in series with an idealized double-layer-free capacitor Cs containing 436

Relative Permittivity and Refractive Index

437

the solution. At high and moderate frequencies, the impedance of the double-layer can be ignored. However, if it is ignored as the frequency is reduced below that where Cs =ðotd Þ2 *Cdl , the apparent value of Cs will increase leading to an apparent increase in e0 . This effect is one of several effects that are called ‘electrode polarization’ and that have complicated dependencies on the properties of the solution, the electrodes, and the amplitude of the applied field. Reference [8] discusses these effects in the context of measuring e0 for water.

9.1.2

Bridge Methods and Resonance Methods

Commercially manufactured, ratio transformer bridges are often used to measure the complex impedance of capacitors at audio frequencies. The four numbers resulting from two bridge balances (one with an evacuated capacitor and the second with the same capacitor filled with a dielectric fluid) lead directly to e0 and e00 . Ratiotransformer bridges are arranged to measure the capacitance Chl between two conductors, conventionally designated h (¼ high) and l (¼ low) and they are essentially insensitive to the capacitances Chg and Clg where g designates grounded conductors that may completely enclose the h and l conductors. Some commercially manufactured bridges can resolve fractional changes in Chl smaller than 106 when Chl is as small as 1 pF. Ordinarily, coaxial cables with grounded shields are used to connect the high and low electrodes to the bridge. The cables reduce the sensitivity of the null detector used to balance the bridge; however, the cables will not affect the point of the balance at low frequencies where the input impedance of the transformer is low. Typically, these bridges are extraordinarily accurate in the range about (0.1 to 10) kHz. Bridges have been designed with reduced accuracy for use at 100 kHz for capacitance measurements. At lower frequencies, the impedance of practical-sized, fluid-filled capacitors becomes so high that the accuracy is reduced; at higher frequencies the output impedance of the transformer increases and the accuracy of the voltage ratios produced by the transformer is reduced. Resonance methods are used to measure capacitance at frequencies over the range (106 to 109) Hz. In this wide frequency range, a fluid-filled capacitor can be connected to an inductor. In a first approximation the resonance frequency is o2 &(LC)1 where C is the capacitance and L is the inductance. The resonance frequency can be counted using a comparatively inexpensive, commercially manufactured frequency counter. Because the capacitor and the inductor play equal roles in determining the resonance frequency, the inductor must be as stable as the capacitor. If both the capacitor and the inductor are immersed in the fluid under study, o2 &(r L0 e0 C 0 )1 , where r is the relative magnetic permeability of the fluid and the subscripts denote the values of the capacitance and inductance when the density is zero. Most dielectric fluids are diamagnetic and the product re0 is well approximated by e0 . If this approximation were used to determine the polarizability of liquid CO2 at 293.15 K for which (e0  1) & 0.60 and (r  1)&  4:7 ? 106 , the relative error would be only (4:7 ? 106 =0:60)&8 ? 106 . If the same approximation were applied to 437

438

Relative Permittivity and Refractive Index

the paramagnetic liquid O2 at 71 K for which (e0  1) & 0.51 and (r  1)&4:2 ? 103 , the relative error would be approximately (4:2 ? 103 =0:51)&0:008. Resonance methods are rarely used to determine e00 . Because e00 reduces the Q of an oscillator, it can be inferred from measuring either the amplitude of the oscillation or from measuring the frequency response of the resonator. However, these additional measurements may require very expensive equipment, particularly at high frequencies. In order to determine e00 from Q, one must account for contributions to Q from the resistance, which is inevitably associated with the inductance, and contributions to Q from the external circuitry.

9.1.3

Designs for Capacitors

We now consider four examples of capacitors that have been used for measuring thermophysical properties. For each example, key features of the construction, instrumentation, and data analysis in the context of the intended application are described. Parallel Plate Capacitor Figure 9.1 shows an unusually compact and mechanically stable, parallel plate capacitor assembly that was designed by Straty and Younglove [17]. The outside

Figure 9.1 Parallel plate capacitor from reference [17].

438

Relative Permittivity and Refractive Index

439

diameter of the assembly was 2.5 cm; the area A of the smaller (lower) circular capacitor plate was approximately 0.8 cm2; and the fluid-filled space between the plates had a thickness h that was, typically, from (0.1 to 0.2) mm. Using the standard formula for parallel plate capacitors, C0 ¼ e0 A=h, the capacitance in vacuum ranged from (3.5 to 7) pF. Because of its compact design, this capacitor fit into a small highpressure cell and only very small samples of fluid were required to fill it. The compact design was advantageous for measuring e0 of pure cryogenic fluids up to high pressures. In other applications, the small thickness between the plates might be a disadvantage. For example, if a fluid mixture were to separate into two phases in the capacitor assembly, it would be difficult to circulate fluid to re-mix the sample. Also, as the thickness of the gap is reduced, fouling, for example, by the deposit of thin films of vacuum pump oil, becomes relatively more important. In the Straty-Younglove design, the capacitor plates, support rings, base, and the shim washers were all made of copper. The use of a single material for these key components minimized the distortions that might result from differential thermal expansion or from differential contraction under hydrostatic pressure. The support plates, shim washers, and base were fastened to each other with three screws. This use of nearly-three-point contact contributes to the stability of the assembled structure. The Straty-Younglove capacitor was designed for use with a ratio-transformer bridge in the frequency range (1 to 10) kHz. The support rings and surrounding pressure vessel were connected to the guard terminal of the bridge. The support rings were insulated from the capacitor plates by two Kapton1 film layers each of thickness 0.025 mm that had been press-fit into place. Because the Kapton film was located between the capacitor plates and conductors connected to the guard, neither the Kapton’s dimensions nor its dielectric constant affected the measurements. The Kapton filled joint had a gentle taper of 0.6 per cent. By design, the capacitor plates were concentric cylinders with unequal diameters. The capacitor measured the dielectric constant of a coin-shaped ‘active’ volume of fluid. The diameter of the active volume was the diameter of the smaller plate. Thus, small displacements of the axes of the plates did not change either the active volume or the measured capacitance. The thickness of the active volume was the average thickness of the three copper shims between the support plates. Coaxial Cylinder Figures 9.2 and 9.3 show the cross section and an exploded view of a coaxial cylinder capacitor designed by Younglove and Straty for use with a ratio-transformer bridge in the frequency range (1 to 10) kHz [18]. This capacitor was used to measure the dielectric constants of methane, [6] oxygen [19], and fluorine [20] over very wide 1

In order to describe material and experimental procedures adequately, it is occasionally necessary to identify commercial products by manufacturer’s or trade names. In no instance does such identification imply endorsement by the NIST, nor does it imply that the particular product is necessarily the best available for the intended purpose.

439

440

Relative Permittivity and Refractive Index

Figure 9.2 Cross section of coaxial cylindrical capacitor in pressure vessel [18].

ranges of temperature and pressure. The active electrodes were two coaxial copper cylinders. Each of these electrodes was supported at each end by conical surfaces that were held rigidly with respect to each other by a central rod. The central rod, the conical surfaces, and the surrounding pressure vessel were all connected to the guard terminal of the bridge. The coaxial design shares two mechanical features with the parallel plate capacitor described in Section 9.1.3. All the metal components (except for the screws) were made of the same metal (copper) and thin polymer films (0.03 mm thick, polytetrafluoroethylene or polyethylene terephthalate) were used to insulate the active electrodes from the guard electrodes. When the capacitor was assembled, the thread on the central rod, shown in Figure 9.3, was tightened to draw the conical supports firmly against the inner active electrode. Then, two clamping screws, one bridging each end of the longitudinal slot in the outer electrode, were tightened thereby drawing the outer electrode tightly against the conical supports. (The conical support surfaces had an angle of approximately 308 with respect to the cylinder axis). The support surfaces had been cut away on three sides so that both the inner and outer active electrodes were

Figure 9.3 Exploded view of coaxial cylindrical capacitor [18].

440

Relative Permittivity and Refractive Index

441

supported with nearly-three-point contact at each end. Thus, the mechanical design exploited both elastic and nearly-kinematic (three-point) design principles while avoiding extensive machining to very tight tolerances. The active electrodes were used to measure the dielectric constant of the fluid within a cylindrical annulus. The length of the annulus, l & 50 mm, was that of the outer surface of the shorter electrode. The thickness of the annulus was 0.75 mm; it was determined by the difference between the outer diameter of the inner electrode, a & 17.5 mm, and the inner diameter of the outer electrode, b & 19.0 mm. The measured capacitance in vacuum was & 34 pF, as expected from the expression for coaxial cylindrical capacitors, C0 ¼ 2pe0 l= ln (b=a). Because the active electrodes had unequal lengths, the active capacitance was insensitive to small displacements of either electrode parallel to the cylindrical axis. The symmetry of coaxial capacitors insures that if the axes of the active electrodes are displaced a small distance d from co-linearity, the change in the capacitance will be on the order of [d/(b  a)]2. Younglove and Straty showed that the thermal expansion of their coaxial cylindrical capacitor was within 10 per cent of that expected for pure copper. Thus, they expected that the contraction of the capacitor under hydrostatic pressure could be calculated from the isothermal compressibility of copper (See Section 9.1.4) and they were surprised to find evidence that the measured pressure dependence of the capacitance was smaller than that calculated. Recently, Fernandez et al. [21] modified the Younglove-Straty coaxial cylinder design by substituting 1 mm diameter sapphire spheres for the insulating polymer films and by substituting stainless-steel cylinders for copper cylinders. After these substitutions, the coaxial capacitor could be used at higher temperatures, where the polymer films might degrade, and with test fluids, such as low density gases, that might have been contaminated by desorption from the polymer films. Fernandez et al. [21] were led to these substitutions when they recognized that ionic impurities would interfere with their audio-frequency measurements of e0 and e00 for pure liquid water. Re-entrant Coaxial Cylindrical Resonator Figure 9.4 shows the cross section of a re-entrant, radio-frequency resonator. This resonator was designed by Goodwin et al. [22] and used [1] at the frequency f &375 MHz ? (e0 )1=2 to detect phase boundaries near the liquid-vapor critical line in mixtures of carbon dioxide and ethane near 290 K and at pressures in the range (5 to 7) MPa. A resonator of the same design was used to determine the dipole moments of partially fluorinated hydrocarbons from measurements of e0 in the gas phase [23]. Jaeschke et al. has proposed to use a similar resonator to monitor e0 of natural gas mixtures [2] in pipelines. Hamelin et al. [11] extended the results of Goodwin et al. [22] to accurately determine e0 in situations where e441. Hamelin et al. [24] also developed a two-frequency re-entrant resonator for accurately measuring e0 for liquid water at the frequencies 216 MHz ? (e0 )1=2 and 566 MHz ? (e0 )1=2 . Anderson et al. [25] developed a three-frequency re-entrant resonator also for the determination of the permittivity of H2O. A liquid level indicator results if the concentric cylinder is vertically oriented [26–29]. Reentrant resonators, where the capacitor is formed from 441

442

Relative Permittivity and Refractive Index

Figure 9.4 Cross section of re-entrant resonator [22]. Approximate dimensions of the brass resonator in millimeters were: r1 ¼ 4.96, r2 ¼ 23.91, r3 ¼ 25.04, z1 ¼ 17.0, z2 ¼ 20.11, z3 ¼ 19.73, and z4 ¼ 9.55.

two parallel plates form a very sensitive displacement indicator and have been used as an accelerometer, thermometer, and pressure gauge [26,27]. Reentrant cavities with both parallel and concentric cylinder capacitors have been used to measure relative permittivity of gases, liquids, and solids [30–39]. The resonator shown in Figure 9.4 was robust and comparatively simple to construct [22]. The body of the resonator was composed of two metal parts machined out of the same billet. The lower part was a hollow cylinder closed at its bottom. The upper part served as a lid to the cylinder and had a cylindrical neck extended downward into the cavity. The lower portion of the neck expanded into a bulbous, coaxial extension that filled approximately 1/3 of the lower part. These two metal parts were sealed together with a gold O-ring. The only dimensions of the resonator that were held to tight tolerances were those related to the O-ring seal. Because of the coaxial design, the resonance frequency will be only weakly sensitive to a small radial displacement of the axes of the top and bottom parts. However, the thickness of the metal O-ring will affect the inductance and therefore the resonance frequency, as discussed below. The only dielectric parts of the resonator were two, commercially-manufactured, ceramic-to-metal seals. These seals terminated the coaxial cables that coupled the resonator to the external circuitry. The seals had to contain the fluids under study; however, they were located where neither their mechanical nor electrical properties were critical to the operation of the resonator. To facilitate studies of phase equilibria in fluid mixtures at elevated pressures, the resonator was designed to minimize those volumes that could not be mixed well. A pump removed fluid from a small hole in the bottom and returned it to the top. To facilitate the drainage of condensate, tapered surfaces were machined on the lower part and on the bulbous extension. For the same reason, the annular gap was comparatively wide & 1.1 mm. 442

Relative Permittivity and Refractive Index

443

For simplicity, the resonator was built to function as a pressure vessel. However, its re-entrant design led to a comparatively large and difficult-to-calculate pressure dependence of the capacitance. Therefore, the pressure dependence of the capacitance was determined by measuring the resonance frequencies as a function of temperature and pressure applied when the cavity was filled with He. The vacuum capacitance C0 had contributions from several parts of the resonator. Approximately 80 per cent of the capacitance was associated with the annular gap that is indicated in Figure 9.4; approximately 15 per cent of the capacitance was associated with the fringing fields at the top and bottom ends of the gap; the remaining 5 per cent was associated with the volume between the bottom surface of the bulbous extension and the bottom of the resonator. Thus, the resonance frequency depended on the values of e0 in all of these regions. In contrast, the capacitance that is measured with audio-frequency, ratio-transformer bridges is usually very well localized. It is comparatively easy to measure changes of the resonance frequency of a high Q re-entrant resonator with a relative uncertainty of less than +106 . However, the uncertainty of e0 deduced from the frequency changes will almost always be much larger than the uncertainty of the frequency changes. If the uncertainty of e0 is not dominated by the reproducibility of the sample, it is likely to be dominated by imperfections in the model used to interpret the frequencies. Thus, the remainder of this section is devoted to the ideas that enter such models. The references should be consulted for details. The inductance and the capacitance of the re-entrant resonator are integral parts of the assembled structure. The inductance is determined by the toroidal volume surrounding the neck and, as mentioned, the capacitance is determined primarily by the annular gap. The values of the inductance L0 and the capacitance C0 under vacuum can be estimated from the dimensions in Figure 9.4 with the formulas: L0 &

0 z1 lnðr2 =r1 Þ; 2p

C0 &2pe0 z3 =lnðr3 =r2 Þ:

ð9:4Þ

In Equation (9.4), 0 :4p ? 107 N ? A2 is the magnetic constant. Equation (9.4) is suitable for design purposes; more accurate formulas appear in [22]. At radio frequencies, electromagnetic fields decrease exponentially when they penetrate metal surfaces. The exponential decay length is d ¼ (o0 s0 =2)1=2 , where 0 and s0 are the magnetic permeability and electric conductivity of the metal surfaces respectively. Electric currents are confined to this same depth. Thus, d appears in approximate expressions for the resistance R and the quality factor of the resonator Q0 when the resonator is evacuated:   0 doz1 1 1 R& þ ; r1 r2 4p

Q0 :

oL0 0 r1 & 0 lnðr2 =r1 Þ: R 2 d

ð9:5Þ

In order to discuss the operation of the re-entrant resonator, we follow Hamelin et al. [24]. In their first approximation, the resonator is treated as a series lumped 443

444

Relative Permittivity and Refractive Index

LCR circuit immersed in a fluid of complex relative dielectric permittivity er . The impedance of the coaxial capacitor is Z c ¼ 1=(ioer C0 ) and the impedance of the coaxial inductance has the form:2 Z l ¼ ioL0 þ (1 þ i)R. The resonance condition is the vanishing of the series impedance Zc þ Z l ¼ 0 and this occurs at the complex resonance frequency F ¼ fr þ ig, where fr is the resonance frequency and g is the half-width of the resonance. The complex resonance frequency is related to er by:

er ¼

1 þ ð1 þ iÞoRC0 er ð2pFÞ2 L0 C0

:

ð9:6Þ

Because the expressions for C0, L0, and R are only rough approximations, Hamelin et al. [24] replaced these variables in Equation (9.6) with combinations of fr0 and g0 , which they measured accurately (under vacuum) with a commercially manufactured network analyzer. They also assumed that 0 and s0 were independent of frequency to obtain a useful expression relating er to measurements of F:

er ¼

 2 F0 1 þ ð1 þ iÞer ðfr =fr0 Þ3=2 ð2g0 =fr0 Þ : 1 þ ð1 þ iÞð2g0 =fr0 Þ F

ð9:7Þ

Equation (9.7) applies in the limit of weak coupling between the resonator and the external circuitry. If the coupling to the external circuitry is too strong, the Q of the resonator is reduced and Equation (9.7) must be modified to account for this [11]. If e0 =e00 :Qd 441, then e0 can be determined from Equation (9.7) with comparatively simple measurements of fr, the real part of F or from measurements of jFj. In these cases, one may expect errors on the order of (e0  1)(2g0 =fr0 ) ¼ (e0  1)=Q0 . In order to minimize such errors, one makes Q0 as large as possible. Thus, the resonator should be made from good conductors that are non-magnetic. Furthermore, the metal parts (especially the neck), should have a good surface finish on the scale of the exponential decay length d. At 375 MHz, an Inconel1 resonator had Q0 ¼ 240 and a brass resonator had Q0 ¼ 920 [22]. If the interior surfaces of these resonators had been covered with thick (> 3.5 d) gold plating, a value of Q0 & 1600 would be expected. If a re-entrant resonator is made from a magnetic metal, the frequency dependence of the permeability may lead to errors larger than (e0  1)=Q0 . 2 Because R ! do, the imaginary part of Z1 =o has the frequency dependence of d. In this respect, the equivalent circuit of the re-entrant resonator differs from that of a lumpedcomponent LCR resonator. Consequently, in the limit of Q441, the resonance frequency of the re-entrant resonator varies as 1/Q while the resonance frequency of the LCR resonator varies as 1/Q2 .

444

Relative Permittivity and Refractive Index

445

Cross Capacitors The capacitors described above are rugged and versatile. In contrast, cross capacitors tend to be delicate and specialized for specific standards applications. Since the 1960s, national standards laboratories have used cross capacitors and ratio-transformer bridges operating at audio frequencies in electrical metrology. Long, evacuated, cylindrical, cross capacitors serve as very stable impedance standards to help realize the ohm [40]. A gas-filled toroidal cross capacitor serves as a standard for the absolute measurement of the loss angle [41] of conventional capacitors. [For dielectrics, the loss angle is tan1 (e00 =e0 ).] These applications for electrical standards exploit two desirable properties of cross capacitors. First, cross capacitors, in common with the coaxial cylindrical capacitors, are very insensitive to small displacements of their electrodes. Second, cross capacitors are uniquely insensitive to the presence of dielectric films for example, permanent oxide layers, condensed oil films, or adsorbed gas layers, on their electrodes. These desired attributes are offset by two disadvantages. First, the capacitance Cx of cross capacitors is quite small, typically, Cx * 0.5 pF. The accurate measurement of 0.5 pF requires a very-high-quality, and therefore expensive, capacitance bridge. Second, as discussed below, each measurement of Cx requires two independent capacitance measurements; this requires extra time and a set of very well shielded switches. Buckley et al. [42] planned to measure e0 of helium as accurately as possible with the ultimate goal of building a pressure standard based on such measurements [43]. They were led to consider toroidal cross capacitors by their requirements for a capacitor that was stable, compact enough to fit inside a pressure vessel, constructed from materials that would not contaminate helium, have a predictable deformation under hydrostatic pressure, and be insensitive to dielectric films on the electrodes. Their prototype was adapted by Moldover and Buckley [44] to measure e0 of CH4, N2, CO2, and Ar at 323.15 K and pressures up to 7 MPa. The results reported in reference [44] are suitable reference values for testing re-entrant resonators designed to meter natural gas. Figures 9.5 and 9.6 are schematic diagrams of the cross capacitor used by Moldover and Buckley [44]. Their capacitor was composed of four coaxial, electrodes, each with a rectangular cross-section. The two electrodes designated T (for top) and B (for bottom) were washer-shaped, and the two electrodes designated I (for inner) and O (for outer) were tube-shaped. In the notation of Figure 9.5, the weighted cross capacitance Cx (with weight w [45]) is defined as the weighted average of the two capacitances measured between the opposite pairs of electrodes: Cx :wC TB þ ð1  wÞCIO :

ð9:8Þ

When CTB was measured, the inner and outer electrodes were grounded and while CIO was determined, the top and bottom electrodes were grounded. The four electrodes enclosed a toroidal volume with a nearly-square crosssection of height h ¼ 9.5 mm and width w ¼ (rO  rI ) ¼ 10 mm. The theory for such 445

446

Relative Permittivity and Refractive Index

Figure 9.5 Schematic cross section of the toroidal cross capacitor [42]. The washer-shaped top and bottom electrodes form the capacitor CTB & 0.72 pF. The tube-shaped inner and outer electrodes form the capacitor CIO & 0.52 pF. The grounded shield that surrounds these electrodes is shown in Figure 9.6. The dimensions were: s & 0.15 mm; ri & 45 mm; rO & 55 mm; h & 9.5 mm.

Figure 9.6 Cross-section of a cross capacitor within its pressure vessel [44]. The sketch includes a coaxial feed-through A and a part of a coaxial cable within the pressure vessel. The construction materials were: pressure vessel, stainless steel; electrodes and base, superinvar; base, aluminum; insulating spheres, sapphire; gasket seal, copper.

446

Relative Permittivity and Refractive Index

447

capacitors [46] provides an expression for Cx: Cx ¼ 2 ln 2re0 f ðh=r; s=hÞ;

where

2

f ðh=r; s=hÞ ¼ 1  0:040 42ðh=rÞ  0:0017ðs=hÞ2 þ   

:

ð9:9Þ

In the leading order, Cx is proportional to a single length, the average radius r of the tori [r ¼ (rO þ ri )=24450 mm]. The corrections to the leading order are quadratic functions of the curvature of the tori (h/r) and thickness of the gaps (s/h) between the electrodes. Unintentionally, the cross-section of the capacitor deviated from a square. A measure of the deviation is d:f1  h=(rO  ri )g&0:05. The weight w ¼ 0.4476 that was used in Equation (9.8) was chosen by the condition ðqCx =qdÞ ¼ 0 from the theoretical result

Cx ¼ 2 ln 2re0 e 1 þ 3:454d2 þ . . . :

ð9:10Þ

Choosing w by this criterion ensured that small vertical movements of the top or bottom electrodes changed Cx only in the second order, and this was confirmed by measurements [44]. If a dielectric layer, for example, a film of pump oil, were deposited on the bottom electrode, it would tend to increase CTB and decrease CIO, both in proportion to (t/h) where t is the thickness of the layer and h is the distance between opposite pairs of electrodes. The net effect is Cx varies as (t/h)2. In contrast, a dielectric layer would change the capacitance of a coaxial cylindrical capacitor or a re-entrant cavity in proportion to (t/h). For the cross capacitor shown in Figure 9.6, the sensitivity to dielectric layers is even smaller because h & 9.5 mm is much larger than the gap of a typical coaxial capacitor & 1 mm. Shields used vinyl tape as a dielectric film in a cross capacitor; his measurements were consistent with variation as (t/h)2 [41]. As shown in Figure 9.6, the electrodes were insulated from each other and from a grounded base by small sapphire spheres. The electrodes were surrounded by a grounded aluminum shield. The entire electrode structure was enclosed by a heavywalled pressure vessel. If the shield were perfect, the values of CTB, CIO, and Cx would not be affected by the deformation of the pressure vessel as it was filled with gas. To achieve a stable and predictable mechanical design, efforts were made to assemble the electrodes and the shield into a stable configuration with a minimum of force and to protect them from stresses as the temperature, the pressure, and the shape of the pressure vessel changed. The electrodes and the base that supported them were machined from the same ingot of SuperInvar, an alloy chosen for its small coefficient of thermal expansion. After machining, these parts were heat treated. The insulating sapphire spheres were placed in three radial V-grooves that had been electro-discharge machined into the top and bottom electrodes and three mating cavities were electro-discharge machined into the inner and outer electrodes. The assembly was kinematically stable and it was held together by soft springs. Despite 447

448

Relative Permittivity and Refractive Index

these precautions, the ratio CTB/CIO exhibited an unexpected pressure dependence and hysteresis. Typically, CTB/CIO changed from 1.3850 to 1.3857 when the pressure was changed from 0 MPa to 7 MPa. If the molar polarizability p(r, T) of argon had been deduced from either CTB or CIO instead of Cx, it would have been either 0.05% too large or too small. However, the pressure dependence of Cx was measured, as suggested in Section 9.1.4, and found to be consistent with that expected from the decrease of the radii of the SuperInvar1 electrodes under hydrostatic pressure. The reasons for the unexpected pressure dependence and hysteresis of CTB/CIO are under investigation at time of this writing.

9.1.4

Measurements e0 ( p, T) in Gases

To determine the performance of a system designed to measure e0 (p, T) in gases, the entire package composed of a capacitor, bridge, thermostat, and pressure sensor is tested by measuring e0 (p, T) for helium and comparing the results with reference values of e0 (p, T) [42,44]. The reference values of e0 (p,T) for helium were obtained by numerically eliminating the amount-of-substance density r from both the virial expansion for the density

p ¼ rRT 1þBðT Þr þ C ðT Þr2 þ DðT Þr3 þ . . . ;

ð9:11Þ

and the molar polarizability  pðr; T Þ ¼



e1 1 ¼ Ae 1 þ bðT Þr þ cðT Þr2 þ . . . : eþ2 r

ð9:12Þ

In Equations (9.11) and (9.12), the temperature-independent constants are the molar gas constant R ¼ ð8:314 472+0:000 015Þ J ? mol1 ? K1 [47] and the molar polarizability of He Ae ¼ ð0:517 253 9+0:000 001 0Þ cm3 ? mol1 [48,49]. The temperature-dependent parameters at 323.15 K have the values; B ¼ (11.703 + 0.025) cm3 ? mol1 from [50]; C ¼ (102.5 + 2.9) cm6 ? mol2 from [51]; D ¼ (729 + 94) cm9 ? mol3 from [51]; b ¼ (0.06 + 0.01) cm3 ? mol1 [52]; and c ¼ (1.75 + 0.15) cm6 ? mol [53]. For the pressure-dependent performance of capacitors at other temperatures, appropriate values of these parameters may be found in the references cited. The parameters used in Equations (9.11) and (9.12), except for the very small term cr2, result from either quantum mechanical calculations, for Ae, b, and B, or thermophysical measurements that do not involve capacitance, for R, C, and D. Thus, the reference values are indeed independent of systems designed to measure dielectric constants. 448

Relative Permittivity and Refractive Index

449

Recently, Schmidt and Moldover [54] completed cross-capacitor measurements of e(p;T). Their data, together with those of Moldover and Buckley [44] span the range (273 to 323) K, include eight gases (He, Ar, N2, O2, CH4, C2H6, C3H8, CO2), and extend up to pressures of 7 MPa. These data [44, 54] were acquired for use as reference data when calibrating other instruments; thus their reliability was tested in three ways. First the helium data were shown to be consistent with the theoretical values. Second, the argon data were acquired using two, very different, cross capacitors. The data from the two capacitors had small inconsistencies (51076  e) that were within the specifications of the capacitance bridge. Third, the data for Ar, N2, O2, CH4, and C2H6, were shown to agree, within combined uncertainties, with data acquired in other laboratories [6, 19, 55]. e0 (T) in Liquids Recently, e0 (T) was determined for H2O and c-C6H12 with a two-mode re-entrant resonator with frequencies in the ratio 2.6:1 [24]. For both modes and both liquids, the results for e0 (T) are internally consistent to within 104 ? e0 ðTÞ and they are in excellent agreement with the most accurate, albeit less precise, measurements made at much lower frequencies [24,56]. The results for two H2O samples (differing in conductivity by a factor of 3.6) are represented by the polynomial

e0 ðtÞ ¼ 87:9144  0:404 399ðt= CÞ þ 9:587 26 ? 104 ðt= CÞ2  1:328 02 ? 106 ðt= CÞ3 ;

ð9:13Þ

in the temperature range (273 to 418) K. The results for c-C6H12 are represented by e0 (t) ¼ 2:0551  0:001 56 (t= C) in the temperature range (293.15 to 303.15) K. These results may be used to evaluate systems designed to measure e0 (T) in liquids. For the measurement of relative permittivity of dilute solution of polar fluids in non-polar solvents Stokes and Marsh [56] developed a dilution method shown in Figure 9.7. Known volumes of a dilute solution of a polar fluid were injected stepwise from the burette, E into a known volume of liquid contained in the dielectric cell, A and mixing cell, B, with mixing obtained by circulation with the pump C. Measurements with relative precision of 105 were reported on solutions with polar component mole fractions from 3 ? 104 to 0:2. The results were analyzed to derive the composition dependence of apparent dipole moments and Kirkwood ‘g’ factors of the polar solute. This technique overcomes the requirement for the constancy of the value of the relative permittivity of the solvent to relatively 105 for independent fillings of the dielectric cell.

449

450

Relative Permittivity and Refractive Index

Figure 9.7 Dielectric apparatus. A, dielectric cell [56]; B, mixing cell; C, pump; D, mercury pipette; E, piston burette; F, coupling; G, dead volume; H, ball valve; K, tap.

Phase Boundaries Phase equilibria measurements are the primary subject of Experimental Thermodynamics, Volume VII [57]. Measurement of the changes in fluid-filled capacitors (that arises from variations in the fluid relative permittivity) has been used to study phase equilibria. The most extensive studies reported in the literature refer to He3/He4 mixtures [58], but other systems have been studied [59–61], including hydrocarbons. Capacitance measurements have been used [62–66] to determine critical points, boiler water quality [67], and water content of methane (with a mole fraction of water less than 95 ? 106 ) [61]. Recently, a concentric cylinder reentrant cavity was used to determine phase boundaries and co-existing phase densities in the systems (1  x)CO2 þ xC2H6, with x ¼ 0.25, 0.5, and 0.75 [1]. Rogers et al. [68] reported a technique where phase transitions were determined from a shift in microwave resonance. In this experiment a resonant chamber, filled with air maintained at atmospheric pressure, is mounted on a fluid container with a sapphire window separating the two cavities. The microwave resonance in the airfilled-cavity radiates, as an evanescent wave, through the sapphire window into the fluid-filled-cell. Reflection at the sapphire-fluid-interface, determined by the inhomogeneities of the permittivity at the interface, perturbs the resonance frequency of the air filled cavity. Essentially, this is a high frequency (GHz) method of measuring the impedance at an interface and it is particularly useful for conducting fluids. This technique has been used by others [69–71] to determine phase transitions in compositionally complex mixtures, including those involving hydrates.

450

Relative Permittivity and Refractive Index 9.1.5

451

Concluding Remarks

Several physical phenomena may complicate measurements of the impedance of fluid filled capacitors: the deformation of capacitors under applied pressure; the divergence of the coefficient of thermal expansion as the critical point is approached; the evolution of bubbles near the bubble curve (or dew near the dew curve); and the effects of charged particles. Under hydrostatic pressure p, the linear dimensions of all isotropic materials shrink by the factor kT p=3 where kT :  (qV=qp)T =V is the isothermal bulk compressibility and has values on the order of 1011 Pa1 for steels. To account for the shrinkage, one may assume that the linear dimensions of a capacitor, and therefore its capacitance, decrease by the factor kT p=3 and use the working equation for e0 e 0 ð pÞ ¼

C ð pÞ ð1 þ kT p=3Þ: C ð 0Þ

ð9:14Þ

However, capacitors are assembled from several materials, some of which may not be isotropic, and often parts of capacitors are subject to stresses that are more complicated than hydrostatic pressure. As an alternative to relying on Equation (9.14), the procedures described in Section 9.1.4 above are recommended. Near liquid-vapor critical points, the extraordinary sensitivity of the fluid density to small temperature changes has confounded numerous measurements of e0 (p, T). The experimental problem can be visualized by considering a pressure vessel that has been filled with a fluid at its critical density rc and at a temperature well above the critical temperature Tc. If a small temperature gradient is present, the fluid within the vessel will be slightly inhomogeneous. The fluid’s density will be higher than rc in cooler locations; its density will be lower than rc in warmer locations. If a capacitor is located within a pressure vessel, the average density within its active volume will depend upon the details of the apparatus’ geometry and the temperature gradient. In any case, the density inhomogeneities will increase as the critical point is approached, because the thermal expansivity increases in proportion to (T  Tc)1.24. Very near the critical temperature, it is easy to encounter conditions where the density in the capacitor differs from rc by 0.05 rc. Far from critical points, bubbles, presumably derived from dissolved gases, have interfered with measurements of e0 (p, T) in liquids near their saturated vapor pressure [3,24]. It is well known that gas-filled capacitors charged to high voltages, for example Geiger counters, are efficient detectors of charged particles. It is less well known that gas-filled capacitors used at lower voltages will manifest excess electrical ‘noise’ when energetic particles pass through them [72].

451

452

9.2

Relative Permittivity and Refractive Index

Refractive Index K.N. MARSH Department of Chemical and Process Engineering University of Canterbury Christchurch, NZ

In Experimental Thermodynamics Volume II, Smith [5] described selected methods for the measurement of refractive index of gases and liquids as a function of temperature and pressure based on the minimum deviation method. Also given was a brief description of interference methods. For gases, interferometric methods using cells with long path lengths are now the method of choice and recent developments have focussed on the refinement and extension of interferometric techniques. For a gas, the refractive index n at wavelength l0 is related to the amount of substance density rm through an Equation analogous to (9.12) with n2 ¼ e to give the molar refractivity Rm (r; T) Ln rm ¼ Rm ðr; TÞ ¼

n2  1 ¼ AR þ BR rm þ CR r2m þ    ; ðn2 þ 2Þrm

ð9:15Þ

where Ln is the refractivity, and AR, BR, CR, . . . are the first, second, third, . . . refractivity virial coefficients respectively. For simple gases the magnitude of the refractivity virial coefficients are small but they increases with both the polarity and the number of atoms on the molecule. The compressibility of a gas at pressure p is given by: Z¼

p ¼ 1 þ BðT Þrm þ CðT Þr2m þ    ; rm RT

ð9:16Þ

and when combined with Equation (9.15) gives: RTLn =p ¼ AR ðTÞ þ fBR ðTÞ  BðTÞAR gp=RT þ    ;

ð9:17Þ

and limp!0 ðRTLn =pÞ gives AR. The absolute refractive index n(p, T) of a fluid can be obtained interferometrically by counting fringes Kn(p, T) when isothermally reducing the pressure from p to zero and using the relationship nðp; T Þ ¼

Kn ðp; T Þl0 þ 1; l

ð9:18Þ

where l0 is the vacuum wavelength of the radiation and l the path length of the radiation in the cell at temperature T. 452

Relative Permittivity and Refractive Index

453

Buckingham and Graham [73] described a differential method for the determination of BR, CR, . . . which has been extended by Achtermann et al. [74]. In this method the refractive index as a function of pressure is expressed by the refractive index virial expansion 2 ðn  1Þr1 m ¼ An þ Bn rm þ Cn rm þ    ;

ð9:19Þ

where, An, Bn, . . . are the first, second, . . . refractive-index virial coefficients and from Equations (9.15) and (9.19) AR ¼ 2An =3 BR ¼ 2Bn =3  A2n =9:

ð9:20Þ

Thus a knowledge of Bn, Cn, . . . allows the density to be determined from refractive index measurements. The direct determination of Bn involves the measurement of the change in the optical path length as the gas expands from one of two central cells into the evacuated second cell such that the amount of substance density is halved and the path length of the cell doubled. One method [74] involves the coupling of two interferometers shown schematically in Figure 9.8, which consists of eight cells labeled 1 to 7. The absolute measurements of refractive index as a function of temperature and pressure were made in cell 1. Cell 3 was used to measure the pressure by measuring the refractive index of the cell when filled with nitrogen. The relationship between the refractive index and pressure was established via a differential pressure gauge and a dead weight gauge. The differential measurements were made by expansion of the contents of cells 4 and 5 into evacuated cells 6 and 7 connected in series and measuring the fringe count DK. During the expansion great care was taken to maintain the fringe count and a special detector was developed to allow fringe counts to 1/200 of a wavelength. For perfectly matched cells with equal path lengths l, the fringe count is related to the refractivity virials by: l0 DKn ¼ l

     1=2 Bn r2m;1 þ 3=4 Cn r3m;2 þ    ;

ð9:21Þ

where rm;1 and rm;2 are the densities before and after the expansions and l0 is the wavelength of the radiation. The advantage of this expansion method for obtaining the density is that measurements can be made rapidly, only small amounts of gas are required, and closely spaced points can be determined by refilling to different starting pressures. Another major advantage of the method is that the results are relatively independent of the amount of substance absorbed onto the cell walls during the expansion. St Arnaud et al. [75,76], following Buckingham et al. [77], have also employed this expansion technique to determine density and dielectric virial 453

454

Relative Permittivity and Refractive Index

Figure 9.8 Schematic drawing of the two coupled interferometers I and II for isothermal measurements of the refractive index n(p, T) and differential measurements for the determination of BR and CR [74]. Cells 1, 2, and 3 were used for the n(p, T) measurements, and cells 4 and 6, and cells 5 and 7, for the measurements of DKn(p, T). V1 to V14 represent different valves, and 10, two compensating chambers.

coefficients from precise capacitance measurements. Obriot et al. [78] have described a method for density determination by the simultaneous measurement of refractive index and relative permittivity without expansion. For the refractive index of liquids St-Arnaud et al. [79] described the use of a thin wedge combined with two Michelson laser interferometers to measure n compared to that of air, to better than 5 ? 106 . The schematic is shown in Figure 9.9. The wedge, made from optical plates, having an apex angle of 108, contains the liquid. The change in fringe counts as the wedge moved from position L1 to L2 gives the refractive index. An alternative method using two parallel optical plates and two laser interferometers has been described [80].

454

Relative Permittivity and Refractive Index

455

Figure 9.9 Schematic diagram of the wedge cell sliding in one arm of a Michelson laser interferometer [79]. L1 and L2: optical path lengths corresponding to position x1 and x2; LASER 1, He-Ne laser; M1 and M2, cube corner reflector; PM I, photodetector.

9.3

Relative Permittivities of Electrolytes J. BARTHEL and R. BUCHNER Institut fu¨r Physikalische und Theoretische Chemie Universita¨t Regensburg Regensburg, Germany

~ the relative permittivity er is the dielectric For a phase exposed to an electric field E ~ ~ at every intensive variable which links E and the induced dielectric displacement D ~ ~ point of the phase, D ¼ e0 er E , where e0 is the permittivity of free space. The particular properties of electrolyte solutions prohibit the measurement of er by the common static methods used for nonelectrolytes. Frequency dependent permittivity measurements are needed instead from which the interesting phase properties at equilibrium must be derived. Various methods must be used to cover the complete frequency range, which is a prerequisite for reliable data. ~(t), can be The response of a sample to a time-dependent external electric field, E ~ subdivided into the dielectric polarization, P(t), which is common to all material systems and a resistive contribution specific for conducting samples. In electrolyte solutions the latter is due to the migration of the ions and determined by the conductivity k. At thermodynamic equilibrium, that is for t ! ? after a field jump ~, the equilibrium polarization P ~eq at time t ¼ 0 or equivalently for a static field E defines the static relative permittivity, er , as an intensive macroscopic material property by the relation ~eq ¼ e0 ðer  1ÞE ~: P 455

ð9:22Þ

456

Relative Permittivity and Refractive Index

~eq also links er to the dipole moment ~ k and polarizability ak of species i present P with number density ri . Dielectric polarization always comprises a term originating ~int )i acting on from the intramolecular charge distortion induced by the local field (E eq ~ a molecule, the so-called induced polarization Pa . Molecules with ~ i =0 align in the ~dir )i against thermal motion, giving rise to orientational local electric field (E ~eq . The P ~eq is given by polarization P  ~eq ¼ P

  X   2i  ~  ~ ~eq ; ~eq þ P Edir ri ai Eint þ ¼P a  i 3kB T i i

ð9:23Þ

~ and where kB is the Boltzmann constant. The relation between the external field E ~ ~ the average local fields (Eint )i and (Edir )i depends on the theoretical level adopted for ~eq [81,82]. P ~eq may be considerably modified by specific the interpretation of P  intermolecular interactions, like hydrogen bonding, and thus reflects the structure of the liquid. ~eq and hence er are obtained from capacitance For non-conducting samples P measurements at audio frequencies f, as described in Section 9.1. The relative permittivity is independent of f and at low frequencies assumed to be the static permittivity. In an electrolyte solution, there is a significant ohmic current when a voltage is applied so that er is frequency dependent. Thus, the static permittivity is obtained from an appropriate extrapolation of either the time or frequency ~(t). Figure 9.10 shows the dependence of the sample’s response to a variable field E frequency ranges over which various microscopic processes relevant to electrolyte solutions absorb electromagnetic energy [88]. For electrolyte solutions of common solvents around room temperature the appropriate time scale is in the order of (104 to 10) ns, thus frequencies between MHz and THz are required. In this range the ratio of the characteristic dimension, l, of the measurement cell to the wavelength l of the applied electromagnetic radiation changes considerably. At low frequencies, l=l551, broad-band coaxial transmission lines can be applied, and these are described in Section 9.3.2. Broad-band experiments are again possible with free-

Figure 9.10 Frequency scale of microscopic processes contributing to the generalized permittivity b Z(f ) of electrolyte solutions [83].

456

Relative Permittivity and Refractive Index

457

space methods from the far-infrared region upwards where l=l441, and these are described in Section 9.3.4. For the intermediate microwave range, where l=l&1 narrow-band waveguide equipment is necessary, and these are described in Section 9.3.3.

9.3.1

Fundamental Aspects of Dielectric Theory

The interaction of electric and magnetic fields of arbitrary time dependence with an isotropic material systems is described by Maxwell’s equations, which relate the electric and magnetic fields. For the case of a perturbation in the dielectric medium ~(t) ¼ X ~0 cos (ot), X ~¼ E ~; H ~, of frequency f ¼ o=(2p) where E ~ by a harmonic field X ~ are the electric and magnetic field strength, the system response is shifted in and H ~0 ¼ Y ~0 6 phase by an angle d(f ). This is conveniently expressed by components Y 00 ~ ¼Y ~0 sin (ot) out of phase with X ~(t) leading to the cos (ot) in phase and Y definition of complex permittivity ^e( f ) ¼ e0  ie00 , complex permeability ^( f ) ¼ k0 ( f )  ik00 ( f ) which ^( f ) ¼ 0 ( f )  i00 ( f ), and complex conductivity k depend only on frequency. ~(t, z) of a plane wave It can be shown [81,84] that the electric field strength E travelling a distance z in the medium is described by ~¼ E ~0 exp½iot ? exp½  ^gz; E

ð9:24Þ

with complex propagation coefficient ^g given by



1=2 ^g ¼ ^0 io^ k þ ^ee0 o2 ;

ð9:25Þ

where ^e is the complex relative permittivity, ^ the complex relative permeability, and ^ the complex conductivity in the limit of linear response. In Equation (9.25) e0 and k 0 are the permittivity and permeability of free space. The abbreviation a^ ¼ exp [^gz] is called the complex propagation factor. For nonmagnetic samples (^ ¼ 1), such as electrolyte solutions and their solvents, and ^g simplifies to

^g ¼

io pffiffiffiffiffiffiffiffiffiffi ^ Zð f Þ ; c0

ð9:26Þ

pffiffiffiffiffiffiffiffiffi where c0 ¼ 1= 0 e0 is the speed of light in vacuum and ^Z( f ) is the generalized 457

458

Relative Permittivity and Refractive Index

permittivity given by k00 ð f Þ and; e0 o k0 ð f Þ : Z00 ð f Þ ¼ e00 þ e0 o Z0 ð f Þ ¼ e 0 

ð9:27Þ ð9:28Þ

Here the notation of the complex refractive index is introduced, pffiffiffiffiffiffiffiffiffiffi ^ Zð f Þ ¼ nð f Þ  ikð f Þ;

ð9:29Þ

nð f Þ ¼

 h 1=2 i 1 2 2 1=2 ðZ0 Þ þðZ00 Þ þ Z0 ; 2

ð9:30Þ

kð f Þ ¼

 h 1=2 i 1 2 2 1=2 ðZ0 Þ þðZ00 Þ  Z0 : 2

ð9:31Þ

n^ð f Þ ¼ with refractive index n given by

and absorption index k

Equation (9.24) and the above expressions yield ~¼ E ~0 exp½ iðotbzÞ exp½aa z; E

ð9:32Þ

where the phase constant b ¼ on=c0 ¼ 2p=lM , lM the wavelength in the medium, and the attenuation coefficient a a ¼ ok=c0 . Expressions similar to Equations (9.24), (9.25) and (9.32) are obtained for both coaxial transmission lines and waveguides but the additional boundary conditions create infinite series of eigenfunctions (modes) for the propagating electric and magnetic fields. Applications generally are restricted to the fundamental modes [84–86]. The interactions of the electromagnetic wave with the medium leads to a phase shift determined by n, and to an exponential amplitude attenuation when k=0. In infrared, ultraviolet, and visible spectroscopy it is more common to use the power absorption coefficient a ¼ 4p~ vk at wavenumber ~n ¼ f =c0 instead of k( f ). Equations (9.26), (9.27) and (9.28) indicate that for electrolyte solutions, which simultaneously ~=qt, where D ~ is the dielectric displacement, and an exhibit a displacement current qD ~ ohmic current density j , only the total permittivity ^Z( f ) is experimentally accessible and is necessary for a theoretical description of the sample interaction with electromagnetic radiation. 458

Relative Permittivity and Refractive Index

459

^( f ) can be determined from the Equation (9.32) implies that n^( f ) and Z ~ as a function of path-length z with a measurement of the phase and amplitude of E variable-pathlength transmission cell. The advantage of this ‘method of travelling waves’, implemented in some precision waveguide instruments, is that only the phase ~(z  z0 ) relative to E ~(z0 ) at some arbitrary pathand amplitude changes of the field E length z0 must be determined. However, experimentally the variation of z for transmission measurements may be inconvenient. In this case, information on the behavior of the electric field at the boundary between the sample and the confining ‘windows’ is required. For normal incidence of a plane wave on a plane-parallel slab, of thickness l0 , Fresnel’s equation defines the complex reflection coefficient of the sample: r^ij ¼

n^i  n^j ; n^i þ n^j

ð9:33Þ

and the complex transmission coefficient t^ij ¼

2^ ni ; n^i þ n^j

ð9:34Þ

of the wave at the interface ij [87]. Figure 9.11 shows that multiple reflections may occur within the sample so that the experimentally accessible total reflection, R^, and transmission functions, T^, are given by the infinite series R^12 ¼ r^12 þ t^12 a^22 r^23 t^21 þ t^12 a^42 ^ r21 ^r223 t^21 þ    ;

ð9:35Þ

Figure 9.11 The rays reflected from and transmitted through a lamellar specimen of complex refractive index n^2 and thickness l0 between media of complex refractive indices n^1 and n^3

459

460

Relative Permittivity and Refractive Index

and T^123 ¼ t^12 a^2 t^23 þ t^12 a^32 r^21 ^ r23 t^23 þ t^12 a^52 r^221 r^223 t^23 þ   

ð9:36Þ

respectively. From the general Equations (9.35) and (9.36), n^( f ) and ^Z(f ) can either be determined numerically or by assuming an appropriate model, for the sample-filled cell. Spurious signals produced by both electrical and optical components between the sample filled cell and the signal source, and detector, must to be taken into account [88]. However, by appropriate experimental arrangements, it is often possible to neglect higher order terms in Equations (9.35) and (9.36), and treat spurious signals by appropriate calibration procedures [89–91].

9.3.2

Coaxial-line Techniques ( f  20 GHz)

For a coaxial line of outer diameter D and inner conductor diameter d the characteristic impedance is given by sffiffiffi ^ ZL ¼ Z0 ; ^ Z

1 Z0 ¼ 2p

rffiffiffiffiffi 0 D ln ; e0 d

ð9:37Þ

from an equivalent circuit analysis. The fundamental mode of wave propagation has an infinite cut-off wavelength, lc ¼ ?. For a coaxial line filled with a dielectric for whichpffiffiffi^ ¼ 1, ^ Z ¼ ^e, and e0 44e00 , higher order modes appear at 0 fc ¼ 0:6=[p e (D þ d)] GHz ? m, so that in principle broad-band experiments at frequencies from 0 to fc are possible. However, stringent machining tolerances are required to construct the cell and current machining limit applications in dielectrometry to f  20 GHz. Transmission experiments are advantageous for certain applications, especially with samples of low polarity [91–93]. However, reflection techniques, similar to those shown in Figure 9.12, have greater versatility and precision similar to transmission experiments. The ratio of the reflected, r, and applied, v, voltages determines the amplitude and phase of the complex voltage reflection ^ (io) ¼ r(ioÞ=v(io) at the sampling port of the reflectometer. The coefficient, G problem is then to relate the dielectric properties of the sample to the reflection ^ M (io) at the measurement plane, z ¼ z0 ¼ 0, shown in Figure 9.13, coefficient G and to eliminate spurious signals due to line imperfections between z0 and the sampling port. ^ can be obtained by time-domain reflectometry The reflection coefficient G (TDR). In this case a fast rising voltage pulse, v(t), is applied to the sample and the reflected signal, r(t), is recorded as a function of time with an oscilloscope. FourierLaplace transformation [89,94–96] is used to translate v(t) and r(t) to the frequency 460

Relative Permittivity and Refractive Index

461

Figure 9.12 Schematic diagram of a coaxial-line time-domain reflectometer: SO digital sampling scope (Tektronix 11802, 20 GHz bandwidth), SH1, SH2 SD-24 sampling heads, Z matched pairs of cut-off cells (see Figure 23(a), T precision thermostat, R personal computer with access to a work station for data analysis [94]

domain. In principle, such instruments cover the frequency range between 100 kHz to (10 to 20) GHz with a single experiment. However, to optimize signal-to-noise ratio, the apparatus is frequency range limited. A major problem for the accuracy of the TDR instruments is the strong decrease of the spectral density of the signal above a few GHz. This is not a problem with vector network analyzers (VNA), which have signal synthesizers and can determine the reflection coefficient directly in the frequency domain. VNA manufacturers provide complete dielectric measurement systems

Figure 9.13 (a) Discontinuous inner-conductor (cut-off) liquid cell, with semi-infinite circular waveguide section and coaxial section of mechanical length l0 and electrical length l. (b) Open circuit coaxial line dielectric sensor.

461

462

Relative Permittivity and Refractive Index

including the necessary software for instrument control and data processing, for investigation in the frequency range (0.05 to 20) GHz, which is of special interest to electrolyte solution studies. However, to reach a relative uncertainty of + 0.02 for both Z0 ( f ) and Z00 ( f ), which is desirable for electrolyte studies, judicious calibration procedures are required [97]. Among other possible apparatus designs [88], the open-ended discontinuous inner conductor (cut-off) cell, shown in Figure 9.13(a), and the open-ended coaxial line, shown in Figure 9.13(b), are now the most widely used. For the open-ended line it is sufficient to immerse the probe into the liquid or to press it onto a solid surface. Such a method has been used for non-destructive material analysis [98,99] and remote sensing [100,101]. Cut-off cells require a filling procedure and are not so easily used with highly viscous liquids or solid samples. However, by appropriate choice of the mechanical length l0 of the centre conductor and of the diameter ratio D/d the desired frequency and permittivity range can be optimized [94,102–104]. For open-ended coaxial lines l0 ¼ 0 and only D/d can be modified. Measurements with the open-ended coaxial line can be classified as a singlereflection experiment when only the first term of Equation (9.35) has to be considered. For cut-off cells the major contribution to the signal arises from the reflection at the end of the inner conductor, z ¼ l0 , but the reflection at z0 is also significant. Additionally, multiple reflections between z0 and z ¼ l0 may have to be included, which interfere with the propagation term a^2 ¼ exp [2^gl]. The effective length of the cell, l, slightly exceeds the mechanical length due to the fringing field at z ¼ l0 . This is due to the change in geometry and eventually in the dielectric ~=qz in addition to the properties at this point which induces field components qE ~ fundamental mode where qE =qz ¼ 0. In a first approximation the load impedance of the cell ZM ¼ ZL

^M 1þG ; ^M 1þG

ð9:38Þ

can be expressed by an equivalent lumped circuit representation of the line discontinuity at z0 ; ZL is the characteristic impedance of the feeding line. With this approach closed form expressions are obtained for ZM which can be either solved numerically or after further simplifications allow direct calculation of ^Z [88]. However, this procedure can lead to significant errors, which are a function of permittivity and thus difficult to assess. More appropriate is a full modal analysis with point-matching methods, which take into account the evanscent waves at z  l0 . For cut-off cells such an analysis is given by Go¨ttman et al. [104]; Hilland and Friisø [101] compare the different approaches for the open-ended line. The point-matching results can be mapped onto a bilinear calibration model, which also incorporates mismatches arising from an imperfect feeding line, that are unknown at ZL. This leads to the expression

462

Relative Permittivity and Refractive Index ^ Zs ð f Þ ¼

A^ð f Þ^ rð f Þ þ ^ Zr ; 1  B^ð f Þ^ rð f Þ

^ ¼ GM;s =GM;r ; r

463 ð9:39Þ

for the relation between the total permittivity of the sample, Z^s ( f ), and the corresponding voltage reflection coefficients of the sample, GM; s , and the reference fluid, GM; r . Preferably, the reference with known ^Zr has dielectric properties similar to the investigated sample. The complex calibration functions A^( f ) and B^( f ) must be determined with two additional calibration standards [89,90,101]. Generally, vacuum ð^ Z ¼ 1Þ is chosen as one of the standards. For an open-ended line, a short circuit between inner and outer conductor, through a metallic shorting block pressed against the device, is a convenient third standard (ideally with ^Z ! ?). For samples with k > 2 O1 ? m1 this calibration step may be critical [97]. The accuracy of coaxial–line reflection measurements depend on the calibration procedure and the quality of the calibration standards. One of the reference liquids should have dielectric parameters, ^ Zr , which are similar to the sample properties, ^Zs . Preferably the static permittivities of sample and reference should be in the range j(es  er )=er j  0:15 and the relaxation times of the dominating dispersion steps within 0:5  ts =tr  5. It is very important that the experimental arrangement is not perturbed either during or between calibration and experiment because mechanical stress acting on the coaxial lines, and especially on the connectors, may lead to systematic errors which exceed the limits of precision of the experiments. With good instrumentation, suitably chosen cell parameters and judicious calibration an accuracy of 2 per cent in the static permittivity of the sample can be achieved both for Z0 ( f ) and Z00 ( f ). Currently, the number of reference fluids with a complex permittivity spectrum characterized to the required accuracy is rather limited [90,97]. The compilations [106,107] of critically evaluated ^e( f ) data may be used as a guideline for the selection of suitable reference fluids for a given sample.

9.3.3

Waveguide Methods ð5  f =GHz  100Þ

For polar lossy liquids at f > 20 GHz the dimensions required of coaxial cells are too small to be manufactured with sufficient precision to obtain accurate permittivity values. On the other hand, the vacuum wavelengths, l ¼ c0 =f ; at f  80 GHz are too large to use free-space methods, where beam widths of at least 10l are required to minimize the effect of diffraction. Here the mechanical stability of the optical components, especially lenses which must be made from Teflon2 or similar materials, is problematic. Consequently, waveguides are used to obtain permittivity in this frequency range and cover the gap between coaxial cells and free space. Typically, waveguides are metal tubes with cross sections that are either rectangular (with dimensions of ab and often a ¼ 2b) or circular (radius R) for which the complex 463

464

Relative Permittivity and Refractive Index

propagation coefficient takes the form:

^g ¼ 2

b2c

 2 o ^Zð f Þ:  c0

ð9:40Þ

In Equation (9.40) the cut-off constant bc defines the minimum frequency, which can be transmitted by the waveguide; for a rectangular geometry bc ¼ pc0 =a, while for circular waveguides bc ¼ j11 =R where j11 &1:84 is the first root of the first derivative of the Bessel function of first order and first kind. The upper frequency limit is defined by the onset of the higher-order modes of wave propagation. Generally, the range 1:25  f =fc  1:9, where fc ¼ bc c0 =ð2pÞ, is used for waveguides [86]. Several instruments for high precision measurements of the reflection coefficient have been developed [108–110]. However, for lossy liquids like electrolyte solutions best results are obtained with a Mach-Zender type interferometer, shown in Figures 9.14 and 9.15, which typically provide results with precision of +(1 to 2) per cent in Z0 and of + (2 to 4) per cent in Z00 [105,111]. The signal obtained in the apparatus, shown in Figure 9.14, results from interference of the beam passing through the attenuators denoted 1(b), 1(c) and 9. The attenuators and the phase-shifter 4 allow the interferometer to be balanced for complete destructive interference at some arbitrary probe position z0 . Subsequently, the interference pattern around this point is recorded yielding the real and imaginary part of the complex propagation constant with an appropriate fitting procedure from the magnitude S of the interferometer

Figure 9.14 Waveguide apparatus for the determination of ^eðvÞ in the E-band (60 to 90 GHz) range with transmission measurements based on the ‘method of travelling waves’. 1–9: waveguide interferometer with cell C and movable probe P; PLO, PLO-D, PLO-P: microwave signal source and control unit; 8, MMC, S, RE: signal detection unit; HH, MT, SMD, SM, PM, SP: probe position control unit; PD: interface enabling the control of four interferometers (E-, A-, Ku-, X-band) in the frequency range 8.5 to 90 GHz; MC: microcomputer. For details see reference [105]

464

Relative Permittivity and Refractive Index

465

Figure 9.15 A schematic representation of a waveguide variable-path-length cell (left-hand side) and its two-port equivalent circuit (right-hand side): 1, circular cylindrical waveguide with 1(a), dielectric window; 1(b), sample volume and 1(c), surface of sample liquid; 2, circular cylindrical waveguide probe shiftable along the direction of the cell axis, filled with 2(a), solid dielectric and provided with 2(b), contactless reflector ring with 2(c), thin holes and 2(d), discshaped insulator; 2(e), tapered waveguide section; 3, flange; and 4, feeding waveguide [111].

signal as a function of path-length s ¼ zout  zin from        T^  exp ð ^ g s Þ  ? SðsÞ ¼ T^ref  ?  þ expðiDfÞ:  T^ref  1  R^ expð2^gsÞ

ð9:41Þ

In Equation (9.41), T^ ¼ T^in T^out ; R^ ¼ R^in R^out ; Df the phase difference between the reference beam and the sample beam at s ¼ 0, and T^ref the amplitude of the reference beam. According to the two-port equivalent circuit representation of Figure 9.15 the transmission function T^in characterizes the signal losses in the line up to the cell. T^out 465

466

Relative Permittivity and Refractive Index

gives the transmission through the sample after the cell. They are essentially determined by the properties of the interfaces between the cell and feed line at zin, and the cell and probe at zout. R^in and R^out are the reflection functions at zin and zout. By choosing a sufficient minimum path-length, the contribution of multiple reflections between zin and zout can often be suppressed for lossy liquids, which considerably simplifies the evaluation of Equation (9.41).

9.3.4

Free-Space Methods ð f > 60 GHzÞ

In the far-infrared (FIR) region of the spectrum at frequencies from 90 GHz (corresponding to a wave number of 3 cm1) to 3000 GHz (corresponding to a wave number of 100 cm1) the lack of powerful signal sources and sensitive detectors has made it difficult to investigate ^ Z( f ) [88]. However, recent developments in the field of 1012 s pulsed optoelectronic FIR antennas [112–114] may overcome this situation in the future. Figure 9.16 shows the THz time domain reflectometer used by Rønne et al. for the infrared investigation of liquid water at frequencies in the range (100 to 2000) GHz and temperature from (271 to 366) K [115]. The THz radiation is generated by a pulse from a Ti sapphire laser, which is split by a beam splitter, with one half of the pulse focused onto the emitting dipole antenna, 3. In this biased emitter the pulse induces a time-dependent photocurrent with less than 1 ps rise time which acts as the source for the THz pulse. The emitted radiation is focused into the sample cell, 1, with silicon window, 2, with a paraboloidal mirror, 7, and the beam

Figure 9.16 Terahertz time domain reflectometer with 1, sample cell; 2, silicon window; 3, THz-emitter; 4, THz-detector; 5, delay line; 6, THz-beam splitter; 7, paraboloidal mirror [115].

466

Relative Permittivity and Refractive Index

467

splitter, 6. The beam is reflected at the air/Si and the air/sample interfaces, then collimated with a second mirror onto the THz-detector, 4, creating the signatures 1 and 2 shown in Figure 9.17(a) for the photocurrent induced at the THz-detector. The latter is biased by the radiation through the sample beam and gated by the second half of the laser pulse, which is scanned in time with a delay line. Fourier transformation of the signatures 1 and 2 of Figure 9.17(a) yields the amplitude, shown in Figure 9.17(b), and phase information necessary to determine ^Z( f ) of the sample. Reflection measurements are suited to the determination of permittivity for solvents with a large FIR absorption, for example, water, alcohols or amides. For less absorbing samples transmission experiments with a variable-pathlength cell are more convenient [116]. In the infrared, ultraviolet, and visible range of the electromagnetic spectrum the complex refractive index of electrolyte solutions can be determined with the methods used for non-conducting liquids, described in Section 9.1.

Figure 9.17 (a) The induced current I as a function of time difference Dt obtained from the THz time domain reflectometer of the type shown in Figure 9.16 while the sample cell was filled with distilled water. Signature 1 is the reflection at the air/silicon interface, 2 is the signal of the silicon/water interface. (b) The spectral amplitude, in arbitrary units, of the reflection from the air-silicon interface used as the reference pulse [115].

467

468

Relative Permittivity and Refractive Index

Figure 9.18 Dielectric permittivity, e0 ð f Þ, and loss spectrum, e00 ð f Þ, of 0.0345 mol dm3 La[Fe(CN)6] in water at 25 8C. Experimental data (coaxial line TDR: r, waveguide: .) are fitted to a superposition of two Debye relaxation processes, solid line, attributed to the ionpair (IP) and to the solvent (H2O), broken lines. Also indicated is the total loss, Z00 ð f Þ, of the solution [117].

9.3.5

Data Analysis

^( f ) and ^e( f ), it is assumed that k ^( f ) ¼ lim k0 ( f ) ¼ k which To separate ^ Z( f ) into k v!0 gives Z0 ð f Þ ¼ e0 ;

and k : Z00 ð f Þ ¼ e00 þ e0 o

ð9:42Þ ð9:43Þ

A typical electrolyte spectrum is shown in Figure 9.18 [117]. The static permittivity of electrically conducting fluids can only be determined from a fit of ^e( f ) to a suitable relaxation model for the orientational polarization based on Equation (9.44). The choice of model is far from trivial and partly dictated by the experimental accuracy and the frequency range covered by the data [83]. The major criterion in favor of a particular relaxation model over another is the variance of the fit. However, model selection should be guided by the possible relaxation processes, shown in Figure 9.10, because only in this way can a molecular interpretation of the fitting parameters be obtained and a comparison with results from other methods becomes feasible. For electrolyte solutions at temperature far from the glass-transition the superposition of n individual relaxation processes j of amplitude Sj and relaxation function F~( f ) is given by

^eðf Þ ¼

n X

Sj F~j ðf Þ þ e? ;

j¼1

468

ð9:44Þ

Relative Permittivity and Refractive Index

469

where er ¼

n X

S j þ e? :

ð9:45Þ

j¼1

Equation (9.45) defines the relative static permittivity of the sample and er  e? is ~eq . The ‘infinite frequency’ permittivity e? expresses the proportional to P  ~eq . FIR data are required to independently contribution of the induced polarization P a determine e? and, in the absence of such data, e? is treated as an adjustable parameter which may incorporate contributions from fast relaxation processes. The relaxation functions F~j ( f ) in Equation (9.44), of the individual dispersion steps, may be represented by modifications of the Havriliak-Negami equation: h

1aj ibj F~j ð f Þ ¼ 1 þ i2pf tj ;

ð9:46Þ

with relaxation time tj and relaxation time distribution parameters, 0  aj  1 and 0  bj  1. Special cases of Equation (9.46) are the asymmetric Cole-Davidson relaxation time distribution, aj ¼ 0, and the Cole-Cole equation, bj ¼ 1. A Debye relaxation process is determined by a single relaxation time, where aj ¼ 0 and bj ¼ 1. For Debye and Cole-Cole equations with their symmetrical loss curves, relaxation time tj and frequency of maximum loss fj;m are related by tj ¼ (2pfj;m )1 . The amplitudes, Sj, so determined, can be used to calculate ion-pair concentrations or solvation numbers. The relaxation times, tj , and the distribution parameters, aj and bj , yield the corresponding information on the dynamics of the species involved. For further information concerning the interpretation of dielectric relaxation spectra of electrolyte solutions, the reader is referred to reference [118,119], which also review recent experimental investigations. For a comprehensive survey of aqueous electrolytes ^e( f ) data up to 1994, the interested reader should consult reference [106], while for nonaqueous electrolytes ^e( f ), literature data are reviewed in reference [107] up to 1995. For electrolyte solutions e depends on concentration, and thus has an effect on their thermodynamic and transport properties [120].

References 1. Goodwin, A.R.H. and Moldover, M.R., J. Chem. Thermodyn. 29, 1481, 1997. 2. Jaeschke, M., Schley, P. and Janssen van Rosmalen, R., Int. J. Thermophys. 23, 1013, 2002. 3. Greer, S.C., J. Chem. Phys. 84, 6984, 1986. 4. Zhong, F. and Meyer, H., Phys. Rev. E 53, 5935, 1996; and A.B. Kogan and H. Meyer, J. Low Temp. Phys. 112, 419, 1998.

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Relative Permittivity and Refractive Index

5. Smith, B.L., in Experimental Thermodynamics, Vol II, Experimental Thermodynamics of Non-Reacting Fluids, B. Le Neindre and B. Vodar eds., For IUPAC, Butterworths, London, pp. 579–606, 1975. 6. Straty, G.C. and Goodwin, R.D., Cryogenics 13, 712, 1973. 7. Debye, P., Phys. Z 13, 97, 1912. 8. Fernandez, D.P., Goodwin, A.R.H., Lemmon, E.W., Levelt Sengers, J.M.H. and Williams, R.C. J. Phys. Chem. Ref. Data 26, 1125, 1997. 9. Campbell, A.N., and Anand, S.C., Canadian J. Chem. 50, 1109, 1972. 10. Marshall, D.B., J. Chem. Eng. Data 32, 221, 1989. 11. Hamelin, J., Mehl, J.B. and Moldover, M.R., Rev. Sci. Instrum. 69, 255, 1998. 12. Hasted, J.B., Aqueous Dielectrics, Chapman and Hall, London, 1973. 13. Thoen, J., Kindt, R., van Dael, W., Merabet, M. and Bose, T.K., Physica A 156, 92, 1989. 14. Robinson, R.A. and Stokes, R.H., Electrolyte Solutions, 2nd edition, Butterworths, London, p. 87, 1959. 15. Braunstein, J. and Robbins, G.D., J. Chem. Education 48, 52, 1971. 16. Buck, R.P., J. Electroanal. Chem. 23, 219, 1969. 17. Straty, G.C. and Younglove, B.A., Rev. Sci. Instrum. 50, 1309, 1979. 18. Younglove, B.A. and Straty, G.C., Rev. Sci. Instrum. 41, 1087, 1970. 19. Younglove, B.A., J. Res. Nat. Bur. Stand. 76A, 37, 1972. 20. Straty, G.C. and Younglove, B.A., J. Chem. Phys. 57, 2255, 1972. 21. Fernandez, D.P., Goodwin, A.R.H. and Levelt Sengers, J.M.H., Int. J. Thermophys. 16, 929, 1995. 22. Goodwin, A.R.H., Mehl, J.B. and Moldover, M.R., Rev. Sci. Instrum. 67, 4294, 1996. 23. Goodwin, A.R.H. and Mehl, J.B., Int. J. Thermophys. 18, 795, 1997. 24. Hamelin, J., Mehl, J.B. and Moldover, M.R., Int. J. Thermophys. 19, 1359, 1998. 25. Anderson, G.S., Miller, R.C. and Goodwin, A.R.H., J. Chem. Eng. Data 45, 549, 2000. 26. Van Degrift, C.T., Rev. Sci. Instrum. 46, 599, 1975. 27. Van Degrift, C.T., Rev. Sci. Instrum. 45, 1171, 1974. 28. Van Degrift, C.T. and Love, D.P., Rev. Sci. Instrum. 52, 712, 1981. 29. Van Degrift, C.T., Proc. 3lst Freq. Control Symp. IEEE, New York, p. 375, 1977. 30. Nakayama, S., Jap. J. Appl. Phys., 26, 1356, 1936. 31. Kaczkowski, A. and Mileski, A., IEEE Trans. Mic. Theory Tech. MTT-28, 225, 1980. 32. Kaczkowski, A. and Mileski, A., IEEE Trans. Mic. Theory Tech. MTT-28, 228, 1980. 33. Xi, W., Tinga, W.R., Voss, W.A.G. and Tian, B.Q., IEEE Trans. Mic. Theory Tech. 40, 747, 1992. 34. Hollway, K.L. and Cassidy, G.J.A., Proc. IEE. 99, 364, 1952. 35. Reynolds, S.L., General Electric Review 35, 34, 1947. 36. Works, C.N., Dakin, T.W. and Boggs, F.W., Proc. I.R.E. 33, 245, 1945. 37. Works, C.N., J. Appl. Phys. 18, 605, 1947. 38. Parry, J.V.L., Proc. I.R.E. 98, 303, 1951. 39. Scott, A.H., Proc. I.S.A. 11, 2, 1956. 40. Shields, J.Q., Dziuba, R.F. and Layer, H.P., IEEE Trans. Instrum. Meas. 38, 249, 1989; and Cutkowsky, R.D., IEEE Trans. Instrum. Meas. IM-23, 305, 1974. 41. Shields, J.Q., IEEE Trans. Instrum. Meas. IM-27, 464, 1978. 42. Buckley, T.J., Hamelin, J. and Moldover, M.R., Rev. Sci. Instrum. 71, 2914, 2000. 43. Moldover, M.R., J. Res. Natl. Inst. Stand. Tech. 103, 167, 1998. 44. Moldover, M.R. and Buckley, T.J., Int. J. Thermophys. 22, 859, 2001.

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45. Makow, D. and Campbell, J.B., Metrologia 8, 148, 1972; and Campbell, J.B. and Makow, D., J. Comput. Phys. 12, 137, 1973. 46. Heerens, W. Chr., Cuperus, B. and Hommes, R., Delft Progress Report 4, 67, 1979. 47. Mohr, P.J. and Taylor, B.N., J. Phys. Chem. Ref. Data 28, 1713, 1999. 48. Pachuck, K. and Sapirstein, J., Phys. Rev. A63, 012504, 2000. 49. Cencek, W., Szalewicz, K. and Jeziorski, B., Phys. Rev. Lett. 86, 5675, 2001. 50. Hurly, J.J. and Moldover, M.R., J. Res. Nat. Inst. Stand. Tech. 105, 667, 2000. 51. Blancett, A.L., Hall, K.R. and Canfield, F.B., Physica 47, 75, 1970. 52. Koch, H., Ha¨ttig, C., Larsen, H., Olsen, J., Jørgensen, P., Ferna´ndez, B. and Rizzo, A., J. Chem. Phys. 111, 10108, 1999. 53. White, M.P. and Gugan, D., Metrologia 29, 37, 1992. 54. Schmidt, J.W. and Moldover, M.R., Int. J. Thermophys. 24, 375 2003. 55. Weber, L.A., J. Chem. Phys. 65, 446 (1976); Ewing, M.B. and Royal, D.D., J. Chem. Thermodyn. 34, 1089, 2002. 56. Stokes, R.H. and Marsh, K.N., J. Chem. Thermodyn. 8, 709, 1976. 57. Experimental Thermodynamics, Vol VII, Measurement of the Thermodynamic Properties of Multiple Phases, R.D. Weir and T.W. de Loos eds., For IUPAC, Elsevier, Amsterdam, 2002. 58. Chan, M., Ryschkewitsch, M. and Meyer, H., J. Low Temp. Phys. 26, 211, 1977. 59. Burfiled, D.W., Richardson, H.P. and Guereca, R.A., AIChE. J. 16, 97, 1970. 60. Steiner, R. and Schadow, E., Z. Phys. Chem. 63, 297, 1969. 61. St-Arnaud, J.M., Bose, T.K., Okambawa, R. and Ingrain, D., Int. J. Thermophys. 13, 685, 1992. 62. Hocken, R., Horowitz, M.A. and Greer, S.C., Phys. Rev. Lett. 37, 964, 1976. 63. Chan, M.H.W., Phys. Rev. B 21, 1187, 1980. 64. Doiron, T. and Meyer, H., Phys. Rev. B 17, 2141, 1978. 65. Early, M.D., J. Chem. Phys. 96, 641, 1992. 66. Dombro, D.A. Jr., McHugh, M.A., Prentice, G.A. and Westgate, C.R., Fluid Phase Equilib. 61, 227, 1991. 67. Mulev, Yu.V., Thermal Eng. 37, 438, 1990. 68. Rogers, W.J., Holste, J.C., Eubank, P.T. and Hall, K.R., Rev. Sci. Instrum. 56, 1907, 1985. 69. Fogh, F. and Rasmussun, P., Ind. Eng. Chem. Res. 28, 371, 1989. 70. Frorup, M.D., Jepsen, J.T. and Fredenslund, A., Fluid Phase Equilib. 52, 229, 1989. 71. Goodwin, A.R.H., Frøup, M.D. and Stenby, E.H., J. Chem. Thermodyn. 23, 713, 1991. 72. Small, G.W., McGreggor, M.C. and Lee, R.D., IEEE Trans. Instrum. Meas. 38, 372, 1989. 73. Buckingham, A.D. and Graham, C., Proc. Roy. Soc. Lond. A366, 275, 1974. 74. Achtermann, H.J., Baehr, H.D. and Bose, T.K., J. Chem. Thermodyn. 21, 1023, 1989. 75. St.-Arnaud, J.M., Bose, T.K., Okambawa, R. and Ingrain, D. Fluid Phase Equilib. 88, 137, 1993. 76. Okambawa, R., St.-Arnaud, J.M., Bose, T.K. and Le Noe¨, O., Fluid Phase Equilib. 134, 225, 1997. 77. Buckingham, A.D., Cole, R.H. and Sutter, H., J. Chem. Phys. 52, 5960, 1970. 78. Obriot, J., Ge, J., Bose, T.K. and St.-Arnaud, J.-M., Fluid Phase Equilib. 86, 315, 1993. 79. St.-Arnaud, J.M., Ge, J., Orbriot, J., Bose, T.K. and Marteau, Ph., Rev. Sci. Instrum. 62, 1411, 1991. 80. Marteau, Ph., Montixi, G., Obriot, J., Bose, T.K. and St.-Arnaud, J.M., Rev. Sci. Instrum. 62, 42, 1991.

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81. (a) Bo¨ttcher, C.F.J., Theory of Electric Polarization, Vol 1, 2nd Ed., Elsevier, Amsterdam, 1973. (b) Bo¨ttcher, C.F.J. and Bordewijk, P., Theory of Electric Polarization, Vol 2, 2nd Ed., Elsevier, Amsterdam, 1978. 82. Scaife, B.K.P., Principles of Dielectrics, Clarendon, Oxford, 1989. 83. Barthel, J., Bachhuber, K., Buchner, R., Hetzenauer, H., Kleebauer, M. and Ortmeier, H., Pure. Appl. Chem. 62, 2287, 1990. 84. Grant, I.S. and Phillips, W.R., Electromagnetism, 2nd Ed., Wiley, Chichester, 1990. 85. Davis, J.L., Wave Propagation in Electromagnet. Media, Springer, Berlin, 1990. 86. Sander, K.F. and Reed, G.A.L., Transmission and Propagation of Electromagnetic Waves, 2nd Ed., Cambridge University Press, Cambridge, 1986. 87. Birch, J.R. and Parker, T.J., in Infrared and Millimeter Waves, Vol 2, K.J. Button, ed., Academic Press, New York, p. 137, 1979. 88. Buchner, R. and Barthel, J., Annu. Rep. Prog. Chem., Sect. C 91, 71, 1994; 97, 349, 2001. 89. Cole, R.H., Berberian, J.G., Mashimo, S., Chryssikos, G., Burns, A. and Tombari, E., J. Appl. Phys. 66, 793, 1989. 90. Evans, S. and Michelson, S.C., Meas. Sci. Technol. 6, 721, 1995. 91. Folgerø, K., Meas. Sci. Technol. 7, 1260, 1996. 92. Vincent, D., Jorat, L., Monin, J. and Noyel, G., Meas. Sci. Technol. 5, 990, 1994. 93. Pelster, R., IEEE Trans. Microwave Theory Tech. 43, 1494, 1996. 94. Buchner, R. and Barthel, J., Ber. Bunsenges. Phys. Chem. 101, 1509, 1997. 95. Bertolini, D., Cassetari, M., Salvetti, G., Tombari, E. and Veronesi, S., Rev. Sci. Instrum. 61, 450, 1990. 96. Feldman, Y., Andrianov, A., Polygalov, E., Ermolina, I., Romanychev, G., Zuev, Y. and Milgotin, B., Rev. Sci. Instrum. 67, 3208, 1996. 97. Buchner, R., Hefter, G.T. and May, P.M., J. Phys. Chem. A 103, 1, 1999. 98. Jenkins, S., Hodgetts, T.E., Clarke, R.N. and Preece, A.W., Meas. Sci. Technol. 1, 691, 1990. 99. Naito, S., Hoshi, M. and Mashimo, S., Rev. Sci. Instrum. 67, 3633, 1996. 100. Jakobsen, T. and Folgerø, K., Meas. Sci. Technol. 8, 1006, 1997. 101. Hilland, J. and Friisø, T., Meas. Sci. Technol. 9, 790, 1998. 102. Berberian, J.G. and Cole, R.H., Rev. Sci. Instum. 63, 99, 1992. 103. Folgerø, K., Friisø, T., Hilland, J. and Tjomsland, T., Meas. Sci. Technol. 6, 995, 1995. 104. Go¨ttmann, O., Kaatze, K. and Petong, P., Meas. Sci. Technol. 7, 525, 1996. 105. Barthel, J., Bachhuber, K., Buchner, R., Hetzenauer, H. and Kleebauer, M., Ber. Bunsenges. Phys. Chem. 95, 853, 1991. 106. Barthel, J., Buchner, R. and Mu¨nsterer, M., in Electrolyte Data Collection, Part 2: Dielectric Properties of Water and Aqueous Electrolyte Solutions, G. Kreysa, ed., Chemistry Data Series, Vol XII, DECHEMA, Frankfurt, 1995. 107. Barthel, J., Buchner, R. and Mu¨nsterer, M., in Electrolyte Data Collection, Part 2a: Dielectric Properties of Nonaqueous Electrolyte Solutions, G. Kreysa, ed., Chemistry Data Series, Vol XII, DECHEMA, Frankfurt, 1996. 108. Alison, J.M. and Sheppard, R.J., Meas. Sci. Technol. 1, 1093, 1990. 109. Richards, M.G.M. and Sheppard, R.J., Meas. Sci. Technol. 2, 975, 1991. 110. Mattar, K.E. and Buckmaster, H.A., Meas. Sci. Technol. 2, 891, 1991. 111. Kaatze, U., Pottel, R. and Wallusch, A., Meas. Sci. Technol. 6, 1201, 1995. 112. Pastol, Y., Arjavalingam, G., Halbout, J.-M. and Kopcsay, G.V., Appl. Phys. Lett. 54, 307, 1989. 113. Capps, C.D., Falk, R.A., Ferrier, S.G. and Majoch, T.R., IEEE Trans. Microwave Theory Tech. 40, 96, 1992.

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114. Pedersen, J.E. and Keiding, S.R., IEEE J. Quantum. Electron. 28, 2518, 1992. 115. Rønne, C., Thrane, L., A˚strand, P.O., Wallqvist, A., Mikkelsen, K.V. and Keiding, S.R., J. Chem. Phys. 107, 5319, 1997. 116. Flanders, B.N., Cheville, R.A., Grischowsky, D. and Scherer, N.F., J. Phys. Chem. 100, 11824, 1996. 117. Buchner, R., Barthel, J. and Gill, B., Phys. Chem. Chem. Phys. 1, 105, 1999. 118. Kaatze, U., J. Sol. Chem. 26, 1049, 1997. 119. Barthel, J., Buchner, R., Eberspa¨cher, P.N., Mu¨nsterer, M., Stauber, J. and Wurm, B., J. Mol. Liq. 78, 83, 1998. 120. Barthel, J., Krienke, H. and Kunz, W., Physical Chemistry of Electrolyte Solutions, Steinkoppf/Springer, Darmstadt, 1998.

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10

Extreme Conditions H. SUGA Kinki University Higashi-Osaka, Japan G. POTTLACHER Institut fu¨r Experimentalphysik Technische Universita¨t Graz, Austria I. EGRY DLR, Institute for Space Simulation Cologne, Germany 10.1 Low Temperatures 10.1.1 Adiabatic Calorimeter as an Ultra-Low-Frequency Spectrometer 10.1.2 Calorimetry at Very Low Temperatures 10.2 High Temperatures 10.2.1 Resistive Pulse Heating 10.2.2 Laser Pulse Heating 10.2.3 Levitation Techniques 10.2.4 Electromagnetic Levitation 10.2.5 Future Directions 10.3 Molten Metals 10.3.1 Containers 10.3.2 Pyrometry 10.3.3 Calorimetry 10.3.4 Thermal Expansion and Density 10.3.5 Surface Tension

Measurement of the Thermodynamic Properties of Single Phases A.R.H. Goodwin, K.N. Marsh, W.A. Wakeham (Editors) # 2003 International Union of Pure and Applied Chemistry. All rights reserved

476 476 483 488 489 497 501 502 502 504 505 509 514 520 526

10

EXTREME CONDITIONS

To determine the thermal functions and phase transitions of materials, it is essential to have a knowledge of their heat capacities at temperatures from T & 0 and up. Recent developments in low temperature calorimetry to measure these properties are described in Section 10.1. Accurate thermophysical property measurements are also required at high temperature to: (i) model liquid metal processing operations, such as casting and welding, with the methods of finite elements, shown in Figure 10.10; (ii) understand, simulate and design new processing equipment such as for the growth of silicon single crystals from the melt; (iii) obtain phase diagrams; (iv) obtain temperature reference points; (v) accurately assess the design of nuclear reactors; (vi) develop aerospace techniques; and (vii) develop basic theory for critical points of metals. Various limitations of traditional methods at high temperatures (T > 103 K), that result from finite measurement times, arise from chemical interaction of the specimens with the containers, loss of mechanical strength, heat transfer, evaporation, and electrical insulation. In Section 10.2, fast dynamic methods, with measurement time intervals between (1 and 1012) s, that have been developed to avoid these difficulties are described. Section 10.3, which attends to molten metals, describes measurement techniques, such as non-contact levitation methods, capable of operating at high temperatures with highly reactive materials.

10.1

Low Temperatures H. SUGA Kinki University Higashi-Osaka, Japan

The study of the thermodynamic properties at low temperatures is indispensable for the characterisation of any new materials. The entropy is a particularly important quantity in clarifying the nature of any disorder in a system. Most of the materials studied at low temperatures are solids, either in a crystalline or non-crystalline state. Considerable progress has been achieved in the heat capacity measurement since the publication of the previous IUPAC book on Experimental Thermodynamics [1]. Advanced instrumentations and the treatment of non-equilibrium states by irreversible thermodynamics are described.

10.1.1

Adiabatic Calorimeter as an Ultra-Low-Frequency Spectrometer

Figure 10.1 is a schematic drawing of an adiabatic calorimeter with an intermittent heating mode [2]. The calorimeter was placed inside a Dewar vessel with an appropriate coolant. The sample cell A was surrounded by the inner B and outer C 476

Extreme Conditions

477

Figure 10.1 A low-temperature adiabatic calorimeter. A, calorimeter cell; B, inner adiabatic shield; C, outer adiabatic shield; D, Vacuum jacket; E, thermal anchor; T, transfer siphon.

adiabatic shields and the vacuum jacket D. The copper block E was a thermal anchor for lead wires coming from room temperature. Since the heat can transfer through various routes (conduction, convection, and radiation), realisation of an adiabatic condition is very important. This can be achieved by evacuation of the calorimetric space and rigorous temperature control of the adiabatic shields and thermal anchor. The transfer siphon T was used only when a disordered phase stable at high temperatures is quenched by pouring a coolant directly into the calorimetric space. The principle of determination of the heat capacity of condensed matter is, as shown schematically in Figure 10.2, simple. A constant temperature of a sample cell under adiabatic conditions is the necessary condition for thermal equilibrium. This can be confirmed during the equilibration period composed of fore- and after-rating periods. A certain amount of electrical energy E is supplied to the cell. Under 477

478

Extreme Conditions

Figure 10.2 Schematic of a temperature versus time curve obtained in a heat capacity measurement.

adiabatic conditions, the whole of the supplied energy is used to raise the sample temperature. A new equilibrium temperature is determined, after the addition of energy, during the after-rating period. The ratio E=DT ¼ E=(Tf  Ti ) gives an average heat capacity at temperature (Tf þ Ti )=2, where the subscripts f and i refer to the final and initial conditions respectively. The after-rating period becomes the forerating period for the subsequent heat-capacity determination and the processes are repeated to cover a wide range of temperature. Adiabatic calorimeters are connected to automated data acquisition systems [3,4], and are maintained adiabatic with a set of computer driven PID controllers. Automatic ac bridges are used to determine the resistance of platinum resistance thermometers. The supplied electrical energy is determined with digital multimeters. The data so acquired are used to calculate the heat capacity data in real-time. Thus, any anomalous behaviour in the observed heat capacity can be verified by further measurement before dismantling the apparatus. The calorimeter can be operated continuously in automatic mode. If a sample is in a frozen-in disordered state, as in the case of glasses, structural relaxation takes place around the glass transition temperature Tg. An excess amount of enthalpy stored in the sample tends to relax toward the equilibrium state and the released energy induces a spontaneous temperature rise of the cell during the equilibration period. This is shown in Figure 10.2 as a dotted line. The rate of temperature rise depends on the excess amount of configurational enthalpy Hconf and the structural relaxation time t, both of which are dependent on temperature. At low temperatures, where Hconf is large and t long, the drift rate dT/dt is also small. Near Tg, where t is short but Hconf small, the drift rate dT/dt is also small. There is a temperature at which the rate of spontaneous temperature drift rate reaches a maximum. Above Tg, where the sample can be in a temporarily super-heated state, the temperature can be observed to spontaneously fall during the equilibration time. 478

Extreme Conditions

479

The enthalpy, which is constant under adiabatic conditions, can be divided into two parts; a configurational Hconf and a vibrational Hvib part. The latter responds rapidly to the variation of temperature. Thus: Dconf H ðtÞ þ Dvib H ðtÞ ¼ constant:

ð10:1Þ

The configurational enthalpy is thought to relax according to the relationship proposed by Kohlrausch [5], and Williams and Watts [6]: Dconf H ðtÞ ¼ Dconf H ðt ¼ 0Þ expðt=tÞb :

ð10:2Þ

where b is the non-exponential parameter. Differentiation of Equation (10.2) with respect to time t, we obtain: dðDconf H Þ=dt ¼ ½dðDvib H Þ=dT ½dT=dt ¼ Cvib ðdT=dtÞ:

ð10:3Þ

Thus the change in calorimetric temperature, T(t), during the equilibration period can be expressed by [7]: T ðtÞ ¼ a þ bt þ c expðt=tÞb ;

ð10:4Þ

where the second term bt is a constant temperature drift rate owing to a residual heat leak and, under favourable conditions, is of the order of several mK ? h1 . For many glass-forming liquids, the parameter b ranges between 0.5 and 0.9 in magnitude for the volume and enthalpy relaxation. The relaxation process is known to be non-linear in that the rate depends on the sign and magnitude of initial departure of a sample from the equilibrium state. The process is known to show nonArrhenius behaviour depending on the fragility of the liquid [8]. The time domain covered by this calorimetric method is (102 to 106) s and is complementary with dielectric and NMR spectroscopy studies that can cover the short-time regime. A comparison of the rates in the enthalpy and volume relaxation of the same substance has been done in some systems. In view of the nature of the relaxation, it is necessary to simultaneously measure the enthalpy and volume of the sample as a function of time because the same non-equilibrium state cannot be realised in separate experiments. For this purpose, an adiabatic calorimeter for the simultaneous measurement of volume and enthalpy under pressure has been developed [9]. Glass transition phenomena have long been considered to be a characteristic nature of liquids. In fact glass transitions can be observed even for crystalline substances possessing some frozen orientational disorder. The freezing-in process can occur, with respect to molecular orientation, while keeping the translational invariance with respect to centres-of-masses of the constituents. The first example of this kind of glassy state in crystals was provided by cyclohexanol in which the freezing-in of the reorientational motion of the molecule in the undercooled high479

480

Extreme Conditions

Figure 10.3 Spontaneous temperature drift rates dT=dt ¼ T_ as a function of temperature T for tetrahydrofuran ? 17H2 O hydrate crystal observed in the glass transition region. , cooled at 0:04 K ? s1 ; 4, annealed at 80 K for 30 h.

temperature face-centred-cubic phase occurred at T & 150 K. The relaxation process is, in every respect, exactly the same as that observed for liquids [10]. Figure 10.3 shows one example of a spontaneous change in the calorimetric temperature observed for a clathrate hydrate crystal [11]. The ordinate is an average temperature drift rate observed during the initial 0.5 h of each equilibration period. The enthalpy relaxation arises from freezing out of reorientational motion of the host water molecules. If we observe the calorimetric temperature over a long period, we can analyse the temperature-time curve to determine the parameter b with Equation (10.4). At temperatures just above Tg, the configurational enthalpy is located below the equilibrium value, and the spontaneous drift rate of temperature changes its sign from a positive to a negative value. The situation occurs whenever the enthalpy of the non-equilibrium sample crosses the equilibrium curve during heating. Glass transitions that have been observed in many liquids have turned out to be just one example of the freezing-in processes, which occur widely in condensed matter irrespective of the translational disorder of the constituent molecules. The existence of some kinds of disorder is necessary for observing a glass transition which occurs when the relaxation time for a motion to realise the equilibrium state becomes comparable to the experimental time. Thus, the development of calorimetry in this direction has broadened the realm of chemical thermodynamics to include the study of kinetics of cooperative molecular motion that has not been the subject of calorimetric measurement hitherto. Miniaturisation of Adiabatic Calorimeters It has been a standard practice of low temperature calorimetry to use large amounts of sample, typically (30 to 50) g, for the measurement of thermodynamic data with 480

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481

high accuracy and precision. It is important to increase the ratio of the heat capacity of the sample to that of the total heat capacity including that of the sample cell. It is easy to prepare large amounts of samples for readily available substances. However, it is increasingly difficult to obtain generous portions of fascinating but expensive substances in a highly purified state. The necessary consequence of this situation is to attempt to miniaturise the calorimeter without loss of the accuracy and precision traditionally achieved [12–15]. An adiabatic calorimeter for small samples has been described [16] and is shown in Figure 10.4. In this twin-type calorimeter, the sample cell A was equipped with only an electric heater for Joule heating. The thermometer, normally attached to the sample cell, was removed in order to reduce the heat capacity of the empty cell and to avoid the self-heating effect required for the resistance measurement. The thermometer was

Figure 10.4 An adiabatic calorimeter for small samples. A, calorimeter cell; B, thermometric block; C and D, inner adiabatic shields; E, outer adiabatic shield; F, thermal anchor; G and H, heat exchangers; I, inner vacuum jacket; J, radiation shield; K, outer vacuum jacket.

481

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instead placed in the second cell B made of a copper block wound with another heater. The temperature difference between the two cells was monitored by three pairs of fine thermocouples along with a nanovolt amplifier. Any deviation signal was fed into the heater of the second cell after power amplification so as to keep it at the same temperature. For this purpose, the temperature of inner adiabatic shield D was kept at a temperature a little lower than that of the sample cell. In this way, the sample temperature was transferred to the second cell possessing a large heat capacity and its temperature was measured by standard thermometry. The precision of the temperature measurement in this new method depends on the reliability of the nanovolt amplifier and temperature servomechanism. The output signal of the nanovolt amplifier during the heating period can exhibit hunting with a large amplitude which arises from a rapid, about 5 mK ? s1 , change in temperature. It is not necessary, however, to follow precisely the sample temperature during this period. The average temperature drift during the equilibration period was (0 to 10) mK ? h1 , and the servomechanism functioned well in this period. The average noise amplitude during the equilibriation period was +5 nV for a single ChromelConstantan thermocouple. The thermoelectric power of the thermocouple was (8.4, 27, and 61) mV ? K1 at (20, 80 and 300) K respectively. Thus, the temperature transfer was achieved with the temperature resolution of (+0.20, +0.062, and +0.027) mK at the respective temperatures. For the calibration of this microcalorimeter, 0.8288 g of benzoic acid crystal, designated as a 1965 Calorimetry Conference Standard, was used. Deviations of the heat capacity data from those recommended by Furukawa et al. [17] and by Robie and Hemingway [18] are shown in Figure 10.5. Since the present data are based on the International Practical Temperature Scale of 1968 (IPTS-68) but the literature values are based on National Bureau of Standards 1955 temperature scale (NBS-55)

Figure 10.5 Deviations of the present heat capacities of benzoic acid crystal from literature values. , Furukawa et al. [17]; , Robie and Hemingway [18].

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483

and the International Practical Temperature Scale of 1948 (IPTS-48), literature data were converted to the IPTS-68 for the comparison. The conversion procedure has been described in detail by Bedford et al. [19], and by Douglas [20]. The figure shows that the differences between the present data and the literature values are approximately +0.3 per cent above 60 K. The increase in the discrepancies at lower temperatures are entirely due to the decreased sensitivity of the thermocouple used for the transfer system. In view of the small amount of the sample, the results can be accepted as satisfactory. The reader should consult references [21,22] to calculate the changes in heat capacity arising from a change in temperature scale from IPTS-68 to The International Temperature Scale of 1990 (ITS-90).

10.1.2

Calorimetry at Very Low Temperatures

Adiabatic demagnetisation techniques, that have been used for the production of very low temperatures in a limited number of laboratories, have been replaced almost entirely by 3He cryostats [23] or 3He-4He dilution refrigerators [24]. The former produces cryogenic temperatures as low as 0.3 K and the latter as low as 5 mK over a long time period. Thanks to the rapid development of cryogenic techniques, calorimetry in the very low temperature region has become a standard method. These calorimeters are used to detect phenomena arising from weak intraand inter-molecular interactions with low-lying excited levels. The associated heatcapacity anomalies appear clearly at low temperatures where the lattice heat capacity becomes negligibly small. Tunnelling leads to splitting of rotation-vibrational levels of a methyl or ammonia group coupled with inter-conversion of nuclear spin is one example of research that can be undertaken in this temperature region. Figure 10.6 shows an isoperibol calorimeter with a built-in 3He evaporator. The calorimeter was immersed in a cryostat filled with liquid 4He at 4.2 K, so that the calorimeter always faced cold surfaces [25]. All the electrical leads entering the system are thermally anchored at low temperatures before reaching the sample cell. The whole apparatus was located on rubber-dampers to reduce heat generated by mechanical vibration. A part of the liquid 4He was introduced into the 1.2 K reservoir A through a needle valve. 3He gas, from an external storage tank, was liquefied at the 3He reservoir C. A charcoal adsorption pump B, preheated to 30 K by an electric heater, was cooled to 4 K by liquid 4He to induce adsorption of 3He. The liquid 3He evaporated rapidly by adsorption and the temperature of the 3He evaporator decreased spontaneously to about 0.3 K. This temperature could be maintained for about 10 h in an experiment involving a single-shot liquefaction of 3 He. Figure 10.7 is a sketch of the calorimeter inside the 1.2 K shield. A gold-plated copper jaw D thermally anchored to the 3He evaporator by copper braids, was used to cool the calorimetric cell E by clamping to a post of the cell. Rapid operation of the mechanical thermal switch induced undesirable heat generation on release of a contact pressure. The sample cell, shown in Figure 10.8, consists of two parts; the upper part equipped with a clamping post D for the thermal switch and a germanium 483

484

Extreme Conditions

Figure 10.6 An isoperibol calorimeter for the temperature range 0.3 to 20 K. A, 1.2 K reservoir; B, charcoal adsorption pump; C, 3He reservoir (evaporator).

resistance thermometer C, while the lower part has a sample container of 40 cm3 capacity. The cell is firmly suspended by fine nylon threads in a high-vacuum space at a pressure of about 104 Pa. Electrical power was supplied by a cell heater wound non-inductively on the surface of the cell and the energy supplied was measured potentiometrically. The resistance of the thermometer was measured with an ac automatic Cryobridge (Automatic System Laboratory Ltd., UK). Its temperature had been calibrated against the NBS Provisional Scale between (2 and 20) K, while in the range below 2 K a magnetic temperature scale, based on measurements of the magnetic susceptibility of a KCr(SO4)2 ? 12H2O crystal, was used. For the latter, a single spherical crystal of KCr(SO4)2 ? 12H2O was placed inside a nylon bobbin 484

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485

Figure 10.7 Sketch of the calorimeter. A, 1.2 K reservoir; B, tag for leads; C, 3He evaporator; D, mechanical thermal switch; E, calorimeter cell; F, sample mounting flame.

around which the primary and secondary coils of superconducting wire are wound. The primary coil was connected to an audio-frequency generator and the secondary coil to a Hartshorn mutual inductance bridge. The real part of the magnetic susceptibility was measured by a variable mutual inductor. The calibration data are fitted to a polynomial of the form lnðR=OÞ ¼

X

Ai ½lnðT=K Þi :

ð10:5Þ

i

With fourteen parameters, Equation (10.5) can reproduce the calibration points to within +0.3 mK at T < 1 mK, within +2 mK at temperatures between (0.8 and 6) K, and within +5 mK at T > 6 K. In low temperature calorimetry, the rate with which the cell attains thermal equilibrium is critical to the performance of the apparatus and the results obtained. Thus, when developing a new calorimeter, considerable attention is given to heat exchange between the sample and the cell. 3He gas, when used as a heat exchange medium between sample powder and cell, can cause serious heat effects that are associated with adsorption or desorption of the gas. Apiezon N and silicone vacuum greases have been recommended as the heat exchange media at low temperatures. The thermal conductivity of these glass-forming media are not high but, because the heat capacities of the sample and cell are also low at low temperatures, thermal 485

486

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Figure 10.8 Sketch of the calorimeter cell. A, flange; B, fine nylon thread; C, germanium thermometer; D, clamping post for thermal switch; E, connectors; F, capillary; G, lid; H, rod; I, indium gasket; J, heater; K, sample with silicon-oil.

equilibrium is attained within a reasonable time. A silicone oil has also been found a good heat transfer medium. A high-purity copper designated as the 1965 Calorimetry Conference Standard was used to check the accuracy and precision of a new calorimeter that operates in the temperature range (0.4 to 20) K. The sample had a cylindrical form of 53 mm in height and 31 mm in diameter, so that thermal equilibrium with a thermometerheater unit could be attained easily. The heat-capacity data obtained from this instrument agreed well with those reported elsewhere [26]. The imprecision of the instrument was found to be within +0.6 per cent at T > 4 K, +1.2 per cent in the temperature range (1.5 to 4) K, +1.5 per cent in the temperature range (1.5 to 0.9) K, and +2.5 per cent at T < 0.9 K. At lower temperatures, a 3He-4He dilution refrigerator is required for the thermostat. A detailed description of the principles and operation of this refrigerator can be found in reference [24]. Liquid 3He and 4He exhibit a positive enthalpy change on mixing, resulting in a lowering of the temperature under adiabatic conditions. As the temperature is lowered, the liquid mixture exhibits a phase separation. At about 0.1 K, one phase has a composition of (0.93 4He þ 0.07 3He) and the other is pure 3 He. The heavy component of the liquid mixture is fed to an evaporator, where the composition of the gas phase in equilibrium with the liquid mixture at T & 0.6 K is 486

Extreme Conditions

487

almost pure 3He gas. Thus 3He condenses, mixes, phase separates, and evaporates to maintain the mixing chamber at temperatures as low as few mK. This cycle can be maintained as long as the pre-coolants, N2 (l) and 4He (l), are preserved. Figure 10.9 shows a schematic diagram of a cryostat designed for calorimetry and constructed by Oxford Instruments Ltd. [27]. Many vacuum pumping systems are necessary for the operation of the refrigerator and it is essential that mechanical vibrations arising from these pumps are well isolated from the calorimeter. Rotary vacuum pumps were mounted on rubber dampers and all the pipes to the cryostat were connected via bellows. Four plinths of a metal-frame supporting the cryostat were firmly fixed on two cement tiers, which were submerged in sand. The coldest part of the cryostat is the mixing chamber at T & 6 mK. A gold-plated copper shield attached to the bottom of the mixing chamber contains the calorimeter chamber in which a sample cell was housed. The calorimeter cell is essentially the same as that used at 3He temperatures except for the cooling procedure where a superconducting thermal switch is used in addition to a mechanical switch. The latter can be used to cool the cell down to 100 mK and the former is necessary for cooling to lower temperatures. The operation

Figure 10.9

Schematic diagram of a cryostat with a built-in (3He þ 4He) dilution refrigerator.

487

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Extreme Conditions

of the superconducting switch depends on the difference in thermal conductivity of the superconductor in the normal kn and superconducting ks states. The ratio ks/kn is approximately proportional to (T/Tc)2, where Tc is the critical temperature of the superconductor at zero magnetic field. The superconducting thermal switch consists of a thick copper filament interposed by an indium wire, 2 mm in length and 1 mm in diameter. One end of the wire was connected to the mixing chamber and the other end to the sample cell. A switching coil made of Nb-Ti wire was wound in 5600 turns on a copper bobbin that surrounds the indium wire. A dc current of about 80 mA was passed through the coil produces a magnetic field beyond the critical field of indium to break the superconducting state. The enhanced thermal conductivity of the wire works to cool the sample cell. After the cell reached the lowest temperature, the coil current was decreased slowly to recover a low thermal conductivity state which quasiisolates the cell from its thermal surroundings during the heat capacity measurement.

10.2

High Temperatures G. POTTLACHER Institut fu¨r Experimentalphysik Technische Universita¨t Graz, Austria

A general description of pulse heating techniques for measurements at high temperatures was given by Cezairliyan and Beckett in Experimental Thermodynamics, II [28]. All the considerations given there regarding design of pulse heating systems and phenomena such as skin-effects and magnetic forces are valid and, therefore, these topics will not be discussed here. During the last two decades subsecond techniques have matured because of improved data acquisition. Now, at the beginning of the twenty-first century a very marked improvement in the measurement techniques has been achieved. Fast and reliable electronic dataacquisition devices are commercially available for measurements, as predicted in [28]. In the early 1980s one still had to deal with photographic methods, taken from fast oscilloscope traces and digitise them for further evaluation. Solid state detectors like Si-photodiodes and InGaAs photodiodes have become accurate and fast enough to serve as detectors in high speed pyrometry. To reach high temperatures, in addition to resistive pulse heating, new techniques such as laser pulse heating and levitation (which use either laser pulse heating or volume heating by eddy current) have been developed and will be described in this chapter. Thermophysical properties at high temperature can be determined using a variety of pulse heating techniques. The emphasis will be on measurements at very high temperatures. Pulse heating techniques are described for the determination of thermophysical properties of matter from temperatures of about (1000 up to 10 000) K. The time regime covered is from (1 to 1012) s. However, finite element 488

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489

analysis has been applied to thermal processes. For example, as shown in Figure 10.10, the cooling of a molten metal within a mould to form an axle. The heating rates used for pulse heating experiments vary between (104 and 10 10 ) K ? s1 and require recording equipment with very short response time. The scope of this review is restricted to resistive self heating methods, to laser heating techniques and to electromagnetic levitation techniques with high sample temperatures. Other techniques for measurements of physical properties at temperatures between (5000 to 10 000) K, such as chemical flames, shockwaves, solar heating, fission, fusion, and high energy electron or neutron heating, will not be considered here.

10.2.1

Resistive Pulse Heating

Resistive pulse heating or rapid volume heating methods are limited to electrically conducting materials. They involve passing an electrical current pulse through the specimen to give resistive self-heating. The temperatures of these experiments cover the range 293 K to T > Tm, where Tm is the specimen melting temperature. Depending on the heating rate required, the energy is stored in either a bank of batteries (for ms experiments) or a bank of capacitors (for ms experiments). The metal samples, which are either wires, foils or tubes, are contained in a controlled environment chamber. Measurements can be performed at pressures in the range 0.1 MPa to 0.7 GPa, at nearly isobaric conditions, with argon or water. The data recording equipment is placed in an electrically shielded room. The data obtained are evaluated after the experiment has been completed.

Figure 10.10 Simulation of the solidification of an axle obtained with the finite element code MAGMAsoft.

489

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Extreme Conditions

A functional diagram of a typical resistive pulse heating system is shown in Figure 10.11. The details of the components used depend on the time scale of the measurement to be performed. The discharge circuit typically consists of the energy storage system, a switch, pressure controller, and the specimen chamber which is fitted with windows to observe the experiment. For an experiment powered by a battery pack, a semiconductor based switch is used, while a mercury vapour ignition tube is used for a capacitor bank. Measurement techniques used for both ms and ms for experiments are similar and these time regimes are considered together while also describing the minor differences. For ms experiments [29], the current I(t) is determined from the voltage drop across a standard resistance R placed in series with the specimen while for ms experiments either a precision current probe or inductive coil and subsequent electrical integration is used [30]. The voltage drop along the sample is measured with two knife-edge probes and ohmic voltage dividers. Inductive contributions to the measured voltage are compensated for either numerically or experimentally with a coil [31] to produce the true voltage V(t) [30]. The specific enthalpy h(t) is determined from measured V(t) and I(t) with,

1 DhðtÞfh ¼ hðtÞ  hðT ¼ 298 KÞg ¼ m

ZT IðtÞVðtÞ dt;

ð10:6Þ

0

where m is the mass of the sample and t the time dependence. The electrical resistivity rel (t) of the sample can be determined from V(t) and I(t) using

rel ðtÞ ¼

Figure 10.11

VðtÞprðtÞ2 ; IðtÞl

Schematic of a resistive pulse heating system.

490

ð10:7Þ

Extreme Conditions

491

where r is the sample radius, obtained from an expansion measurement, and l the length of the specimen. The thermal conductivity k can be estimated [32,33], from the temperature T and the electrical resistivity rel by means of the Wiedemann-Franz-law, k¼

LT ; rel

ð10:8Þ

assuming a constant, theoretical value of the Lorentz number L ¼ 2:45 ? 108 V ? K2 . The thermal diffusivity a can be estimated from heat capacity cp and electrical resistivity from the definition [32,33]: a¼

k LT ; ¼ cp r cp rrel

ð10:9Þ

where r, is the mass density and cp the specific heat capacity at constant pressure. As discussed in [32], the principal carriers for thermal conduction in solid metals are electrons and lattice waves. However, for temperatures close to the melting point of the pure metal, electronic conduction is the predominant mechanism and lattice conduction is negligible. The Lorentz number for most pure metals is close to the theoretical value of L ¼ 2:45 ? 108 V2 ? K2 . Therefore, Equation (10.8) can be used as a tool for obtaining thermal conductivity in the vicinity of the melting point. For alloys, for example, Inconel 718, lattice contributions must also be considered [34]. Contributions owing to convection may lead to large uncertainties in thermal conductivities. However, in pulsed experiments, owning to the fast heating rate, convection is negligible and suppressed at the onset of the liquid phase. The surface radiation intensity of the hot sample is detected by fast pyrometers fitted with either Si (operating at wavelengths l in range (600 to 900) nm) or InGaAs photodiodes (operating at l & 1500 nm) and with interference filters with a bandwidth of (10 to 20) nm. Photomultiplier tubes can be used but introduce higher uncertainty because of the drift during the course of the experiments. The temperature of the sample is obtained with a calibrated pyrometer. The known melting temperature of the specimen can be used as a calibration point [35,36]. These pyrometers are sensitive to temperatures above 1000 K and the temperature is calculated from the measured radiance intensity I(T) which is given by Planck’s law, Z? IðTÞ ¼ g

sðlÞtðlÞeðl; TÞ l¼0

c1 dl; l5 ½expðc2 =lTÞ  1

ð10:10Þ

where T is the temperature, g the geometric factor of the pyrometer-system, l the wavelength, s the spectral sensitivity of the detector, t the transmittance of the 491

492

Extreme Conditions

optical systems, and e the normal spectral emissivity. In Equation (10.10), c1 and c2 are the first and second radiation constants. The unknown temperature T is determined from a ratio of the measured radiance intensity at the melting temperature Tm and the measured radiance intensity I(T) at a temperature T with T¼

n

l ln 1 þ

c2 Im ðTm Þeðl;TÞ IðTÞeðl;Tm Þ

h

 exp

c2 lTm



1

io ;

ð10:11Þ

where e(l, T) is the emissivity of the liquid sample and e(l, Tm ) is the emissivity at Tm. It is assumed that the emissivity of the liquid metal is the same as that at the melting temperature [36]. Recent developments in the measurement of temperature from emissivities in pulse heated liquid metals are described in Section 10.2.2. Accurate pyrometry requires careful calibration and data reduction procedures. Calibration procedures are generally based on the use of tungsten filament lamps. In some cases it may be desirable to perform pyrometer calibration directly with blackbody radiation sources [37]. The reader is referred to [38–40] and Section 10.3.2 for more information regarding high-temperature measurement and calibration. The sample radius can be determined, in microsecond and submicrosecond experiments, as a function of temperature from photographs of the radially expanding sample; the same measurements are also used to determine the stability of the specimen. The short-time pictures can be taken once during an experiment with a Kerrcell shutter that has 30 ns exposure time [30]. A high speed camera, which is capable of exposing 150 frames at a maximum rate of 3 ? 106 frames per second, can also be used [41]. More recently, a fast framing charge coupled device (CCD) based camera [42], which can take pictures of the diameter of a small part of the sample every 9 ms with a minimum exposure time of about 300 ns, have been used. This camera uses a CCD, with an array of 576 ? 384 pixels. A gateable multichannel plate in front of the chip operates as an amplifier and shutter. The chip is masked with a metal foil, which covers most of the sensitive area of the chip, and only a window with 32 ? 384 pixels remains open. These 32 pixel rows are sequentially exposed at a rate of one every 9 ms to give 18 exposures in 162 ms. Commercially available highspeed streak cameras (e.g. the Hadland, Imacon 500) or fast framing cameras (such as the Hadland, Imacon 468) have been used [43] that are capable of obtaining four exposures with an exposure time of 10 ns each. Alternately, for millisecond experiments the dimensional measurement of an expanding sample can be determined with a Michelson-type polarised beam interferometer by measuring the shift in the fringe pattern in the interferometer. In this measurement, a polarised beam from a He-Ne laser is split into two component beams, one which undergoes successive reflections from optical flats on opposite sides of the specimen, and one which serves as the reference beam. The thermal expansion of the specimen is then determined from the cumulative fringe shift [44]. Some microsecond pulse heating systems are able to measure the speed of sound in the liquid metal sample as a function of temperature. In this experiment, a laser 492

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493

driven stress wave is induced into the sample by focusing the output of a ruby laser pulse on one side of the liquid wire. This disturbance travels through the liquid sample, and the motion caused by the emergence of this wave on the opposite side of the sample into the surrounding medium, is detected by an image converting streaking camera, and the transit time through the sample can be determined. The sample diameter is monitored continuously by the streak camera and thus the average sound speed through the sample can be calculated [45,46]. Determination of Emissivity Values of Pulse-Heated Samples As mentioned above, it is usual to assume that the emissivity of the liquid phase [47] is independent of the temperature [e(l, T)=e(l, TM ) ¼ 1]. The emissivity of metals is often known at the melting point and it is also known that large changes in specific volume occur between the melting point and the maximum experimental temperatures. Since electronic structures, optical properties, and emissivities have a large dependence on the electron density of a material, it follows that the emissivity cannot be assumed constant and equal to the value at the melting point a priori. The lack of appropriate emissivity data results in increased uncertainties of the temperature measurements, leading to uncertainties of up to 20 per cent at the maximum temperature in the liquid phase. The most direct, and accurate, method for determining normal spectral emissivity utilises measurements of both the radiance from the surface of the material of interest and that of a blackbody cavity at the same temperature. The measurements are performed on tubular specimens with a small, carefully machined, rectangular hole in the wall of the specimen, which approximates a blackbody for optical temperature measurements [29]. This technique is limited to the solid phase (and thus to ms experiments). Surface tension effects result in a blackbody hole in a liquid metal sample that is usually unstable. The normal spectral emissivity at the melting point can also be determined from measurements of the radiance temperature with a calibrated pyrometer and the spectral emissivity obtained from Planck’s law at the known melting temperature [37]. Another method, with at least ms resolution, is to measure the normal spectral emissivity of solid strip specimens with an integrating sphere reflectometer operated under pulse heating conditions [48]. This method uses a high speed comparative integrating sphere reflectometer in which the reflectivity of the sample, which undergoes pulse-heating, is compared to the reflectivity of a barium sulphate reference. A modulated laser diode beam (at l & 900 nm) strikes the side of the sample facing the sphere. The reflected beam is collected hemispherically by the integrating sphere. At high temperatures, a technique based on a lock-in amplifier is used to discriminate between the reflected laser radiation and the continuous component generated by the specimen itself. Normal spectral emissivity measurements, with < 1 ms, resolution, have been performed with laser polarimetry [35,49] on ms pulse-heated liquids; this ellipsometric method has also proved reliable in ms pulse-heating experiments for the solid phase [35]. In this approach, a polarised laser light is reflected from the pulse-heated liquid 493

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wire, and the change in polarisation upon reflection determined at a series of intensities. From these measurements, and the normal working equations of ellipsometry, the Stokes optical constants and the spectral emissivity of the material are determined. Emissivity measurements in conjunction with the radiometric measurements are used to derive the absolute temperature of the liquid material of interest. Figure 10.12 shows a schematic of a division-of-amplitude photopolarimeter (DOAP) used for emissivity measurements. It includes the laser light source and the incident polarisation optics. The specimen is depicted as a thin wire. The reflected light is collected by the DOAP. The four exiting light beams are detected by four silicon photodiodes. The coated beamsplitter is designed to provide amplitude and phase changes in the reflected and transmitted components such that the complete polarisation state can be accurately determined from the four intensity measurements. The incident light beam is amplitude modulated. Phase-locked detection schemes are used to reject the background light due to stray or incandescent radiation. The DOAP contains a narrow band-pass interference filter centred at the optical wavelength of interest. This serves as an additional means of rejecting the incandescent background radiation emitted by the specimen. A beam splitter reflects a portion of the collected light to a part of the sensor used for alignment. The current electronic and optical design of the DOAP provides measurements every 0.5 ms. At the onset of melting all high temperature melting materials, investigated recently [50] at the Technical University in Graz, Austria, exhibited a sudden drop in normal spectral emissivity at a wavelength of 684.5 nm. These observations are consistent with both a decrease in surface finish and evaporation of the oxide layer. Materials can be categorised according to the behaviour exhibited by their normal spectral emissivity in the liquid phase, as follows: materials whose normal spectral emissivity increased with temperature, materials whose normal spectral emissivity decreased with temperature, and materials whose normal spectral emissivity remained constant. Additional experimental details can be found in references [49,50].

Figure 10.12 A division-of-amplitude photopolarimeter (DOAP). LD, laser diode; LP, linear polarizer; L, lenses; S, wire sample; FS, field stop; BS, beam splitter; GTP, Glan–Thompson prism; D1–4, detectors.

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495

Table 10.1 Thermophysical properties that may be measured by dynamic (subsecond) heating methods obtained by different dynamic resistive pulse heating methods. Time Resolution

Solids

Liquids

ms

heat capacity thermal expansion electrical resistivity hemispherical total emittance normal spectral emittance melting temperature (s þ s) transformation temperature (s þ s) transformation energy phase transitions apparent temperature at melting point deduced from radiance thermal conductivity mechanical propertiesa speed of sounda

heat capacity thermal expansion electrical resistivity hemispherical total emittance normal spectral emittance melting temperature enthalpy of fusion surface tension

ms

enthalpy of fusion heat capacity cp electrical resistivity thermal expansion speed of sound melting temperature apparent temperature at melting point deduced from radiance phase transitions thermal conductivity thermal diffusivity normal spectral emittance

enthalpy of fusion heat capacity cp, cv electrical resistivity thermal expansion speed of sound melting temperature apparent temperature at melting point deduced from radiance Gru¨neisen parameter g thermal conductivity thermal diffusivity adiabatic compressibility isothermal compressibility adiabatic bulk modulus isothermal bulk modulus normal spectral emittance critical pressure critical volume critical temperature surface tensiona optical constantsa enthalpy of fissiona

a

Indicates a method under development.

495

496

Extreme Conditions Experiments with Millisecond Time Resolution

In the 1 s to 1 ms time regime the measurement techniques are mature and provide accurate temperature measurements and these are described in Section 10.3.2. The temperature range from about 1000 K up to the melting temperature of the specimen is covered with ms time resolution experiments. The loss of stability of the sample at Tm and the resultant collapse in the presence of a gravitational field, normally defines the upper temperature limit in this time regime. In both argon and air atmospheres the pressures at which measurements are performed cover the range from high vacuum up to 0.1 MPa. The experimental quantities can be recorded about every 0.4 ms with a digital data acquisition system of 14 bit resolution. For a solid material the measurements yield specific heat capacity, normal spectral emittance, hemispherical total emittance, and thermal conductivity. The thermophysical properties that may be obtained with millisecond time resolution experiments are listed in Table 10.1. Techniques have been developed to permit measurements with ms resolution of current and voltage with an uncertainty of 0.1 per cent. Temperature measurements are uncertain to about +5 K at 2000 K. Thus, measurements of the specific heat capacity will be uncertain to within 2 per cent at 2000 K, while the reproducibility of measurements will be about 0.5 per cent at T ¼ 2000 K. For liquid phase samples and ms time resolution experiments a microgravity environment is required to obtain results with this level of uncertainty. The absence of gravity enables the experiment to enter the first 50 K of the samples liquid phase. Pseudo microgravity experiments in (e.g. those conducted with NASA’s KC-135 aircraft) have been performed by Cezairliyan and co-workers [51]. Measurements performed in the ms regime have been reported in references [52–58]. Experiments with Microsecond Time Resolution Depending on the material investigated with ms experiments, the temperature range covers (1000 to 6000) K and thus extends well into the liquid metal region. The stability limit for this type of experiment is determined by the onset of a complex sample geometry arising from boiling. The heating rates used in these experiments are typically in the range from (107 to 108) K ? s1 . To achieve the highest temperatures, static pressures up to 0.7 GPa must be applied to the sample. Experiments investigating actinide metals, for example, must be performed in a glove box [59]. Microsecond techniques have reached a high level of development and accurate measurements have been made on liquid metals (106 samples per second). During one fast pulse experiment, values of the enthalpy, temperature, electrical resistivity, and density of the specimen, as it rapidly passes through a wide range of states, are obtained. In addition, the speed of sound can be determined in the liquid phase. From these quantities the specific heat capacity at constant pressure, thermal expansion coefficient, thermal conductivity and thermal diffusivity may be calculated along with the thermophyiscal quantities listed in Table 10.1. 496

Extreme Conditions

497

With microsecond resolution, the measured current are uncertain to within 1 per cent, voltage within 2 per cent, and temperature uncertain to at least 2 per cent. Therefore, the specific heat capacity has an uncertainty of 5 per cent at 2000 K that increases to 10 per cent at T & 5000 K. At 2000 K the reproducibility of measurements will be approximately 2 per cent. The reader interested in ms time resolution experiments should consult references [31,45,59–62]. Experiments with Submicrosecond Time Resolution The temperature range covered at t < 1 ms lies from (1000 to 10 000) K and depends on the material investigated. The stability limit in this time regime is determined by the spinodal line in the p, V plane of the classical van der Waals equation of state. Further heating will produce rapid volume expansion and so-called ‘wire explosion’ occurs, interrupting the electrical circuit and terminating the experiment. The heating rates are between (109 and 1011) K ? s1 . These high heating rates can achieved at a sample pressure of about 0.1 MPa, but more accurate results are obtained at pressures up to 0.7 GPa. Submicrosecond techniques allow the investigation of liquid metals and (lþg) transition temperatures. Achieving heating rates greater than 109 K ? s1 requires the discharge circuit has low inductance and extremely well shielded cables to transmit the required signals. These requirements lead to a coaxial mounted discharge system with double coaxial shielded cables leading from the apparatus to a screened room, which includes the data acquisition equipment. The sample may be maintained within a capillary tube [63] and measurements are conducted until the capillary is filled, because of thermal expansion, with sample. In this time regime ( 109 samples per second) over a wide temperature range, the results obtained from these experiments are usually considered of an exploratory nature and, depending on the property measured, uncertainties in the measured properties will be between (10 to 20) per cent. In this time regime, results are usually in agreement with those obtained from microsecond techniques, but are of lower accuracy. Investigations with submicrosecond time resolution have been reported in the literature [64–68].

10.2.2

Laser Pulse Heating

Numerous laboratories perform laser-pulse heating experiments at high temperatures and this method has a wide range of application. Therefore, only a general outline of the measurement techniques used for the determination of thermophysical properties at high temperature are reviewed. The main advantage of laser pulse heating is the ability to investigate all kinds of materials including metals, ceramics, nuclear fuel materials and non-metals. The energy is delivered to the sample in a localised way so that efficient heating results and hence small specimens can be used. The use of ultra-short laser pulses results in very fast heating, thus extremely high temperatures may be achieved. With this 497

498

Extreme Conditions

technique, the amount of matter heated can be considerably less than required of other techniques. The main disadvantage is that, in most cases, the laser-energy deposition rate cannot be determined precisely. In this experiment, a short pulse of laser radiation heats a small part of the surface of a plain sample from room temperature up to either the melting point or even vaporises part of the surface. The samples are usually inside a controlled environment chamber, and the measurements are performed either in laser heating vacuum (LHV) or at pressures in the range 0.1 MPa up to 0.2 GPa with laser heating pressure (LHP). The pressurising medium is usually an inert gas. A schematic of a laser pulse heating system along with the main instrumentation components is shown in Figure 10.13. The details of the components vary for different measurements and for different techniques. The experimental setup for performing laser heating measurements is centred on a pulsed laser source, usually a Nd:YAG mode-locked laser. The output radiation (rectangular laser pulse, for example, l ¼ 1.06 mm, 90 fs to 1 ms pulse duration, with Gaussian spatial power profile and power densities up to 107 W ? cm2 ) may be frequency converted by harmonic generation in non-linear crystals, reaching visible and UV wavelengths. The laser pulse is focused on the polished surface of the sample and provides the excitation up to temperatures of (5000 to 10 000) K. The heated area of the target ranges from (1 to 3) mm diameter. The sample itself is inside a controlled environment chamber (not indicated in Figure 10.13). The light emitted from the centre of the target is detected optically by a highspeed pyrometer; the calculation of the temperature of the target surface is possible by making certain assumptions, similar to those made with temperature determination from spectral and total emittance. It is also possible to determine ultra-fast phase transitions. A mass-spectrometer can be used to determine the partial pressures of the various neutral and ionised gaseous species of the vapour jet. Spectroscopic

Figure 10.13

Schematic of a laser pulse heating system.

498

Extreme Conditions

499

temperature measurements and high speed photography of the evaporating jet are required as input data to determine the gas dynamic expansion mechanism at extreme rates of evaporation and to estimate back-scattering effects. A probing laser beam can be focused on the centre of the heated area of the sample, and the reflected and/or transmitted beam energies can be measured by light detectors. These signals, once normalised, provide reflectivity and/or transmission for the probed wavelength at a certain time. From the instantaneous values of reflectivity and transmission, one can deduce the complex refractive index of the material during the interaction. After acquiring the measurements with fast digital equipment, the results can be analysed, in order to obtain properties such as temperature, emissivity, and complex refractive index, provided additional information, such as, optical pyrometry, energy balance of the sample, and Fresnel formulae are known. A combination of the complex calibration and the analytical procedures, implies that the uncertainty of the thermophysical properties obtained will be larger than those obtained with techniques operating at lower heating rates. The thermophysical properties obtained with laser pulse heating are listed in Table 10.2. Further experimental and theoretical information on LHV and LHP Table 10.2 Thermophysical properties that may be measured by laser heating methods. Time Resolution

Solids

Liquids

ms

enthalpy of fusion normal spectral emittance hemispherical total emittance melting temperature vapour pressure partial pressures reflectivity phase transitions thermal conductivity thermal diffusivity heat capacity cp transmission

enthalpy of fusion normal spectral emittance hemispherical total emittance melting temperature vapour pressure partial pressures reflectivity phase transitions thermal conductivity thermal diffusivity heat capacity cp density

ns

phase transitions reflectivity transmission thermal conductivity

phase transitions reflectivity

ps

phase transitions reflectivity speed of sound EOS parametersa transmission

phase transitions reflectivity speed of sound EOS parametersa

a

Indicates a method under development.

499

500

Extreme Conditions

experiments can be found in review articles [69–71]. Recently, laser pulse experiments, producing lase shocks at very extreme conditions, have been reported, for example, equation of state parameters including speed of sound [72–74]. Laser pulse heating of very small samples in a diamond anvil cell [75] will not be discussed here. Millisecond and Microsecond Heating Pulses In the millisecond range, laser heating is used to measure thermal diffusivity and heat capacity of materials in the solid and liquid phase near the melting point. In the microsecond range, pulsed laser heating is used to obtain thermophysical properties of high melting materials, such as those used in nuclear reactors. In an acoustic levitation experiment, spherical samples of fuel material in a pressure vessel (at pressures from 0 up to p ¼ 0.4 GPa) are pulse heated by means of four symmetrically arranged laser beams (100 ms pulse duration). Time-resolved electrical, optical and X-ray measurements are performed, from which temperature, spectral and total emittance, density, thermal expansion, enthalpy, and heat capacity of the sample can be obtained to temperatures above 5000 K [76]. Measurements of the dependence of melting temperatures on pressures have been reported at pressures up to 0.25 GPa [77]. In another experiment, the sample surface is heated by microsecond laser pulses and simultaneously characterised by photoelectron spectroscopy (ultraviolet photoelectron spectroscopy and X-ray photoelectron spectroscopy) in a time resolved mode yielding spectra, in one measurement cycle, that can be associated with different sample temperatures or phases. This method can be applied to various research fields, previously inaccessible to standard surface analytical techniques. Photoelectron spectroscopy can be combined with other experimental techniques such as time-of-flight mass spectrometry. Potential applications of these new methods include surface investigations under extreme conditions including material degradation or decomposition and reactivity studies at high temperatures, photochemical reactions or the determination of physical properties at high temperatures [78,79]. Nanosecond Heating Pulses There is some advantage in using laser pulses in the ns regime in terms of both efficiency of heating and stability of the heated area after melting. During heating and rapid cooling, time resolved ellipsometry and transient reflectivity measurements are used to determine the electronic structure of molten materials. These techniques are limited to excitations not involving excessive material evaporation which might affect optical and electrical measurements. In a typical experiment, excimer laser pulses with 15 ns duration induce melting and evaporation of thin layers in vacuum. Sample surface modification in the nanosecond timescale can be monitored in situ with optical probes. Surface temperatures up to 10 000 K can be reached [80]. Laser pulse induced melting has 500

Extreme Conditions

501

also been studied in the ns timescale. Time resolved reflectivity measurements are used to investigate fast phase transitions [81,82]. Femtosecond Heating Pulses The ultimate regime in surface heating of solids occurs with femtosecond pulses. Here, the laser pulse duration reaches the characteristic times for energy relaxation of the photoexcited electron gas to the lattice. The heat is deposited in such a short time that the heat conduction out of the light-absorbing volume does not occur within the laser pulse duration. Extremely high temperatures are generated in a shallow layer which is still in a quiescent state and can be observed before matter escapes the irradiated area by evaporation or ablation. These circumstances are extremely attractive for the study of condensed matter at high-temperature including thermophysical property measurement. Properties of condensed matter under planetary interior conditions can be measured with femtosecond spectroscopy. When an intense femtosecond pulse interacts with a solid target, energy is deposited into solid density electrons within a skin depth about 10 nm thick before surface thermal gradient form. Thus the electrons, then the ions, are heated at constant volume, generating internal pressures up to 100 GPa. Consequently ‘deep earth’ pressure-temperature conditions are created. A time window exists in which a femtosecond probe pulse of any desired wavelength, incidence angle, and polarisation may reflect from this heated surface to perform spectro-ellipsometry measurements [83,84]. The dynamics of melting and evaporation with picosecond laser pulses has been studied by means of reflected probe pulses [85].

10.2.3

Levitation Techniques

Containerless processing techniques provide non-contact conditions in which most interactions between liquid samples and the environment are eliminated. In this method the liquid sample is suspended or floated in a levitation cell by an external force. This is accomplished by one or more of the following levitation techniques: electromagnetic levitation, aerodynamic levitation, acoustic levitation, electrostatic levitation and microgravity levitation in space. A summary of levitation techniques is given in Table 10.5. Heating is achieved with induction from an ac power source (operating at frequencies between (0.1 to 1) MHz), an incandescent radiator or laser radiation [86]. Levitation techniques are used at high temperatures, but they are not necessarily limited to subsecond measurements. Therefore, no timescale of experiments will be given here. Temperatures are usually measured with a pyrometer. Levitation techniques are required for the determination of thermophysical properties such as surface tension and viscosity that cannot be determined with resistive pulse heating methods. The measurement of surface tension is discussed in Section 10.3.5. For more detailed information on different levitation techniques, review articles [86,87] should be consulted. 501

502 10.2.4

Extreme Conditions Electromagnetic Levitation

For metallic melts, electromagnetic levitation provides an elegant method of noncontact containerless measurement. Electromagnetic levitation has mostly been used for the study of highly reactive melts at high temperature. In these experiments, the sample assumes a simple spherical shape, is contained in a clean environment, and can be studied over a large temperature range. Electromagnetic levitators employ inhomogeneous radio-frequency electromagnetic fields to heat and position the samples. Such a field has two effects on a conducting, diamagnetic body. First, it induces eddy currents within the material, which, owing to ohmic losses, inductively heats the sample, and second, it exerts a Lorentz force on the body, displacing it towards regions of lower field strength. The latter effect can be used to compensate for the gravitational force [88]. On Earth, strong magnetic fields are needed to compensate for the gravitational force. Special arrangements (e.g., as provided by microgravity facility known under the acronym TEMPUS [89]) have been built to investigate materials in a low gravity environment on board a space shuttle. A microgravity environment offers the possibility to minimise the magnetic positioning force. Therefore, in the cooling phase, after melting, only negligible forces act on the sample, and the spherical shape is maintained. With this approach, liquid metal properties such as surface tension and viscosity can be determined with the oscillating drop technique by monitoring the sample with a fast video-camera [88,90]. Recently, the potential of levitation techniques for the study of undercooled melts has been recognised and exploited. Due to the absence of the container walls, which are usually the major source of heterogeneous nucleation sites, high-purity conditions can be maintained. This allows undercooling of the melt, and the liquid is more quiescent than in equilibrium. This quiescence renders the undercooled liquid suitable for structural studies [91], with, for example, X-ray-absorption, for fine structure determination. A schematic display of an electromagnetic levitation system used for emissivity measurements is given in Figure 10.14. The sample is levitated in a high frequency coil, driven by an RF generator, that is usually water-cooled. The levitated sample is placed in a vacuum chamber with quartz windows for various types of optical diagnostics. The ambient cooling-gas also has to be supplied to the experimental chamber. Techniques proposed for the measurement of electrical conductivity and other physical properties within microgravity are described in reference [89]. Noncontact measurements of electrical conductivity of a liquid or solid material can also be based on electromagnetic induction. Reliable conductivity data, obtained from a ground based levitation facility, have recently been published [92].

10.2.5

Future Directions

Future progress in high temperature technologies depends strongly on the availability of reliable data on the thermodynamic and transport properties of 502

Extreme Conditions

Figure 10.14 ments.

503

Schematic of an electromagnetic levitation experiment for emissivity measure-

candidate materials. Such experimental data are of extreme importance at very high temperatures, where the values can differ dramatically from predictions based on extrapolation of low temperature data. Numerical simulations of fluid flow, heat transfer, solidification or thermal induced stresses have become of great significance

Table 10.3 Thermophysical properties that may be measured with electromagnetic levitation methods. Solids

Liquids

melting temperature normal spectral emittance hemispherical total emittance reflectivity heat capacity thermal expansion density enthalpy of fusion vapor pressure resistivitya thermal conductivity phase transitions transmission growth phenomena (dendritic, eutectic) grain refinement, nucleation

melting temperature normal spectral emittance

non equilibrium phases metastable phases quasicrystals metallic glasses a

reflectivity heat capacity thermal expansion density enthalpy of fusion vapor pressure resistivitya thermal conductivity structure of undercooled liquids surface tension viscosity partial pressures optical constants, dielectric constants, refractive indices undercooling

Indicates a method under development.

503

504

Extreme Conditions

in various industries. With the advent of adequate computing power, full threedimensional calculation of the determining physical equations has become possible. The major drawback to these simulation techniques is the lack of accurate thermophysical property data. For example, an important input parameter for the heat transfer equation is the thermal conductivity. Since direct measurements of thermal conductivity of alloys are almost impossible, it is often estimated from the electrical conductivity using the Wiedemann-Franz-law. Future research on subsecond thermophysics should be directed towards new and more accurate dynamic measurement techniques. For example, the melting point of carbon is uncertain to within +500 K. Measurements at higher temperatures and pressures should lead to critical point data of all metals, which, except for a few metals like lead, indium, zinc, gold, and iron, only can only be estimated at present. Generation of new materials with higher melting temperatures than pure elements could also be studied with these techniques, for example, hafnium nitride is a candidate compound to study, as it reaches a melting temperature of 4600 K whereas the melting temperature of pure hafnium is only 2500 K. In addition, these techniques can assist in the formulation and study of new alloys and compounds with improved properties compared with pure elements and compounds. For example, CMSX-4, which is a aluminium compound may be studied with the techniques described herein. Comparisons between the results obtained from equilibrium and transient techniques will clarify the limits of validity of ultrafast heating techniques. For example, the microgravity facility TEMPUS provides the opportunity to compare electrical conductivity data measured on liquid samples obtained from pulse heating with those obtained by means of the quasi-static induction method used in TEMPUS. Combining results obtained from the techniques described here with theory will significantly contribute to our understanding of matter under extreme conditions of temperature and pressure and under conditions far removed from thermodynamic equilibrium.

10.3

Molten Metals I. EGRY DLR, Institute for Space Simulation Cologne, Germany

Molten metals can be categorised, with respect to temperature, into three different classes: low melting point metals with melting temperatures Tm less than 800 K, high melting point metals with Tm between (800 and 2000) K, and refractory metals with Tm > 2000 K. For low melting point metals, the conventional measurement techniques, as described in this book, can be applied. Therefore, this section is restricted to high-temperature and refractory molten metals. While Section 10.2 on high temperatures discussed mainly the different heating methods, this section will concentrate on measurement techniques for specific thermophysical properties. 504

Extreme Conditions

505

From a technological standpoint, there are at least two important classes of metals: low density materials and high temperature (and corrosion) resistant materials. Aluminium, with (Tm ¼ 933 K), and magnesium (Tm ¼ 922 K) belong to the first class, whereas iron (Tm ¼ 1808 K), and nickel (Tm ¼ 1726 K), are representative of the second class. Titanium (Tm ¼ 1933 K) is a member of both classes. Recently, alloys, which form bulk metallic glass even at moderate cooling rates have received much attention. These alloys have Tm around 1000 K, and contain zirconium and, sometimes, beryllium as major components and are, therefore, highly reactive. It is rather difficult to find suitable containers and sensors which remain inert when in contact with these liquid metals. On the International Temperature Scale of 1990 (ITS-90) any temperature above the melting point of silver (Tm ¼ 1234.93 K) is defined in terms of blackbody radiation [93], which is a non-contact measurement. In addition, some thermophysical properties, in particular surface tension, are very sensitive to contamination [94]. For example, a mole fraction of a few 106 of oxygen can reduce the surface tension of some molten metals by 10 per cent. As a consequence, much effort has been expended on the development of suitable containers, and, as a radical solution, in the development of containerless measurement techniques. These will be discussed in the following subsections. In writing this chapter, existing excellent review papers of the topic have been extensively used [39,95–98], and the reader is referred to these articles for further details. In contrast to existing literature, this contribution focuses on the measurement of thermophysical properties of molten metals with a particular emphasis on recent developments, such as the use of levitation methods. The utilisation of the microgravity environment, on board an orbiting spacecraft, for thermophysical property measurements will not be discussed here. The interested reader is referred to two related review articles [99,100].

10.3.1

Containers Crucibles

In the study of molten metals it is crucial that an appropriate crucible material is selected. The crucible should mechanically withstand high temperatures and also not react with the molten metal sample. Depending on the application either perfect wetting or perfect non-wetting of the crucible by the liquid metal is desired. Here the term ‘crucible’ refers not only to a container, but to any material in contact with the sample, for example, the gas tube immersed in the melt in maximum bubble pressure experiments. Clearly there is not one ideal crucible material for high temperature applications and the selection of suitable crucible material can only be determined when the material to be contained is specified. Generally speaking, one can define two classes of crucible materials. The first class of crucible materials constitute refractory metals: platinum (Tm ¼ 2045 K), molybdenum (Tm ¼ 2890 K), tantalum (Tm ¼ 3270 K), graphite (Tm ¼ 3920 K) and 505

506

Extreme Conditions

tungsten–rhenium alloys (Tm > 3000 K). When crucibles constructed from these materials are used with liquid metals, the phase diagram for the metals should be consulted to insure that there is no eutectic or low melting composition, which would lead to partial melting and alloying of the sample and the crucible. Obviously, the use of a graphite crucible is inadvisable for the study of materials that form carbides, such as liquid silicon. The second class of crucible materials are ceramics, for example, alumina (Al2O3, Tm ¼ 2300 K), zirconia (ZrO2, Tm ¼ 2890 K), boron nitride (BN, Tm & 3000 K), silicon nitride (Si3N4) and silicon carbide (SiC). The process used to manufacture the ceramic crucible material, is as important as their chemical reactivity with the substance under study. These crucible materials are usually hot pressed or sintered, and, depending on the process, have different porosities. In addition, these materials can contain binders and gaseous inclusions as a result of the manufacturing process. When used under vacuum conditions, this leads to unacceptable outgassing, which makes these crucible material inappropriate for some specific applications. A novel, and promising, class of crucible materials are aerogels [101]. These materials are not wet by most liquid metals and provide, owing to their low thermal conductivity, nearly perfect thermal isolation. In addition, most aerogels are transparent for visible and infrared light, so that optical methods, such as pyrometry can be used to measure the temperature of the contained liquid sample. Commercially available aerogels are silica-based and therefore not applicable at high temperatures, but recent research has developed high-temperature graphitebased aerogels [102]. Cammenga et al. [103] have examined the compatibility between conventional crucible materials and selected calibration substances. Part of their results, relevant to liquid metals, are reproduced in Table 10.4 and in general, the ceramic crucibles seem to be the better choice. The quality of a crucible with respect to wetting phenomena and temperature resistance can be further improved by coating the inner wall of the crucible. For high temperature applications and for highly reactive materials, such as titanium alloys, yttria (Y2O3, Tm & 2680 K) has proven a good coating material [104]. When selecting Table 10.4 Compatibility between crucible and sample material. [, compatible; (þ), partial solution with negligible effect; 6, melt dissolves crucible; ?, compatibility unknown. Crucible/Sample

Al

Ag

Au

Al2O3 BN C Pt Mo Ta W

[ [ [ 6 ? 6 (þ)

[ ? [ 6 ? [ [

[ ? 6 6 6 6 (þ)

506

Extreme Conditions

507

a suitable coating material, the different thermal expansion of coating and substrate must be taken into account. Even if the crucible is compatible with the melt, the choice of the crucible material may still influence the measured thermophysical properties. Indeed, there is some evidence that wetting between crucible and sample may affect the measurement. For example, Ruppersberg and Speicher [105] report that reproducible density measurements, using the maximum bubble pressure method, can only be achieved when the melt wets the bubble tube. An even more pronounced effect of the wettability on the results has been reported for viscosity measurements of liquid silicon with an oscillating viscometer [106]. The apparent viscosity when measured in a silicon carbide crucible is about 60 per cent higher than that determined in a boron nitride crucible. Sato et al. [107] repeated this experiment and performed a parametric study using boron nitride, alumina, silicon nitride and graphite crucibles. They found reasonable agreement between the results obtained with all of the crucibles materials used as well as with data obtained using a boron nitride crucible previously. Therefore, it is plausible that the data from the silicon carbide crucible are affected by a chemical reaction between container wall and silicon melt. Materials science research on metallic glasses has traditionally studied the glass transition by approaching the transition from low temperatures and taking advantage of the solid-like behaviour of the glass described in Section 10.1. More recently, the glass temperature has also been approached from the undercooled liquid region, with moderate cooling rates between (1 to 10) K ? s1 [108,109]. If these measurements are performed in a crucible the catalytic potency of the container wall for heterogeneous nucleation and consequent crystallisation must be suppressed. Bardenheuer and Bleckmann [110], achieved this by embedding the sample into an oxide glass flux. The glass oxide inhibits the contact between crucible wall and sample, but, in addition, dissolves potential metal oxide impurities in the sample, thereby stabilising the undercooling. Suitable flux materials are boric oxide (B2O3) and boron-silicate glasses, such as either Duran2 or Pyrex2. Willnecker et al. [111] used these oxide glass fluxes in a differential scanning calorimeter and were able to investigate a wide range of metallic alloys at temperatures as high as 1600 K; they obtained an undercool of more than 350 K for Co80Pd20, which exceeds the limit of hypercooling. Containerless Techniques As mentioned in the introduction, the use of containerless methods is an elegant way to avoid reactions between the crucible and sample. Handling and melting of a sample without a container requires complex instruments, which are also not easy to combine with established diagnostic tools. Levitation techniques were discussed in Section 10.2.3 and further information can be found in [87]. In Table 10.5 the characteristic features of the levitation methods are compared and a brief description of these methods follow. Aerodynamic levitation (ADL) is probably the simplest of the levitation methods [112]. In this method a conical nozzle is placed below the sample, and a 507

508

Extreme Conditions

stream of inert gas blown through it. The gas flows around the sample levitating it and, owing to the nozzles’ conical shape, also provides lateral stability. The sample is usually heated with a laser. The sample size is limited to less than 1 mm in diameter. ADL is used to study high temperature ceramics, such as alumina, and could, in principle, be applied to liquid metals. Electrostatic levitation (ESL) uses, as shown in Figure 10.15 for the DLR apparatus, electric fields between several electrodes to position the sample [113]. Two electrodes, one placed atop and one below the sample, are used to provide the field required to overcome gravity. A further set of electrodes, placed horizontally around the sample, provides lateral stability. The sample is heated by a laser. ESL can be used for all kinds of materials, but it only operates under UHV conditions and, consequently, is unsuitable for materials with a high vapour pressure. ESL is intrinsically unstable and requires an active electronic feedback control loop. This is achieved by illuminating the sample with auxiliary lasers, detecting the shadow of the sample with positioning sensors, and in real-time feeding the sensor signals back into the high voltage power supplies that provide the positioning fields. The full potential of ESL has yet to be determined, but it appears to produce the most stable levitation of all techniques. For electrically conducting materials, such as liquid metals and liquid semiconductors, electromagnetic levitation (EML) is the most widely used levitation technique. Electromagnetic levitation is intrinsically stable. In this method, the sample is placed inside a levitation coil, which is connected to an rfgenerator. A current of a few hundred amps is used to produce an inhomogeneous electromagnetic field in the sample, which, in turn, induces eddy currents. The interaction of these eddy currents with the field produces a Lorentz force which is used to compensate the gravitational force. The eddy current also has ohmic losses, owing to the finite electrical conductivity of the metal, that heat, and eventually

Figure 10.15 Electrostatic levitation device showing the top and bottom electrodes, with ceramic discs in the centre and lasers for the position sensing devices.

508

Extreme Conditions

509

melt, the specimen. This inductive heating is so efficient that it is possible to melt most refractory materials [114] and it is one of the major drawbacks of terrestrial electromagnetic levitation devices for temperatures below 1200 K. The power required to overcome gravity with an electromagnetic field is so high, and the heat produced so great, that most of the heat has to be removed from the sample by convective gas cooling. In principle, electromagnetic levitation is independent of a gas atmosphere. It has been used under ultra high vacuum (UHV) conditions in two microgravity experiments [115,116]. Two different designs are used for EML: glass-tube and vacuum chamber. In the former design, the coil is wound around a glass tube, whereas the latter design consists of a vacuum chamber with the coil inside. Examples of the two designs are shown in Figures 10.16 and 10.17, respectively. So called drop tubes are also used to provide containerless conditions. In this approach, a molten drop is released at the top of an evacuated tube and allowed to fall under the influence of gravity. During free fall, which is typically less than 1 s, the liquid drop cools by radiation and eventually solidifies, either during the fall or on impact with the bottom of the tube. This method has been used for solidification and nucleation studies of refractory metals. During cooling, the specific heat capacity divided by the total hemispherical emissivity can be determined. In summary, drop tubes are of limited use for thermophysical property measurements.

10.3.2

Pyrometry

For high temperature measurements thermocouples, which undergo chemical reactions with the molten metal, cannot be used. For containerless techniques,

Figure 10.16

Glass tube for electromagnetic levitation.

509

510

Extreme Conditions

Figure 10.17

Electromagnetic levitation vacuum chamber.

they cannot be used at all. Therefore, non-contact temperature measurement techniques, such as pyrometry, are applied. Pyrometry (or radiation thermometry also described in Section 2.6) is based on Planck’s law of blackbody radiation, which relates the emitted power to the true thermodynamic temperature of a black body. In Wien’s approximation of blackbody radiation, the spectral radiance per unit area, Lb(l, T), of a black body is given by [117]:  c  c1 2 exp  ; where lT l5 c1 ¼ 1:19 ? 108 W ? mm4 ? m2 ? sr1

Lb ðl; TÞ ¼

and

ð10:12Þ

c2 ¼ 1:44 ? 104 mm ? K: In Equation (10.12), l is the wavelength and T the temperature. The total black body radiance Lb(T) emitted is obtained by integration over l and yields the StefanBoltzmann law: Lb ðTÞ ¼ ðs=pÞT 4 ;

ð10:13Þ

where s is the Stefan-Boltzmann constant. Temperature measurement can be based on both Equations (10.12) and (10.13), the former is a spectral pyrometer, the latter a total radiance pyrometer or bolometer. Unfortunately, the surface of a molten metal is far from being a black body. To account for this difference, the emissivity, which is e < 1, of a real body must be 510

Table 10.5 Summary of the levitation methods, environment, and materials along with general comments concerning containerless techniques. Levitation

Environment

Comments

Material

511

Positioning

Heating

Sample visibility

Atmosphere

Temperature Range

Liquid/ Solid

Conductive/ Nonconductive

Sample mass

Acoustic

Sound waves

Laser

Good

Gas

Both

Both

mg

Aerodynamic

Gas jet

Laser

Poor

Gas

Both

Both

mg

Drop tube

Free fall

None

Poor

Both

Both

mg

Electromagnetic

Lorentz force

Inductive

Good

1000–3000 K

Both

Coulomb force

Laser

Excellent

2000 K

Optical

Light pressure

Both

Both

mg

Aeroacoustic

Hybrid

Not applicable Limited by material property and heating device

Both

Both

mg

Good Laser

Poor

Gas

mg

Isothermal conditions only Mainly applied to oxide glasses Short-duration experiments Also used in microgravity Active position control required Very limited size Suited for glass and ceramics

Extreme Conditions

Method

511

512

Extreme Conditions

introduced into Equations (10.12) and (10.13) to give: Le ðl; TÞ ¼ en ðl; TÞ Lb ðl; TÞ;

and

Le ðT Þ ¼ eh ðT ÞLb ðT Þ;

ð10:14Þ ð10:15Þ

where en(l,T) is the normal spectral emissivity and eh(T) the total hemispherical emissivity. (Strictly speaking, the radiation emitted from a real surface may not be isotropic, and e will also depend on the angle of radiation. It is usually assumed that the radiation emitted normal to the surface is detected.) Conventional pyrometers are usually single-colour devices, based on Equation (10.14), operating in a narrow band around a central wavelength l0. If the spectral emissivity is known from independent measurements, for example, from ellipsometry [118], Equations (10.12) and (10.14) allow a direct determination of the true temperature [119] from: 1 1 l0 ¼ þ ln½eðl0 ; TÞ; T Tb c2

ð10:16Þ

where Tb is the temperature of the pyrometer calibrated with a black body. However, in practice the emissivity is not known, and the pyrometer is calibrated at a temperature T0, which in most cases is taken to be the melting temperature of the sample. This temperature is known from independently determined thermodynamic data, such as phase diagrams, and is easily recognisable, from the thermal arrests, in either heating or cooling curves. In this calibration, the pyrometer must point at a completely molten surface, because the emissivity of the solid at the same temperature may be significantly different. Formally, the emissivity e(l0 , T0 ) can be obtained from:  eðl0 ; T0 Þ ¼ exp

  c2 1 1  : l0 T0 Tb

ð10:17Þ

Once e(l0 , T0 ) is determined, it is used throughout the entire temperature range. In view of the fact that the emissivity depends on temperature, this is of course an approximation. It has been shown however [118], that the emissivity of pure metals depends only slightly on temperature. For alloys, in particular, if there is surface segregation, this assumption may not hold. Independent measurements of the normal spectral emissivity of the liquid phase, as a function of temperature, will improve the accuracy of all thermophysical data, obtained with both resistive heating experiments and levitation techniques. The emissivity introduced above is in fact an effective emissivity. In addition to the true emissivity of the radiating body, it contains an additional factor for the transmissivity of the optical path between the object and the pyrometer. If the 512

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513

radiation has to pass through windows or is reflected by mirrors, this transmissivity will be less than unity. In the case of molten metals, especially when studied under vacuum conditions, there is considerable evaporation from the sample. This metal vapour will be deposited on cool surfaces such as mirrors and viewports within the optical path leading to a change, in most cases to a decrease, in the transmissivity and, consequently, to an apparent decrease in temperature. Some precautions have to be taken to minimise this effect. One possible solution consists of placing an exchangeable protective window in front of the optics of the pyrometer; another method, which has proven effective, is to place a double-mirror periscopic system in front of the pyrometer [120] as shown in Figure 10.18. Although metal vapour is deposited on the primary mirror, the overall reflectivity of such a system is less affected than the transmissivity of a window. In any case, it is advisable to repeat the calibration measurement from time to time to compensate for degradation of the optical path. Numerous approaches have been developed to eliminate the emissivity from pyrometric temperature measurements. A popular approach is the use of the socalled ratio or two-colour pyrometer [121]. This instrument measures the radiation at two different wavelengths l1 and l2 and determines the temperature from the ratio of the two radiance signals. By analogy with the derivation of Equation (10.16), the

Figure 10.18

Periscopic double mirror system to prevent coating of pyrometer window.

513

514

Extreme Conditions

following equation for the temperature is obtained:   1 1 l1 l2 eðl2 ; TÞ ¼ ln þ : T Tb eðl1 ; TÞ c2 ðl2  l1 Þ

ð10:18Þ

The argument for ratio pyrometry is that the term containing the logarithm of the two emissivities will, usually, be small, and, moreover, the temperature dependence of the emissivities at different wavelengths will also be small, so that, by taking the ratio, the logarithm will be temperature independent. In practice, unfortunately, these assumptions often do not hold, because the effective emissivities also contain the transmissivity of the optical path. If there is some degradation due to evaporation, this will usually affect the longer wavelength more strongly and the temperature ratio will drift from its original value.

10.3.3

Calorimetry Differential Scanning Calorimetry

The principle of differential scanning calorimetry (DSC) has been described in detail in Chapter 7.3 of this book and elsewhere [122]. Therefore, in the following, the focus is on those aspects of DSC which are specific to molten metals at high temperature. These are common to differential thermal analysis (DTA), differential heat flux calorimetry (Heat Flux DSC) and power-compensated differential scanning calorimetry (Power-DSC) and all these instruments are included in our discussion. In order to distinguish between DTA and DSC, it suffices to say that DTA can be used to identify temperatures of exo- or endothermic reactions, such as phase transformations, while DSC allows the quantitative determination of heat capacity and enthalpy of fusion. The term differential refers to the fact that the sample is measured relatively to a second reference sample of approximately the same, but known, heat capacity. One experiment consists usually of three separate runs each with the same reference crucible and the same heating rate: a scan with two empty pans; a scan with one pan containing the unknown sample; and, a scan with a reference sample (usually sapphire) replacing the unknown. Occasionally, a fourth run, which repeats the first scan with two empty pans, is included in the measurement procedure. The specific heat capacity of the sample, csp , is obtained from Equation (7.62): csp ðTÞ ¼ crp ðTÞ

mr DT3  DT1 ; ms DT2  DT1

ð10:19Þ

where DT1, DT2, and DT3 are the DSC signals with empty pans, of the reference sample, and of the sample respectively. In Equation (10.19), mr and ms are the masses of the reference and sample respectively, and crp is the heat capacity of the reference 514

Extreme Conditions

515

sample. In DSC the baseline must be determined along with the effect of finite heating and cooling rates by measurements at different heating rates and extrapolating to zero heating rate. A specific problem of high-temperature DSC is related to the choice of crucible with respect to both chemical reactions and thermal properties. For compatibility reasons, one is forced to use ceramic crucibles. In the derivation of the working equation for DSC, Equation (10.19), it is assumed that heat flow is controlled through the pans and registered by the instrument sensors. However, at high temperature, radiative heat loss from the ceramic crucibles is not accounted for. These losses are proportional to surface area, rather than volume, and the DSC signal is no longer directly proportional to the sample mass. This effect can be minimised by use of metallic, usually platinum, crucibles coated with, for example, yttria or zirconia ceramics. Alternatively, the effect may be minimised by encapsulating the ceramic crucible in a metallic crucible, which acts as a heat shield, and also has no direct contact with the sample. Commercially available DSC’s are, because of these issues, usually limited to temperatures below 1900 K. Only a few investigations of glass transition at high temperatures have been reported [104,111,123]. For higher temperatures, alternative techniques must be used which will be discussed in the remaining sections.

Drop Calorimetry In drop calorimetry, the heat content of a specimen dropped into a calorimeter is determined from measurements of the temperature rise of the calorimeter. The calorimeter may be either placed into a thermostat (isothermal) or isolated from its surroundings (adiabatic). If the thermal mass of the sample is much less than the thermal mass of the calorimeter, the temperature rise is proportional to the heat content QK of the sample at the time when it enters the calorimeter: QK ¼ CK DT:

ð10:20Þ

Here, CK is the calibration constant of the calorimeter, and DT the temperature rise measured in the calorimeter corrected for inevitable heat losses, as described in reference [122]. The temperature of the drop is measured before it is released and the heat losses Qf during the fall must be accounted for with the relationship: QT ¼ Qf þ QK ;

ð10:21Þ

where QT is the heat content of the sample at release. If the drop calorimeter operates under vacuum, the heat loss Qf is entirely due to radiation, and the falling time tf is 515

516

Extreme Conditions

given by: sffiffiffiffiffi 2h : tf ¼ g

ð10:22Þ

In Equation (10.22), h is the distance the drop falls and g the local gravitational acceleration of free fall. However, in most experiments the calorimeter is filled with inert gas, such as helium or argon, and there are additional heat losses and time delay arising from the drag on the falling sample. In this case, the heat loss Qf is given by: Ztf Qf ¼

Ztf AaðT  T0 Þ dtþ

0

Aeh sSB ðT 4  T04 Þ dt;

ð10:23Þ

0

where a is an empirical heat transfer coefficient of the gas, A the surface area of the drop, eh the total hemispherical emissivity, sSB the Stefan-Boltzmann constant, T the sample’s temperature at time t, and T0 the ambient temperature. If the falling time tf is short, it is often assumed that the sample temperature does not change during fall, and the integration may be replaced by a multiplication. At high temperatures, characteristic of liquid metals, radiative heat loss dominates, conductive heat loss may be neglected, and the sample temperature is always much greater than ambient temperature (T04 55T 4 ) so that Equation (10.23) can be simplified to give: sffiffiffiffiffi 2h Aeh sSB T 4 : Qf ¼ g

ð10:24Þ

Neglecting the heat loss due to heat conduction underestimates Qf, the neglect of the backradiation from the environment overestimates Qf. If the fall time is short enough, Qf is only a small correction to the total heat content QT, which is simply related to the enthalpy H of the sample through: QT ¼ n½H ðT Þ  H ðT0 Þ;

ð10:25Þ

where n is the amount-of-substance of the sample and T0 the temperature of the calorimeter. Combining Equations (10.20) and (10.24) with (10.25), the relationship: sffiffiffiffiffi 2h Aeh sSB T 4 CK DT ; DH ¼ HðTÞ  HðT0 Þ ¼ þ g n n 516

ð10:26Þ

Extreme Conditions

517

is obtained. The determination of DH requires knowledge of sample mass, sample surface area, total hemispherical emissivity, and the temperature of the drop at release. The calorimeter must also be calibrated, and the temperature rise DT corrected. Measurements of DH as a function of T are used to determine the heat capacity at constant pressure Cp(T) with the definition: Cp ¼ ðqH=qTÞp :

ð10:27Þ

The measured DH(T) are differentiated and this operation tends to decrease the accuracy of the Cp so determined. Drop calorimetry is a popular technique, and, in the case of liquid metals, it is often combined with electromagnetic levitation [124,125]. An extensive review of ‘Thirty years of levitation melting calorimetry’ has been published recently by Frohberg [126]. The levitated liquid metal droplet is maintained at a constant temperature, which is determined with pyrometry (described in Section 10.3.2). Turning off the levitation field, releases the sample and it falls into the calorimeter block (typically made from copper) which has a small hole to receive the liquid drop. A thin foil is fitted to the cavity to avoid splashing of the metal drop as it enters the hole within the Cu calorimeter block. Heat losses are reduced by closing the cavity with a shutter as soon as the drop has fallen into the calorimeter. Levitation drop calorimetry can be used at temperatures up to 3000 K, but the correction term Qf becomes significant at such high temperatures. The containerless nature of this technique provides access to the undercooled state and to data in this regime. A variant of levitation calorimetry was introduced by Qin et al. [127], who termed it levitation alloying calorimetry, that can be used to measure the enthalpy of mixing of liquid alloys. To measure the enthalpy of mixing of alloy AB in the high A mole fraction side of the phase diagram, a sample of pure A of mass mA is levitated and melted. The levitation device is modified in such a way that small samples of pure B of mass mB can be dropped onto the levitated and molten drop of A, leading to the formation of a liquid drop of composition AxBy, with x ¼ nA =(nA þ nB ) and y ¼ nB =(nA þ nB ), where nA and nB are the amount-of-substance of A and B. Under the condition of energy conservation, the enthalpy balance of the mixing process can be expressed by: Dmix Hm ¼ nB ½Hm ðA; T Þ  Hm ðB; T0 Þ þ ðnA þ nB ÞCp ðAx By ÞDT;

ð10:28Þ

where T and T0 are the temperatures corresponding to samples A and B before release respectively, and Cp(AxBy) is the heat capacity of the alloy. If the enthalpy of substance B, Hm(B, T), and the heat capacity of the alloy are known, the enthalpy of mixing can be derived from the measured temperature change DT, which is usually determined pyrometrically, of the levitated drop. By subsequent addition of sample B to the alloy AxBy, the enthalpy of mixing can be obtained as a function of xA. Of course, eventually, the levitated drop will become too heavy for the levitation field to hold. 517

518

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Unfortunately, the accuracy of the results obtained from this very elegant approach are controlled by the determination of DT and knowledge of Cp(AxBy). The measured value of DT must be corrected for the additional heat input into drop B from the electromagnetic levitation field. The group of Frohberg has given some rules how to do this. In addition, there is a change in emissivity as a result of the alloying process and this variation must be taken into account when DT is determined pyrometrically. The quantity Cp(AxBy) is essentially unknown and to overcome this limitation Lin and Frohberg [128] assumed that the Neumann-Kopp rule holds and they neglected the excess heat capacity, CpE in the relationship:

Cp Ax By ¼ nA Cp; m ðAÞ þ nB Cp; m ðBÞ þ CpE :

ð10:29Þ

For ideal solutions, this is a reasonably good approximation. In a later paper, reporting application of the method to titanium and zirconium alloys, Qin et al. [127] took the excess heat capacity into account with a value obtained from an association model [129].

AC Calorimetry of Levitated Drops Fecht and Johnson [130], have shown that the principle of ac calorimetry, described in Chapter 7.2, can be applied to levitated drops. In this experiment the power absorbed by the sample must be precisely known as a function of the power supplied to the levitation coil. To determine the heating efficiency, an electromagnetic calculation is used, which requires knowledge of the exact field geometry along with the shape and position of the sample within the electromagnetic field. For nearly homogeneous magnetic fields and a spherical sample, the heating efficiency has been calculated by Loho¨fer [131]. The main parameter in these calculations is the penetration depth of the electromagnetic field into the sample and is given by the skin depth d: d ¼ ðpf m0 sÞ1=2 ;

ð10:30Þ

where f is the (radio-)frequency of the magnetic field, m0 the magnetic permeability, and s the electrical conductivity of the sample. The electrical conductivity must be known in order to calculate the heating efficiency and hence the heat capacity. In ac calorimetry the power input into the sample is sinusoidally modulated &DPf cos (2pft), and the temperature response DTf observed at the same frequency. For ac calorimetry the thermal environment is characterised by two relaxation times, discussed in Section 7.2, an external relaxation time te, and an internal relaxation time ti. Under ultra-high-vacuum (UHV), the power loss within the experiment is 518

Extreme Conditions

519

entirely radiative, and the relaxation time te is given by: te ¼

cp ; 16pR2 sSB eh T 3

ð10:31Þ

where cp is the (unknown) specific heat capacity, R the radius of the drop, sSB the Stefan Boltzmann constant defined in Equation (10.13), eh the total hemispherical emissivity, and T the sample temperature. The internal relaxation time ti which is governed by heat conduction (and possibly also convection) in the molten metallic sample, is given by: ti ¼

cp fV ; keff

ð10:32Þ

where, keff is the (effective) thermal conductivity of the liquid metal (including convective contributions) and fV the volume fraction heated by the rf-field (fV & 3 d/ R). Because of the high thermal conductivity of metals, te 44ti , which means that a modulation frequency f for ac calorimetry can be found, which satisfies both the adiabatic and the isothermal condition given by Equations (7.7) and (7.15): 1=te 55f 551=ti

ð10:33Þ

In practice, the modulation frequency is between 0.1 and 1 Hz and, under these circumstances, the heat capacity can be obtained from Equation (7.8) with:

cp ¼ DPf = 2pf DTf :

ð10:34Þ

In most levitation devices, the power input, Pin, into the sample is controlled by a voltage Uc so that: Pin ¼ aUc2 ;

ð10:35Þ

where a is a constant characterising the rf-circuit including the sample. When the control voltage is modulated according to: Uc ¼ U0 þ Um cosð2pftÞ;

ð10:36Þ

the following power modulation results:   2 2 P ¼ a U02 þ 12Um þ 2U0 Um cosð2pftÞ þ 12Um cosð4pftÞ :

ð10:37Þ

The first term in Equation (10.37) is the unmodulated rf power and it determines the 519

520

Extreme Conditions

equilibrium temperature of the sample before modulation. The modulation produces a periodic temperature response, described by the 3rd and 4th terms in Equation (10.37) superimposed over an increase in the average sample temperature, which is the 2nd term in Equation (10.37). Figure 10.19 shows the temperature response of a liquid Ni24Zr16 sample to modulation in the IML-2 microgravity control voltage (power) mission reported in reference [115]. In Figure 10.19, T0 is the equilibrium temperature before modulation, DTav the increase in the average temperature due to the second term in Equation (10.37), and DTmod the amplitude of the temperature modulation.

10.3.4

Thermal Expansion and Density

There are numerous techniques for the determination of density, but not all are well suited for high temperature systems, in particular metallic melts. The most important techniques are: archimedean, dilatometric, gamma-ray attenuation, levitated drop, maximum bubble pressure, pycnometric and sessile drop. An extensive review of the densities of liquid metals and alloys and their measurement has been given by Crawley [132] and Iida and Guthrie [96]. The Handbook of Physico-Chemical Properties at High Temperatures is an excellent data compilation [133], which also cites values of the thermal expansion albeit only at one temperature. The temperature dependence of the density may be non-linear. From the tables in reference [133], we statistically evaluated different experimental techniques for liquid metals and the results of this analysis are shown in Figure 10.20. According to this analysis, the two most popular techniques are the maximum bubble pressure (MBP) and the pycnometric method. The latter has probably the highest accuracy of all techniques, while the former is probably the easiest to use. The Archimedean and

Figure 10.19 Temperature response of a liquid Ni24Zr16 sample arising from modulation of the voltage applied to the heater (line below T0) as a function of time t. The equilibrium temperature before modulation is denoted by T0 (- - - -), the increase in the average temperature is DTav, and the amplitude of the modulation is DTmod.

520

Extreme Conditions

521

Figure 10.20 The different techniques for density measurements and their frequency of use for molten metals [133].

pycnometric methods have been discussed extensively by Crawley [132]; consequently, we refer the reader to his review and provide here only a short summary. In the Archimedean method, described in Chapter 5, a bob of known volume V and mass m1 is submerged into a pool of the liquid metal. The bob is connected through a wire of radius r to a balance and the mass m2 of the submerged bob is measured. The density r is obtained from r¼

m1  m2 þ ð2prg cos yÞ=g ; V þ pr2 h

ð10:38Þ

where g is the local acceleration of free fall and y the angle of contact. The last term in the numerator accounts for the surface tension acting on the wire and the last term in the denominator accounts for the additional volume displaced by the wire submerged to a depth h. In this experiment the largest uncertainty arises from the correction for the surface tension. In the pycnometric method the volume V is predetermined by the shape of the pycnometer, and the mass m occupying this volume is measured. The density is then obtained from r¼

m : V

ð10:39Þ

In the following, we will briefly discuss the maximum bubble pressure, dilatometric and the levitated drop method of determining density. The latter is least used but may have the highest potential. The sessile drop technique is primarily used for surface tension measurements and will be discussed in Section 10.3.5. Maximum Bubble Pressure In the maximum bubble pressure (MBP) technique, shown schematically in Figure 10.21, the liquid metal of unknown density is placed in a thermostated vessel. A capillary tube is immersed into the melt at a depth h and pressurised with an inert 521

522

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gas. The pressure required to produce a spherical bubble of radius r is given by: p1 ¼ rgh þ 2g=r;

ð10:40Þ

where g is the surface tension of the liquid metal. The measurements are performed at two different depths and the unknown surface tension can be eliminated from Equation (10.40) to give: r¼

p1  p2 : gðh1  h2 Þ

ð10:41Þ

In the derivation of Equation (10.41) it is assumed that both g and r are independent of the immersion depth. For g this means that the temperature inside the melt must be homogeneous. The radius r is that of the maximum bubble which is maintained by the capillary tube and is assumed to be determined solely by the geometry of the capillary’s tip. However, as has been shown by several authors [105,134], it is important that the liquid metal wets the capillary tube; otherwise the bubble may spread along the outer surface of the capillary [105,134]. At high temperatures, the capillary material may become porous, allowing the bubbling gas to leak out. Crawley [132] documents corrections that must be applied for both the thermal expansion of the capillary tube and the volume of liquid displaced by it. To avoid surface oxidation of the liquid metal, which could stabilise the bubble and lead to erroneous results, the inert gas used to pressurise the sample must be pure.

Figure 10.21 Maximum bubble pressure setup. A vessel is filled with the liquid metal of unknown density and a capillary tube is inserted, which is connected to a pressurised gas reservoir. The pressure before the bubble detaches is measured.

522

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523

The MBP technique, in contrast to the pycnometer, allows the determination of both density and thermal expansion as a function of temperature in a single run. Once the density is known, Equation (10.40) can be used to determine the surface tension of the liquid metal. In practice, measurements are performed at different heights and the pressure plotted against height. This yields a linear relation, with the slope given by rg, and the intercept by 2g/r. This technique requires comparatively large quantities of the liquid metal which may be prohibited by cost. Dilatometers In contrast to pycnometers, dilatometers measure volumes and volume changes. A known mass of liquid metal is placed into a dilatometer, the volume it occupies is measured and the density is obtained from Equation (10.39). Both capillary and push-rod dilatometers are used. Capillary dilatometers consist of a long, thin capillary tube connected to a reservoir of known volume. If the capillary is exactly cylindrical, that is there are no variations of the radius r as a function of height h, the volume V of a liquid column inside the capillary is given by: V ¼ V0 þ pr2 h:

ð10:42Þ

The sensitivity of the apparatus is given by dh/dV ¼ 1/pr2, and explains the use of narrow capillaries. Capillary dilatometers are well suited to measure the volumetric thermal expansion of liquid metals by observing the height of the meniscus as a function of temperature. Corrections must be made for the thermal expansion of the capillary tube and for the meniscus shape. For liquid metals at high temperatures, transparent tubes cannot be used and the height of the liquid column cannot be determined by visual inspection. In this case, the height is measured by electrical contacts [135] that penetrate the tube at known heights. The mercury-in-glass thermometer is a liquid metal capillary dilatometer. The density of mercury is extremely well known and it is often used to calibrate a dilatometer’s internal volume. Most commercially available dilatometers apply the push-rod technique. This method was originally developed for solid samples, but it can be extended to liquids when special crucibles are combined with the piston-cylinder device. In this apparatus, the sample is placed between a plate and a moving piston, also called the push-rod, and from measurements of the change Dl in piston position the linear thermal expansion of solid samples is determined. There are both single and dual push-rod dilatometers available, the first delivering an absolute measurement, while the latter compares variations in Dl between an unknown and a reference material. Both horizontal and vertical configurations of this instrument are used, with the latter being preferred for high temperature sintering studies. Obviously, the push-rod must be kept in contact with the sample without exerting a force (strain) on it. Dilatometers with a maximum operating temperature over 2000 K are commercially 523

524

Extreme Conditions

available with the specific value being determined by the furnace used and the crucible material. Ruffino [136] has described a push-rod dilatometer that uses an optical interferometric system to detect either the movement of the push-rod or the plates enclosing the specimen. In the solid phase, push-rod dilatometers provide the linear thermal expansion, while in the liquid phase these dilatometers determine the volumetric thermal expansion [123]. Thus, when push-rod dilatometers are used to obtain liquid thermal expansions, the measurements require careful analysis, particularly at the liquidsolid phase transition. This is because the liquid fills the container, and linear thermal expansions in the directions perpendicular to the push-rod movement are converted into linear movements along the push-rod direction. In the case of an isotropic expansion, the linear thermal expansion al and volumetric thermal expansion av are related by

av ¼ 

1 qr 1 qV 1 ql ¼ ¼ 3al ¼ 3 : r qT V qT l qT

ð10:43Þ

Corrections are also required to account for the thermal expansion of the container itself arising from changes with temperature in both cross-sectional area and increase in piston length. Push-rod dilatometers also have the following limitations: the solid test-piece must be a loose fit inside the cylinder at the start of the test, but should completely fill it at the melting point, so that, any discontinuity in volume on melting is correctly determined; the piston must be a good sliding fit inside the cylinder to avoid leakage, yet slide smoothly with minimal friction; the piston can jam because of oxide entrapment of solid metal; the force applied to the piston must be adequate to counteract the tendency of surface tension to round the ends of the molten metal; and, where there is a melting range, the results obtained on heating and cooling may differ. Indeed, when the sample is cooled the volume changes may be followed correctly by the rod only down to the point where the volume fraction of solid becomes rigid. Below this temperature, the true shape of the cylinder cannot be maintained since there is no method of counteracting shrinkage on solidification. Nevertheless, liquid thermal expansion determined with this technique is in good agreement with that obtained from direct density measurements.

Levitated drop Density of levitated samples can be obtained with videography. In terrestrial experiments, the samples are elongated spheroids (not perfect spheres) due to the action of gravity and the electromagnetic field used for levitation. However, the static equilibrium shape of the sample is, to a good approximation, rotationally symmetric about the vertical axis (parallel to the gravity vector). Therefore, images taken perpendicular to this axis can be used to determine the volume V, of a 524

Extreme Conditions

525

rotationally symmetrical body, with:



2p 3

Z1 r3 ðuÞ du;

ð10:44Þ

1

where u ¼ cos y and y is the angle measured from the vertical axis and r(u) the angledependent radius of the sample. The mass m of the sample is determined before and after the measurement and the sample density obtained from Equation (10.39). The images are obtained at constant temperature and analysed off-line by a digital image processing software. The edge of the incandescent sample is determined and then averaged over approximately 100 additional images so as to remove the effect of asymmetrical dynamic surface oscillations from the result. The shape of the averaged image is then fitted to a series of Legendre polynomials and the coefficients of this series expansion are used to calculate the volume and thence density. A detailed description of this algorithm has been given in reference [137]. This method suffers from three sources of systematic error: calibration, edge detection, and sample movement. The radius of the sample is obtained in pixel units with the image analysis software and is converted to a physical length by comparison with images of a calibration body of known dimensions. To obtain a fractional precision in length l of Dl/l ¼ 104, with l ¼ 10 mm, the calibration body has to be manufactured with a precision of 1 mm. At this scale, the thermal expansion of the solid calibration body is significant, and a material with low thermal expansion should be chosen. In addition, this body has to be placed at exactly the same position as that of the levitated sample. The problem of detecting the edge of the incandescent object and thus determine the size of the drops has been studied and reported in reference [138]. For any horizontal line within an image which contains the drop, there are two regions where the intensity changes from dark to bright and then again from bright to dark. One option, used by some experimentalists, is to use the midpoint between 10 per cent and 90 per cent brightness as the edge position. However, there is always some noise in the signal, either from the electronics or from scattered light, and, consequently, the lower limit of the signal is not well defined. Alternatively, one may first eliminate the noise by introducing a threshold and use a definition of the edge which is independent of the lower limit. As such, one can use the turning points in the brightness profile. These points are defined locally, and do not depend on the upper and lower limit. Numerically, they are found by taking the second derivative of the brightness profile. Numerical differentiation works well only when the original signal is not too noisy. Whichever approach one takes, it is essential to confirm that the edge detection algorithm is independent of the brightness of the liquid drop, since this changes with temperature. If this condition is not satisfied, one introduces a systematic error into the thermal expansion measurement. There are two different kinds of sample movements: translational oscillations of the entire drop and surface oscillations. Lateral translational oscillations perpendi525

526

Extreme Conditions

cular to the direction of view have no effect; oscillations along the direction of view change the amplification factor and hence the apparent size of the drop. This effect can be minimised by using a telecentric optical system, where the amplification is, within limits, independent of object distance. As mentioned before, the surface oscillations of an axisymmetric drop are generally not axisymmetric. Therefore the area of the visible cross section may change according to the surface oscillation being excited. To eliminate these effects, two different approaches have been used. In the setup used by Shiraishi and Ward [139], a second camera is installed perpendicular to the first one viewing the sample from the top. The two cameras are synchronised in such a way that they take pictures simultaneously. For data analysis, only those pairs of pictures are retained where the top camera shows a well centred spherical cross section. Then the volume is calculated from the side view taken at the same instant. The system used at DLR [137] uses an averaging procedure to eliminate the effects of sample movement, the idea being that the drop is axisymmetric and well centred on the average. In principle, each image should be analysed with respect to volume and an average should be taken over the volumes thus obtained. In practice, image analysis is a time-consuming process and it is not feasible to perform it on typically 100 images for every temperature step. Therefore, these images are superimposed creating an average image which is then used to calculate the volume. Some of the problems discussed above may be alleviated by using backlight illumination and applying a shadowgraph technique rather than direct imaging of the incandescent sample. There are two major advantages to this approach: the contrast between shadow and background is temperature independent and the size of the shadow is independent of translational oscillations, if parallel light is used for illumination. The density of levitated drops has been measured using the method described above, both terrestrially [140], and in microgravity [116]. The microgravity environment offers the additional advantage that the drops are perfectly spherical. In Figure 10.22, the side view of a molten PdCuSi sample levitated in microgravity is shown. The dark shadows (top left and right, bottom centre) are caused by the wire cage surrounding the sample, the vertical bar is an electronically created code, indicating the average brightness of a single line. Bright spots on the sample are mirror images of the levitation coils.

10.3.5

Surface Tension

The surface tension of liquid metals is an important parameter in metallurgical processes, such as welding and casting. It is also one of the physical properties most sensitive to contamination, particularly by oxygen and sulfur. For example, Goumiri and Joud [141] report that one monolayer of oxygen reduces the surface tension of liquid aluminium by 20 per cent. The available surface tension data of pure metals have been reviewed and critically assessed by Keene [94]. A review of the various measurement techniques and a survey of high temperature tensiometry can be found in reference [142]. More recently, Eustathopoulos et al. [143] have published a 526

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527

Figure 10.22 Side view of a molten PdCuSi sample levitated in microgravity. The dark shadows (top left and right, bottom centre) are caused by the wire cage surrounding the sample, the vertical bar is an electronically created code, indicating the average brightness of a single line. Bright spots on the sample are mirror images of the levitation coils. Note the perfect spherical shape of the drop.

monograph on wettability at high temperatures, including a careful discussion of the sessile drop measurement technique. The following methods have been used to determine surface tension: maximum bubble pressure (MBP), maximum drop pressure (MDP), capillary rise (CR), sessile drop (SD), pendent drop (PD), drop weight (DW), levitated drop (LD), and maximum pull (MP). For high melting point materials, only four of these methods are relevant: MBP, SD, PD and LD. Among these, SD is by far the most popular method. This is borne out by a statistical analysis of the data within Keene’s compilation [94], which is shown in Figure 10.23. Pendent drop and drop mass are experimentally quite similar and these are listed as one entry in Figure 10.23. For a discussion of the PD technique, the reader is referred to Keene’s article [94] for further details. The other three methods MBP, SD and LD are discussed below.

Figure 10.23 Frequency of measurement methods for surface tension of pure liquid metals [94].

527

528

Extreme Conditions Maximum bubble pressure

This method can also be used for density measurements and it has been discussed in Section 10.3.4. One major advantage of MBP for surface tension measurements is the fact that for each measurement, a fresh, uncontaminated bubble surface is created. In the derivation of Equation (10.40) it was assumed that the bubble is spherical and its size independent of the immersion depth, however, these assumptions are a crude approximation for surface tension measurements. The hydrostatic pressure on the top of the bubble is less than on the bottom which creates a non-spherical bubble and leads to depth-dependent bubble deformation. This effect can be minimised by operating with small bubbles, formed with narrow capillaries, but it cannot be neglected. Thus, the pressure drop across a non spherical surface is given by (2g/r)F, where F is a correction factor for a non-spherical surface. Several authors have given expressions for F and these are discussed in reference [144]. A correction formula derived by Schro¨dinger [94] is: " g ¼ geff

 2 # 1 r2 rg 1 r2 rg 1  ; 3 geff 24 geff

ð10:45Þ

where geff ¼ (ph  rgh)r=2, that is, the surface tension value which one would obtain from Equation (10.40) directly. Equation (10.45) is a series expansion in the Bond number, Bo ¼ r2 r g=geff and is valid only for Bo 551, which is a condition usually satisfied. In this experiment, the higher the bubbling rate, the more data that can be acquired. However, the fluid must be at rest before a new bubble is created and for liquids with high viscosity, the bubble movement is very slow, and it will take several seconds for the bubble to disappear; Keene [94] reports typical bubbling rates between (0.5 to 1) Hz. Sessile Drop The sessile drop technique is often used for measuring both surface tension and density. An example of a sessile drop is shown in Figure 10.24. It is important in this experiment that the substrate does not react with the liquid drop and the drop should not wet the substrate, that is, the contact angle y should be larger than p/2 and the same remarks apply here as those discussed for crucibles in Section 10.3.1. Graphite, boron nitride, zirconia and alumina have been used as substrate material. In addition to being inert and non-wet, the substrate should also be microscopically smooth to avoid pinning of the triple line between substrate, drop and environment. Eustathopoulos et al. [143] claim that a surface roughness of less than 0.1 mm is required. If an oxide forms on the metal, the drop will not acquire its equilibrium shape, and the data analysis will lead to erroneous results. Pure helium or argon atmospheres are usually used to prevent oxidation; operation under ultra-high vacuum has been attempted, but evaporation can be a significant source of error. 528

Extreme Conditions

529

Figure 10.24 Picture of a sessile liquid Pd43Cu27Ni10P20 drop of 5 mm diameter at 1050 K. Picture is taken with backlight illumination.

The calculation of drop shape and volume usually assumes a rotationally symmetric drop, and this must be experimentally realised and verified. To do so requires the substrate be perfectly horizontally aligned and held on a vibration isolation mount. If the substrate is smooth and flat, positioning the drop inside the observing optics field of view can be difficult and, therefore, a slightly conical substrate shape is preferred to centre the drop. The image processing software was described in Section 10.3.4. The volume of the levitated drop is calculated from the meridional profile of the drop, using Equation (10.44). The density then follows from Equation (10.39). For surface tension measurements, the meridional profile, obtained from image processing, must be compared with the theoretically predicted shape. This shape can be derived from the Laplace equation and is given, in non-dimensional form, by: b=r1 þ b=r2 ¼ 2 þ Bo z=b;

ð10:46Þ

where b is the radius of curvature at the apex of the drop, r1 and r2 are the two principal radii of curvature at point [z, x(z)], and z is the vertical depth measured from the apex. The coordinate system and geometry of the sessile drop are shown schematically in Figure 10.25. Bo is the Bond number, defined by Bo ¼ rgb2 =g:

ð10:47Þ

The principal radii of curvature can be expressed as second order derivatives, d2x/ dz2, and consequently, Equation (10.46) is a second order, second degree differential equation that does not possess a closed analytical solution and must be solved numerically with the appropriate boundary conditions. Approximate explicit expressions have been used [143], that avoid numerical integration of Equation 529

530

Extreme Conditions

Figure 10.25 Schematics of a sessile drop. The contact angle is y, and the radius of curvature at the apex is b.

(10.46). When the profile is determined by digital image processing, this approach is not required and can be replaced by fitting procedures as described by Passerone and co-workers [142,145]. As a result of the fitting procedure, one obtains the meridional profile x(z) as a function of the two parameters b, which is a scaling factor, and Bo, which contains the surface tension g, in the term rg/g. The density can be simultaneously obtained in this experiment by determining the volume as described above, and the value of the surface tension can be deduced from Bo.

Levitated Drop The levitated drop technique is an elegant way to measure both surface tension and viscosity. It employs digital image processing for frequency analysis of surface waves. The radius r of a droplet undergoes oscillations of the form: drl;n ðW; j; tÞ ! Yl;n ðW; jÞ cosðol;n tÞeGl;n t ;

ð10:48Þ

where, Yl,n are spherical harmonics. The angular frequency ol,n is related to the surface tension, while the damping Gl,n of the waves arises from viscosity. If the equilibrium shape of the droplet is spherical, the formulae of Rayleigh and Kelvin can be used to relate frequency o and damping G of the oscillations to surface tension g and viscosity Z, respectively. Rayleigh’s formula is given by: o2r ¼ 32pg=3m;

ð10:49Þ

while the expression derived by Kelvin is GK ¼ 20paZ=3m; 530

ð10:50Þ

Extreme Conditions

531

where m is the mass of the droplet and a its radius. Equations (10.49) and (10.50) apply to the fundamental mode of oscillation, which corresponds to l ¼ 2. For spherical drops, frequencies and damping constants do not depend on n(jnj  2). A spherical shape is obtained only if the droplet is free of external forces, which is well approximated in microgravity. Under terrestrial conditions, the above relations are not valid and corrections must be made for these external forces. For electromagnetic levitation these corrections have been calculated by Cummings and Blackburn [146]. They take into account both the splitting of the peaks arising from the breaking of symmetry and the shifting of the peaks due to magnetic pressure. For the Rayleigh formula the Cummings correction is:  2 32p g 1 X 2 ¼ o2;n  1:9 O2tr  0:3 ðO2tr Þ4 g=r ; 3 m 5 n

ð10:51Þ

where, O2tr is the mean of the translational frequencies of the sample in the potential well of the levitation field, and g the local gravitational acceleration. It has been shown that applying the Cummings correction to surface tension data obtained with the oscillating drop technique on earth, the spurious mass dependence can be eliminated [147]. For gold, the value thus obtained agrees with data determined using the sessile drop technique. In addition, Egry et al. [88] have performed microgravity experiments on gold and a gold-copper alloy. These experiments clearly show a single peak in the oscillation spectrum that means the frequencies do not depend on n and, furthermore, they yield values of the surface tension which are in excellent agreement with terrestrial data, provided the latter are corrected according to Equation (10.51). The surface oscillations are recorded by a high-speed video camera and evaluated with digital image processing software. Pictures can be taken from the top (along the symmetry axis of the drop) or from the side. The top view is generally preferred, because it is not obstructed by the windings of the levitation coil. The following parameters are usually determined frame by frame: horizontal radius, vertical radius, area of the cross section, and the centre of mass of the cross section. In contrast to density measurements, only relative changes are important, and edge detection is not a critical issue. The oscillations usually have frequencies between (50 to 100) Hz, which implies that a frame rate of at least 200 Hz must be used to avoid aliasing effects. The analysis uses a fourier transform and, to obtain sufficient accuracy, a sequence of (211 to 212) frames are required. The centre of mass of the cross section is required to determine Otr. The signal derived from the cross sectional area contains the frequencies ol,n. If all 5 peaks corresponding to l ¼ 2 are clearly visible in this signal, then there is sufficient information to apply Equation (10.51). If not all frequencies are present then the analysis of two perpendicular radii is required for mode identification. A signal equivalent to the cross sectional area, can be obtained by measuring the variation in intensity with a photodiode focussed at the edge of the oscillating drop [148]. This method avoids image processing and can be applied in real-time, but some frequencies may be omitted and the correction 531

532

Extreme Conditions

formula, given by Equation (10.51), cannot always be applied, which can lead to errors of about 5 per cent in the derived quantity. The Cummings formula, Equation (10.51), was derived with a number of assumptions: both the amplitude of the oscillations and the deviation from sphericity of the drop’s equilibrium shape are small; the magnetic field inside the levitation coil has a linear gradient; and, the drop is inviscid. Corrections for the frequency shift due to viscous effects have been calculated [149] and found negligibly small for liquid metals. Whether or not the field geometry of the levitation coil meets the assumptions of Cummings’ theory can be determined from the ratio between the frequency Otr;\ , of the translational oscillations in the plane perpendicular to the symmetry axis of the coil, and that along the coil axis, Otr;k , which should equal 1=2. If the amplitude of the oscillations increases, the frequency may become amplitude-dependent. Soda et al. [150] claim an increase in frequency, while Asakuma et al. [151] find a decrease. However, a large amplitude does improve the image processing, and as a compromise, amplitudes of about 3 per cent are used. Finally, the surface tension is obtained from the frequencies by multiplying by the mass m of the drop. Therefore, evaporation losses during processing must be minimised by operating within an inert, or slightly reducing, gas atmosphere. The sample should be weighed before and after levitation. In view of the total accuracy of the method, mass losses of less than 1 per cent are acceptable.

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536

Extreme Conditions

121. Tenney, A.S., in Theory and Practice of Radiation Thermometry, D.P. deWitt and G.D. Nutter eds., Wiley, New York, 459, 1989. 122. Ho¨hne, G., Hemminger, W. and Flammersheim, H.-J., Differential Scanning Calorimetry, Springer, Berlin, 1996. 123. Blumm, J., Henderson, J. and Hagemann, L., High Temp. High Press. 30, 153, 1998. 124. Chekhovskoy, V., in Theory and Practice of Radiation Thermometry, D.P. deWitt and G.D. Nutter eds., Wiley, New York, p. 555, 1989. 125. Ohsaka, K., Gatewood, J. and Trinh, E., Scripta Met. and Mat. 25, 1459, 1991. 126. Frohberg, M.G., Thermochim. Acta 337, 7, 1999. 127. Qin, J., Ro¨sner-Kuhn, M., Schaefers, K. and Frohberg, M.G., Z. Metallkd. 85, 10, 1994. 128. Lin, R. and Frohberg, M.G., High Temp. High Press. 24, 543, 1992. 129. Sommer, F., Z. Metallkd. 73, 72, 1982. 130. Fecht, H.J. and Johnson, W., Rev. Sci. Instr. 62, 1299, 1991. 131. Loho¨fer, G., SIAM J. Appl. Math. 49, 567, 1989. 132. Crawley, A., Int. Met. Rev. 19, 32, 1974. 133. Watanabe, S., Ogino, K. and Tsu, Y., in Handbook of Physico-Chemical Properties at High Temperatures, Y. Kawai and Y. Shiraishi eds., ISIJ, 1988. 134. Irons, G. and Guthrie, R., Can. Met. Q. 19, 381, 1981. 135. Keskar, A. and Hruska, S., Met. Trans. 1, 2357, 1972. 136. Ruffino, G., in Compendium of Thermophysical Property Measurement Methods, Vol 1: Survey of Measurement Techniques, K.D. Maglic, A. Cezairliyan and V.E. Peletsky eds., Plenum Press, New York, p. 689, 1984. 137. Gorges, E., Racz, L., Schillings, A. and Egry, I., Int. J. Thermophys. 17, 1163, 1996. 138. Choi, M.Y. and Dryer, F., in Drops and Bubbles, T. Wang, ed., AIP, New York, p. 338, 1989. 139. Shiraishi, S.Y. and Ward, R.G., Can. Met. Quarterly 3, 117, 1964. 140. Langen, M., Hibiya, T., Eguchi, M. and Egry, I., J. Cryst. Growth 186, 550, 1998. 141. Goumiri, L. and Joud, J., Acta Met. 30, 1397, 1982. 142. Passerone, A. and Ricci, E., in Drops and Bubbles in Interfacial Research, D. Moebius and R. Miller eds., Elsevier, Amsterdam, p. 475, 1998. 143. Eustathopoulos, N., Nicholas, M. and Drevet, B., Wettability at High Temperatures, Pergamon, Oxford, 1999. 144. Fainerman, V. and Miller, R., in Drops and Bubbles in Interfacial Research, D. Moebius and R. Miller eds., Elsevier, Amsterdam, p. 279, 1998. 145. Liggieri, L. and Passerone, A., High Temp. Techn. 7, 82, 1989. 146. Cummings, D. and Blackburn, D., J. Fluid Mech. 224, 395, 1991. 147. Egry, I. and Sauerland, S., Mat. Sci. & Engng. A178, 73, 1994. 148. Keene, B., Mills, K. and Brooks, R., Mater. Sci. Tech. 1, 568, 1985. 149. Suryanarayana, P. and Bayazitoglou, Y., Int. J. Thermophys. 12, 137, 1991. 150. Soda, H., McLean, A. and Miller, W., Met. Trans. 9B, 145, 1978. 151. Asakuma, Y., Hirata, T., Tsukada, T., Hozawa, M., Nogi, K. and Imaishi, N., J. Chem. Eng. Japan 33, 861, 2000.

536

Index 1/f noise, 56 1965 Calorimetry Conference Standard, 482, 486 benzoic acid crystal, 482 high-purity copper, 486 absolute refractive index, 452 absorbed amount of substance, 452 absorption, 191, 240, 244 classical, 244 absorption coefficient, 267 absorption index, 458 absorption measurements, 272 accelerometer, 269 accommodation lengths, 247, 263 viscous thermal, 263 acoustic admittance, 240 acoustic filter, 255 acoustic impedance, 245, 249, 274, 280, 287, 291, 299 acoustic noise, 255 acoustic pressure, 248, 259, 283 cavity wall, 248 acoustics of, 239 porous media, 239 acoustic thermometry, 256, 304 adiabatic demagnetisation, 483 adiabatic shields, 477 admittance circles, 270 adsorption, 87, 100, 133, 139, 143, 181, 199, 240, 257, 397–8, 483 adsorption dryer, 100 adsorption isotherm, 240 adsorption pump, 484 AGA8, 210 air-buoyancy, 61 alternative refrigerants, 389, 393, 395 mixtures, 389, 393, 395 amount-of-substance, 240 amplitude modulated frequency, 268 annular resonator, 275–8

boundary layer contributions, 276 inlet tube, 277 thermally relaxing gases, 275 working equation, 275 annular shell, 277 wall compliance, 277 annular slots, 250, 295 effect on half width, 250 effect on radial resonance frequency, 250 API, 209 Archimedean, 520–1 Archimedes, 131, 140 Archimedes principle, 127, 133, 143, 210 magnetic float, 127 Arrhenius, 479 atomic absorption spectroscopy, 421 atomic force microscopy, 376 attenuation, 272 beam spreading, 272 attenuation coefficient, 458 Avogadro constant, 192, 195, 203 X-ray crystal, 192 azimuthal modes, 261, 262, 276, 275, 277 back electrode, 295 back plate, 294, 296 Balling, 209 Barkometer, 209 Baume, 209 beam spreading, 287 bellows volumetry, 168 bellows volumometer, 168–71, 174 bellows, 171–2, 175 effective area, 171, 172 variable volume, 175 Bessel functions, 152, 246 BET adsorption isotherm, 397 BET adsorption model, 398

538

Index caloric properties, 166 calorimeter, 21, 166, 327–8, 331, 335–7, 341, 347–50, 363, 368–70, 408–10, 412–13, 476–81, 483–6, 516–17 absorption, 335 ac, 327, 336–8, 340, 343–4, 346–7, 359, 518–19 adiabatic, 21, 410, 412–14, 476–81 aqueous electrolyte solutions, 327 combustion, 409 counter-current heat exchange, 328 coupled cell, 363 differential heat flow, 349 differential scanning, 327, 350 drop, 516 enthalpies of dilution, 327 enthalpy, 335 enthalpy-increment, 328, 331 excess enthalpy, 335 flash vapourising, 328 flow, 327–9, 331–2, 335 gas mixing, 329 heat capacity, 327, 328–31 heat flux, 327, 412–13 heat flux flow, 327 heat flux scanning, 349 high pressure, 327–8 high-temperature, 413 ice, 368 isoperibol, 410, 483–4 isoperibol solution, 409 isothermal dilution, 328 Joule-heating, 340, 347 Joule Thomson, 330–1 Joule-Thomson coefficients, 328 levitated drops, 518 light-irradiation, 343, 347 liquids, 344 low-temperature, 477 membrane type, 370 MEMS, 369 miniature, 480 mixing mixing, 166, 328–31 near critical locus, 335

bimorph discs, 292 binary gaseous mixture, 305 composition, 305 bio-nanocalorimeter, 371 Biot theory, 239 black body, 512 blackbody cavity, 493 black body radiance, 510 blackbody radiation, 291, 492, 505, 510 bolometer, 510 Boltzmann constant, 456 bond number, 529 boundary layer corrections, 269 Brillouin scattering, 290 Brix, 209 bubble curve, 107 bubble pressure, 104, 107, 111 buffer rods, 284, 287–91 aluminium oxide, 291 sapphire, 287 bulk attenuation, 263, 276, 279 bulk micromachining, 375 bulk modulus, 311 buoyancy, 104 buoyancy corrections, 102 buoyancy force, 192, 201 burette, 405–6 Burnett, 137, 146, 181–4, 190–1, 397–8, 399, 401–3 analysis, 182 apparatus, 395–6 differential, 181, 257, 395–6 expansion, 183, 184, 191, 252, 257, 389, 393 isochoric, 181, 183, 397, 399, 401–3 isochoric coupled, 391, isochoric methods, 398 method, 191, 392–3 refractive-index, 137 technique, 393 calibration, 163 vibrating tube densimeter, 163 538

Index

539

Clausius-Mossotti relation, 434 coefficient of thermal expansion, 272, 312 Cole-Cole equation, 469 Cole-Davidson relaxation, 469 combustion calorimetry, 411–12 complex amplitude, 258 complex conductivity, 457 complex dielectric constant, 435–6 complex heat capacities, 345, 347, 360–1 complex impedance, 437 complex permeability, 457 complex relative permeability, 457 complex transmission coefficient, 459 composition fluid, 105 compressibility factor, 308, 389–93, 397, 452 natural-gas, 391 compression wave, 312 compressive stress, 311 consolute points, 434 constructive interference, 272, 284, 288 contact angle, 528, 530 corning glass, 297 corona discharge, 293 coulometric titration, 418, 422 coupled Burnett-isochoric, 393 cricondentherm, 106–7 critical points, 238, 476 metals, 476 critical pressure, 222 critical temperature, 222 cross capacitor, 445, 446, 449 coaxial cylindrical, 445 cylindrical, 445 rectangular cross-section, 445 toroidal, 445–6 crucible, 349, 350–2, 354–8, 361, 363–7, 369, 415, 505–7, 515 aerogels, 506 ceramics, 506, 515 metallic, 349 non-wetting, 505 reference, 350, 354, 356

non-adiabatic scanning, 337 Peltier, 338 plug-in, 328–30 plug-in gas flow, 328 power-compensated, 348 scanning adiabatic, 413 single-stage gas mixing, 328 solution, 410 twin heat-flux, 410 thermoelectric, 332 twin type, 409, 481 ultra-low-frequency spectrometer, 476 calorimetric signal, 350, 353, 356, 359– 62, 364, 366–7, 409, 479, 514 kinetic, 362 thermal, 362 three-dimensional, 353 calorimetric temperature, 479–80 calometring, solution, 327, 409 Calvet type calorimeter, 349, 353 Calvet type microcalorimeter, 349, 354 capacitance, 437, 443, 456 non-conducting, 456 pressure dependence, 443 capacitance bridge, 54, 445 capacitance diaphragm gauge, 75 capacitors, 293, 376, 435–41, 447–8, 450–1 coaxial cylinder, 439–41, 447 energetic particles, 451 gas-filled, 451 parallel plate, 293, 439 pressure-dependent, 448 capsule platinium thermometer, 265 carbonates, 411, 422 cell reaction, 415 ceramic crucibles, 506 chemical compatibility, 251, 292, 294 chemically inert, 291 chemical potential, 414 circulation pump, 249 electromagnetic, 249 clathrate hydrate crystal, 480 Clausing factor, 420 539

540

Index cylindrical geometry, 215 differential, 157 double-sinker, 139 element, 178 expansion, 180 flow, 165–6 high-temperature flow, 166 hydrostatic balance, 127, 128, 130 hydrostatic, 211 hydrostatic weighing, 149 in situ, 210 magnetic float, 129–30 magnetic suspension, 127, 129–30, 148, 151, 178, 203–4 method, 180 neutron, 219 nuclear, 218 nuclear attenuation, 217 phase equilibria, 166 pressure difference, 217 refractive index, 217 single-sinker, 140–5, 147–9 single tube, 213 thin walled cylinders, 214 ‘tuning fork’, 215–17 twin tube, 213 two-sinker, 131–40, 145–6 Ultrasonic, 217 uncertainty, 131 U-shaped tube, 149 vibrating, 178 vibrating objects, 212–14 vibrating plate, 149, 159, 160, 162, 166, 215, 402 vibrating-tube, 149, 158, 164–5, 167– 8, 202, 204, 310, 404 vibrating U-tube, 148 vibrating wire, 149–50, 154–7 Weighing, 210 density, 78, 80, 99, 101–2, 105, 128–31, 134–5, 138–140, 143–4, 148–51, 157, 162, 165, 169, 177, 179, 185, 187, 189, 191–2, 194–5, 200, 202– 10, 214, 220, 222, 262, 301, 339,

refractory metals, 505 sample, 354, 356 wetting, 505–6 Curie temperature, 287, 292 cylinder, 261 azimuthal, 261 longitudinal, 261 radial modes, 261 cylindrical Bessel function, 262, 277 cylindrical cavities, 261 time-of-flight, 261 cylindrical Neumann functions, 277 cylindrical resonators, 261, 263–4, 268 fixed path-length, 261 high temperature, 268 long, 268 longitudinal modes, 264 motion shell, 263 sound speed, 268 steady state, 261 transient, 261 transport coefficient, 268 variable-frequency, 261 variable path length, 261 dead weight gauge, 182, 187–8 gas lubricated, 188 Debye, 469 Debye-Drude model, 436 Debye equation, 435 Debye relaxation, 469 Decker’s equation, 69 densimeter, 127–51, 154–60, 162, 164– 8, 178, 180, 202–5, 210–19, 223, 310, 404 acoustic, 217 applications, 219 attenuation, 217 buoyancy, 210 Burnett, 180 calibrating, 202 calibration, 204 centrifugal, 216 crude oil, 223 540

Index

541

relative, 209 relative permittivity, 434 saturated liquid, 128, 130–1, 138, 144, 169, saturated-vapour, 131, 134–5, 138, 149 sessile drop, 528, 530 silicon crystals, 192, 194, 202 SMOW, 207 stardard, 209, 221–2 uncertainty, 135, 140, 149, 157, 187, 189, 202–3, 207 vibrating bodies, 149–50 vibrating-wires, 151 water, 205–7 X-ray crystal, 203 density standard, 205, 209, 221–2 water, 205 liquid, 205 desorption, 139 destructive interference, 281, 288 detectors, 290 capacitive, 290 piezoelectric, 290 Dewar, 476 dew curve, 101, 107 dew pressure, 111, 115 diamond-anvil cell, 69, 71 dielectric constant, 169, 435–6, 438–9, 441–3, 445, 448–50, 453 capacitance, 453 coaxial cylinder, 439 cross capacitors, 445 gases, 448 liquids, 449 parallel plate, 438 phase boundaries, 450 re-entrant resonator, 441–3 dielectric loss, 435 dielectric permittivity, 467 loss spectrum, 467 dielectric polarization, 455 dielectric relaxation, 436, 469 differential apparatus, 395 differential Burnett, 257

345, 391–3, 398, 404, 406, 434, 448, 451–3, 491, 496, 500, 507, 520–6, 528, 530–1 absolute of silicon crystals, 195 absolute standards, 191 air, 102 alloys, 520 amount-of-substance, 101, 398, 448, 452–3 aqueous solutions, 165 Archimedean, 520–1 capacitance, 454 clay slurries, 219 dielectric-constant, 434 dilatometer, 523 electrolytes, 165 expansion method, 452 flotation, 202 flow metering, 209 fluids, 129, 130, 150 gas, 140, 143, 148, 220, 398 gas mixture, 392 high temperature, 520 hydrostatic weighing, 200, 204 in situ, 208–9 isochoric, 185, 187, 392 levitated drops, 524, 526, 531 liquefied gas, 222 liquid, 128–31, 148, 169, 214, 404 liquid metals, 520 liquid-vapor critical points, 451 low-boiling fluids, 135 magnetic suspension, 203 mass, 491 maximum bubble pressure, 520–2, 528 mercury, 192, 208 mixture, 391 molar, 393, 434 natural gas, 139, 149, 392 nonelectrolytes, 165 precision, 162 pressure, 202 pure gases, 392 pycnometer, 520, 523 refractive index, 453 541

542

Index solids, 408 solute-solvent, 411 solution, 408–9, 423 Sondak’s method, 368 temperature-modulated, 347–8, 359 model, 359 very low temperatures, 483 differential thermal analysis, 349, 353, 514 diffraction corrections, 289 diffraction effects, 283 diffraction, guided mode dispersion, 271 diffusion coefficients, 262 diffusivities, 243, 251 thermal, 243, 251 viscous, 243, 251 digital counter, 162 dilatometers, 404, 406, 519–20 capillary, 523 dilution, 404 liquid metal capillary, 523 push-rod, 523–4 single shot, 404 dilatometric, 520–1 dilatometry, 187, 242 diamond, 71 mercury, 187 Raman line, 71 dilution dilatometer, 404–5 407 tilting, 407 dilution refrigerators, 483, 486–7 dimensional microwave measurements, 242, 252 dimensional stability, 255 dipole moment, 435–6, 454 direct weighing, 401–3 dispersion curve, 268 division-of-amplitude photopolarimeter, 494 drill-bit lubricant, 113 drilling fluid, 113 oil-based, 113 water-based, 113 drilling lubricant, 112, 114–15, 119

differential pressure transducer, 182 differential scanning calorimeters, 327, 347–8, 350, 352–3, 355–8, 361–3, 369, 413, 507, 514 adiabatic, 358 base line, 363 heat flux, 347–8, 353 high temperature, 514 model, 355 molten metals, 514 power-compensated, 350 phase transitions, 363 single cell, 358 solution, 327, 409 specialised, 361 differential scanning calorimetry (DSC), 310, 347–8, 356, 358–9, 368, 408–14, 423–24, 476, 480, 483, 485, 487, 514–15, 517 adiabatic, 358, 409–14 baseline, 515 combustion, 411, 423 combustion reaction, 408 cryostat, 487 differential heat flux, 514 differential scanning, 348, 514 direct reaction, 412, 423 drop, 414, 515 drop-solution, 410–11 heat capacity, 409, 413, 423 heat flux, 413, 355–7, 362, 514 equivalent circuit, 355–7 high-temperature, 413, 515 levitation, 517 levitation alloying, 517 levitation drop, 414, 517 low temperature, 476, 480, 485 modulated, 359, 361 heat capacity, 361 glass transitions, 361 nano, 368 power compensated, 350, 352, 358, 363, 514 pulse, 414 Skau’s method, 368 542

Index

543

emissivity, 32–33, 491–3, 499, 502–3, 510, 514 normal spectral, 492 spectral, 32, 33 energetic properties, 408 solids, 408 energy transfer, 243 translational-to-vibrational, 243 vibrational-to-vibrational, 243 enthalpy of, 371–2, 478–80, 486, 496, 500, 516–17 adsorption, 335 configurational, 478–80 chemical reactions, 411 formation, 409, 411–13, 424 fusion, 495, 514 mixing, 328, 330, 410, 517 gases, 328 partial molar, 410 oxidation, 412 reaction, 408–9, 411 relaxation, 479–80 solution, 328, 335, 409–411 enthalpy increment calorimeter, 334 critical region, 334 enthalpy increment flow calorimeter, 332, 333 entropy, 476 entropy increment, 424 equation of state, 68, 120, 136–7, 207, 238, 240, 301–3, 306–7, 389, 497, 500 amount-of-substance, 301 empirical, 307 gaseous, 303 van der Waals, 497 virial, 277, 303, 306, 307, 389, 392 Euler algorithm, 309 European Gas Research Group, 393 eutectic, 506 EXAFS, 72 excess enthalpies, 328, 390 gas, 328 gaseous, 390 gas mixtures, 328

contamination, 119 oil based, 114, 115, 118–19 drop calorimetry, 515 drop tubes, 505 DTA, 359 eddy currents, 488, 508 eigenfunctions, 458 eigenvalues, 250, 260 elastic compliance, 246 elastic constants, 204, 311–12 from acoustic measurements, 312 silicon crystals, 204 elastic constants of solids, 312 elastic moduli of solids, 288 elastic properties, 239 fluids, 239 solids, 239 electret, 293, 296 electret microphone, 279, 293, 295 electric field, 455, 457 electrical conductivity, 434–5 443, 502, 508, 518 electrical crosstalk, 293 electrical dissipation, 435 electrical double-layers, 436 electrical resistivity, 490–1, 495–6 electrochemical cells, 417–18 construction, 417 electrochemical potential, 415–16 ‘electrode polarization’, 437 electrodes, 418 electrode-solution interface, 436 electrolytes, 414, 417–18, 434 electrolyte solutions, 464, 468 electromagnetic field, 518 penetration depth, 518 electron beam, 293 electron beam welded, 255, 298 electronic conduction, 491 electrostatic transducers, 267 elements, 102 atomic masses, 102 ellipsometry, 198, 494, 500, 512 EMATS, 292 543

544

Index fluid mixtures, 103 intermediate, 103 fluids, 105 flow, 105 porous media, 105 metering, 105 fluid sampling, 105–6, 109–18, 121 advances, 118 bottom hole, 110, 112–13, 115, 120 drill stem test, 112–14, 120 modular dynamics tester, 121–2 open, 120 pressure compensation, 119 representative, 106, 109–11, 115 slick-line, 112 surface, 110 wire-line, 112–14, 119, 121 Fourier transformation, 466 Fourier-Laplace transformation, 460 free field diffraction, 284 free field diffraction equation, 283 freezing-point cell, 18 metal, 18 frequency synthesiser, 257, 293, 296 Fresnel’s equation, 198, 459 fringe count, 453 fused quartz sphere, 206

liquid mixtures, 328 excess half width, 242–3 excess losses, 252 excess volumes, 390, 404, 406 mixture, 390 partial molar, 404 excimer laser, 500 Fabry-Perot etalon, 192 femtosecond spectroscopy, 501 finite element analysis, 59–60, 64, 488 pneumatic, 64 finite element code, 489 finite elements, 476 first order perturbation theory, 276 fixed points, 11–12 apparatus, 11 flexible bellows, 168 Flexural bimorphs, 292 edge-clamped circular metal plate, 292 flow calorimetry, 327 flow impedance, 73 flow-meter, 221 gas, 221 flow metering, 220 gas, 220 fluid cylinder, 104, 116–18 buffer, 118 movable separator, 116–17 movable separator or piston, 104 nitrogen, 118 nitrogen buffer, 116–17 piston, 116–17 fluid mixing, 104, 116, 395 circulation pumps, 104 convection, 104 heating, 104 inverting, 104 magnetic, 395 mechanical agitation, 104, 116 metal foil, 104 pump, 395 rocking, 104 rolling, 104

galvanic cell, 416 g-ray, 218–19 gamma-ray attenuation, 516 gas, 99, 103 cylinder, 99 gas chromatographic, 100 gas constant, 242, 245, 252, 270, 293, 304 acoustic, 270 acoustic determination, 252, 293 error, 245 gas contamination, 255 gas imperfection, 277 gas inlet tubes, 263, 246 gas-liquid chromatography, 362 gas mixture, 98–100, 102–3, 244–5, 249, 388–9, 392–5, 398–9, 401 544

Index

545

fluids, 327 high temperature, 361 isobaric, 300 isochoric, 300 metals, 336 molar, 246, 348 perfect gas, 242, 249, 267, 304 pure fluids, 327 reference, 514 reference material, 351 reference state, 301 sample, 351 specific, 339, 345, 346, 491, 496–7 spectroscopy, 345 vibrational contribution, 243 heat flow, 515 heat flux, 337–40, 343, 345–6, 354 three-dimensional, 354 heating, 489 resistive, 489 laser, 489 heating rates, 496 Helmholtz function, 302, 307 Helmholtz resonance frequency, 279 Helmholtz resonator, 252, 262, 279, 296, dumbbell-shaped, 262, 279 speed of sound, 279 viscosity, 279 hemispherical resonator, 259–60 hemispherical total emittance, 496 high temperature, 488, 496–7, 504 ms experiments, 496 ms time resolution, 497 pulse heating, 488 hydrates, 411, 480 hydrostatic weighing, 192, 208

compositionally complex, 103 differential, 394 gravimetric preparation, 244 low pressure, 394 measurements, 399 methods, 392 moderate pressure, 393 natural, 102 natural gas, 389, 401 non-corrosive, 99 pressure change on mixing, 395 gas-oil ratio (GOR), 111 gas volumetric methods, 422 Gaussian, 70, 494 Geiger counters, 451 generalized permittivity, 458 Gibbs energies, 408, 424 Gibbs energy of formation, 415 Gibbs energy of reaction, 424 gravitational acceleration of free fall, 516 gravitational force, 502, 508 gravity, 203, 524 Greenspan viscometer, 262, 279 Groupe Europe´en de Recherches Gazie`res, 393 guided mode dispersion, 270, 284 half width, 247, 258, 265 shell motion, 247 Havriliak-Negami equation, 469 head corrections, 73 Heat, 375 heat capacities, 238, 242–3, 246, 249, 262, 267, 277, 300–1, 304, 310, 327, 336–7, 339, 340, 343, 344–9, 351–2, 355–7, 360–1, 365–7, 372, 376–8, 403, 408, 413–14, 424, 476– 8, 481–3, 491, 495–7, 500, 514, 517–19 benzoic and crystal, 482 condensed matter, 477 constant pressure, 246, 267, 337, 339, 366–7, 491, 517 fluid mixtures, 327

ice-point apparatus, 16–17 impact momentum sensors, 420 impedance, 245, 435–7, 450 fluid, 450 to flow, 245 impedance circles, 284 inductance, 437, 443 545

546

Index International Practical Temperature Scale of 1948, 483 International Practical Temperature Scale of 1968, 482–3 International System of Units, 191 International Temperature Scale of 1990, 9–10, 22, 26–8, 34, 37–8, 40–1, 208, 256, 483, 505 International Union of Pure and Applied Chemistry, 2, 13 intramolecular relaxation, 274 isentrope, 308 isentropic bulk modulus, 312 isentropic compressibility, 300, 310 isobar, 311 isobaric molar heat capacity, 301 isoperibol heat flux, 409 isothermal bulk modulus, 312 isothermal Burnett method, 396 isothermal compressibility, 203, 208, 300 isothermal dilution calorimeter, 328, 335 isotropic elastic solids, 300 isotropic solid, 311 elastic properties, 311 Deviation Functions, 28 error, 37–38 Reference Functions, 26–27 uncertainties, 37 water, 37

inelastic X-ray scattering, 290 infrared spectroscopy, 362 inlet tubes, 253, 255, 260 integrated circuits, 369 interdiffusion, 73 interferometer, 197, 239, 257, 268–71, 279, 284, 292, 452 air, 268 variable path length, 268 cylindrical, 242, 271 cylindrical acoustic, 266 double transducer, 271 Fabry-Perot, 279 fixed-frequency, 292 optical, 193, 195–6, 205–6, 269, 284, 524 path-length, 292 reflector, 268 scanning-type, 195 single transducer, 268 single-transducer variable path, 269 spherical cavity acoustic, 245 transducers, 292 two-transducer, 270 variable path, 257, 270, 271, 287 variable path-length, 280, 283–4, 286–7 dual transducer, 283, 286 Liquids, 280 single transducer, 283, 286 intermolecular potential, 305–6, 389 attractive well, 306 hard-core-square-well, 306 non-additivity, 306 square-well, 307 three-body, 307 intermolecular potential-energy functions, 306 International Association for the Properties of Water and Steam, 207 -95, 207 -IF97, 207 International Confederation of Thermal Analysis, 363

JFET, 295–6 Johnson, 56 Johnson noise, 57 Joule-heating, 341, 481 Joule-Thomson effect, 329 Kelvin, 530 Kelvin formula, 530 Kerrcell shutter, 492 Kirchoff-Helmholtz boundary layer, 250, 271 tube attenuation constant, 250 Kirkwood ‘g’ factor, 449 546

Index

547

local gravitational acceleration, 74 lock-in amplifier, 154, 156, 257, 368, 336, 342–3, 344, 493 longitudinal modes, 262, 267 longitudinal sound, 300 longitudinal sound waves, 312 longitudinal wave speed, 249 Lorentz force, 502, 508 Lorentzian, 70, 258 Lorentz number, 491

Knudsen cell, 420–1 twin-type, 421 Knudsen effusion, 419, 421 Knudsen effusion mass spectrometry, 421, 424 Lame´ constants, 311 Laplace equation, 529 laser, 279, 508 argon-ion, 279 ruby, 493 lattice conduction, 491 Legendre function, 194 Legendre polynomials, 521 levitated drop, 520–1, 524, 529–30 levitation, 291, 476, 488–9, 500–3, 505, 507–11, 517–19, 524, 531 acoustic, 500–1, 511 aeroacoustic, 511 aerodynamic, 501, 507, 511 coil, 531 drop tube, 511 electromagnetic, 291, 489, 501–3, 508–11, 518, 524, 531 electrostatic, 501, 508, 511 liquid metal droplet, 517 magnetic, 130–1 magnetostrictive, 291 microgravity, 502 non-contact, 476 optical, 511 linear thermal expansion, 524 solid phase, 524 linear variable differential transformer, 89, 173, 182, 210 liquid, 103–4 compressibility, 406 liquid metal, 476 liquid mixture, 98, 104, 403–6 enthalpy of mixing, 404 excess enthalpy, 404 excess volume, 404 volume of mixing, 404 local acceleration of free fall, 201, 406, 521

magnetic permeability, 437, 443, 518 relative, 437 magnetic susceptibility, 484–5 magnetic suspension couplings, 132–4, 139–43, 147–8 gas, 134 liquid, 134 magnetostrictive solids, 292 longitudinal, 292 Man, 99 top loading electronic scale, 99 manometer, 59, 61, 72–9, 80, 87, 208, 329, 394–5 capacitance, 394 cat’s-eye, 77 cat’s-eye reflector, 76–7 compression, 73 Hg, 74–7, 208 laser interferometry, 72–4, 76–7, 79 liquid column, 73–74, 80 multiple column, 75 non-interferometric, 73 oil, 78, 80 optical interferometry, 78 Schwien, 76 tilt errors, 75, 79 two-column, 74, 79 ultrasonic interferometric, 76–80 vibration, 77 vibration isolation, 75 white light interferometry, 78 manometric methods, 422 mass, 99, 102, 191 equal arm balance, 99, 102 547

548

Index molecular slip, 263, 276 molecular thermal dispersion, 243–4, 258 molten metals, 504 refractory metals, 504 momentum accommodation coefficient, 260 momentum accommodation length, 260 monocrystalline silicon, 297 MST, 368 mutual inductance bridge, 485 Hartshorn, 485

mass spectrometer, 362, 421, 498, 500 time-of-flight, 500 maximum bubble pressure, 520–2 maximum bubble pressure method, 507 Maxwell’s equations, 457 Maxwell-Wagner effect, 434 Maxwell-Wagner relaxation, 436 melting line, 367 melting temperature, 489 MEMS, 215, 298, 368, 371 Michelson Interferometer, 76, 192, 208, 286, 454–5, 492 microbalance, 374 microcalorimeter, 482 ac, 344 flow, 327 micro electro-mechanical system, 215, 298, 369 capacitive transducer, 298 microgravity, 496, 502, 505, 509, 520, 526–7, 531 micro-system technology, 369 microwave thermal expansion, 256 mixing pump, 249, 253 electromagnetic, 249 electromagnetically activated, 253 mixture preparation, 98–104 blending, 99 component purity, 100, 104 gas, 98–99, 101 gravimetric, 98–9, 102–4 liquid, 98, 104 material compatibility, 101 oxygen analyzer, 100 partial pressure, 98 procedures, 98 volumetric, 98, 104 water content, 100–1 mixtures, 103 molar mass, 301 molecular absorption, 299 molecular collisions, 241 molecular sieve, 100

nanocalorimeter, 368–76 dielectric membrane, 374 gas detector, 374 membrane, 371–3, 374 membrane coatings, 373 membrane-type, 370 MEMS, 369, 374–6 portable, 375 thermal analysis, 376 National Bureau of Standards 1955 temperature scale, 482 natural gas, 257, 392, 441, 445 speed of sound, 257 Navier–Stokes equation, 150–1, 153 Neumann–Kopp rule, 518 NMR spectroscopy, 479 non-linear regression, 258 normal density, 209 normal spectral emissivity, 512 normal spectral emittance, 492 nuclear reactors, 476 numerical integration, 307, 308, 310 gases, 307 initial conditions, 308 liquids, 310 orientational polarization, 467 oscillating, 151–2 forced mode, 151 transient decay mode, 151 wire, 152 548

Index

549

piezoelectric elements, 292 piezometer, 58, 168, 174–5, 178–80, 183–4 coupled experiments, 183 density, 184 expansion, 179 fixed volume, 175, 178 multiple expansion, 180 single expansion, 179 variable volume, 179 piston-cylinder, 60–1, 65–6, 83–4 annular gap, 60, 83 clearance, 60 materials, 61 modules, 66 mounting, 66 oil lubricated, 66 size, 65 piston gauge, 58–66, 68, 72–3, 80–8 aerodynamic effect, 64 automated, 59–65 characterisation, 61 controlled clearance, 59 cross float procedure, 59 digital, 59, 66, 98, 83 effective area, 59, 60, 62–4, 68, 82, 86 force-balance, 73, 81–3 gas-operated, 80–1 geometry, 60 high pressure, 66 hydraulic, 59 inclined, 81 large diameter, 61 manufacturing, 59 oil-operated, 80 pneumatic, 59 reference, 84 repeatability, 85 single, 73 twin, 73, 81, 86 two, 85–6 uncertainties, 59 vibrations, 80 Planck’s law, 32, 491, 493, 510 blackbody, 32

oscillating drop, 502, 531 oscilloscope, 265, 268, 281, 289 paramagnetic liquid, 438 partial entropies of a solution, 414 partial entropy of oxygen, 414 partial Gibbs energy, 418 partial molar entropy of oxygen, 419 partial pressure, 420–2 path length, 285, 292 penetration lengths, 263, 277 thermal, 263, 277 viscous, 263, 277 perfect-gas, 301, 304 perfect-gas heat capacities, 308 permeability of free space, 457 permittivity, 457, 464 complex, 457 relaxation model, 464 permittivity of free space, 455 perturbation theory, 246 phase advance, 283 phase behavior, 105, 108 phase boundaries, 120, 135, 419 phase cancellation, 288 phase comparison pulse echo, 273 phase diagrams, 476 phase envelope, 106 phase equilibrium measurements, 179 saturated phase, 179 phase sensitive detection, 284 phase separation, 434 phase transformation, 510 phase transition, 113, 121, 238, 337, 349, 365–6, 495, 501 enthalpy, 337, 366 photoelectron spectroscopy, 500 ultraviolet, 500 X-ray, 500 photolithography, 46, 58, 370 photopolarimeter, 494 division-of-amplitude, 494 Picker, 327 piezo-crystal, 279 piezoelectric disk, 265 549

550

Index isolating, 89 MEMS, 46, 88 optical, 69 piezoresistive, 88 piston, 58 pneumatic, 58, 59 quartz Bourdon tube, 84, 87–8, 90 quartz Bourdon-type, 190 quartz resonant, 88, 90 reference, 83, 89 resolution, 46 resonant, 88 ruby fluorescence, 69–72 sensitivity, 85 standard, 89 static expansion, 72, 87 strain gauge, 46 pressure generator, 73, 87 static expansion, 73, 87 pressure release boundary condition, 280 pressure standard, 69 quartz, 69 calibrant, 69 calibration, 68 long-term stability, 51 mercury, 208 oscillating quartz crystal, 188 oscillation quartz, 214 piston barometer, 144 piston manometer, 144 precision, 52 refractive index, 453 relative dead weight gauge, 135 standard, 208 uncertainty, 66, 76, 79, 82, 84, 87 pressure transducer, 45, 46–58, 73, 85, 86, 88, 90, 135 absolute, 46–7 capacitance, 45 capacitive, 53–56, 58 CMOS, 49, 55 compensation, 48 differential, 46, 73, 85–6, 89 drift, 47

blackbody radiation, 33, 510 radiation, 32 platinum resistance, 30–1, 36 capsule, 30, 31, 36 platinum resistance thermometer, 12, 29, 196, 329–30, 342, 352, 478 capsule, 12, 29 high-temperature, 12, 29 long-stem, 12 point-like sources, 290 laser, electron beam, 290 X-rays, 290 Poisson’s ratio s, 47, 63, 311 polarizability, 434–6, 437, 448, 456 induced, 456 molar, 434, 448 molecular, 435 polarization, 456 orientational, 456 porosity, 368 positive displacement pump, 104 power modulation, 519 Prandtl number, 262 precondensation, 240, 242–3, 257 frequency, 240 predictor-corrector method, 309 pressure, 51–2, 66, 68–9, 76, 79, 82, 84, 87, 135, 144, 188, 208, 214, 453 pressure balance, 58, 72, 80, 86 liquid-column, 72 twin, 86 pressure change on mixing, 397 pressure gauge, 45–6, 58–9, 68–9, 70, 71–2, 75, 83–5, 87–90, 190, 395, 453 calibrants, 71 capacitance, 75, 88–9 capacitive, 89 Diamond-Anvil, 68 diamond anvil cell, 72 differential, 395, 453 dynamic range, 46 electronic, 45 hydraulic, 58–9 infrared absorption, 72 550

Index

551

pulse-heated, 492–3 liquid metal, 492 liquids, 493 pulse heating, 488–90, 492–501 1 ms, 497 femtosecond, 501 heating rates, 489 laser, 488, 497–500 microsecond, 492, 500 millisecond, 493, 496, 500 ms experiments, 496 nanosecond, 500 resistive, 488–90, 495, 501 pulse methods, 272 beam spreading, 272 elastic moduli of solids, 272 multiple reflections, 272 two reflectors, 272 two transducer, 272 pulse technique, 284 purging, 255 pycnometer, 175–8, 208, 224, 392, 523 accuracies, 175 continuously-weighed, 175 pycnometric, 516–17 pyrometer, 488, 491–3, 498–9, 501, 506, 509–10, 512–14, 517 double-mirror periscopic, 513 radiation, 510 ratio, 514 two-colour, 513

FET, 49 gauge, 46 hysteresis, 48, 50, 52 linearity, 50 mechanical resonator, 50 MEMS, 45, 58 MEMS strain gauge, 54 micro electro-mechanical systems, 45 micro-machined resonator, 50 MOSFET, 49 piezoresistive, 47, 54, 57–8 piezoresistive silicon, 45 pressure non-linearity, 48 quartz bourdon, 45 quartz Bourdon tube, 90 quartz resonant, 45, 50–1, 90 quartz, 52 reference, 87 reproducibility, 52 resistive strain gauge, 46 resolution, 52 resonant silicon gauges, 45 resonator, 52 sensitivity, 54, 57 silicon capacitive, 55 strain gauge, 55 temperature, 48 Transfer standard, 73 primary thermometer, 245, 256 spherical resonator, 245 propagation coefficient, 250, 457, 464 complex, 457 propagation constant, 250–1 propagation factor, 457 complex, 457 propagation speed, 300 properties of mixing, 408 solids, 408 provisional low temperature scale, 38 PLTS-2000, 38 pseudo isochore, 183, 185, 190–1 pulse-echo, 280, 290 detect phase boundaries, 280 pulse-echo method, 282 speed of sound, 282

quartz buffer, 284 quartz crystal, 273 quartz pressure transducer, 53, 187 drift, 53 quartz resonance, 88 quartz transducer, 268 Quevenne, 209 radial mode, 248, 250, 257, 261–2, 275– 6 radiance intensity, 492 radiation, 248 551

552

Index capacitance, 455 coaxial-line techniques, 460 complex, 457 cross capacitors, 445 electrolyte solutions, 455 electrolytes, 455 free-space, 465 nonelectrolytes, 455 parallel plate, 438 re-entrant resonator, 441, 444 waveguide, 462–3 relaxation time, 478 structural, 478 reservoir fluid, 105–10, 114–16, 119–21 acoustic properties, 280 asphaltenes, 107–8, 115–16, 118 black oil, 106–7, 110, 114, 120 composition, 109, 116 condensate, 114, 120 contamination, 120 dry gas, 106 gas condensate, 110, 115 heavy oil, 106–7 hydrates, 107–8, 116 hydrocarbon, 105 retrograde gas, 107–9 transfer, 116 volatile oil, 106–7, 110, 115, 120 wax, 107–8, 116 wet gas, 106–7 reservoir formation, 106 resistance bridge, 135 resistive pulse heating, 489 resistive self-heating, 489 resonance, 248, 298, 444 half-width, 444 microwave, 298 radial, 248, 250 resonance frequencies, 246, 250, 257, 260, 265, 444 complex, 246, 444 measurement, 257 radical, 250 resonance half-width, 243–4, 246 resonance methods, 287

radiometer, 32, 40 Rankine-Hugoniot curves, 68 rapid volume heating, 489 ratio transformer bridge, 439, 445 Rayleigh’s formula, 530 reaction, 371–2 re-entrant cavity, 442, 447, 450 phase boundaries, 450 co-existing phase densities, 450 re-entrant resonator, 441–5, 449 accelerometer, 442 pressure gauge, 442 thermometer, 442 three-frequency, 441 two-frequency, 441 two-mode, 449 reference electrodes, 415, 417 referred density, 209 reflection coefficient, 459, 464 complex, 459 reflection measurements, 466 reflectometer, 460, 493 time-domain, 460 reflectors, 261, 274 refractive index, 79, 183, 196, 218, 279, 434, 451–4, 458, 499 complex, 458, 499 four-column, 79 gases, 452 interferometers, 453–4 interferometric methods, 452 liquids, 452, 454 refractivity, 452 molar, 452 refractory materials, 509 refractory metals, 509 solidification, 509 nucleation, 509 molten metals, 504 relative dielectric permittivity, 434 complex, 444 relative electrical permittivity, 434–5 relative permittivity, 241, 434, 438, 441, 444–5, 449–50, 455–7, 460, 462–5, 463–4, 474 552

Index

553

shell correction, 248 shell motion, 251, 260 shell perturbation, 249 sikes, 209 silicon crystal, 192 silicon single crystals, 476 silicon sphere, 198, 200, 201 mean diameter, 198 single-crystal silicon sphere, 194 single-sinker densimeter, 144 uncertainty, 144 skin depth, 443, 518 skin-effects, 488 slot, 250–2 open ended, 251 terminated at one end, 250 SMOW, 205–7 uncertainty, 207 solid electrolyte, 415–17 solid-liquid phase behavior, 107 solid mixtures, 407–8, 414, 418, 421, 423 calorimetric methods, 408 electrochemical method, 414 equilibration methods, 421 measurement uncertainty, 423 special methods, 418 vapour pressure, 419 solids, 476 crystalline, 476 non-crystalline, 476 sorption, 148 sound absorption, 268 sound reflection, 280 sound speed, 78, 80, 217, 238–45, 272, 277–8, 280, 282–3, 286–91, 300, 302, 305, 307–8, 310–11, 368, 280– 1, 307, 310–13, 489, 492, 494–6 absorption, 238, 240–1, 243, 268, 280 amount of substance, 240 annular resonator, 238, 277 attenuation, 80, 238, 240–1 boundary layer, 241 Brillouin frequency shift, 279 Brillouin scattering, 279, 291

resonant ultrasound, 238 elastic constants, 238 spectroscopy, 290 resonator, 241–2, 244–7, 251–2, 254–5, 259, 261, 265, 274, 296, 441 annular, 241, 244, 274–5 coaxial cylindrical, 441 cylindrical, 241, 244, 261, 265 hemispherical, 259, 261, 296 longitudinal modes, 265 opening in, 244, 246 pressure compensated, 247, 251 as the pressure vessel, 251 spherical, 241–2, 244. 251–2, 261, 296 spherical geometry, 245 unguided, 242 variable-frequency fixed-cavity, 244 variable-frequency fixed-geometry, 242 variable path-length fixed-frequency, 244 retrograde condensation, 107 retrograde gas condensate, 107, 115 reverberation method, 258 reverberation technique, 258 Reynolds number, 152 Richter, 209 ruby pressure scale, 69 scanning electron microscope, 375 scattering, 239, 287, 291 Brillouin, 239, 287, 291 inelastic neutron, 291 neutron, 239, 287 X-ray, 239, 287 Scholte, 239 Schottky effect, 56 second radiation constants, 492 Seebeck coefficient, 374–5, 419 sessile drop, 520–1, 527–8, 530–1 shear boundary layer, 259 shear modulus G, 311 shear stress, 312 shear viscosity, 267, 269 553

554

Index systematic errors, 240 thermal boundary layer, 241 thermally relaxing, 241 thermodynamic, 242 thermodynamic properties from, 287, 300 thermodynamic quantities, 239 through transmission, 289 time of flight, 238 time-of-flight measurements, 238–9, 242, 272 transient decay, 289 ultrasonic frequencies, 238 uncertainty, 241, 242, 245, 286, 288– 9, 291, 313 variable-frequency, 239 variable path interferometers, 238 variable path interferometry, 238 variable path-length, 239 viscothermal, 242 viscous, 241 water, 239, 272 working equations, 242 zero-density, 308 zero-frequency, 244 zero-frequency limit, 300 sound propagation, 240 phase-separating fluids, 240 specific gravity, 221 specific volume, 311 spectral emissivity, 493–4, 510 spectral radiance, 510 spectro-ellipsometry, 501 spherical cavities, 249, 280 gas port, 249 opening, 249 reverberation methods, 280 sound absorption, 280 sound speed in liquids, 280 spherical resonator, 245–6, 250, 253–5, 294–295 non-radial resonance, 246 radial modes, 245, 246 spherical shell, 247 motion, 247

cavity, 239 composition, 305 compressive, 312 continuous wave, 238, 289 critical, 241 cylindrical resonators, 239 diffraction of, 283 diffraction corrections, 272, 282 dispersion, 238, 240–1, 268 elastic constants, 287 fixed-cavity resonator, 239 fixed-frequency, 239 fluids near critical, 279 gases, 239–40, 302 gas-imperfections, 242 high pressures, 238 interferometers, 239 isotropic elastic solids, 300 isotropic Newtonian fluids, 300 liquid metal, 492 liquid phase, 496 liquids, 239, 242, 242, 280, 310 liquid water, 280 longitudinal, 288, 312 molecular thermal, 240 neutron scattering, 291 non-rigid boundary, 242 numerical integration, 307, 310 phase-sensitive pulse, 281 point sources and point detectors, 290 precondensation, 240 pulse, 77, 238 pulse-echo overlap, 281, 289 Pulse methods, 272 relaxation, 240 relaxing gases, 278 resonant ultrasound spectroscopy, 291 shear, 288 shell motion, 241 sing around, 289 solids, 239–40, 280, 287, 311 speed of light, 458 spherical resonator, 238, 245 554

Index

555

temperature-modulated calorimetry, 347 differential scanning, 347 temperature scale, 8, 484 Celsius, 8 Kelvin, 8 magnetic, 484 tensiometry, 526 high temperature, 526 terminal admittance, 250 terrestrial, 524, 531 thermal accommodation length, 260 thermal boundary layer, 243, 246, 247, 260 losses, 280 thermal, 280 viscous, 280 thermal conductivity, 247, 261–2, 267, 269, 339, 346, 349, 361–2, 370–1, 376–7, 485, 488, 491, 495–6, 504, 519 thermal diffusivity, 243, 246, 337, 339, 340, 344, 376, 377, 491, 496, 500 thermal effusivity, 346–7 thermal expansion, 208, 255, 311, 439, 492, 495–6, 500, 520, 522–3 microwave resonance frequencies, 255 thermal gravimetric analysis, 355 thermal gravimetric techniques, 362 thermal penetration depth, 260 thermal penetration length, 251 thermal relaxation, 240 polyatomic gas, 240 translational to vibrational, 240 thermal switch, 488 superconducting, 488 thermal transpiration, 73, 89 thermally relaxing gases, 241, 259 compact, 241 thermocouples, 187, 354 differential, 187 thermodynamics, 8 Second Law, 8 Zeroth Law, 8

radiative losses, 247 standard mean ocean water, 205 static permittivity, 456, 467 electrically conducting fluids, 467 statistical mechanics, 304, 388 Stefan Boltzmann constant, 510, 516, 519 Stefan-Boltzmann law, 409 Stokes optical constant, 494 Stonely, 239 surface acoustic wave, 239, 287, 374 surface micromachining, 375 surface separation, 110–11 recombination sampling, 111 surface tension, 61, 75, 78, 79, 493, 495, 501–12, 521–2, 524, 526–12 angle of contact, 521 capillary rise, 527 drop weight, 527 levitated drop, 527, 531 liquid metals, 522, 526 maximum bubble pressure, 527–8 and maximum pull, 527 pendent drop, 527 sessile drop, 527–8 synthesiser frequency, 265 Takaishi-Sensui equation, 89 Taylor expansions, 311, 309 Taylor’s series, 258 temperature, 8–9, 11, 12, 14–16, 22–5, 37, 40–1, 135–6, 144, 307, 496 absolute, 8 critical, 136 fixed points, 11, 23–5 ice point, 16 precision, 41 thermodynamic, 8, 12, 22, 37, 40, 249, 304 triple-point, 15, 135 triple point of water, 14 uncertainties, 37, 41 temperature-jump, 247, 263, 276 thermal accommodation coefficient, 247 555

556

Index three-omega method, 347 time domain reflectometer, 461, 466–7 coaxial-line, 461 time-of-flight, 239, 242, 244, 271–3, 280, 284, 287–8, 290, 292 diamond anvil, 290 diffraction error, 272 liquids, 280 multiple path, 280 phase comparison, 288 pulse-echo overlap, 288 pulse shape distortion, 273 pulse superposition, 288 sing around, 288 single path, 280 speed of sound, 271 tone burst, 281 total hemispherical emissivity, 505, 208, 515 total permittivity, 458, 463 coaxial–line reflection, 463 total radiance pyrometer, 506 total reflection, 459 Tralles, 209 transducers, 244, 252, 255, 267, 269–70, 280, 284, 287, 290, 294, 297 acoustic, 252, 255–256, 259, 287, 291–3, 295, 296–9 back plate, 293 buffer rods, 287 capacitance, 295 capacitive, 293, 296–8 electromagnetic, 292, 299 electrostatic, 293 impedance, 267, 294 laser, 292, 299 microwave, 255, 297 moving-coil, 269, 299 piezoelectric ceramic, 280 piezoelectric, 290, 292 polymer film, 293 ports, 252, 294–5 probes, 259 quartz crystal, 270, 284 solid dielectric, 293

thermodynamic temperature, 304 thermogravimetry, 421, 422 thermometer, 12, 22, 29, 31–7, 40–1, 50, 88, 182, 187–9, 252, 255–65, 297, 402, 486, 509, 523 acoustic, 37, 252, 255–6, 297 capacitance, 37 cryogenic, 34 electrical, 36 gas, 12, 34–7, 77 germanium, 486 germanium resistance, 36 high-temperature, 31 long stem, 30–1 mercury-in-glass, 523 noise, 38, 40 nuclear orientation, 38 platinium resistance, 12, 22, 135, 144, 182, 189, 187–8 quartz, 50, 88 radiation, 9, 12, 31–4, 40, 509 resistance measurements, 29 rhodium-iron resistance, 36 SPRT, 12 standard platinium resistance, 12, 22, 31, 135, 256 thermocouples, 41 thermoelectric effect, 37 vapour-pressure, 12, 34–6 white light interferometer, 77 thermometer immersion depth, 20 corrections, 20 thermometry, 11, 20, 26 fixed points, 10–11, 17–18, 22, 33, 36 blackbody cavity, 33 freezing-points, 10, 17 ITS-90, 11, 22 Melting Points, 10, 17 Other, 22 triple-points, 10 low-temperature, 20 platinum resistance, 26 triple-point, 11 thermomicroscopy, 362 556

Index

557

tube, 212 tuning fork, 212 vibrating object, 214–15 cylindrical, 215 cylindrical geometry, 214 vibrating tube, 161, 214 densimeter, 161 working equations, 161 vibrational period, 162, 214 vibrational relaxation time, 241, 244 attenuation, 244 virial coefficient, 180, 247, 257, 260, 303, 305–7, 388–94, 397–9, 452 acoustic, 247, 257, 260, 303 acoustic, 303 binary mixtures, 389 cross second, 393 cross third, 398 dielectric, 454 excess, 390, 393, 395 excess second, 394 fourth, 398 interaction, 390–1, 393, 397 mixtures, 390–1, 393, 397, 399 pure gases, 388 pure substance, 389 refractive-index, 453 refractivity, 452–3 refractivity virials, 453 second, 303, 306, 389, 391, 392–3, 394–5, 397–8 third, 303, 306, 389, 392–3, 398 temperature change method, 392 volumetric, 257 virial expansion, 448, 453 refractive index, 453 viscometer, 78, 105, 150–1, 261–2, 497–8, 503 oscillating, 503 vibrating-wire, 150–1 viscous boundary layer, 243, 263 viscous, 263 thermal, 263 viscous diffusivity, 243

solid dielectric capacitance, 267, 279, 299 sound, 244, 287 tensioned metal foil, 293 transient reflectivity, 500 transit time, 283 transmission functions, 459 transmissivity, 513–14 transpiration method, 422 vapour pressure, 422 transport coefficients, 243 transport properties, 239, 261 acoustic measurement, 261 triple point cell, 8, 12–13, 15–16, 20–1, 26, 29, 35, 38 argon, 20 gas, 20–1 hydrogen, 20 neon, 20, 35 oxygen, 20 water, 13, 15–16, 29, 8, 12, 26 truncation error, 304 tube, 250, 259 open flanged, 250 perturbation, 259 sealed cavity, 250 termination, 250 Twaddle, 209 two matched transducer, 280 two-sinker densimeter, 146 uncertainty, 146 ultrasonic cylindrical resonators, 284 ultrasound, 78 uncertainty densities, 146 universal gas constant, 249 vacuum sublimation, 100 vapour pressure, 78, 100, 408, 419, 421 Knudsen cell, 419 variable-path, 285 Venturi, 218 vibrating cylinder, 150 vibrating element, 212 cylinder, plate, 212 557

558

Index waves, 292, 312 longitudinal, 312 shear, 292, 312 transverse, 312 wettability, 527 high temperatures, 527 Wheatstone bridge, 48, 56, 342 Wiedemann-Franz-law, 491, 504 Wien’s approximation, 510 blackbody radiation, 510 Wollaston wire, 377 working equations, 244 x-cut quartz, 272 x-cut quartz crystal, 281

viscous penetration depth, 260 viscous penetration length, 251 viscous and thermal boundary layer, 244 voltage, 257 in-phase, 257 quadrature, 257 volume of mixing, 405 volumetric properties, 166 volumetric thermal expansion, 523–4 liquid phase, 524 volumometer, 169, 171–4 wall compliance, 244 water, 108 connate, 108 wavefunction, 259, 260, 278 waveguide, 264, 296, 298–9, 459, 465–6 relative permittivity, 462 variable-path-length, 465 waveguides transducer, 265 permittivity, 461 wavelength, 250, 268, 270, 284–5, 289, 312

X-rays, 68, 500 diamond-anvil, 68 diffraction, 68 X-ray-absorption, 502 X-ray absorption fine structure, 72 X-ray diffractions, 362 Young’s modulus E, 47, 63, 311–13, 406

558