Physico-Chemical Analysis MOLTENELECTROLYTES

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PHYSICO-CHEMICAL ANALYSIS OF MOLTEN ELECTROLYTES

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PHYSICO-CHEMICAL ANALYSIS OF MOLTEN ELECTROLYTES

Vladimír Daneˇ k† Institute of Inorganic Chemistry Slovak Academy of Sciences Bratislava, Slovak Republic

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

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© 2006 Elsevier B.V. All rights reserved. This work is protected under copyright by Elsevier B.V., 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 Rights Department in Oxford, UK: phone (+44) 1865 843830, fax (+44) 1865 853333, e-mail: [email protected]. Requests may also be completed on-line via the Elsevier homepage (http://www.elsevier.com/locate/permissions). In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 20 7631 5555; fax: (+44) 20 7631 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 the Publisher 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 prior written permission of the Publisher. Address permissions requests to: Elsevier’s Rights Department, at the 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 diagnoses and drug dosages should be made. First edition 2006 British Library Cataloguing in Publication Data A catalogue record is available from the British Library. Danek, Vladimir Physico-chemical analysis of molten electrolytes 1. Electrolyte solutions I.Title 541.3 72 ISBN-10: 0-444-52116-X

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

Dedicated to my wife Maria

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Publisher’s Note Vladimír Danˇek, 6 April 1940 – 28 November 2005 It is our sad duty to inform you that after submitting the manuscript for this book, Dr. Vladimír Daneˇ k passed away after a short illness in November 2005. In 1963 Dr. Daneˇ k joined the Institute of Inorganic Chemistry of the Slovak Academy of Sciences in Bratislava, of which he was the director in the period 1991–1995. His main field of interest was the physical chemistry of molten salts systems; in particular the study of the relations between the composition, properties, and structure of inorganic melts. He developed a method to measure the electrical conductivity of molten fluorides. He proposed the thermodynamic model of silicate melts and applied it to a number of two- and three-component silicate systems. He also developed the dissociation model of molten salts mixtures and applied it to different types of inorganic systems. More recently his work was in the field of chemical synthesis of double oxides from fused salts and the investigation of the physicochemical properties of molten systems of interest as electrolytes for the electrochemical deposition of metals from natural minerals, molybdenum, the synthesis of transition metal borides, and for aluminium production. During his career Dr. Daneˇ k was also head of the Department of Molten Salts, chairman of the Scientific Board of the Institute, chairman of the Scientific Collegium for Chemical Sciences of the Slovak Academy of Sciences, and member of the Presidium, the Executive Board, and chairman of the Commission for Chemical Sciences of the Scientific Grant Agency of the Ministry of Education of the Slovak Republic and the Slovak Academy of Sciences. In 1994 he was elected an active member of the New York Academy of Sciences. His numerous achievements were also recognized with the Award of the Czechoslovak Academy of Sciences, 1968; the Award of the Slovak Academy of Sciences, 1972, 1986 and 1989; the Dyonýz Štúr Silver Medal for Achievements in Natural Sciences, 1990; and the Dyonýz Štúr Golden Medal for Achievements in Natural Sciences, 2000. In the course of his distinguished career he published more than 240 papers in international journals and conference proceedings. We extend our condolences to his wife Maria and further family on their sad loss.

Acknowledgement Ing. Marián Kucharík, PhD, took on himself the task of correcting the proofs. Ing. Zden˘ek Pánek, DrSc, was responsible for compiling the index. vii

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Preface The idea of this book came to me after seeing a multitude of young scientists carrying out their PhD studies on molten salts and attending various conferences, many of whom were eager for any new knowledge on molten salts. Many of them focused their study on problems that would contribute a little more to the understanding of the structure of the system of interest using sophisticated and very expensive equipment. On the other hand, while being mostly interested in the quantum chemistry and molecular dynamics simulation calculations, their knowledge of the physical chemistry of inorganic melts is often poor. At the universities, we can also see another very encouraging development. After the resumption of relations between East and West at the end of the 1990s, a lot of young scientists from Eastern Europe and Asia came to the universities in Western Europe for a shorter or longer stay, or a PhD study. While in Eastern Europe and Asia, the measurement of physico-chemical properties is more frequent due to the lack of other expensive sophisticated physical equipment, in Western Europe and the United States, the emphasis is rather on the use of high-tech investigations. This book includes selected topics on the measurement and evaluation of physicochemical properties of molten electrolytes. It describes the features, properties, and experimental measurement of different physico-chemical properties of molten salt systems used as electrolytes for the production of different metals, metallic layer deposition, as a medium for reactions in molten salts, e.g. precipitation of double oxide powders used for functional and construction ceramics, special parts for steel and copper production, etc. The physico-chemical properties such as phase equilibria, density (molar volume), enthalpy (calorimetry), surface tension, vapor pressure, electrical conductivity, viscosity, etc. are the most important parameters of electrolytes needed for technological use. For each property, the theoretical background, experimental techniques, as well as examples of the latest knowledge and the processing of most important salt systems will be given. Most of the examples are among the published works of the author. The aim of the book is not only to present the state of the art studies on different properties of molten salt systems and their measurement, but also to present the possibilities of modeling molten salt systems, to be able to forecast the properties of an electrolyte mixture from the properties of the pure components in order to avoid experimentally demanding, and in most cases, also expensive measurements. Some direct methods of study on the structure (ionic composition) of molten electrolytes are also presented.

ix

x

Preface

During the last decades of the twentieth century, a substantial development of experimental and especially, computational techniques took place. Thus, new measurement methods and mathematical processing of data appeared. This new information is not given anywhere in a comprehensive form and therefore, a lot of people working in this field are not acquainted with it. This book thus fills a substantial gap in this field of science. The book also documents the latest research in molten salt chemistry and brings new results and new insights into the study of molten salt systems using the results of X-ray diffraction and XSAF methods, Raman spectroscopy, and NMR measurements. On the other hand, it is not the goal of this book to present an exhaustive review of the measurement of physico-chemical properties carried out in the last two or three decades. This would certainly exceed the framework of this book. The electrochemistry of molten salts is not included in this book either, since in my opinion, this topic would need a separate book due to its complexity and wealth of applications. The quantum chemistry and molecular dynamics simulation methods are omitted as well. However, recent books of this kind have been written by the Russian authors Antipin and Vazhenin (1964), and the American authors Blander (1964), Sundheim (1964), Mamantov et al. (1969), Lumsden (1966), and by Lowering and Gale (1983). Since then, new information on physical properties of molten salt systems has been disseminated only in papers in different journals. This book should serve as a textbook for all people working in the field of molten salt chemistry. It is aimed at undergraduate students working on their diploma work, for students working on their PhD theses, and also for other graduates, e.g. teachers at universities, scientists in academic institutes, research institutes, and in industry, i.e. to all who need up-to-date information on this subject. It is a pleasure for me to express my thanks to my teachers Professor Milan Malinovský, who taught me the theory of molten salts, Professor Kamil Matiašovský, who taught me his experimental skills, and Dr. Ivo Proks, who helped me to understand a little of thermodynamics. Professor Pavel Fellner of the Slovak Technical University is greatly acknowledged for his advice and comments which helped improve this book. I am very thankful to my colleagues at the Institute of Inorganic Chemistry of the Slovak Academy of Sciences for their help and good friendship during my whole professional career. Finally, my warm thanks and great admiration are devoted to my wife. Without her timeless patience, systematic support, and help, this book could not have been written. Vladimír Daneˇ k 2005

CONTENTS

Publisher’s Note .......................................................................................................... vii Preface.......................................................................................................................... ix 1

Introduction ........................................................................................................

2

Main Features of Molten Salt Systems............................................................ 5 2.1. Structure of melts containing metals of various valence ......................... 10 2.1.1. Systems of univalent electrolytes ................................................ 10 2.1.2. Systems containing divalent cations ........................................... 18 2.1.3. Systems containing trivalent cations ........................................... 26 2.1.4. Systems containing tetravalent cations ....................................... 41 2.1.5. Systems containing pentavalent cations ...................................... 47 2.1.6. Systems containing hexavalent cations ....................................... 50 2.1.7. Systems containing halides and oxides ....................................... 56 2.1.8. Systems containing jumping electrons ........................................ 79 2.1.9. Systems of silicate melts ............................................................. 101 2.1.10. Systems of molten alkali metal borates ...................................... 102 2.1.11. Systems of metallurgical slags .................................................... 104

3

Phase Equilibria ................................................................................................. 3.1. Thermodynamic principles........................................................................ 3.1.1. Gibbs’s phase law ........................................................................ 3.1.2. Lever rule..................................................................................... 3.1.3. Thermodynamics of solutions ..................................................... 3.1.4. Thermodynamic models of molten salts ..................................... 3.2. Phase diagrams of condensed systems ..................................................... 3.2.1. Binary systems............................................................................. 3.2.2. Ternary systems ........................................................................... 3.2.3. Quaternary systems...................................................................... 3.2.4. The CaO–Al2 O3 –SiO2 system .................................................... 3.3. Experimental methods............................................................................... 3.3.1. Thermal analysis .......................................................................... 3.3.2. Cryoscopy .................................................................................... 3.3.3. Differential thermal analysis ....................................................... xi

1

107 107 107 108 110 119 155 155 167 184 186 189 189 191 205

xii

Contents

3.4.

Calculation of phase diagrams .................................................................. 208 3.4.1. Coupled analysis of thermodynamic and phase diagram data.... 208 3.4.2. Calculation of the phase diagram of the quaternary system KF−KCl−KBF4 −K2 TiF6 ............................................... 213

4

Enthalpy .............................................................................................................. 4.1. Thermodynamic principles........................................................................ 4.1.1. Hess’ law...................................................................................... 4.1.2. Kirchoff’s law .............................................................................. 4.1.3. Enthalpy of reaction .................................................................... 4.1.4. Estimation of enthalpy of fusion ................................................. 4.1.5. Enthalpy balance.......................................................................... 4.2. Experimental methods............................................................................... 4.2.1. Calorimetry ..................................................................................

221 221 222 223 223 225 231 231 232

5

Density................................................................................................................. 5.1. Theoretical background............................................................................. 5.1.1. Molar volume............................................................................... 5.1.2. Partial molar volume ................................................................... 5.1.3. Application to binary and ternary systems.................................. 5.2. Experimental methods............................................................................... 5.2.1. Method of hydrostatic weighing ................................................. 5.2.2. Maximum bubble pressure method .............................................

255 255 255 256 257 266 266 268

6

Surface Tension .................................................................................................. 6.1. Thermodynamic principles........................................................................ 6.1.1. Gibbs equation ............................................................................. 6.1.2. Surface adsorption of ideal and strictly regular binary mixtures ............................................................................ 6.1.3. Surface tension in ternary systems.............................................. 6.1.4. Surface tension models................................................................ 6.2. Experimental methods............................................................................... 6.2.1. Capillary method ......................................................................... 6.2.2. Maximum bubble pressure method ............................................. 6.2.3. Detachment methods ................................................................... 6.2.4. The drop methods ........................................................................ 6.3. Contact angle............................................................................................. 6.3.1. Contact angle measurement......................................................... 6.4. Interfacial tension ...................................................................................... 6.4.1. Experimental methods .................................................................

271 271 273 278 284 286 290 290 292 296 303 305 305 306 307

Vapor Pressure ................................................................................................... 7.1. Thermodynamic Principles........................................................................ 7.1.1. Gas mixtures ................................................................................ 7.1.2. Liquid–gas equilibrium................................................................

313 313 313 314

7

Contents

7.2.

xiii

Experimental methods............................................................................... 315 7.2.1. The boiling point method ............................................................ 315 7.2.2. The transpiration method............................................................. 322

8

Electrical Conductivity...................................................................................... 8.1. Theoretical background............................................................................. 8.1.1. Electrical conductivity of “ideal” and real solutions .................. 8.1.2. Electrical conductivity in ternary systems .................................. 8.2. Experimental methods............................................................................... 8.2.1. Capillary cells .............................................................................. 8.2.2. Conductivity cell with continuously varying cell constant......... 8.2.3. Two-electrode cell........................................................................ 8.2.4. Four-electrode cell .......................................................................

327 328 329 345 346 349 352 354 356

9

Viscosity............................................................................................................... 9.1. Theoretical background............................................................................. 9.1.1. Viscosity of “ideal” and real solutions........................................ 9.1.2. Application to binary and ternary systems.................................. 9.2. Experimental methods............................................................................... 9.2.1. Method of torsional pendulum .................................................... 9.2.2. Falling body method.................................................................... 9.2.3. Rotational method........................................................................

359 359 360 361 369 369 377 380

10

Direct Methods of Investigation ....................................................................... 10.1. X-ray diffraction and XAFS measurements ............................................. 10.2. Raman spectroscopy.................................................................................. 10.2.1. Theoretical background ............................................................... 10.2.2. Characteristic features of Raman scattering in melts ................. 10.2.3. Experimental techniques of measurement................................... 10.2.4. Raman spectroscopy in various systems ..................................... 10.3. Nuclear magnetic resonance ..................................................................... 10.3.1. Theoretical background ............................................................... 10.3.2. Experimental technique of measurement .................................... 10.3.3. High temperature NMR measurement ........................................

385 385 388 388 390 391 395 402 402 405 406

11

Complex Physico-chemical Analysis ................................................................ 423 11.1. Dilute solutions ......................................................................................... 423 11.2. Whole systems........................................................................................... 424

References .................................................................................................................... 429 Index............................................................................................................................. 445

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Chapter 1

Introduction Inorganic ionic melts represent a group of waterless solvents and solutions, which is interesting both from the point of view of fundamental research as well as with regard to their present and prospective use in technical practice. At present we find them used, e.g. in the metallurgical production of aluminum, magnesium, alkali, and refractory metals, where they are used as electrolytes; in black metallurgy, where they form slags gathering unwanted admixtures and reaction products in iron and steel production. Molten electrolytes are used in galvanic metal plating in melts, e.g. in aluminum plating, boriding, or in the deposition of refractory metal layers (Ti, Mo, Nb, etc.). Molten mixtures of alkali metal halides and zirconium, thorium, beryllium, and uranium are used as heat-bearing medium in primary circuits of nuclear plants. Mixtures of molten alkali metal halides and hydroxides have a potential use in energy storage, where the relatively high value of the enthalpy of fusion is used. Molten carbonates of alkali metal halides are used as electrolytes in molten carbonate fuel cells. A big industrial field, where oxide melts are predominantly used, is the glass industry. Here, the high affinity of these melts to under-cooling and glass formation is exploited. Recently, there has also been an increase in the importance of melts in their use as a reaction medium for chemical and electrochemical synthesis of compounds for functional and construction ceramics, e.g. double oxides with spinellitic and perowskite structure and binary compounds with prevailing covalent bond character, mainly borides and carbides of transition metals. In order to decide the suitability of a certain melt in technical practice, an in-depth knowledge of its physico-chemical properties is unavoidable. The present database of the properties of inorganic melts is relatively broad. Many properties are known, such as phase equilibria, enthalpies of fusion, heat capacities, density, electrical conductivity, viscosity, surface tension, emf of galvanic cells of many molten systems, the measurement of which was stimulated first by their technological application. Nevertheless, the published data on the physico-chemical properties of the molten systems are often incomplete and in many cases the results given by different authors may more or less differ. The reason is that the experimental measurement of the physicochemical properties of inorganic melts is sometimes inadequate. First, because of the shortage of expensive construction material. Second, due to the relatively high costs connected with the construction of unique measurement devices. It is obvious that the choice of a suitable melt for a concrete application, which is often a multi-component mixture, is given by the requisite optimum physico-chemical 1

2

Physico-chemical Analysis of Molten Electrolytes

properties at a given temperature. With regard to the experimental difficulty of direct measurement, in choosing a concrete melt it is much cheaper to use convenient structural models that enable us to forecast the values of the physico-chemical properties of multi-component molten systems on the basis of knowledge of the properties of the pure components. It is then necessary to know the mathematical description of the functional property – composition and property – temperature dependences. The definite form of such dependences is given first by the structure (ionic composition) of the molten systems, which in many cases is still not satisfactorily known. Recently, research on the structure of inorganic ionic melts has rapidly advanced. A substantial improvement in the experimental techniques, especially in high-technology electronics and computer techniques, has contributed to this. However, the present knowledge of the physico-chemical properties of molten systems is in many cases on a higher level than the possibility of their interpretation. First, it follows from the lack of an adequate knowledge of their structure. High-temperature X-ray diffraction analysis applied to the liquid phase has not principally contributed to the classification of the structure of melts, mainly due to the lack of suitable approaches in structural analysis. More success was attained recently by the exploitation of sophisticated methods of high-temperature infrared and Raman spectroscopy, NMR, MAS NMR, and some numerical methods, especially methods of quantum chemistry and of molecular dynamics. While for the last mentioned methods of investigation the measuring devices could be acquired, with only small adaptations, devices for measurement of physico-chemical properties are not available on the market. The measurement, especially its precision depend on the skills of the scientists and the workers in the laboratory. For instance, the construction of a high-temperature torsional pendulum viscosimeter requires in-depth knowledge of many features of the technique. It should also be mentioned that for scientific purposes, the accuracy and precision of results must be at least one order higher than for industrial purposes. On the other hand, using generalized knowledge many industrial measurements could be avoided as necessary data could be estimated with good accuracy. Generally, any electrolyte is composed of a mixture of alkali metal halides, which serve as solvent, and the compound of the deposited metal. In addition, there may be other additives, which may improve the properties of electrolyte or enhance the metal deposition. For a certain purpose a concrete molten system must be used. For example, in the electro-deposition of metals from molten salts several types of molten systems were tested as electrolytes. From the analysis of literature and on the basis of the electro-active species used, they can be divided into two principal groups: • •

systems containing halo-complexes of deposited metals, systems containing oxides or oxy-complexes of deposited metals.

Introduction

3

In all of the investigated systems, one of the most important tasks to be solved is to find the proper composition of the electrolyte with regard to both the suitable physico-chemical properties and the desired character of the electrodeposited product. Both problems are closely related to the actual structure, i.e. the ionic composition of the melt. Quite recently attention was paid to the role of oxides, either as electro-active species, as impurities or as additives in the electro-deposition of transition metals. This may be demonstrated, e.g. in the case of electro-deposition of molybdenum, where the electrolysis of neither pure K2 MoO4 , nor the KF–K2 MoO4 mixture yields a molybdenum deposit. However, introducing small amounts of boron oxide, or silicon dioxide to the basic melts, smooth and adherent molybdenum deposits may be obtained. Also, in the case of niobium and tantalum deposition, the presence of oxygen either from the moisture or added on purpose leads to the formation of oxohalo-complexes, which due to their lowered symmetry and thus lower energetic state, decompose easier at the cathode yielding pure metal. It is thus the aim of the present book to serve as a guide in the measurement of the physico-chemical properties of molten salts and to characterize briefly the properties and the structure of different types of molten salt systems. In this book, only direct methods of measurements and different methods of processing the measured data are discussed. Computer simulation methods are not considered.

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Chapter 2

Main Features of Molten Salt Systems The transition from the solid to the liquid state is so commonplace that ancient scholars tried to explain this phenomenon. The development of the natural sciences in the sixteenth and seventeenth centuries enabled its deeper understanding, especially in connection with the development of suitable methods for the measurement of volume and heat. The first experiments to apply thermodynamic principles to the melting process of substances arose approximately at the time of the formulation of the second law of thermodynamics, i.e. in the middle of the nineteenth century. As it is known, the main advantage of classical thermodynamics lies in the fact that it is not necessary to know the structure of the investigated system. This enabled some of the most important classical thermodynamic conclusions on the melting process to be made before the general idea of the atomic structure of a substance was accepted. However, as the latter observations showed, melting depends on the structure of the crystalline state of matter and due to the great variability of the crystal structure of matter, different features of this phase transformation have to be investigated. Today it is obvious that, without the thorough knowledge of the crystal structure of solid substances, it is not possible to study the structure of the melt. It should be, however, also emphasized that X-ray structural analysis, which was broadly exploited for the study of the crystal structure of solids, did not enable explanation of all structural aspects of the molten state. For deeper understanding of the structure of melts obviously, new experimental and theoretical approaches are needed. From the general structural classification, the crystalline substances can be divided into four groups: (a) (b) (c) (d)

molecular, ionic, metallic, network forming.

At melting, each group forms its own type of melt. Of course, melts having the features of two of these groups may also occur. For example, silicate melts belong to the network forming as well as to the ionic melts.

5

6

Physico-chemical Analysis of Molten Electrolytes

The structure of inorganic melts is, in spite of their relative simplicity, not completely understood. The earlier calculations and simple models of molten salts were built up rather on intuition. However, they were the necessary first step for more sophisticated approaches. In general, inorganic melts increase their volume at melting. The number of substances that decrease their volume at melting is surprisingly low. From among the more common substances, we have e.g. Sb, Bi, Ga, H2 O, and RbNO3 . These substances are characterized by a low coordination of atoms, which indicates an “open” structure of the solid. At melting, their structure partially collapses, which is followed by a volume contraction. At a sufficiently high temperature, molten salts are miscible in any ratio. At lower temperatures, a limited miscibility can be observed. Partially miscible salts are in general mixtures of halides of multivalent metals with prevailing covalent bonds, like e.g. Al, Bi, and Sb, with halides having a high ratio of ionic bonds, e.g. CaCl2 and KCl. The shape of the dependence of the given property on composition gives certain information about the structure of these mixtures. The comparison of the values of molar volumes of mixtures of alkali metal halides and the mixtures of alkali metal halides with PbCl2 can be mentioned as an example. In the former case, the molar volume changes almost additively, which indicates that in the mixtures, the cations and anions are arranged more or less equally as in the pure salts. In the latter case, the change in molar volume on composition is almost additional only in the system NaCl–PbCl2 , and from potassium to cesium, there is always a more expressive minimum. This trend indicates that in these melts, structural changes causing deviation from ideal behavior take place. However, the molar volume of molten salt mixtures is in general not influenced much by structural changes, since the changes in the ionic ordering cannot be great. More sensitive in this direction is the electrical conductivity, which shows larger changes with the change in composition. An additive change of the electrical conductivity on composition is not known. For instance in the systems of alkali metal halides, in contrast to the molar volume, negative deviations in the concentration dependence of conductivity from the additivity can be observed. This effect supported the supposition of the formation of associates, leading to lowering the number of conducting ions in the melt. Even though until now, no further data supporting these ideas are available, it is clear that between cations of uneven size, an important interaction exists. The concept of “complexing” in molten salt mixtures is somewhat different from that for solutes in aqueous or organic solvents. In water solutions, every ion is separated from the other ions by the solvatation wrapping of water. Ions, simple or complex, are mutually influenced only by weak forces and each ion behaves independently. In molten salts, the positively and negatively charged ions are in close contact and the interaction forces are great. The heats of formation of complex anions, except for the stabile ions like [SO4 ]−2 , [NO3 ]− , etc., are in the order of a few kJ · mol−1. These values are lower than that of the activation energy of diffusion, which in these melts is

Main Features of Molten Salt Systems

7

relatively fast. This leads to the idea of the fast formation and disintegration of the complex ions in halide melts. If the complex ion in the melt could be detected, its lifetime must be much longer than the vibration period of the heat vibrations of individual ions (10−13 s), and longer than the mean time of contact between individual cations and anions (10−10 s). The mean lifetime of the ion [CdBr 3 ]− in the system KBr–CdBr2 is around 10−2 s, thus this ion can be really regarded as a complex ion. In every pure alkali metal halide or earth alkali metal halide melt, the cation is surrounded by anions in the first coordination sphere and has cations in the second coordination sphere. This arrangement is caused by the coulombic forces between cations and anions. In simple binary univalent systems AX–BX (e.g. NaCl–KCl), the A+ cations are surrounded by X− anions in the first coordination sphere and by A+ and/or B+ cations in the second. However, the individual arrangement changes very quickly and its lifetime is very short (<10−13 s). Thus, it cannot be detected by spectroscopic methods. The large deviation from ideal behavior, the minimum in the enthalpy of mixing, and the inflection points on the partial molar entropy of charge-unsymmetrical binary systems have been attributed to special ordering effects. On the basis of simple structural models, it had been suggested that these effects could be explained by real “complex” formation. For binary mixtures containing divalent cations, complexes with the stoichiometry [MeX4 ]2− were proposed. On the basis of suitable structural models, S-shaped curves for the partial molar entropy and the minimum in the enthalpy of mixing at x(MeX2 ) = 0.33 were calculated. In systems containing trivalent cations, the formation of the [MeX6 ]3− complex anion has been suggested from the minimum in the enthalpy of mixing. A series of spectroscopic investigations support this view of “complexing.” Every component of the MX–MeX2 mixture in its pure molten state possesses a certain structure, defined by an average number of atoms in the first and second coordination spheres. The M+ cations are surrounded in the first coordination sphere by X− anions and have M+ cations in their second coordination sphere. The same situation applies to the molten MeX2 component. In the molten mixture, the Me2+ and M+ cations are in the first coordination sphere again surrounded by common X− anions but have M+ and/or Me2+ cations in their second coordination sphere. Mixtures dilute in MeX2 have an excess of M+ cations. The divalent Me2+ cations have higher field strength in comparison with the M+ cation. They thus attract more X− anions in order to supplement their coordination sphere to the maximum. This effort is compensated by the presence of M+ cations in the second coordination sphere. On the other hand, the probability of having two or more Me2+ cations in the second coordination sphere of M+ is small. Thus the first and second coordination spheres define the shape of the “complex” anion. When the concentration of MeX2 is sufficiently high, there are not enough M+ cations for the second coordination sphere to surround the “complex” anion and a new

8

Physico-chemical Analysis of Molten Electrolytes

microstructure of the melt arises. This leads to the change in the course of the enthalpy of mixing. For the MX–MeX2 systems, this change occurs in the composition range 0.3 < x(MeX2 ) < 0.35 while for the MX–MeX3 , it occurs in the range 0.2 < x(MeX3 ) < 0.25. However, the absence of the additive M2 MeX4 compound in the phase diagram of some systems does not necessarily imply the absence of the complex [MeX4 ]2− anions in the melt. The presence of complex anions in the melt has been proved by Raman and FTIR spectra. Divalent cations like Mn, Fe, Co, Ni, Cu, Zn, etc., due to the splitting of the d-electrons of the Me2+ cations in the environment of the X− anions, at mixing with alkali metal halides, show a certain degree of covalence of bonds. The splitting of d-electrons leads to an increase of energy, which is called the ligand field stabilization energy (LFSE). This splitting varies with the symmetry of the coordination sphere of the cation. The change of the cation coordination and its thermodynamic stability is also affected by the LFSE. This effect can be seen on phase diagrams of various systems. For instance in the systems MF–MgF2 , where M = Li, Na, K, Rb, and Cs, the ionic strength and the repulsive forces of the alkali metal cation in the second coordination sphere affect the stability of the complex compounds formed in the individual systems. The lower the polarization ability of the M+ cation, the more stable are the complex anions formed in the system and the more complex anions are formed. The Li+ cation does have the highest polarization ability among the alkali metals. Therefore in the system LiF–MgF2 , no complex compound is formed and the system shows continuous solid solutions. In the system NaF–MgF2 , two compounds – NaMgF3 and Na2 MgF4 – are formed, which melt incongruently due to the still high polarization ability of the Na+ cation. In the system KF–MgF2 again, two compounds KMgF3 and K2 MgF4 , are formed, however, due to the lower polarization ability of the K+ cation, they melt already congruently. The same situation also applies to the system RbF–MgF2 , but the fields of primary crystallization of both compounds, RbMgF3 and Rb2 MgF4 , are more extensive compared with those in the system KF–MgF2 . The Cs+ cation shows the lowest polarization ability. In the system CsF–MgF2 , thus four complex compounds are formed. There are three congruently melting compounds Cs3 MgCl5 , Cs2 MgCl4 , and CsMgCl3 , and one incongruently melting compound CsMg3 Cl7 . From the structural measurement, it follows that MgF2 crystallizes in the tetragonal system. The most expressive complex compounds MMgF3 (M = Li, Na, K, Rb, Cs) also crystallize in the tetragonal system. On the other hand, the complex compounds M2 MgF4 (M = Na, K, Rb, Cs) crystallize in the cubic system with an octahedral space group. Similar thermodynamic behavior can be observed also in other MCl–MeCl2 systems. It may be thus concluded that the cohesive energy of pure transition metal chloride melts is affected by LFSE. Upon mixing with alkali metal chlorides, the coordination of the transition metal cation does not change, i.e. its coordination in pure MeCl2 and in the MCl–MeCl2 mixtures is the same.

Main Features of Molten Salt Systems

9

Regarding the thermodynamic behavior of the charge-non-symmetrical binary systems, the following general rules can be deduced. (i)

With regard to the Temkin’s model of ionic melts, the MX–MeXn systems show negative deviation from ideal behavior, which increases with: • • • •

increasing size of the alkali metal cation (e.g. by substituting Na+ with K+ ), decreasing size of the multivalent cation (e.g. by substituting Ca2+ by Mg2+ ), increasing polarizability of the alkali metal and/or multivalent cation (e.g. by substituting Mg2+ by Al3+ ), increasing charge of the multivalent cation (e.g. by substituting Ca2+ by Ti4+ ).

Some empirical rules can also be deduced regarding the behavior of the common anion. • • (ii)

(iii)

(iv)

the deviation from the Temkin’s ideal behavior decreases with increasing size of the anion (e.g. by substituting F− by Cl− ), the enthalpy of mixing increases in the sequence Cl < Br < I.

The appearance of inflection points on partial enthalpies of mixing and of minima in the interaction parameter indicates a possible formation of the complex anion in the mixture. In spite of the very high polarization ability of the Li+ cation and no minimum in the interaction parameter of the LiX–MeXn system, tetrahedral, octahedral, and/or a distribution of geometries has been observed by spectroscopic methods. In order to elucidate the structure of the melt, i.e. the ionic composition, besides the calculation of the thermodynamically consistent phase diagram, supplementary spectroscopic measurement is needed. Raman and IR spectroscopic studies have confirmed the appearance of MeX2− 4 complexes at x(MeX2 ) < 0.33 and of the MeX3− complexes at x(MeX ) < 0.25 with tetrahedral or octahedral symmetries, 3 6 respectively, for these complexes. The observed values and the shape of the enthalpies of mixing are not, or only very little affected by temperature.

At present, only the crude structural concept of Førland (1954) and conclusions made by Lumsden (1964) have described the above listed conclusions on a semi-quantitative level. The recent molecular dynamics computer simulations made by Liška et al. (1995b) and Castiglione et al. (1999) for the system NaF–AlF3 resulted in a non-realistic dependence of structure on concentration. Obviously the polarization forces of atoms, which have not been taken into account, seem to be of great importance. The most direct proof of the existence of complex ions was obtained using spectroscopic measurements. The presence of [CoCl4 ]−2 , [NiCl4 ]−2 , [CuCl4 ]−2 , [PbCl3 ]− , and

10

Physico-chemical Analysis of Molten Electrolytes

[PbCl4 ]−2 in mixtures of CsCl and the halide of the respective metal was confirmed using ultraviolet absorption spectroscopy. By means of the Raman spectroscopy, the presence of complex anions [AlF4 ]− , [AlF5 ]2− , and [AlF6 ]−3 in mixtures of AlF3 with NaF or KF was confirmed. On the basis of the conductivity, the enthalpy of mixing, as well as of other indices, the existence of a number of complex anions like [CdCl3 ]− , [CdCl4 ]−2 , [CdCl6 ]−4 , [ZnCl3 ]− , [ZnCl4 ]−2 , [PbCl4 ]−2 , [PbCl4 ]−4 , [MgCl4 ]−2 , [CuCl3 ]−2 , etc. in the systems of KCl and the chloride of the respective metal were foreseen. In the case of the systems KCl–CdCl2 , KCl–PbCl2 , and RbCl–PbCl2 , the dependence of the electrical conductivity on composition minima can be observed. They can be reasonably explained by the formation of the anions of the type [CdCl4 ]−2 , [CdCl3 ]− , [PbCl4 ]−2 , etc. The existence of associates such as [SO4 F]−3 , [MoO4 F]−3 , [TiF6 Cl]−3 , etc., which undergo at melting a more or less thermal dissociation, while thermodynamically proved, is under lively discussion.

2.1.

STRUCTURE OF MELTS CONTAINING METALS OF VARIOUS VALENCE

In general, the ionic composition of molten salt systems depends on the solvent used for the dissolution of the compound, which contains the metal to be deposited, and the chemical nature of this compound. Usually, chemical reactions take place between this compound and the solvent. At these chemical reactions, new complex anions are formed, atomic composition and stability of which depend on the electronic state of the central metallic atom and the polarization ability of the alkali metal cations. The chemical nature of the anions present also plays a non-negligible role. The above-mentioned phenomena will be explained in the following chapters. It is not the goal of this book to describe the behavior of all molten salt systems in question. This can be found in the textbooks of Inorganic Chemistry. The features of only the technologically most important systems will be described in the following chapters.

2.1.1. Systems of univalent electrolytes

Alkali metal halides, mainly sodium and potassium fluorides and chlorides, are usually used as solvents for salts of multivalent metals, which are deposited on electrolysis at the cathode. Alkali metal halides improve the physico-chemical properties of the electrolyte and in many cases, it is the only possibility to deposit the desired metal altogether. 2.1.1.1. Single salts

Alkali metal halide melts are characterized by a quasi-crystalline structure originating by dilatation of the crystal structure and by the occurrence of different kinds of positional disordering. Cations and anions are preferably surrounded by ions of the opposite charge

Main Features of Molten Salt Systems

11

forming two one-into-another inserted quasi-crystalline structures. Forces of mutual ionic interactions are electrostatic and due to the above-mentioned ordering, they are centrosymmetrical. Due to the positional randomness, the geometrical compensation of the electrostatic forces is, with regard to the crystalline state, reduced as also the electrostatic polarization effects of ions. This leads to shortening of the inter-ionic distances and lowering of the coordination number of ions (Table 2.1). The statistical consequence of such mutual interactions is the formation of ionic pairs or associates. The positional disordering probably causes association of ions due to the polarization forces, whereby the extent of association increases with the increasing atomic number. Alkali metal halides increase in general their volume at melting (Table 2.2), which is due to the lowering of atomic coordination and the formation of a free volume. Several attempts to calculate the properties of alkali metal halides on the basis of the equation of state can be found in the literature. Reiss et al. (1959, 1960) estimated the reversible work needed for creation of a spherical cavity in liquids of rigid spheres and derived the equation of state for these liquids pV 1 + Y + Y2 = RT (1 − Y )3

(2.1)

Table 2.1. Cation–anion distances of alkali metal halides in the solid, liquid, and gaseous states, and their coordination numbers Salt LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCl KBr KI RbF RbCl RbBr RbI CsF CsCl CsBr CsI

d(s) (nm)

Coord. number

d(l) (nm)

Coord. number

0.201 0.257 0.275 0.300 0.231 0.282 0.299 0.329 0.267 0.314 0.331 0.354 0.282 0.330 0.343 0.367 0.301 0.359 0.362 0.385

6 6 6 6 6 6

0.195 0.247 0.268 0.285 0.230 0.280 0.298 0.315 0.266 0.310 0.330 0.353 0.273 0.329 0.342 0.366 0.282 0.353 0.355 0.383

3.7 4.0 5.2 5.6 4.1 4.7 – 4.0 4.9 3.7 – – – 4.2 – – – 4.6 4.6 4.5

6 6 6

6

6 8 8

d(s) - at ambient temperature, d(l) - at 1.05 × Tfus , d(g) - at boiling temperature.

d(g) (nm) 0.153 0.203 0.217 0.239 0.184 0.236 0.250 0.271 0.213 0.267 0.282 0.305 0.225 0.279 0.295 0.318 0.235 0.291 0.307 0.332

12

Physico-chemical Analysis of Molten Electrolytes Table 2.2. Thermodynamic parameters of melting of some alkali metal halides Sfus (J/mol · K)

Salt

Tfus (K)

Vfus /Vs (%)

LiF LiCl LiBr

1121 883 823

29.4 26.2 24.3

24.2 22.6 21.3

NaF NaCl NaBr NaI

1268 1073 1020 933

27.4 25.0 22.4 18.6

26.3 25.9 25.5 25.1

KF KCl KBr KI

1131 1043 1007 954

17.2 17.3 16.6 15.9

25.1 25.5 25.5 25.1

RbCl RbBr CsCl

995 965 920

14.3 13.5 10.0

23.8 24.2 22.2

where p is the pressure, V is the molar volume, T is the temperature, and Y =

π a3N 6V

(2.2)

In Eq. (2.2), a is the diameter of the rigid spheres and N is the Avogadro number. Wertheim (1963) and Thiele (1963) showed that the equation of state derived by Reiss et al. (1959, 1960) is identical with the equation of state derived by Percus and Yevick (1958), who presented the exact solution of the integral equation for the radial distribution function. Lebowitz et al. (1965) generalized later the validity of this theory for the mixture of simple liquids. The same equation of state was derived by Stillinger (1961), who showed that Eq. (2.1) can be applied to fused salts also if we replace the fused salts by a rigid sphere fluid in which the particle diameter a equals the sum of the cation’s and anion’s radii. Later Reiss and Mayer (1961), Mayer (1963), and Yosim and Owens (1964) calculated some thermodynamic properties of fused salts (entropy, heat capacity, entropy of fusion, compressibility, and surface tension) on the basis of this theory. The agreement between the calculated and experimental data is reported as good and in some cases as very good. However, this model fails in the calculation of transport properties. Vasu (1972a,b) used the results of the above-mentioned works for the calculation of contact correlation function in molten salts on the basis of a double hard core model, which describes better the real situation in molten salts. This contact correlation function was applied in the calculation of viscosity and electrical conductivity of molten alkali

Main Features of Molten Salt Systems

13

metal halides. In this model, the coulombic interactions among particles in the melt were taken into account, replacing them by double hard core interactions. This respects the fact that the ions of the same sign cannot come closer than to a certain distance, which in the √ first approximation, equals the minimum distance in the NaCl type crystal lattice, i.e. a 2, where a is the cation–anion distance. In the melt, the distance can be even shorter because the regular arrangement of particles in the lattice disappears after melting and the particles can be deformed and compressed. Further, it is known from the radial distribution functions that the coordination number decreases after melting from 6 to 3.5–5.6. Therefore, Fellner and Daneˇ k (1974), for molten alkali metal halides, assumed that the minimum distance between the ions of the same sign√could be expressed in terms of aF, where the value of the factor F will be lower than 2. For the contact correlation function g(a) of oppositely charged ions, which is important in the calculation of the transport properties, it holds Y2 Y (1 − Y )−3 g(a) = 1 − + 2 4

(2.3)

where Y =

π a3N 3 F +1 6V

(2.4)

√ Vasu (1972a) used for the geometrical factor the value F = 2. Using Thorne’s equation for the viscosity of rigid spheres given by Chapman and Cowling (1960) and Eqs. (2.3) and (2.4), Vasu (1972a) derived for the viscosity of molten alkali metal halides, the equation η=

0.419Y 2 g(a) m1 m2 kT 1/2 m1 + m 2 a2 m 1 m2 0.475 0.993 + 1 + 0.375 × 1+ Y g(a) (m1 + m2 )2 [Y g(a)]2

(2.5)

where m1 and m2 are the masses of particles, k is the Boltzmann constant, and T is the absolute temperature. On the basis of the kinetic theory of liquids, Vasu (1972b) also derived the following equation for the electrical conductivity of molten alkali metal halides

κ = 3e

2

4g(a)a

2

2π kT M1 M2 M1 + M 2

−1 (2.6)

14

Physico-chemical Analysis of Molten Electrolytes

where κ is the electrical conductivity, e is the electronic charge, M1 and M2 are the atomic masses of particles, and the other symbols express the above mentioned meaning. In Table 2.3 the comparison of the calculated and experimental values of the viscosity and electrical conductivity at the temperature of T = 1.05 Tfus is given. As it follows from the table, the agreement between the calculated and the experimental results is good and in some cases, even excellent. One value of the factor F is applicable in the calculation of both the viscosity and electrical conductivity. The values of the molar volume, as well as of the viscosity and the electrical conductivity were taken from Janz et al. (1968). Experimental data for cation–anion distance as they were found in the diffraction studies of molten alkali metal halides and collected by Janz (1967) were used in the calculation. Since these data are known for 13 alkali metal halides only, the rest of the data was estimated from the gas phase as was recommended by Yosim and Owens (1964). It was found that the value of the factor F decreases in the series Li+ , Na+ , K+ , Rb+ , + Cs and attains the mean values 1.45, 1.36, 1.31, 1.29, and 1.26, respectively. This can be explained by the dependence of this factor on the size √ and polarizability on ions. For lithium salts, the value of the factor F is higher than 2. This is not surprising since in this case the effect of the anion–anion repulsion should be considered. Similarly, the

Table 2.3. Calculated and experimental viscosity and electrical conductivity of molten alkali metal halides (T = 1.05 Tfus ) MX

T (K)

a (nm)

F

ηexp (cP)

ηcalc (cP)

κexp (S/cm)

κcalc (S/cm)

LiF LiCl LiBr LiI

1174 927 864 758

0.195 0.247 0.268 0.285

1.46 1.45 1.42 1.47

2.05 1.49 1.55 2.19

2.33 1.58 1.41 1.68

8.80 5.95 4.92 3.96

10.03 5.92 4.84 3.41

NaF NaCl NaBr NaI

1331 1127 1074 982

0.230 0.280 0.298 0.315

1.38 1.35 1.34 1.38

1.65 1.19 1.23 1.31

2.07 1.26 1.34 1.31

5.10 3.74 3.06 2.40

5.55 3.98 2.98 2.49

KF KCl KBr KI

1185 1095 1058 1006

0.270 0.310 0.330 0.353

1.31 1.33 1.30 1.31

– 1.02 1.04 1.43

1.72 1.21 1.32 1.43

3.73 2.29 1.74 1.38

3.60 2.77 2.02 1.48

RbF RbCl RbBr RbI

1100 1037 1001 959

0.273 0.330 0.342 0.366

1.44 1.29 1.29 1.29

– 1.19 1.33 1.27

2.04 1.51 1.58 1.56

– 1.62 1.20 0.94

2.99 1.84 1.47 1.15

CsF CsCl CsBr CsI

1003 964 954 939

0.282 0.353 0.355 0.385

1.44 1.24 1.29 1.26

– 1.17 – 1.66

2.08 1.44 1.61 1.70

2.53 1.25 0.91 0.73

2.63 1.49 1.25 0.88

Main Features of Molten Salt Systems

15

effect of the cation–cation repulsion must be taken √ into account in the case of RbF and CsF, where the value of the factor F is higher than 2 as well. A similar argument was given by Yosim and Owens (1964) to explain differences in the calculation of the absolute entropy of these melts. 2.1.1.2. Binary systems

Calorimetric measurements indicated that the enthalpy of mixing is frequently negative (the mixture has a lower enthalpy than the pure components). This is due to the following reasons: (a) (b) (c) (d)

the change in the repulsion energy of evenly charged ions is an important part of the enthalpy of mixing, structural changes in the process of mixing are small, the number of neighbors in the nearest coordination sphere is not changed, the change in the state of ion polarization.

On the other hand, van der Waals–London forces show generally small positive contribution to the enthalpy of mixing. Førland (1954) illustrated the change in the cation–cation repulsion by a crude model shown in Figure 2.1. The arrangement of ions before mixing is on the upper side, while the arrangement after mixing is given below. Since the arrangement of anions has changed, the change in the repulsion energy is approximately given by the relation 2e2 E ≈ − d1 + d 2

F−

Rb+ d1

Cs+

Rb+

d1 − d 2 d1 + d 2

(2.7)

Cs+

F− d2

Rb+ d1

2

Cs+

d1

F− d2

Cs+

d2

Rb+

F− d2

d1

Figure 2.1. Cation–cation repulsion energy change at the RbF–CsF mixing.

16

Physico-chemical Analysis of Molten Electrolytes

where e is the electronic charge. This crude estimation yields a negative enthalpy of mixing and indicates its variation with the size of the ions. In the extreme case, it can lead to the formation of binary compounds. In a similar manner, Lumsden (1964) has calculated the contribution to mixing if the anion is polarizable. His result for this contribution to energy is very similar to Eq. (2.7). The consequence of the above considerations is that binary systems of alkali metal halides form different types of phase diagrams, starting with the simple eutectic ones through the solid solution eutectic ones, the phase diagrams with the formation of a binary compound up to those with complete solid solubility. Tables 2.4 and 2.5 summarize the main features of individual phase diagrams. As it can be seen from the tables, binary compounds are formed only in systems with a common anion. The compounds originate in the systems Li+ , Rb+ //X− and Li+ , Cs+ //X− , where X = F, Cl, Br, I. On the other hand, phase diagrams with continuous solid solutions can be observed predominantly in systems with “soft” cations. These facts are connected with the value of the repulsion energy of evenly charged ions (see Eq. (2.7)). The higher the repulsion energy, the higher is the probability of binary compound formation. Table 2.4. Characteristic features of binary systems of alkali metal halides with a common anion System

Type of phase diagram

Compound

System

Type of phase diagram

Compound

LiF–NaF LiF–KF LiF–RbF LiF–CsF

lss ses ewc ewc

– – LiF·RbF LiF·CsF

LiCl–NaCl LiCl–KCl LiCl–RbCl LiCl–CsCl

css ses ewc ewc

NaF–KF NaF–RbF NaF–CsF KF–RbF KF–CsF RbF–CsF

lss ses ses css lss css

– – – – – –

NaCl–KCl NaCl–RbCl NaCl–CsCl KCl–RbCl KCl–CsCl RbCl–CsCl

css lss ses css css css

– – Lil·RbCl LiCl·CsCl LiCl·2CsCl – – – – – –

LiBr–NaBr LiBr–KBr LiBr–RbBr LiBr–CsBr NaBr–KBr NaBr–RbBr NaBr–CsBr KBr–RbBr KBr–CsBr RbBr–CsBr

css ses ewc ewc css ses ses css lss lss

– – LiBr·RbBr LiBr·CsBr – – – – – –

LiI–NaI LiI–KI LiI–RbI LiI–CsI NaI–KI NaI–RbI NaI–CsI KI–RbI KI–CsI RbI–CsI

css ses ewc ewc? css lss ses css lss lss

– – LiI·RbI 3LiI·2CsI – – – – – –

ses – simple eutectic system; lss – limited solid solutions; css – continuous solid solutions; ewc – eutectic system with compound(s); * – solid–solid transformation; ? – no experimental data, tentative PD.

Main Features of Molten Salt Systems

17

Table 2.5. Characteristic features of binary systems of alkali metal halides with a common cation System

Type of phase diagram

System

Type of phase diagram

LiF–LiCl LiF–LiBr LiF–LiI LiCl–LiBr LiCl–LiI LiBr–LiI

ses ses ses css ses css

NaF–NaCl NaF–NaBr NaF–NaI NaCl–NaBr NaCl–NaI NaBr–NaI

ses ses ses css lss css

KF–KCl KF–KBr KF–KI KCl–KBr KCl–KI KBr–KI

ses ses ses css lss css

RbF–RbCl RbF–RbBr RbF–RbI RbCl–RbBr RbCl–RbI RbBr–RbI

ses ses ses? css css? css

CsF–CsCl CsF–CsBr CsF–CsI

ses* ses ses

CsCl–CsBr CsCl–CsI CsBr–CsI

lss lss css

ses – simple eutectic system; lss – limited solid solutions; css – continuous solid solutions; ewc – eutectic system with compound(s); * – modification transformation; ? – no experimental data, tentative PD.

For a molten alkali metal halide mixture with a common cation, the Gibbs energy of mixing is given by the equation

mix G = RT xA+ ln xA+ + xB+ ln xB+

(2.8)

where xA+ and xB+ are the cation fractions of A+ and B+ , respectively. In systems with a common anion, the Gibbs energy of mixing is given by the equation

mix G = RT xX− ln xX− + xY− ln xY−

(2.9)

where xX− and xY− are the anion fractions of X− and Y− , respectively. From the above equations it follows that for all systems with a common ion, the value of the Gibbs energy of mixing should be equal. However, in the general case the Gibbs energy of mixing is also affected by the enthalpy of mixing and by the non-ideal entropy of mixing. Comprehensive cryoscopic, calorimetric, and galvanic cell studies of alkali metal halide systems with a common anion and a common cation have been carried out. These measurements have shown that the enthalpy of mixing for these systems can, in general, be expressed by a parabolic type equation mix H = xi xj a + bxj + cxj2

(2.10)

18

Physico-chemical Analysis of Molten Electrolytes

where xi and xj are the mole fractions of the two components and a, b, and c are coefficients determined by a least-squares fit of the experimentally determined data. Rearranging Eq. (2.10) we get mix H /xi xj = λ = a + bxj + cxj2

(2.11)

where λ is the “enthalpy interaction parameter.” λ varies slowly with composition, revealing details that cannot be easily recognized from the dependencies of the enthalpy of mixing on composition. The limiting values of the interaction parameters for x2 → 0 or x2 → 1 are of particular interest since they are related to the partial enthalpies of solution of the salt components lim λ = λ0 = λ(x2 = 0) = H 2 (x2 = 0) x2 =0

lim λ = λ1 = λ(x1 = 0) = H 1 (x1 = 0)

(2.12)

x1 =0

2.1.2. Systems containing divalent cations

As it follows from numerous experimental measurements, some pure halides of divalent metals partially dissociate in the molten state according to the scheme MeX2 ⇔ MeX+ + X−

(2.13)

Other divalent cations have the tendency to form complex anions of the type [MeX3 ]− and/or [MeX4 ]−2 not only in the presence of the alkali metal halides, but also in the pure state. Some pure halides of divalent metals show a tendency to form auto-complexes. The ability to form auto-complexes according to the scheme 2MeX2 ⇔ MeX+ + MeX− 3

(2.14)

is demonstrated by e.g. mercury halides, MgCl2 , MgBr2 , and MgI2 , which is marked at melting by a significant increase in the electrical conductivity. A typical example of a salt that forms auto-complexes is ZnCl2 . The formation of complexes at melting takes place according to the scheme 2ZnCl2 = [ZnCl4 ]2− + Zn2+

(2.15)

According to Mackenzie and Murphy (1960), the high viscosity of about 500 Pa · s and the low conductivity of about 10−3 S · cm−1 of liquid ZnCl2 near its melting point,

Main Features of Molten Salt Systems

19

indicate strong differences in structure compared with other typical fused salts. This is due to the presence of network structures in the melt. Hefeng et al. (1994) have found that the tetrahedrons consisting of four Cl atoms around Zn (ZnCl2− 4 ) are dominant and stable in molten ZnCl2 . Ballone (1986) pointed out that the local structural arrangement around the zinc ion is not significantly different in the molten or solid state. The network structure is often referred to as polymeric. As it follows from conductivity measurements, at increasing temperature a fast disintegration of the ligand structure takes place in the melt. The polymeric structure rapidly breaks down as the temperature increases and the proportion of ions, i.e. Zn2+ , ZnCl+ , 2− ZnCl− 3 , and ZnCl4 becomes higher, increasing the conductivity and decreasing the viscosity. The classic ionic melt originates, in the case of ZnCl2 , already at temperatures far above the melting point. Polyanions are probably also present in molten SnBr2 and PbCl2 . In general, the earth alkali metal halides increase their volume at melting (Table 2.6). This is due to the lowering of atomic coordination and the formation of free volume. Yamura et al. (1993) found that depolymerization also occurs with the addition of alkali metal halides, increasing the conductivity and decreasing the viscosity of the melt. These authors proposed that the addition of alkali metal chlorides to ZnCl2 induces depolymerization as expressed by the following equations [ZnCl2 ]n + Cl− = [ZnCl2 ]n−m + [ZnCl2 ]m Cl− [ZnCl2 ]m Cl− + Cl− = [ZnCl4 ]2− + [ZnCl2 ]m−1 Table 2.6. Thermodynamic parameters of melting of some earth alkali metal halides Salt

Sfus (J/mol · K)

Tfus (K)

Vfus /Vs (%)

MgF2 MgCl2 MgBr2

1536 987 984

– – –

37.6 43.5 35.1

CaF2 CaCl2 CaBr2 CaI2

1687 1045 1015 1052

– 0.9 4.0 –

17.6 27.2 28.8 39.7

SrF2 SrCl2 SrBr2 SrI2

1673 1146 930 811

– 4.1 2.1 –

10.9 13.4 11.3 24.2

BaF2 BaCl2 BaBr2

1593 1233 1130

– 3.6 11.8

13.4 13.4 28.4

(2.16) (2.17)

20

Physico-chemical Analysis of Molten Electrolytes

Von Bues (1955) found evidence of a polymer (ZnCl2 )n using Raman spectroscopy. After an addition of 33 mole % KCl, ZnCl− 3 species were suggested. When the ZnCl2 :KCl 2− molar ratio was 1:2, tetrahedral ZnCl4 species were mentioned. During the past four decades, numerous investigations of systems with divalent cations have been performed and a large area of molten salt chemistry was discovered. The main results were obtained by EMF, calorimetric, and other physico-chemical methods, and accurate enthalpies of mixing and chemical potentials are available. A brief summary of the results was given by Østvold (1992). Electrical conductivity at salt mixing is generally not additive. Duke and Fleming (1957) found for a mixture of KCl and ZnCl2 , negative deviation of the conductivity, which was explained by ion interactions or the formation of complex ions. Maximum conductivity was reached for the highest measured temperature and the largest addition of KCl. In general, the increase in conductivity is largest for the addition of alkali metal chloride with the smallest cation. Thus, lithium chloride will favor the conductivity most. Benhenda (1980) determined the conductivity of ZnCl2 –LiCl mixtures in the temperature range from 320 to 450◦ C. Furthermore, the effect of LiCl addition to ternary mixtures of ZnCl2 , KCl, and NaCl on the conductivity was determined by Driscoll and Fray (1993). A significant increase in conductivity was reached when 20 mole % LiCl was added. When mixing zinc chloride with different alkali metal chlorides, viscosity does not generally obey the additive rule. Though, for higher contents of zinc chloride, the viscosity will increase. The viscosity of four melts was determined by Driscoll and Fray (1993); the temperature dependence of viscosity showed Arrhenius behavior. Generally, the surface tension of the mixtures of ZnCl2 with alkali metal chlorides increases with decreasing amounts of ZnCl2 . Copham and Fray (1990) measured surface tension of four ternary ZnCl2 –NaCl–KCl melts and found that the surface tension decreases linearly with increasing temperature. Near the primary crystallization temperature of the mixture, the surface tension attains low values, indicating that the ratio of temperature to the temperature of primary crystallization (T/Tpc ) has a significant effect on the surface tension. Driscoll and Fray (1993), who determined the surface tension of another four melts, also came to the same conclusion. The vapor pressure of pure zinc chloride was measured by Meyer et al. (1989). However, a more accurate value for vapor pressure at 450◦ C, i.e. 55.5 Pa, was given by Anthony and Bloom (1975), who determined the vapor pressure in the temperature range 450–625◦ C. Bloom et al. (1970) had determined that the addition of NaCl or KCl reduces the vapor pressure of ZnCl2 . The vapor pressure of these binary mixtures was used to determine the activity coefficients of zinc chloride and alkali chloride. Haver et al. (1976) reported the weight loss of the bath for different electrolyte compositions and observed that when pure ZnCl2 was used, there was more loss; the weight loss decreases with additions of LiCl, NaCl, and KCl.

Main Features of Molten Salt Systems

21

LiCl-MeCl2

λ /(kJ/mol)

0

−40

0.33

−80

−120

↓

0

0.33 ↓

x(MgCl2) 1 0 x(MnCl2) 1 0

0.33 ↓

x(CeCl2)

CsCl-MeCl2

0.33 ↓

1 0 x(ZnCl2)

1

Figure 2.2. Enthalpy interaction parameter in charge-unsymmetrical molten binary systems.

Most of the systems containing divalent cations show negative enthalpy of mixing, however, certain weakly interacting systems have shown positive values. The enthalpy interaction parameter λ depends strongly on composition. For a family of systems MX–MeX2 (M = Li, Na, K, Rb, Cs) the variation of λ with the mole fraction of MeX2 , x2 , may be a smooth but fast-changing function or a more complex function showing a minimum. The shape in the interaction parameter for different univalent–divalent systems shows variations depicted schematically in Figure 2.2. For the LiX–MeX2 systems almost linear behavior has been observed, whereas for the CsX–MeX2 binaries, a minimum of λ appears at x2 ≈ 0.33. For the remaining alkali metal halides (Na, K, Rb) intermediate behavior has been found. It was found out that the interaction parameter, λ, is in general not a linear function of the distance parameter δ12 . However, in the series with one salt as a common salt, the λ versus δ12 plot is linear. 2.1.2.1. Systems with common earth alkali metal halide

Let us consider the energy change that follows the substitution of the alkali metal cation by the earth alkali metal in a liquid alkali metal–earth alkali metal halide mixture. If the

22

Physico-chemical Analysis of Molten Electrolytes

structural changes by this process are minor, so that the number of neighbors in the nearest coordination shells is not changed, the change in the cation–cation repulsion energy is an important part of the energy change. When the anion–cation distances can be assumed to be constant and equal to the sum of the ionic radii, the change in energy following the substitution due to the change in cation–cation repulsion may be described by a crude model introduced by Førland (1964) and illustrated in Figure 2.3. Let us now consider the reaction, when one earth alkali metal cation is substituted by another

2 Me2+ X− MI+ mix(l) + MII+ X− MII+ pure(l) (2.18)

= 2 Me2+ X− MII+ mix(l) + MI+ X− MI+ pure(l) with the change in energy E(MeX2 ) = 2EMeXMII + EMI XMI − 2EMeXMI − EMII XMII

(2.19)

The change in coulombic energy following the substitution may be described roughly by the equation EC (MeX2 ) = e2

4 1 4 1 + − − dMeX + dMII X 2dMI X dMeX + dMI X 2dMII X

(2.20)

where e is the electronic charge and d with subscript is the respective interatomic distance.

X–

Me2+

d1

Me2+

d1

X–

M+

d2

M+

d2

d1

M+

d2

X–

Me2+

X–

Me2+

d1

M+

d2

Figure 2.3. Change in the cation–cation repulsion energy, E C , at mixing in univalent–divalent salt systems.

Main Features of Molten Salt Systems

23

In addition to the change in coulombic interaction occurring when an earth alkali metal cation is substituted by another, according to reaction (2.18), there is also a change in polarization of the common anion due to the change in the cation environment. Førland (1964) and Lumsden (1964) pointed out that the anion would become polarized by the unsymmetrical electric field due to the different size and charge of the two cations on opposite sides of the anion (Figure 2.3). In the group (Me2+ X− M+ ), the anion X− is subjected to an electric field F =e

2

1

−

2 dMX

(2.21)

2 dMeX

If α is the polarizability of the anion, the polarization energy of the (Me2+ X− M+ ) group is αe2 αF 2 EP = − =− 2 2

2 2 − dMX 2dMeX

2 (2.22)

4 4 dMeX dMX

The change in polarization energy following reaction (2.18) can thus be approximated by the expression EP (MeX2 ) = −αe

2

1 4 dMe II X

−

1 4 dMe IX

−

4

2 dMX

1 2 dMe II X

−

1 2 dMe IX

(2.23)

The polarization energy of the pure salts can be neglected. From the above considerations it can be concluded that the slopes of the λ versus δ 12 plots for the common earth alkali halide systems should increase in the sequence Mg < Ca < Sr < Ba. This was indeed experimentally observed. Except for the lithium halide–earth alkali metal halide systems, the data are well represented by the λ versus δ 12 plot for each common earth alkali metal halide system. The slopes of λ versus δ12 plot increase when the radius of the common earth alkali metal cation is increasing. 2.1.2.2. Systems with common alkali metal halide

For these systems let us consider the following reaction 2+ − 2 M+ X− Me2+ mix(l) + Me2+ pure(l) I II X MeII 2+ − 2+ = 2 M+ X− Me2+ pure(l) II mix(l) + MeI X MeI

(2.24)

24

Physico-chemical Analysis of Molten Electrolytes

with the change in energy E(MX) = 2EMXMeII + EMeI XMeI − 2EMXMeI − EMeII XMeII

(2.25)

According to Førland’s model illustrated in Figure 2.3, the change in coulombic energy following the substitution may roughly be described by the equation EC (MX) = e

2

4 1 4 1 + − − dMX + dMeII X 2dMeI X dMX + dMeI X 2dMeII X

(2.26)

The changes in coulomb energy, when substituting one alkali metal cation by another one in a binary alkali metal–earth alkali metal halide melt, will tend to give negative λ versus δ 12 plot slopes, which are decreasing when the size of the common alkali metal cation is increasing. The change in polarization energy, which follows the substitution of one alkali metal halide cation by another one, can be estimated in the same way as outlined previously and can be approximated by the expression EP (MX) = −αe

2

4 4 dM II X

−

4 4 dM IX

−

4 2 dMeX

1 2 dM II X

−

1 2 dM IX

(2.27)

2.1.2.3. Influence of the common anion on the enthalpy of mixing

It is obvious from the foregoing discussion that the enthalpies of mixing for chargeunsymmetrical systems do not follow the simple conformal solution theory. When the anion in a strontium halide–alkali metal halide mixture from chloride to bromide and from bromide to iodide is changed, the enthalpy of mixing is decreasing. For all systems, the enthalpy interaction parameter, λ, is a linear function of δ 12 with the usual exception for lithium-containing systems. Two important features of the λ versus δ 12 plot should be emphasized: (1) (2)

the enthalpy of mixing is decreasing in the sequence common chloride, common bromide, common iodide, the slope of the λ versus δ 12 plot is also decreasing in the above sequence.

Strontium- and barium-containing systems do not deviate much from regular solution behavior. The relatively simple nature of these melts therefore indicates that the variation in the enthalpy of mixing with the common anion must be explainable by a model taking into account coulombic and polarization interactions between ions in a fairly simple manner. This behavior is also evidenced by the phase diagrams of the binary systems

Main Features of Molten Salt Systems

25

M2 F2 –SrF2 (M = Li, Na, K). All three binary systems are the simple eutectic. Also in the same series of strontium chlorides, there are only simple eutectic systems for lithium, sodium, and potassium, but systems containing bigger rubidium and cesium cations with low polarization ability already form binary compounds. 2.1.2.4. Partial molar quantities in charge-unsymmetrical systems

Measurements of the chemical potential for each of the component salts in chargeunsymmetrical binary systems yield negative values, which become more exothermic with increasing concentration and by substituting the M+ cation from Li to Cs. Changes of the divalent salt in the MX–MeX2 systems give rise to more negative entropy and sharper S-curves for some systems, whereas binary systems having no minimum in the enthalpy interaction parameter λ show a smoother dependence of the partial entropy of MX on x(MeX2 ). This is schematically depicted in Figure 2.4 for three representative binary systems KCl–CaCl2 , KCl–MgCl2 , and KCl–CoCl2 .

∆Sparc(J/mol.K)

10

id.

5 0 –5

∆Hparc(kJ/mol)

id.

id.

KCl-MgCl2

KCl-CaCl2

KCl-CoCl2

0 –10 –20 –30 –40

0

10

x(MeCl2)

10

1

Figure 2.4. Schematic presentation of partial molar quantities in different univalent–divalent chloride systems.

26

Physico-chemical Analysis of Molten Electrolytes

2.1.3. Systems containing trivalent cations

Among the systems containing a trivalent cation, technologically most important are the cryolite-based melts used as electrolytes for aluminum production. Very important also are systems containing BF3 , which are used as electrolytes for boriding of steels and as heat-bearing medium in secondary circuits of nuclear power plants. To the melts containing a trivalent cation belong a great number of systems of alkali metal halide–rare earth metal halide systems. Lanthanide halides play an important role in the production of lanthanide metals by molten salt electrolysis and they are also used in a number of applications ranging from lighting to catalysis, through pyrochemical reprocessing of nuclear fuel. The above-mentioned three kinds of systems containing a trivalent cation will be described in the following chapters. 2.1.3.1. Structure of melts of the systems MF–AlF3 (M = Li, Na, K)

In spite of extensive investigations over six decades, the structure of these melts is still not sufficiently understood. The most accepted electrolytic dissociation scheme of molten cryolite is as follows: Na3 AlF6 = 3Na+ + AlF3− 6

(2.28)

By comparing the measured liquidus curve of cryolite with the theoretical one, Grjotheim et al. (1977) concluded that the AlF3− 6 anion undergoes further a partial thermal dissociation according to the scheme − − AlF3− 6 = AlF4 + 2F

(2.29)

with the equilibrium constant K=

aAlF− · aF2− 4

aAlF3− 6

=

4α 3 (1 − α)(1 + 2α)2

(2.30)

The degree of dissociation α was found to be 0.3. The scheme of the thermal dissociation was modified recently by Dewing (1986) in order to explain some discrepancies in the heat capacity of the NaF–AlF3 mixtures, and the dissociation constant was calculated on the basis of thermodynamic and spectroscopic data. Dewing proposed the following dissociation mechanism 2− − AlF3− 6 = AlF5 + F

(2.31)

− − AlF2− 5 = AlF4 + F

(2.32)

Main Features of Molten Salt Systems

27

He also found that AlF2− 5 is the most abundant anionic species at x(AlF3 ) = 0.333 composition. The distribution of individual anions in the system NaF–AlF3 according to Dewing (1986) is shown in Figure 2.5. Recent Raman spectroscopic studies performed by Gilbert and Materne (1990) con2− − firmed the existence of the AlF3− 6 , AlF5 , and AlF4 anions in the melt. The existence 2− of the AlF5 anions in the liquid state was also postulated by Xiang and Kvande (1986) on the basis of liquidus temperature calculations. An attempt to confirm the presence of the AlF2− 5 anions in the MF–AlF3 (M = Li, Na, K) melts on the basis of vapor pressure measurements was also made by Zhou (1991). The study of structure and thermodynamics of the MF–AlF3 (M = Li, Na, K) melts using vapor pressure, solubility, and Raman spectroscopy measurements was conducted by Olsen (1996). From the Raman spectroscopic studies several conclusions could be drawn.

1.0 AlF4 − F−

Activity

0.8

0.6 AlF52−

0.4 AlF63−

0.2

0.0

NaF

0.1

0.2

0.3 x(AlF3)

0.4

0.5

0.6

Figure 2.5. Distribution of individual anions in the system NaF–AlF3 according to Dewing (1986).

28

• • •

Physico-chemical Analysis of Molten Electrolytes

all three systems have the principal group of bands located between 500 and 650 cm−1 , the bandwidths decrease when the size of the cation increases, the intensities of the bands change with composition.

A detailed interpretation of the spectra of the NaF–AlF3 and KF–AlF3 melts has shown that they are made up of three complex anionic species. This is most clearly seen in the KF–AlF3 system. The distribution of anions in the system KF–AlF3 according to the Raman spectroscopic and vapor pressure studies of Robert et al. (1997b) is shown in Figure 2.6. For Li-containing melts, it is not obvious from the spectra that there are three different bands. However, when the amount of LiF increases, there is a shift of frequency for the main band at around 570 cm−1 . It is assumed that in the system LiF–AlF3 , due to the strong polarizing ability of the Li+ cation, the AlF2− 5 anions are not present. More information on cryolite-based melts can be found in specialized books by Grjotheim et al. (1982) and Thonstad et al. (2001) devoted to the fundamentals of aluminum electrolysis.

1.0

F−

Activity

0.8

AlF4−

0.6 AlF52−

0.4

0.2 AlF63−

0.0

KF

0.1

0.2 0.3 ) x(AlF3

0.4

0.5

Figure 2.6. Distribution of individual anions in the system KF–AlF3 according to Robert et al. (1997b).

Main Features of Molten Salt Systems

29

2.1.3.2. Systems containing boron trifluoride

Boron trifluoride, BF3 , is the source of boron in thermochemical and electrochemical boriding of steels and cutting tools. In the presence of alkali metal halides BF3 forms tetrafluoroborates, the stability of which depends on the alkali metal cation present. As a consequence of the high polarization ability of the Li+ cation, LiBF4 does not exist. NaBF4 is relatively stable only up to approximately 500◦ C, and KBF4 up to approximately 900◦ C. Above the mentioned temperatures, these compounds decompose to the respective alkali metal fluorides and BF3 (g). In exploitation of systems containing KBF4 , the appropriate choice of electrolyte composition prevents the formation of volatile compounds, as BF3 or BCl3 , which leads to undesirable emissions. Reactions of potassium tetrafluoroborate with molten alkali chlorides was investigated by Daneˇ k et al. (1976) using the cryoscopic method. The main aim of this work was to − study the stability of the BF− 4 anion in the presence of Cl anions and a different cationic environment. In molten alkali chlorides, the exchange reaction 4MCl + KBF4 ↔ 4MF + KBCl4

(2.33)

(M = Li, Na, K) may occur. In some cases, the originating tetrachloroborate anion may further decompose under the formation of gaseous BCl3 . It can be assumed that reaction (2.33) proceeds by successive exchange of fluorine in the BF− 4 anion for chlorine through − reaction steps BF3 Cl− , BF2 Cl− , and BFCl . Such a mechanism, however, cannot be 2 3 determined by means of the cryoscopic method. In Figures 2.7 – 2.9, the experimentally determined dependencies of the melting point depression of LiCl, NaCl, and KCl on composition, respectively, are compared with the theoretical course of liquidus curves for different values of the Storkenbeker’s correction factor kSt . From Figure 2.7, it follows that in molten LiCl, the BF− 4 anion undergoes exchange − reaction under the formation of BCl4 ions. At concentrations surpassing 0.05 mol · kg−1 KBF4 , the originating tetrachloroborate anions decompose owing to the polarization effect of the Li+ cations according to the equation − BCl− 4 = BCl3 + Cl

(2.34)

and gaseous BCl3 escapes from the melt. Consequently, the number of new particles in the melt decreases (kSt < 6) and the remaining melt corresponds to the ternary system LiF–LiCl–KCl. From the cryoscopic study in the system NaCl–KBF4 , it follows that in the concentration range of 0.02–0.25 mol · kg−1 KBF4 , the number of new particles is reduced from six to two, which indicates the change in the direction of the reaction (2.33) in this concentration range. At concentrations below 0.02 mol · kg−1 KBF4 , the reaction (2.33)

30

Physico-chemical Analysis of Molten Electrolytes

0 2

1 ∆T(K)

4

2 6

3

8

4

10

5

12

LiCl

0.05

0.10 0.15 m(KBF4)

0.20

0.25

Figure 2.7. LiCl melting point depression in the system LiCl–KBF4 . 1–kSt = 2; 4–kSt = 5; 5–kSt = 6.

2–kSt = 3;

3–kSt = 4;

0 2

1

∆T(K)

4 6 8

2 10 12

NaCl

0.05

0.10 0.15 m(KBF4)

0.20

0.25

Figure 2.8. NaCl melting point depression in the system NaCl–KBF4 . 1–kSt = 2;

2–kSt = 6.

Main Features of Molten Salt Systems

31

0

∆T(K)

1

1

2 3 4

2

5 6 0.05

0.10 0.15 m(KBF4)

0.20

0.25

Figure 2.9. KCl melting point depression in the system KCl–KBF4 . 1–kSt = 1;

2–kSt = 2.

proceeds under the formation of BCl− 4 anions, however, owing to the low concentration of these anions and to the lower polarization ability of Na+ cations, compared with that of the Li+ , there are no conditions favoring the decomposition of the BCl− 4 anions. Above the concentration of 0.25 mol · kg−1 KBF4 , practically only tetrafluoroborate anions are present in the melt and the exchange reaction (2.33) does not proceed. In the system KCl–KBF4 , the Stortenbeker’s correction factor equals 1, which corresponds to the introduction of one new particle into the KCl melt with the addition of KBF4 . This indicates that the simple dissociation of KBF4 into K+ and BF− 4 is not accompanied by the exchange reaction (2.33). The probability of the exchange reaction (2.33) in the alkali metal chlorides could also be estimated using the thermodynamic calculation of the degree of conversion. For the sake of simplicity, let us assume that after introducing KBF4 into the MCl melt, only − the BF− 4 and BCl4 complex anions and no intermediate products will be present in the melt. If we add n mol of KBF4 into 1 mol of MCl (n 1) then, according to Temkin, xCl− ∼ = 1 and for the equilibrium constant of reaction (2.33), the simplified equation can be derived K=

xF4− xBCl− 4

4 x − xCl − BF 4

= 256n4

α5 1−α

(2.35)

32

Physico-chemical Analysis of Molten Electrolytes

where α is the degree of conversion of reaction (2.33). The values of the Gibbs energies and thus of the equilibrium constants in molten LiCl, NaCl, and KCl at the temperature of fusion of the respective alkali metal chlorides were calculated on the basis of the values of the Gibbs energy of the reactants presented in the JANAF Thermochemical tables (1971). Since the formation Gibbs energy of KBCl4 is not known, its value was estimated from the enthalpy of reaction KCl(s) + BCl3 (g) = KBCl4 (s)

(2.36)

determined by Titova and Rosolovkii (1971). The estimated formation Gibbs energy of liquid KBCl4 at temperatures of fusion of LiCl, NaCl, and KCl are supposed to be: f G0883 K ∼ = −800 kJ · mol−1 , f G01044 K ∼ = −783 kJ · mol−1 , f G01073 K ∼ = −1 −779 kJ·mol , respectively. The error in the estimation is approximately ± 10 kJ·mol−1 . For the exchange reaction (2.30) in the system LiCl–KBF4 , the value of the Gibbs energy is r G883 K ∼ = 35.2 kJ · mol−1 , which corresponds to the value of the equilibrium constant KLiCl ∼ = 8 × 10−3 . Hence, according to Eq. (2.35), the degree of conversion at −2 n < 10 is α ∼ = 1. This means that the exchange reaction (2.33) is completely shifted − to the right side. The BF− 4 anions are converted quantitatively into BCl4 , which is in agreement with the experimental findings. The Gibbs energy for the exchange reaction (2.33) in the system NaCl–KBF4 , r G1073 K ∼ = 189.4 kJ·mol−1 , corresponds to the equilibrium constant KNaCl ∼ = 2×10−9 . −3 −3 ∼ At n < 10 the degree of conversion is α = 1 and at n ≈ 10 (i.e. at m = 0.25 mol·kg−1 ) the value is α ∼ = 0.5. This indicates that with increasing the concentration of KBF4 , the equilibrium of the exchange reaction shifts from the right to the left side − and, besides BF− 4 , also BCl4 anions are present in the melt. In the system KCl–KBF4 , for the Gibbs energy and of the equilibrium constant of the exchange reaction, the values r G1043 K ∼ = 302 kJ · mol−1 and KKCl ∼ = 10−15 , respectively, were calculated. Thus, for n > 10−3 , the degree of conversion is α ∼ = 0. Hence, the calculation confirms the experimental results that the exchange reaction in this system does not take place. From the results of both the cryoscopic measurements and the thermodynamic calculation it follows that the stability of BF− 4 in molten alkali metal chlorides increases in the series LiCl < NaCl < KCl. In molten LiCl, the KBF4 anions are unstable and decompose under the formation of gaseous BCl3 . In molten NaCl, the exchange reaction between KBF4 and Cl− anions under the formation of KBCl4 proceeds only at very small concentrations of KBF4 , while no reaction occurs in molten KCl, where the BF− 4 anion is ◦ relatively stable up to approximately 900 C. The equilibrium constant of the exchange reaction (2.33) and the unknown value of the formation Gibbs energy of KBCl4 at 1073 K were calculated from the results of the cryoscopic measurement in the system NaCl–KBF4 . In this system, the equilibrium of the

Main Features of Molten Salt Systems

33

exchange reaction, and thus also the depression in the melting point of NaCl, depends on the concentration of KBF4 . The equation for the melting point depression can be derived from the boundary conditions: Let for α = 0,

T = T1 = Kcr mB kSt, 1 = Kcr mB 2

and for α = 1,

T = T2 = Kcr mB kSt, 2 = Kcr mB 6

(2.37)

where Kcr is the cryoscopic constant of NaCl and mB is the molality of KBF4 added. Omitting again the intermediate products (BF3 Cl− , etc.), the number of new particles can be expressed as kSt = 1 + (1 − α) + α + 4α = 2 + 4α

(2.38)

The dependence of the melting point depression on the conversion degree can then be described by the equation T = Kcr mB (2 + 4α)

(2.39)

After rearranging and substituting, we get α=

T − Kcr mB 2 T − Kcr mB 2 T − T1 = = Kcr mB 4 Kcr mB 6 − Kcr mB 2 T2 − T1

(2.40)

Using values of α calculated in this way and by means of Eq. (2.35), the value of the equilibrium constant for any arbitrary concentration of KBF4 can be calculated. The results of this calculation are shown in Figure 2.10. The arithmetical mean of the values of the equilibrium constants is K = 7 × 10−7 , which corresponds to the reaction Gibbs energy r G1073 K ∼ = 188 ± 10 kJ·mol−1 . For the formation Gibbs energy of liquid KBCl4 , the value f G01073K ∼ = −769 ± 10 kJ · mol−1 was then calculated, which is in very good agreement with the estimated value of −779 kJ · mol−1 . As a possible electrolyte for electrochemical boriding of steels and cutting tools, the molten system KF−KCl−KBF4 studied by Daneˇ k and Matiašovský (1977) and Matiašovský et al. (1978), could be used. The phase diagrams of the boundary binary system KF–KBF4 have been studied by Barton et al. (1971) and Daneˇ k et al. (1976), while the binary system KCl–KBF4 was studied by Samsonov et al. (1959) and Daneˇ k et al. (1976). It was found that both the binary systems are simple eutectic. In the phase diagram of the system KCl–KBF4 , determined by Samsonov et al. (1959), the formation of the congruently melting compound 11KBF4 ·KCl was suggested. However, the existence of this compound was not confirmed in the later investigations made by Barton et al. (1971) and Daneˇ k et al. (1976).

34

Physico-chemical Analysis of Molten Electrolytes

1.0 α

0.8 0.6 0.4 0.2 0.0 –4 10

10–2 10–3 n(KBF4)

10–1

Figure 2.10. Dependence of the degree of conversion on the content of KBF4 in the system NaCl–KBF4 .

The phase diagram of the ternary system KF−KCl−KBF4 was measured by Patarák and Daneˇ k (1992). The system is simple eutectic with the coordinates of the eutectic point of 19.2 mole % KF, l8.4 mole % KCl, 61.4 mole % KBF4 , and the temperature of eutectic crystallization of 422◦ C. Among the physico-chemical properties, the density of the ternary system KF−KCl−KBF4 was measured by Chrenková and Daneˇ k (1991), the surface tension by Lubyová et al. (1997), the electrical conductivity by Chrenková et al. (1991b), and the viscosity by Daneˇ k and Nguyen (1995). From the results of the physico-chemical analysis, it follows that deviations from the ideal behavior were observed in all the boundary binaries as well as in the ternary system. With regard to the fact that the investigated system has a common cation, the observed deviations from the ideal behavior have to be a consequence of the anionic interaction only. The observed interaction of components could be of different origin. A different character of interaction has to be considered in the boundary binaries KF–KBF4 and KCl–KBF4 . In the pure KBF4 melt the BF− 4 tetrahedrons tend to link, forming relatively weak B–F–B bonds. The strength of this bond depends considerably on temperature. Introducing F− ions into the KBF4 melt by addition of KF, the B–F–B bridges break off, which lowers the viscosity and leads to negative deviation from the ideal behavior in

Main Features of Molten Salt Systems

35

the KF–KBF4 system. Besides, the mixing of small anions F− with relatively large BF− 4 takes place. In systems of this type, the deviation from additivity is proportional to the fractional difference in the radii of different anions. Therefore a relatively large deviation from ideality may be expected, which was confirmed by all measured physico-chemical properties. An opposite interaction effect takes place in the system KCl–KBF4 where the mixing of − two relatively large and polarizable anions BF− 4 and Cl occurs. As was found by Sangster and Pelton (1987), no important deviations from additivity were found in systems of alkali metal chlorides, bromides, and iodides with a common cation. Also, such behavior was confirmed by Chrenková and Daneˇ k (1990) in the system KCl–KBF4 by density, conductivity, and viscosity measurements. By introducing Cl− ions into the KBF4 melt, the substitution of fluoride atoms in the BF− 4 tetrahedrons by chloride takes place according to the general scheme − − − BF− 4 + nCl ⇔ [BF4−n Cln ] + nF

(2.41)

and [BF4−n Cln ]− mixed anions are probably present. Consequently, the lower stability of the B−Cl−B bridges and the lower concentration of the B−F−B lead to negative deviation of the properties in the KCl−KBF4 system. This explanation is supported also by the asymmetric course of the excess viscosity curve, which is due to the reaction (2.41) shifted to the right side in the region of high concentration of KBF4 . The negative deviations of the properties, found in the ternary system KF–KCl–KBF4 , have obviously the same origin as described for the boundary binary systems. As it follows from the course of the excess molar Gibbs energy of mixing (Figure 2.11), the excess molar volume (Figure 2.12), the excess molar conductivity (Figure 2.13), and the excess viscosity (Figure 2.14) of the system KF–KCl–KBF4 , the maximum interaction effect is localized near the KF–KBF4 boundary. The anionic interaction according to the reaction (2.41) was confirmed by means of infrared spectroscopy by Daneˇ k et al. (1997a). The IR spectra of KBF4 and of the quenched molten KBF4 −KCl (1:1 molar ratio) mixture are shown in Figure 2.15. With the exception of the 600–900 cm−1 region, the mid infrared spectra of both samples are almost identical. In agreement with the earlier study of the alkali metal tetrafluoroborates by Bates and Quist (1975), common vibrations can be assigned to crystalline KBF4 . Significant differences have been observed only in the 600–900 cm−1 region. It is obvious that beside the ν(1) vibration corresponding to KBF4 , the quenched molten KBF4 −KCl mixture produces two additional peaks at 760 and 796 cm−1 with a shoulder at 770 cm−1 . It may be assumed that these peaks arise due to the different B–F and B–Cl valence vibrations in the anions [BF4−n Cln ]− . The substitution of the fluorine atom by chlorine in the coordination sphere of the BF− 4 anion in the molten KF–KCl–KBF4 mixtures was thus confirmed. However, the type of the mixed anion was determined neither by physico-chemical analysis nor by spectroscopic methods.

36

Physico-chemical Analysis of Molten Electrolytes

KBF4 1

0.8

0.4

0.6

X(

) F4

KB

-450

X(

KF )

0.2

0.4

0.6 300

-300

150

0.8

0.2

0 -150

150

0

0 KF

0.2

0.6

0.4

1 KCl

0.8

X(KCl)

Figure 2.11. Excess Gibbs energy of mixing of the system KF–KCl–KBF4 . Values are in J · mol−1 .

KBF4 1

0.8

0.4

0.4

0.6

X(

X(

0.6 1.2

1 0.8

0.4

0.4

0.6

F 4)

KB

KF )

0.2

0.2

0.8 0.2

0 KF

0.2

0.4

0.6

X(KCl)

0.8

1 KCl

Figure 2.12. Excess molar volume of the system KF–KCl–KBF4 at 1100 K. Values are in cm3 · mol−1 .

Main Features of Molten Salt Systems

37

KBF4 1

0.8

-12

-9

-6

-3

0.6

0.4

F 4)

X(

-15

KB

KF

0.6

0.4

X(

)

0.2

0.2

0.8 -3

0 KF

0.2

0.6

0.4

0.8

1 KCl

X(KCl)

Figure 2.13. Excess molar conductivity of the system KF–KCl–KBF4 at 1100 K. Values are in S · m2 /mol.

KBF4 1

0.8

0.4

0.6

X(

F 4) KB

X(

KF

)

0.2

0.4

0.6 -0.2

0.8

0.2

-0.15 -0.1

-0.1

-0.05

0 KF

0.2

0.4

0.6 X(KCl)

0.8

-0.05

1 KCl

Figure 2.14. Excess viscosity of the system KF–KCl–KBF4 at 1100 K. Values are in mPa · s.

Physico-chemical Analysis of Molten Electrolytes

Transmittance (a.u.)

38

600

700

v (cm−1)

800

900

Figure 2.15. IR spectra of KBF4 and of the quenched molten KBF4 –KCl (1:1 molar ratio) mixture.

2.1.3.3. Systems containing halides of rare earth metals

Lanthanide halides are used in a number of applications ranging from lighting to catalysis, through pyrochemical reprocessing of nuclear fuel. However, many of these industrial processes are still under development due to relatively little knowledge of the properties and the behavior of systems used. Different experimental methods have been used in order to fully characterize pure compounds and binary systems, such as DTA, X-ray, electrochemical techniques, etc., the latter making it possible to identify the nature of phase transitions that take place in the solid state. They have enabled distinguishing between the formation of compounds in the solid state (reconstructive phase transition) and their structural transition (nonreconstructive phase transition). Several reports have been published on the structure of rare earth chlorides studied by Mochinaga et al. (1991a) and Iwadate et al. (1994) using X-ray diffraction, by Saboungi et al. (1991) using neutron diffraction, and by Papatheodorou (1975, 1977) and Matsuoka et al. (1993) using Raman spectroscopy. These investigations revealed that the octahedral complex anion LnCl3− 6 and a loose disordered network of corner- or edge-sharing octahedral units exist in the melt. Moreover, the measurements of conductivity helped to study the dynamics of molten salts, reflecting the average dynamic structure. The molar conductivity of several rare earth chlorides was measured by Iwadate et al. (1986), Mochinaga et al. (1991b, 1993), and Fukushima et al. (1991), and the results suggested that the existence of dimmers or more polymeric complex anions might be presumed in

Main Features of Molten Salt Systems

39

these melts. The structure of molten ErCl3 , Na3 ErCl6 , and K3 ErCl6 was investigated by Iwadate et al. (1993, 1994) and the existence of octahedral complex ions ErCl3− 6 was confirmed. Some clustering of distorted edge-sharing octahedrons was expected. The temperatures, enthalpies of phase transitions, and heat capacities of pure lanthanide chlorides (LaCl3 , CeCl3 , PrCl3 , NdCl3 , GdCl3 , DyCl3 , ErCl3 , and TmCl3 ) and of the M3 LnCl6 compounds (Ln = La, Ce, Pr, Nd; M = K, Rb, Cs) were determined by Gaune-Escard et al. (1994a). As far as thermodynamics is concerned, much attention has been paid to the M3 LnCl6 stoichiometric compounds that exist in most of the LnCl3 –MCl systems and have a more extended stability range than those of a different stoichiometry. Gaune-Escard and Rycerz (2003) determined the temperatures and enthalpies of the solid–solid phase transition, as well as of fusion of M3 LnCl6 compounds (Ln = La, Ce, Pr, Nd, Tb; M = K, Rb, Cs). They are presented in Table 2.7, together with the data of the enthalpy of formation taken from the literature. From the table, it follows that, the smaller the ionic lanthanide radius, the lower the formation temperature and the higher the melting temperature. This means that these compounds are not stable at low temperatures, which is in agreement with the results of thermochemical calculation made by Seiffert (2002). They originate already at higher temperatures from M2 LnCl5 and MCl. During cooling, they decompose back to the above-mentioned components. At 0 K they can exist only as a metastable phase. From the values given in Table 2.7 it may be concluded that the formation of the M3 LnCl6 compounds is associated with large enthalpy changes (45–55 kJ · mol−1 ) Table 2.7. Temperatures of formation and fusion of congruently melting M3 LnCl6 compounds Compound

Tform (K)

δform H (kJ/mol)

Tfus (K)

δfus H (kJ/mol)

K2 LaCl5 K3 CeCl6 K3 PrCl6 K3 NdCl6 K3 TbCl6

– 811 768 724 –

– 55.4 52.6 46.3 –

906 908 944 973 1049

78.1 39.1 48.9 48.0 53.2

Rb3 LaCl6 Rb3 CeCl6 Rb3 PrCl6 Rb3 NdCl6 Rb3 TbCl6

725 651 598 547 –

48.4 – – – –

978 1016 1037 1060 1049

50.2 52.4 54.0 58.8 –

Cs3 LaCl6 Cs3 CeCl6 Cs3 PrCl6 Cs3 NdCl6 Cs3 TbCl6

– – – – –

1055 1078 1093 1103 1153

1055 1078 1093 1103 1153

58.7 67.4 61.1 66.4 –

40

Physico-chemical Analysis of Molten Electrolytes

due to the so-called “reconstructive” phase transition, while their solid–solid transformation corresponds to more modest enthalpies of 7–8 kJ · mol−1 due to the structural (“non reconstructive”) phase transition. The linear dependence of both Tform and Tfus on lanthanide ionic radius allows estimation of the formation and fusion temperatures of the other K3 LnCl6 compounds. However, it should be stressed that the temperatures reported in the literature as formation temperatures do not always correspond to the formation of the compound. For the Rb3 CeCl6 , Rb3 PrCl6 , and Rb3 NdCl6 compounds, an additional thermal effect could be observed at significantly lower temperatures. This effect may arise most probably from the transition of the metastable phase at low temperature. From among the Cs3 LnCl6 compounds, only Cs3 LaCl6 , Cs3 CeCl6 , and Cs3 PrCl6 do exist at temperatures above 0 K. They originate from Cs2 LnCl5 and CsCl at 462, 283, and 143 K. Due to the low enthalpies of formation, decomposition does not occur upon cooling and metastable phases are formed. All Cs3 LnCl6 compounds undergo a structural phase transition at a nearly identical temperature of about 670–680 K. K3 CeCl6 , K3 PrCl6 , K3 NdCl6 , and Rb3 LaCl6 originate at higher temperatures and the formation of these compounds corresponds to a “reconstructive” phase transition. The crystal structure of these compounds is of the elpasolite-type and no other structural phase transition takes place before melting. For the other compounds, e.g. those that originate at lower temperatures (e.g. K3 TbCl6 , Rb3 PrCl6 , Rb3 NdCl6 , Rb3 TbCl6 , and all Cs3 LnCl6 ) and before melting, they undergo a solid–solid transformation, two different crystal structures are characterized: low-temperature monoclinic Cs3 BiCl6 -type and the high-temperature cubic elpasolite-type. In the electrical conductivity versus temperature plot of the first group of compounds, a significant (two orders of magnitude) conductivity jump is observed at the same temperature as determined for the “reconstructive” phase transition. A second, but smaller jump happens approximately at an identical temperature for all compounds (835–845 K). This effect is not observed in the DSC thermograms. For the second group of compounds formed at lower temperature, two conductivity regimes can be observed. The first conductivity break appears at a different temperature for each compound, has a smaller magnitude, and also spreads over a 40–50 K temperature range. Mutual solubility of molten lanthanide chlorides and alkali chlorides has been recently investigated intensively by Seifert et al. (1985–1988, 1990, 1991, 1993), Thiel and Seifert (1988), Mitra and Seifert (1995), Roffe and Seifert (1997), and Zheng and Seifert (1998). Numerous investigations of the composition and thermochemistry of lanthanide chlorides and bromides and their binary systems with the alkali halides have been carried out by Papatheodorou (1974a,b), Gaune-Escard et al. (1994a,b, 1995, 1996a,b), Takagi et al. (1994), and Rycerz and Gaune-Escard (1998).

Main Features of Molten Salt Systems

41

Several investigations of the enthalpy of mixing of liquid rare earth halides with alkali metal halides were carried out. The sytems NdC13 –MCI (M is Na, K, Rb, Cs), NdBr3 –MBr (M = Li, Na, K, Rb, Cs), NdI3 –MI (M = Li, Na, K, Cs), LaCl3 – MCl (M = Li, Na, K, Rb, Cs), and TbCI3 –MCI (M = Li, Na, K, Rb, Cs), the NdF3 –KF, PrCI3 –NaCI, PrCI3 –KCI, DyCI3 –NaCI, DyC13 –KCl, LaCl3 –CaCl2 , PrCI3 – CaCl2 , NdCl3 –CaCl2 , and Ln(1)Cl3 –Ln(2)Cl3 binaries were studied by Hatem and Gaune-Escard (1993), Gaune-Escard et al. (1994a,b, 1996c), Gaune-Escard and Rycerz (1997), and Rycerz and Gaune-Escard (2002). The LnCl3 –MCl binary systems have relatively simple phase diagrams for the light alkali metal chlorides (LiCl and NaCl), while those including KCl, RbCl, and CsCl exhibit several compounds of stoichiometry M3 LnCl6 , M2 LnCl5 , and MLn2 Cl7 . All the M3 LnCl6 compounds melt congruently, whereas M2 LnCl5 and MLn2 Cl7 can melt congruently or decompose peritectically. Congruently melting MLnCl5 compounds exist only in the systems with lanthanum and cerium chlorides, while M2 Ln2 Cl7 melt congruently in the sequence K < Rb < Cs at increasing lanthanide atomic numbers, i.e. at decreasing lanthanide ionic radii. Accordingly, the congruently melting CsLn2 Cl7 exist in all chloride systems starting from cerium chloride, the RbLn2 Cl7 starting from samarium chloride and KLn2 Cl7 from europium chloride. The LaCl3 –KCl system forms an exception to the previous description, since it includes a single congruently melting compound, K2 LaCl5 .

2.1.4. Systems containing tetravalent cations

The most important representatives of tetravalent elements are titanium and zirconium because of their uses as advanced construction materials in different technologies. Systems of titanium and zirconium fluorides and chlorides with alkali metal fluorides and chlorides, respectively, can be found in the Phase Equilibria Diagrams Database (1993). A number of compounds with various stoichiometry are formed in these systems, depending on the size of the titanium and zirconium cation and the polarization ability of the alkali metal. Some of these compounds are very stable and melt congruently even at relatively high temperatures. On the other hand, some compounds melt incongruently and their existence is limited to a certain temperature range. In the systems LiF–TiF4 and NaF–TiF4 there are no compounds present, while in the systems MF–TiF4 (M = K, Rb, Cs), only two congruently melting M3 TiF7 and M2 TiF6 compounds are formed. In the systems LiCl–TiCl4 , NaCl–TiCl4 , and RbCl–TiCl4 , no compounds are formed, while in the systems KCl–TiCl4 and CsCl–TiCl4 , the congruently melting compounds K2 TiCl6 and Cs2 TiCl6 originate respectively. The systems MF–ZrF4 (M = Li, Na, K, Rb, and Cs) are very abundant in compounds. In all MF–ZrF4 systems, the congruently melting M3 ZrF7 and the incongruently melting M2 ZrF6 compounds originate. In the systems MF–ZrF4 (M = K, Rb, and Cs), there

42

Physico-chemical Analysis of Molten Electrolytes

exist the congruently melting compounds MZrF5 . Besides these, the following congruently melting Rb5 Zr4 F19 and incongruently melting Li4 ZrF8 , M5 Zr2 F13 (M = Na, K), M3 Zr2 F11 (M = Na, K), M3 Zr4 F19 (M = Li, Na), MZr2 F9 (M = Na, Rb), and LiZr4 F17 compounds are formed. In the last group of the zirconium tetrachloride containing systems, the congruently melting NaZrCl6 , M2 ZrCl6 (M = K, Cs), Na7 Zr6 Cl31 and the incongruently melting Na3 ZrCl7 and Na3 Zr4 Cl19 compounds are present. However, among all of the above mentioned compounds, the technologically most important are the K2 TiF6 and K2 ZrF6 , because of their frequent use in the electrochemical deposition of titanium and zirconium and the electrochemical synthesis of titanium and zirconium diborides. Titanium diboride exhibits a high melting point, electronic conductivity, wetability by molten aluminum, and a resistance towards chemical attack of aluminum and molten fluorides. Due to these properties, TiB2 is considered to be the most promising material for inert cathodes in aluminum electrolysis. Also zirconium diboride belongs to the category of promising constructive materials due to its favorable properties. Different methods used to prepare titanium diboride have been reviewed by Samsonov et al. (1975). At present, it is mainly produced as a powder by thermochemical reduction of boron and titanium oxides followed by hot pressing and sintering to process the final product. The less costly alternative appears to be to coat suitable substrate materials with TiB2 or TiB2 -based composites by hot pressing, plasma spraying, chemical vapor deposition, etc. The promising alternative for TiB2 preparation is the high-temperature electrochemical synthesis in molten salts. The electrolytes employed can essentially be divided into two groups according to the type of electro-active components. (a)

(b)

Molten systems containing oxygen-containing compounds of titanium and boron, e.g. Me2 B4 O7 , MeBO2 , B2 O3 , TiO2 , Me2 TiO3 (where Me denotes alkali metal): To improve the physico-chemical properties of the electrolyte, alkali metal halides, alkali earth metal oxides, or cryolite are added to the melt. In some cases, natural minerals such as ilmenite or rutile replace titanium oxide. Systems containing alkali metal fluoroborates and alkali metal fluorotitanates: The electro-active components are dissolved in a supporting electrolyte, which generally consists of a mixture of alkali metal halides. Alkali chlorides, in comparison to fluorides, exhibit some obvious advantages like price, low corrosion of construction materials and easy separation of the product from the cathodic deposit containing solidified electrolyte.

According to Matiašovský et al. (1988), very suitable electrolytes in the electrochemical synthesis of TiB2 are the melts of the quaternary system KF−KCl−KBF4 −K2 TiF6 ,

Main Features of Molten Salt Systems

43

especially when coherent coatings on metallic bases have to be prepared. Besides the mentioned application, the molten system KF−KCl−K2 TiF6 is a promising electrolyte for the deposition of titanium. The knowledge of the structure of these electrolytes is needed for understanding the mechanism of the electrochemical process. The interaction of components and the chemical reactions taking place in the melt affect the ionic composition, determining the kind of the electro-active species. A suitable choice of the electrolyte composition may suppress the formation of volatile compounds, which leads to undesirable exhalation and lowers the efficiency of the process. One of the crucial problems in the chemistry of potassium hexafluorotitanate, K2 TiF6 , is its stability in the environment of different alkali metal fluorides and/or chlorides, since these compounds are usually used as complementary electrolyte components. The cryoscopic measurements in a system containing alkali metal halides and K2 TiF6 make it possible to determine the dissociation scheme of K2 TiF6 , or other reactions taking place between alkali metal halides and K2 TiF6 . The study of some systems of this type can be found in the literature. Janz et al. (1958) determined the freezing point depression in the eutectic mixture LiCl–KCl effected by the addition of M2 TiF6 (M = Li, Na, K) in the concentration range 0.008–0.066 m, and on the basis of the experimentally measured values proposed the following dissociation scheme M2 TiF6 = TiF4 + 2M+ + 2F−

(2.42)

Petit and Bourlange (1953) carried out cryoscopic measurements in the system NaCl–K2 TiF6 , but their conclusions were not unambiguous. They assumed that in the above system TiF4 , formed by dissociation of K2 TiF6 , reacts with chloride anions according to the equation TiF4 + 4Cl− = TiCl4 + 4F−

(2.43)

The behavior of K2 TiF6 in the alkali metal halides LiF, NaF, LiCl, NaCl, KCl, and in the LiCl–KCl and LiF–LiCl eutectic mixtures was studied using the cryoscopic method by Daneˇ k et al. (1975). The experimentally determined dependency of the melting point depression on molality and molar fraction of K2 TiF6 , in the proximity of the melting point of the above mentioned solvents were compared with the liquidus curves calculated according to the equation TA = KAcr mB r

(2.44)

where TA is the melting point depression of substance A, KAcr is the cryoscopic constant of substance A, mB is the molality of component B, and r is the correction factor for

44

Physico-chemical Analysis of Molten Electrolytes

component B in the system A–B introduced by Stortenbeker (1892) and is numerically equal to the number of new (foreign) particles, which are introduced into substance A by addition of one molecule of substance B. The necessary thermodynamical data were taken from JANAF Thermochemical Tables (1971) and from Janz (1967). From the measurements, it follows that in molten LiF and NaF, the correction factor r equals 3, which corresponds to the formation of three new particles: two K+ cations and one Ti-containing particle, most probably in the form of a complex anion (e.g. TiF2− 6 , − 3− TiF5 , or TiF7 ). The existence of the last two ionic species was discussed by Delimarskii and Chernov (1966), but the choice between them cannot be made by the cryoscopic method. In molten LiCl, NaCl, and KCl, the correction factor r was found to be 9, 5, and 3, respectively. Hence it follows that in molten sodium and potassium chloride, K2 TiF6 dissociates most probably according to the scheme K2 TiF6 = 2K + + 2F− + TiF4

(2.45)

As in the previous case, the cryoscopic method does not enable to determine whether Ti is bound in a complex anion (e.g. in TiF4 Cl2− 2 as proposed by Janz et al. (1958)) and, therefore, it is written as TiF4 . However, from the experimental results it can be concluded that in molten NaCl and KCl, no substitution reaction occurs between TiF4 and Cl− anions. On the other hand, the value of the correction factor in molten LiCl (r = 9) indicates that in this solvent titanium fluoride reacts with the chloride ions, thus increasing the number of new particles to 9, and the reaction can be interpreted by the dissociation scheme K2 TiF6 + 4Cl− = 2K + + 6F− + TiCl4

(2.46)

2− There is no evaporation of TiF4 since complex anions TiCl− 5 or TiCl6 are formed in the melt. It can be concluded that the substitution reaction occurs only in molten LiCl. This was also confirmed by the cryoscopic measurement in the LiCl–KCl eutectic mixture, where the value of the correction factor r indicates the formation of 7 new particles. This can be explained only by the dissociation of K2 TiF6 and the subsequent substitution of fluorine in TiF4 by chlorine. The results of cryoscopic measurement in the LiCl–KCl eutectic mixture obtained by Daneˇ k et al. (1975) do not agree with the data presented by Janz et al. (1958), who found only 3 new particles. However, the cryoscopy in eutectic mixtures might be the matter of discussion (see Section 3.3.2.3). The possibility of the substitution of fluorine atoms in TiF4 by the chlorine was confirmed by Daneˇ k et al (1975) and also by thermodynamic calculation of the degree of

Main Features of Molten Salt Systems

45

TiF4 + 4MCl = TiCl4 + 4MF

(2.47)

conversion of the reaction

where M = Li, Na, and K. From the results of calculation, it follows that in molten LiCl for x(K2 TiF6 ) < 10−2 , the degree of conversion is α ∼ = 1, which indicates that the TiF4 is completely converted into TiCl4 . For NaCl as a solvent for x(K2 TiF6 ) > 10−3 the conversion degree is α ∼ = 0, which means that the substitution reaction in molten NaCl practically does not take place. In molten KCl, for x(K2 TiF6 ) > 10−3 is α ∼ = 0. Hence it follows that in this case, the substitution reaction also does not occur. However, if the above calculation is applied for a more concentrated solution, the picture is entirely different. In the case of LiCl, the degree of conversion α ∼ = 0.1 at 9 mole % K2 TiF6 can be expected. Hence it can be assumed that in the system LiF–LiCl–K2 TiF6 , at a sufficiently high concentration of fluoride ions, the equilibrium of reaction (2.47) is shifted completely to the left. The cryoscopic measurement in the eutectic mixture LiF–LiCl fully confirmed the above-mentioned assumption. The correction factor was found to be r = 3, indicating the formation of three new particles due to the dissociation of K2 TiF6 . This means that in the LiF–LiCl eutectic mixture the substitution reaction does not take place. Chrenková et al. (1998) studied the structure or ionic composition of the melts of the system KF−KCl−KBF4 −K2 TiF6 based on the measurements of phase equilibria, density and viscosity and using complex thermodynamic and physico-chemical analysis. From the results of the physico-chemical analysis, it follows that deviations from the ideal behavior were observed in all boundary binary, ternary, as well as in the quaternary systems. With regard to the fact that the investigated system has a common cation, the observed deviations from the ideal behavior have to be a consequence of the anionic interaction only. The phase equilibria and volume properties of the systems KF–K2 TiF6 and KCl– K2 TiF6 were studied in detail by Daneˇ k and Matiašovský (1989). They found that in these binary systems, the compounds K3 TiF7 and K3 TiF6 Cl are formed according to the reactions K2 TiF6 (l) + KF = K3 TiF7 (l) (2.48)

r G01100 K = 0.963 kJ · mol−1 K2 TiF6 (l) + KCl(l) = K3 TiF6 Cl(l) r G1100 K = 0.587 kJ · mol 0

−1

(2.49)

The formation of K3 TiF7 and K3 TiF6 Cl does not substantially affect the volume properties. This indicates that, either these compounds dissociate in the melt to a

46

Physico-chemical Analysis of Molten Electrolytes

3− 3− considerable degree, or the volumes of the anions TiF2− are 6 , TiF7 , and TiF6 Cl fairly similar. However, the formation of both the compounds affects the course of the volume properties differently. While the formation of K3 TiF7 is connected with small volume expansion, the formation of K3 TiF6 Cl is connected with substantially higher volume contraction. The difference probably agrees with different stereochemistry of both complex anions and with its relations to the other anions present. However, the formation of K3 TiF7 and K3 TiF6 Cl also takes place in the quaternary system KF–KCl–KBF4 –K2 TiF6 . The thermodynamic analysis of the phase equilibrium and of the volume properties of the systems KF–K2 TiF6 and KCl–K2 TiF6 , were used for the calculation of the dissociation degree of the K3 TiF7 and K3 TiF6 Cl compounds. The dissociation degree of K3 TiF6 calculated on the basis of the experimentally determined phase diagram is α0 = 0.64, which agrees very well with the values obtained on the basis of the density data, α0 (1000 K) = 0.6 and α0 (1100 K) = 0.7. The corresponding data for K3 TiF6 Cl are α0 = 0.78 from the phase equilibria, and α0 (1000 K) = 0.72 and α0 (1100 K) = 0.81 from the density data. In the binary system KBF4 −K2 TiF6 , at 1100 K the partial molar volume of K2 TiF6 for x(KBF4 ) → 1 is V(K2 TiF6 ) = 129.69 cm3 · mol−1 . This value is higher than the molar volume of pure K2 TiF6 (V 0 = 114.62 cm3 · mol−1 ), which indicates the formation of larger complex ions, e.g. TiF2− 7 . The partial molar volume of KBF4 for x(K2 TiF6 ) → 1 is V(KBF4 ) = 68.08 cm3 · mol−1 . This value is lower than the molar volume of pure KBF4 (V 0 = 75.12 cm3 · mol−1 ). The reason for this lowering is the decomposition of KBF4 and escape of gaseous BF3 from the melt, which was actually observed. On the basis of the given findings, the following chemical reaction may be considered to take place in the binary system KBF4 –K2 TiF6

KBF4 (l) + K2 TiF6 (l) = K3 TiF7 (l) + BF3 (g) 0 −1 r G1100 K = 10.71 kJ · mol

(2.50)

The relatively low positive value of Gibbs energy of the reaction, as well as the observed escape of gaseous BF3 , indicates that the reaction (2.50) probably takes place in the melt. Reaction (2.50) also takes place obviously in the quaternary system KF–KCl–KBF4 –K2 TiF6 . In the quaternary system KF–KCl–KBF4 –K2 TiF6 , the following chemical reaction is probable KCl(l) + 2K2 TiF6 (l) + KBF4 (l) = K3 Ti7 F(l) + K3 TiF6 Cl(l) + BF3 (g) r G1100 K 0 = 12.05 kJ · mol−1

(2.51)

Main Features of Molten Salt Systems

47

The originating additive compounds dissociate thermally according to reactions (2.48) and (2.49). The Gibbs energies of reactions (2.50) and (2.51) were calculated on the basis of the Gibbs energies of dissociations of K3 TiF7 and K3 TiF6 Cl, as well as of the Gibbs energies of formations of KF, KCl, KBF4 , and BF3 taken from JANAF Thermochemical Tables (1971). From the physico-chemical properties of the melts of the system KF–KCl– KBF4 −K2 TiF6 , it follows that the most characteristic feature of these melts is the 3− with formation of the thermodynamically less stable ionic species TiF3− 7 and TiF6 Cl lowered symmetry of the coordination sphere in the melt. The presence of these anions in the melt facilitates most probably the electro-reduction of titanium and thus the formation of titanium diboride on the cathode.

2.1.5. Systems containing pentavalent cations

Among the pentavalent elements, the most important are niobium and tantalum. Niobium is an excellent material for surface treatment of steel materials for chemical industry due to its high hardness and corrosion-resistance in wet acidic conditions. Nowadays, niobium is also used for the preparation of superconductor tapes and it is used in other branches of industry, for instance in nuclear technology and metallurgy. Tantalum is also of similar importance. For these applications, it is necessary to prepare high purity metal. Molten salt electrolysis, as an alternative process to classical thermal reduction, provides niobium and tantalum with required quality. In order to optimize these processes, it is necessary to know details of both complex formation and redox chemistry of the species present in the melts. The possibility of obtaining metallic niobium and tantalum coatings was demonstrated in fluoride, chloride, as well as in mixed fluoride–chloride electrolytes. K2 NbF7 and K2 TaF7 are usually the electrolyte components, which serve as niobium and tantalum source, respectively. From the number of systems used as electrolytes for niobium deposition, the systems LiF–KF–K2 NbF7 and LiF–NaF–K2 NbF7 seem to be the most promising. Due to the high hygroscopic behavior of KF, the alternative NaF was used in the second system. Also from the theoretical point of view, the above mentioned systems are interesting as they are associated with the formation of the congruently melting compound K3 NbF8 , affecting the ionic composition of the melt and thus possibly influencing the mechanism of the electrochemical process on the electrode surface. In a number of papers, the measurement of different thermodynamic and transport properties of the LiF−KF−K2 NbF7 and LiF−NaF−K2 NbF7 melts was performed with the goal to clarify their structure, i.e. the ionic composition. For the study of the electrolyte structure, physico-chemical analysis has been used. The phase diagram of the system LiF−KF−K2 NbF7 was measured by Chrenková et al. (1999), the density by Chrenková et al. (2000), surface tension by Nguyen and

48

Physico-chemical Analysis of Molten Electrolytes

(a)

K2NbF7

(b)

F) 7

2 Nb

x(K

2 Nb

) F7 Nb

2

x(K

x(K

F) 7

0.8

0.8

0.6

0.6 −4

0.4

−3

0. 4

−2

0.2

0.2 −1

LiF

KF

Figure 2.16. Phase diagram ((a), values in ◦ C) and excess molar volume at 1100 K ((b), values in cm3 · mol−1 ) of the system LiF–KF–K2 NbF7 according to Chrenková et al. (1999, 2000).

Daneˇ k (2000b), and viscosity by Nguyen and Daneˇ k (2000c). The X-ray phase analysis and IR spectroscopy measurements of quenched melts were carried out by Van et al. (2000). To draw conclusions on the electrolyte structure from the concentration dependency of the particular property, the following thermodynamic, statistical, and material balance calculations were also used. In Figure 2.16, the phase diagram (a) and the excess molar volume (b) of the system LiF−KF−K2 NbF7 are shown for illustration. The electrochemical behavior of the melts of these systems was studied in detail by Van et al. (1999a,b) and described in a short overview by Daneˇ k et al. (2000a). From the complex physico-chemical analysis of the system LiF–KF–K2 NbF7 , it follows that the behavior and properties of the melts are affected by the formation of the congruently melting compound K3 NbF8 . Regarding the presence of the anion [NbF8 ]3− in the molten state the system behavior is not far from the ideal. The congruently melting compound K3 NbF8 was found to be not quite stable. At melting, it undergoes a partial thermal dissociation according to the scheme α KF + K2 NbF7 K3 NbF8 ←→

(2.52)

The degree of thermal dissociation of K3 NbF8 calculated from the phase diagram was found to be α0 = 0.44. A similar value also was calculated from the density measurements. Also, the formation of K3 NbF8 in the ternary system was confirmed by all the physico-chemical parameters. The system LiF–NaF–K2 NbF7 is the planar section of the quaternary reciprocal system Li2+ , Na2+ , K2+ // F− , [NbF7 ]2− . Both boundary binary systems LiF–K2 NbF7 and

Main Features of Molten Salt Systems

49

NaF–K2 NbF7 form stable diagonals of the appropriate quaternary reciprocal systems. Van and Daneˇ k (2001) calculated the Gibbs energy of formation of K2 NbF7 on the basis of electrochemical measurements in the LiF–NaF–K2 NbF7 melts. However, the Gibbs energies of formation of other fluoroniobates, Li2 NbF7 and Na2 NbF7 , are not known so far. Thus the Gibbs energies of reactions of both ternary reciprocal systems could not be calculated. The phase diagram of the ternary system LiF–NaF–K2 NbF7 was studied by Chrenková et al. (2003a) using the thermal analysis and the subsequent coupled analysis of thermodynamic and equilibrium phase diagram data. The density was measured by Chrenková et al. (2005), the surface tension by Kubíková et al. (2003), and the viscosity by Cibulková et al. (2003). Figure 2.17 shows the phase diagram (a) and the excess molar volume (b) of the system LiF–NaF–K2 NbF7 for illustration. The X-ray phase analysis and the IR spectroscopy measurement were performed by Bocˇ a et al. (2005). Similarly, as in the system LiF–KF–K2 NbF7 , also the system LiF–NaF–K2 NbF7 is characterized by the formation of the complex anion [NbF8 ]3− in the melts, even though no crystallization field of any M3 NbF8 compound is present in the ternary phase diagram. Its presence was, however, confirmed by all the investigated physico-chemical parameters and is the general feature of this system. The formation of the [NbF8 ]3− anions is more pronounced in the melts of the system NaF–K2 NbF7 compared with those of the system LiF–K2 NbF7 . The infrared spectra of 25, 50, 75 mole % K2 NbF7 dissolved in equimolar LiF–NaF melt as well as the spectrum of pure K2 NbF7 were measured by Bocˇ a et al. (2005). Characteristic vibrations of fluoroniobate complex anions [NbF7 ]2− and [NbF8 ]3− are

K2NbF7

70 0

650

0.6

0.6

0.4

0.4

65 0

0.4

0.2

0

0.2 90

0

800

80

75 0

750

70 0

700

0. 8

F7 )

2 Nb

x (K

600

2 Nb

0.6

650

0.4

0.2

0.8

x (K

0.6

0. 8

F 7) Nb 2 x(K

F)

0.8

K2NbF7

(b)

0. 2

85 0

(a)

-0.6

-1.0

-1.8

95

0

-0.2

-1.4

LiF

0.2

0.4

N 0.6

x(NaF)

0.8

NaF LiF

0.2

0.4

0.6

0.8

NaF

x(NaF)

Figure 2.17. Phase diagram ((a), values in ◦ C) and the excess molar volume at 1100 K ((b), values in cm3 · mol−1 ) of the system LiF–NaF–K2 NbF7 according to Chrenková et al. (2003a, 2005).

50

Physico-chemical Analysis of Molten Electrolytes

Nb=O

[NbF8]3−

Nb=O

K 2 NbF 7

Relative transmitance

75%K2NbF7

50%K2NbF7

25%K2NbF7O

-Nb-O

[NbF7]2− 1200

1000

800 Wavenumbers (cm−1)

600

400

Figure 2.18. Infrared spectra of the quenched LiF–NaF melts with various K2 NbF7 content.

found at 546 and 476 cm−1 , respectively. As it follows from Figure 2.18, the intensity of the [NbF7 ]2− vibration increases with an increasing content of K2 NbF7 , while the [NbF8 ]3− vibration increases up to the content of 50 mole % of K2 NbF7 and then it decreases. This is in agreement with the results of the physico-chemical studies, especially those of the excess Gibbs energy of mixing, the excess volume, and the excess surface tension. 2.1.6. Systems containing hexavalent cations

Among the hexavalent elements, molybdenum and tungsten seem to be the most technologically important metals because of their use as deposits on steels in the form of protecting metal or metal boride layers. Since the main features of molybdenum and tungsten electrolytes are similar, the following text will deal with the properties of the former. Analysis of the literature data on the electro-deposition of molybdenum shows that several types of molten electrolytes have been tested. On the basis of the electro-active species used, they can be divided into two main groups: (i)

halide systems containing either K3 MoCl6 or K3 MoF6 dissolved in alkali metal halides, mainly chlorides and fluorides,

Main Features of Molten Salt Systems

(ii)

51

mixed systems containing oxide compounds of molybdenum, such as molybdenum oxide, MoO3 , alkali metal molybdates, and CaMoO4 . Supporting electrolytes include: LiCl–KCl mixtures, sodium and lithium metaborates, KF–Na2 B4 O7 , KF–Li2 B4 O7 , KF–B2 O3 and CaCl2 –CaO.

Comparing results of molybdenum electro-deposition from several types of electrolytes, it was confirmed that the process is most successful in electrolytes consisting of a mixture of alkali metal fluorides and boron oxide (or alkali metal borate), to which molybdenum oxide or alkali metal molybdate is added as the electrochemical active component. From the literature it follows that the electro-deposition of molybdenum from the binary MeF–Me2 MoO4 mixtures is impossible. However, a small addition (1 mole %) of boron oxide or SiO2 to the electrolyte facilitates the electro-deposition of molybdenum. The presence of boron or silicon oxide most probably modifies the structure of the melt, which results in changes in the cathode process. The survey of electrochemistry of molybdenum deposition was given by Daneˇ k et al. (1997). Among the number of electrolytes studied, the most promising seem to be the melts of the system KF–K2 MoO4 –B2 O3 . In a number of papers, the measurement of different thermodynamic and transport properties of the KF–K2 MoO4 –B2 O3 melts was done with the goal to clarify their structure, i.e. the ionic composition on the basis of a complex physico-chemical analysis. From the theoretical point of view, the melts of this system represent very little investigated electrolytes containing both the classical ionic components and the network forming ones. The possible chemical interactions between them have not been well understood till now. The system KF–K2 MoO4 –B2 O3 is a considerably complicated subsystem of the quinary reciprocal system K+ , B3+ , Mo6+ // F− , O2− , in which a number of compounds is formed. The phase equilibrium in the system KF–K2 MoO4 –B2 O3 was studied by Patarák et al. (1993, 1997). The phase diagram of the investigated part of the ternary system KF–K2 MoO4 –B2 O3 up to 30 mole % B2 O3 , is interesting from the electrochemical deposition of molybdenum point of view, and constructed using the coupled analysis of thermodynamic and phase diagram data is shown in Figure 2.19. The very extended plateau on the crystallization surface of K2 MoO4 shifts the boundary line with the primary crystallization field of the additive compound K3 FMoO4 expressively to the KF corner, most probably due to the formation of the [BMo6 O24 ]9− heteropolyanions. The formation of the heteropolyanions [BMo6 O24 ]9− also indicates the excess Gibbs energy of mixing of the system, as shown in Figure 2.20. The enlarged shape of the isotherms of the crystallization surface of K2 MoO4 may also be probably due to the substitution of the oxygen atoms of the co-ordination sphere of molybdenum in the heteropolyanions [BMo6 O24 ]9− by the fluorine atoms. However, the existence of more polymerized heteropolyanions with Mo/B ratio being 9 or 12 cannot be excluded.

52

Physico-chemical Analysis of Molten Electrolytes

0.4

B

2O 3

0.4

0.2

0.2

88 0

0 80

0

0

72

76

720

0

760

72

840

0.0 KF

0.2

0.4 0.6 x(K2MoO4)

0.8

1.0 K2MoO4

Figure 2.19. Phase diagram of the system KF–K2 MoO4 –B2 O3 according to Patarák et al. (1997).

0.4

B

2O 3

0.4

4000

0.2

0.2

3000 2000 1000 0

0.0 KF

0.2

0.4 0.6 x(K2MoO4)

0.8

1.0 K2MoO4

Figure 2.20. Excess Gibbs energy of mixing of the system KF–K2 MoO4 –B2 O3 according to Patarák et al. (1997).

The density of the melts of the molten system KF–K2 MoO4 –B2 O3 was measured by Chrenková et al. (1994). For the concentration dependence of the molar volume in the investigated ternary system at the temperature of 827◦ C, the following equation was obtained V cm3 ·mol−1 = 29.89xKF + 89.50xK2 MoO4 + 44.62xB2 O3

+ xKF xK2 MoO4 8.50−5.28xK2 MoO4 + xKF xB2 O3 11.92−53.22xB2 O3

−xK2 MoO4 xB2 O3 81.31+102.51xB2 O3 + 110.29xKF xK2 MoO4 xB2 O3 (2.53)

Main Features of Molten Salt Systems

53

The first three members represent the ideal behavior, the following three represent the interactions in binary systems and the last one, the interaction of all three components. The calculation of the coefficients was carried out by the method of multiple linear regression analysis and omitting the statistically non-important terms on the 0.99 confidence level. The standard deviation of the fit is 0.404 cm3 ·mol−1 . The different sign of the coefficients in the systems KF–B2 O3 and K2 MoO4 −B2 O3 indicate the different behavior of B2 O3 in KF and K2 MoO4 . The excess molar volume of the melts of the ternary system KF–K2 MoO4 –B2 O3 is shown in Figure 2.21. From the figure it follows that in this system a region of volume expansion exists with the maximum at 10 mole % B2 O3 and 20 mole % K2 MoO4 , and a region of volume contraction with the maximum at 40 mole % B2 O3 and 50 mole % K2 MoO4 . The volume expansion region indicates the formation of larger ions, while the volume contraction region indicates the formation of [BMo6 O24 ]9− heteropolyanions, according to the equation 6K2 MoO4 + 2B2 O3 = K9 [BMo6 O24 ] + 3KBO2

(2.54)

and the probable further polymerization of the melt is assumed. The deviation from the ideal behavior is more pronounced in comparison with the boundary binary systems, which indicates stronger interaction of all the three components. Such ternary interaction may be explained as the substitution of oxygen atoms in the coordination sphere of the [BMo6 O24 ]9− heteropolyanions by fluorine atoms. The viscosity of the melts of the KF–K2 MoO4 –B2 O3 system has been measured by Silný et al. (1995) using the computerized torsion pendulum method. The viscosity of

0.

4

x(B

2O 3)

−2

0.

2

−2

−4

−6

−10 −8

0

0 0 KF

0.2

0.4 x(K2MoO4)

0.6

0.8

1 K2MoO4

Figure 2.21. Excess molar volume of the ternary system KF–K2 MoO4 –B2 O3 at 827◦ C according to Chrenková et al. (1994).

54

Physico-chemical Analysis of Molten Electrolytes

the melts increases with increased contents of both K2 MoO4 and B2 O3 . Within the experimental error the viscosity of melts in the system KF–K2 MoO4 increases linearly with the increasing content of K2 MoO4 . Taking into account this additive behavior in the binary system KF–K2 MoO4 and adopting the formal value of 1000 Pa · s for the viscosity of pure boron oxide, the following final equation was obtained for the concentration dependence of the viscosity in the ternary system KF–K2 MoO4 –B2 O3 at 877◦ C: η (mPa·s) = 1.28xKF + 2.81xK2 MoO4 + 999xB2 O3 2 −xKF xB2 O3 3317−3658xKF + 1336xKF −xK2 MoO4 xB2 O3 3242−3586xK2 MoO4 + 1348xK2 2 MoO4

+xKF xK2 MoO4 xB2 O3 3214 + 4157xB2 O3

(2.55)

The standard deviation of the fit is 0.082mPa · s. The viscosity of the ternary system at 927◦ C is shown in Figure 2.22. From the regression analysis, it follows that the interactions in the binary system KF–K2 MoO4 is statistically non-important, compared with those in the binary systems KF–B2 O3 and K2 MoO4 –B2 O3 . Thus, the values of the standard deviations of approximation relate mostly with the viscosity of the ternary melts. The viscosity of the melts of the ternary system increases very steeply with the increasing content of boron oxide. This observation is, however, not surprising, because of the polymerization ability of the boron oxide. The formation of more voluminous

0.

x(

B

2O 3)

4

0.

2

3

3.2

3.4

2.8 2.6

1.4

0

1.6

1.8

2

1.2

0 KF

0.2

0.4 x (K2MoO4)

2.2

2.4

0.6

0.8

1.0 K2MoO4

Figure 2.22. Viscosity of the ternary system KF–K2 MoO4 –B2 O3 at 927◦ C according to Silný et al. (1995).

Main Features of Molten Salt Systems

55

(B4 O7 )2− and BF4− anions will obviously increase the viscosity of the melt as well. However, the increase in the viscosity is higher than it might be expected from the pure polycondensation of boron oxide. Evidently bigger particles than the boroxol rings are formed in the ternary system, especially near the K2 MoO4 –B2 O3 boundary. According to the conclusions based on the phase equilibrium and density measurements, it may be assumed that these bigger particles are the [BMo6 O24 ]9− heteropolyanions formed in the melt according to Eq. (2.54). The statistically important ternary interaction, however, may be also ascribed to the entry of the fluorine atoms into the co-ordination sphere of molybdenum in the heteropolyanions. Also surface tension measurements in the system KF–K2 MoO4 –B2 O3 , performed by Nguyen and Daneˇ k (2000a), supported the above mentioned conclusions. From the viscosity, as well as the phase equilibrium, surface tension, and density measurements it is evident that the system KF–K2 MoO4 –B2 O3 is very complex. Beside the chemical reactions, the polymerization tendency of the melts, especially in the region of higher contents of boron oxide, makes this system difficult to study. Similar electrolytic preparation of molybdenum coatings has been investigated by Silný et al. (1993) and Zatko et al. (1994) in the molten system KF–K2 MoO4 –SiO2 . From the above system, these authors obtained coherent, smooth and well-adhesive molybdenum layers on electrically conductive substrates in a relatively narrow composition region. The quality of the deposit depends on the silica content in the melt. The authors explained the positive role of SiO2 in the molybdenum electro-deposition due to the change in the structure of the electrolyte and the formation in the melt of heteropolyanions [SiMo12 O40 ]4− according to the reaction 12K2 MoO4 + 7SiO2 + 36KF = K 4 [SiMo12 O40 ] + 6K2 SiF6 + 22K2 O

(2.56)

Such heteropolyanions are rather voluminous and thus, much more polarizable. In the vicinity of the cathode in the electric double layer, this anion is strongly polarized and finally disintegrated into smaller species, from which consecutive molybdenum deposition takes place. The X-ray diffraction analysis of the solid deposit on the top closure and furnace wall proved that the deposit consists of pure K2 SiF6 , which supports the assumption on the formation of the above-mentioned heteropolyanions. Unfortunately, the authors did not study the mechanism of the cathodic process in this system. From the physico-chemical and thermodynamic analysis of the molten systems KF–K2 MoO4 –B2 O3 and KF–K2 MoO4 –SiO2 , it can be concluded that the formation of heteropolymolybdates containing boron, [BMo6 O24 ]9− , and silicon, [SiMo12 O40 ]4− , as a central atom is most probably responsible for an easy molybdenum deposition. Besides, the entry of fluorine atoms into the coordination sphere of molybdenum in the heteropolyanions lowers the symmetry and thus, also the electrochemical stability of such electro-active species.

56

Physico-chemical Analysis of Molten Electrolytes

2.1.7. Systems containing halides and oxides

The presence of oxides and the formation of oxofluoro-complexes in molten electrolytes may be sometimes unwanted, but in many cases they are the fundamental features of the system. For instance, the formation of oxide complexes in alkali–alkaline earth chloride melts may be mentioned. The formation of oxofluoride complexes in molten cryolite– alumina melts, used as electrolytes for aluminum production, is typical as well. On the other hand, the presence of oxofluoride complexes in electrolytes used for niobium production was initially regarded as unwanted. Recently, however, it has been proven that their presence in niobium electrolytes plays an important role in the niobium electrodeposition. In the following, some technologically important examples of systems containing halides and oxides will be described. 2.1.7.1. Oxofluoro-complexes in magnesium electrolytes

The physico-chemical properties such as density, conductivity, and viscosity of the magnesium chloride electrolyte are of substantial importance for the current efficiency of the magnesium electrolysis. The solubility of the reaction products, Mg and Cl2 , in the electrolyte is also important to attain the high current efficiency. Interfacial properties between the Mg electrolyte and the metal may depend significantly on the oxide concentration of the bath. In modern electrolytic cells for magnesium production, the metal is collected above the melt surface, the density of the melt thus being higher than that of the metal. Therefore the composition of the MgCl2 electrolyte is usually adapted by the addition of NaCl and KCl with or without CaCl2 or BaCl2 . Together with MgCl2 , some oxide impurities are also added into the melt mostly in the form of MgO. The very low MgO solubility in halide melts supports the fact that Mg2+ –O2− interaction is very strong. This also indicates a possible Mg–O–Cl complex formation in MgCl2 –alkali chloride melts containing oxide impurities. However, little is known about the Mg–O–Cl complexes, which may form in such melts and in this way increase the MgO solubility. Combes et al. (1980) used calcia-stabilized zirconia electrodes to determine the O2− concentration in the MgCl2 –NaCl–KCl mixtures at 1000 K with different BaO additions. They suggested that the following reactions take place in the melt: 2Mg2+ + O2− → Mg2 O2+

(2.57)

Mg2 O2+ + O2− → 2MgO(s)

(2.58)

The solubility of MgO in the equimolar NaCl–KCl mixture at 1000 K was found to be very low. The solubility of other alkaline-earth oxides in the equimolar NaCl–KCl melts are low, but are considerably higher than that of MgO and do increase in the sequence MgO < CaO < SrO < BaO. However, in earth alkali metal chloride-rich melts,

Main Features of Molten Salt Systems

57

the solubility of the corresponding oxides is considerable. The large difference in the solubility of oxides indicates different dissolution mechanisms of oxides in the alkali metal chloride and earth alkali metal chloride melts. Boghosian et al. (1991) studied the solubility of earth alkali metal oxides in alkali metal–earth alkali metal chlorides and NaCl–MeCl2 melts. They found that the oxide solubility is in general very low and increases markedly with the MeCl2 concentration and with increasing atomic number of the earth alkali metal atom. The very low oxide solubility in earth alkali metal chlorides can be explained by the reaction (e.g. for the MgCl2 melts) MgCl2 (l,mix) + MeO(s) = MeCl2 (l,mix) + MgO(s)

(2.59)

For this reaction, r G0 0, for Me = Ca and Ba. The observed enhanced solubility of MeO is most probably caused by the oxide complex formation according to the reaction mMeCl2 (l) + nMeO(s) = Mem+n On Cl2m (l)

(2.60)

It was found that in the NaCl–MeCl2 mixtures containing 25 mole % MeCl2 , where Me = Mg, Ca, Sr, and Ba, the solubility of MeO at 850◦ C is x(MgO) = 6×10−5 , x(CaO) = 10−3 −10−4 , and x(SrO) = 0.039. In the BaCl2 system, the BaO/BaCl2 molar ratio at saturation equals 1. The oxochloro-complexes formed in these melts were further justified using cryoscopic measurements. An increasing temperature of the primary crystallization of the binary NaCl–MeCl2 melt was observed when the oxide was added. Model calculations in combination with the two sets of experimental data (solubility and cryoscopy) indicated that the Me/O ratios for the different complexes are Mg2 O, Ca3 O, Ca4 O, Sr3 O, Sr4 O, Ba3 O, and Ba4 O. The equilibrium constants for the formation reactions of the neutral complexes, based on the reaction (m+n)M2+ + nO2− + 2mCl− = Mm+n On Cl2m

(2.61)

are very large. Mediaas et al. (1997) studied the effect of fluoride additions to MgCl2 -containing melts on the MgO solubility. It was found that in the MgCl2 –MgF2 melts, and at 840◦ C, the MgO solubility increases linearly with MgF2 concentration. The solubility of MgO in pure MgCl2 melt at 840◦ C was determined to be xsat (MgO) = 0.0061, while in the 60 mole% MgCl2 + 40 mole% MgF2 mixture, it was xsat (MgO) = 0.0109. They determined the effect of temperature on the MgO solubility for pure MgCl2 and for two MgCl2 –MgF2 mixtures with x(MgF2 ) = 0.1816 and x(MgF2 ) = 0.2016 in the temperature range 676◦ C < t < 930◦ C. It was found that for pure MgCl2 the derivative

58

Physico-chemical Analysis of Molten Electrolytes

dln xsat (MgO)/d(1/T ) equals −5200 K and for the two mixtures −4200 K. In the MgCl2 – NaCl–NaF ternary system, three different oxide solubility experiments were performed: (a)

(b)

(c)

NaF and MgCl2 were added to a MgCl2 –NaCl mixture saturated with MgO at 850◦ C such that x(MgCl2 ) = 0.63. The MgO solubility increases with increased NaF content up to x(NaF) ≈ 0.08, thereafter it remains constant at x(NaF) = 0.20. NaF was added to a MgCl2 –NaCl mixture saturated with MgO at 850◦ C. The initial MgCl2 concentration was x(MgCl2 ) = 0.63. An increase in MgO solubility up to about 8 mole % NaF was observed. On further NaF additions to x(NaF) = 0.18, the MgO solubility decreases. A melt closer in composition to the electrolyte used for electro-winning of magnesium was investigated. The MgCl2 mole fraction was kept constant at x(MgCl2 ) = 0.10. The temperature was 900◦ C. No effect could be observed for NaF content on the MgO solubility for x(NaF) ≤ 0.06.

A reasonable thermodynamic model was tried to explain the effect of fluoride concentration on the MgO solubility in the MgCl2 –NaCl–NaF ternary melts. However, both the activity model used to calculate the activity of MgCl2 and MgF2 and the understanding of the Mg–O–Cl(F) complexes formed in the melt seemed to be too simple to give a reasonable mechanism for MgO solubility in these complicated melts. The MgO solubility in technical electrolytes for Mg production is very low (in the ppm range). Together with MgCl2 , BaO may also be introduced into the Mg electrolyte. Even when BaO dissolves readily in NaCl–BaCl2 melts, the solubility of BaO in MgCl2 containing melts is also very low, since BaO(s) reacts with MgCl2 (l) along with the formation of BaCl2 (l) and MgO(s). 2.1.7.2. Oxofluoroaluminate complexes in cryolite–alumina melts

The cryolite–alumina melts, used as electrolytes for aluminum production are technologically very important. Aluminum oxide (alumina) is dissolved on purpose in cryolite–based melts as it is the source of aluminum. The electrolyte composition is one of the most important technological parameters at aluminum production. Specification of the optimum electrolyte composition, mainly concerning the alumina and aluminum fluoride content, characterizes the particular producer and belongs to the priorities at achieving the maximum current efficiency. The melts of the system Na3AlF6 –Al2 O3 are the basis of the electrolyte used in aluminum production. The most accepted phase diagram of this system is that published by Chin and Hollingshead (1966). During approximately half a century of investigations on the structure of the melts of the system Na3AlF6 −Al2 O3 , there have been numerous suggestions on the nature of the possible oxygen-containing species present in the melt. A comprehensive review on this topic was given by Grjotheim et al. (1982). However, in the last two decades, it was

Main Features of Molten Salt Systems

59

generally accepted that complex oxofluoroaluminate anions are the most probable species formed. Dewing (1974) suggested that alumina dissolves predominantly during the formation of AlOF1−x anions, while Førland and Ratkje (1973), Ratkje and Førland (1976) x on the basis of cryoscopic measurements suggested that the reaction 1−y

= AlOFy Al2 OF4−x x

(3−x−y)

+AlFx−y

(2.62)

takes place in the melt. The probable values for x are 6 or 8, while for y, it may be 2, (3−x−y) − is AlF3− indicating that AlFx−y 6 or AlF4 , which are the most important anions in oxide-free cryolite melts. The more recent investigations based mainly on Raman spectroscopic measurements by Gilbert et al. (1976, 1995) indicate that anions with bridging Al–O–Al bonds are present, and have most probably the following structure

F F

Al

O O

Al

F F

F and/or

|

F |

F −Al −O −Al −F | | F F

It was suggested that these species originate in the melt by the reaction of AlF3− 6 with dissolved alumina. Julsrud (1979) proposed a thermodynamic model for cryolite-alumina melts based 4− on cryoscopic and calorimetric measurements and considered the Al2 OF2− 6 , Al2 OF8 , 2− 4− Al2 OF6− 10 , Al2 O2 F4 , and Al2 O2 F6 species to be present in alumina saturated melts. Sterten (1980) developed a model of cryolite melts saturated with alumina on the basis of experimental NaF and AlF3 activities. He found that Al2 OF4−x and Al2 O2 F2−x x x 1−x were the most probable species in these melts. AlOFx was of minor importance. He 4− claimed that in very basic melts, CR > 5, the Al2 O2 F2− 4 and Al2 O2 F6 complexes were 2− dominant, while in acidic melts, CR < 3, the complexes Al2 OF6 and Al2 O2 F2− 4 were the most abundant. Kvande (1980, 1986) suggested that Al2 OF4− 8 is the most abundant species in cryolite melts with low alumina contents. He further claimed that the solubility of alumina in NaF–AlF3 melts attains a maximum at the composition of cryolite. Based on this, he argued that it is reasonable to assume that alumina, when dissolving, reacts predominantly with AlF3− 6 anions according to the following reaction scheme 2− − Al2 O3 + 4AlF3− 6 = 3Al2 OF6 + 6F

(2.63)

60

Physico-chemical Analysis of Molten Electrolytes 4− Al2 O3 + 4AlF3− 6 = 3Al2 OF8

3 2− Al2 O3 + AlF3− 6 = 2 Al2 O2 F4

(2.64) (2.65)

Reactions of alumina with F− ions are less probable, since the alumina solubility in pure molten NaF is very low. Zhuxian and Gang (1989), using the Monte Carlo method, simulated the structure of the oxofluoro-complexes originating in the molten cryolite after the addition of aluminum oxide at 1010◦ C. From the results, it follows that the species AlOF1−x (x = 2−5), x 6− Al2 OF2− , and Al OF are formed in the melt. They also suggested that above 9 mole % 2 6 10 Al2 O3 , besides the mentioned complexes, the Al2 O2 F2− anion may also originate. 4 Bache and Ystenes (1989) used the IR spectroscopy and X-ray powder diffraction analysis in order to determine the qualitative composition of quenched or heated equimolar mixtures of cryolite and chiolite with the addition of 10 mass % Al2 O3 . In the IR spectrum of this quenched mixture, new bands arise at 530, 870, and 1170 cm−1 , which do not belong either to cryolite or to chiolite. In the IR spectra of these samples, these bands do not occur. In the X-ray diffraction pattern of the above-mentioned quenched mixture, only one peak arises, which does not belong either to cryolite or chiolite. For heated samples, all main peaks of α-Al2 O3 were detected. On the basis of Raman spectroscopy investigation, Robert et al. (1993) gave a new insight into the character of the oxofluoroaluminate species. The Raman spectra of Robert et al. did not confirm the presence of the species with non-bridging Al–O bonds. Only those having bridging Al–O–Al bonds, the Al2 OF4−x and Al2 O2 F2−x complexes, were x x present in the melt. The above picture of the cryolite–alumina melts was recently confirmed also by Raman spectra and vapor pressure studies carried out by Gilbert et al.(1995). At low Al2 O3 concentrations, they observed a new band at 450 cm−1 in the Raman spectrum which they ascribed to the anion Al2 OF4− 8 originating in the basic region (CR > 3) or to the anion Al2 OF2− existing in the neutral (CR = 3) and acidic region (CR < 3). With increasing the 6 concentration of Al2 O3 , the 510–515 cm−1 band sharply increases, which they ascribed 2− to the anion Al2 O2 F2− 4 . Based on a new dissociation scheme of cryolite, the AlF5 anion was considered to be the most abundant species in molten cryolite. A new reaction scheme at the dissolution of Al2 O3 in cryolite was therefore suggested 2− − − Al2 O3 + 4AlF2− 5 + 4F = 3Al2 OF6 + 6F

(2.66)

4− − Al2 O3 + 4AlF2− 5 + 4F = 3Al2 OF8

(2.67)

3 2− − Al2 O3 + AlF2− 5 + F = Al2 O2 F4 2

(2.68)

Main Features of Molten Salt Systems

61

Picard et al. (1996) performed theoretical calculations of the stability of the till now presumed oxofluoroaluminate anions. For these anions, they also calculated the vibration frequencies and intensities of bands originating in the infrared region. Their calcula4−x tions indicated that the most stable anion is AlOF− species, the 2 . Among the Al2 OFx 2− 2−x most stable seems to be Al2 OF6 and of the Al2 O2 Fx species, the highest stability have those with x = 2–6. From the calculated vibration frequencies and the intensities of bands, it follows that bands at 460 and 185 cm−1 , which arise in Raman spectra measured by Gilbert et al. (1976), belong to the Al2 OF4−x and Al2 O2 F2−x species. Bands x x −1 observed by Gilbert et al. (1976) at 530 and 310 cm were ascribed only to the Al2 O2 F2−x x complexes. Raman spectroscopy, vapor pressure, and the solubility of alumina in cryolite measurements were used by Robert et al. (1997a) in order to elucidate the structure of the system NaF–AlF3 –Al2 O3 . On the basis of the obtained data, they suggested that at low 4− concentrations of alumina, the first originating species are Al2 OF2− 6 and Al2 OF8 . With 2− increasing the content of aluminum oxide, the anion Al2 O2 F4 is formed, but with further Al2 O3 addition, the intensity of the Al2 O2 F2− 4 band decreases. This is explained by the formation of a next anion with a similar structure, but with weaker bonds. The solubility of Al2 O3 in NaF–AlF3 melts was recently measured by Diep (1998) and the results were explained in terms of possible oxofluoroaluminate species formed in the melt at alumina saturation. In order to explain the maximum alumina solubility at CR = 3, 2− he introduced a new species, Al3 O3 F3− 6 , which consists of three AlO2 F2 tetrahedrons linked by the oxygen corners. Thermodynamic calculations indicated that the oxygencontaining compounds Na2Al2 OF6 , Na6Al2 OF10 , Na2Al2 O2 F4 , and Na3Al3 O3 F6 had to be present to describe the Al2 O3 solubility limit in the NaF–AlF3 –Al2 O3 ternary. He found that when these compounds were present at the composition of alumina saturation, the melts behave approximately ideally. It is thus reasonable to believe that the Al2 OF2− 6 species are dominant at low Al2 O3 concentration while the Al2 O2 F2− species are predominant at high alumina contents. The 4 two species are formed according to the reactions 4Na3 AlF6 + Al2 O3 = 3Na2 Al2 OF6 + 6NaF

(2.69)

At low oxide content and at high alumina concentration, the reaction is 2Na3 AlF6 + 3Al2 O3 = 3Na2 Al2 O2 F4

(2.70)

Direct oxygen determination in MF–AlF3 –Al2 O3 (M = Li, Na, K) melts in dependence on the cryolite ratio and the alumina content was performed using the method of carbothermal reduction by Daneˇ k et al. (2000b). The MF–AlF3 melts with alumina, successively added in 5–7 steps from 0 mole % and beyond the solubility limit, were heated at

62

Physico-chemical Analysis of Molten Electrolytes

9.6 % Al2O3

Al3O3F63–

pure Na2Al2O2F4

Al2OF62–

Relative oxygen content

Al2O2F42–

1000◦ C in an inert argon atmosphere. Samples were analyzed for oxide content using a LECO TC-436 Nitrogen/Oxygen analyzer. During the LECO measurement, the samples are heated with carbon powder in a graphite crucible at a constant heating rate to 2900◦ C over a short time. The oxygen, present in the sample, reacts at specific temperatures with carbon powder resulting in a characteristic LECO curve showing the dependence of the actual oxygen content on temperature. On this curve, several peaks corresponding to the decomposition of individual present oxofluoroaluminates were observed. The temperature at which carbon starts to react with oxygen in the sample depends on the energy with which oxygen is bonded in the present compound, i.e. on the Gibbs energy of formation of the individual compound. The LECO curves obtained in the system NaF–AlF3 –Al2 O3 at 1000◦ C and the cryolite ratio CR = 4 are shown in Figure 2.23. Similar LECO curves were also obtained for the systems LiF–AlF3 –Al2 O3 and KF–AlF3 –Al2 O3 . The LECO curves showed that generally two distinct peaks could be observed. From an energetic point of view, it can be assumed that the double oxygen bridge is more stable, thus requiring higher temperatures to be decomposed, compared with the single Al–O–Al bond. Hence, oxygen-containing species decomposing at lower temperature are most probably those in which two aluminum atoms are linked by one bridging oxygen atom, while the species decomposing at higher temperatures are those having two aluminum

7.2 % Al2O3 5.2 % Al2O3 3.6 % Al2O3 2.1 % Al2O3 1.0 % Al2O3

500

1000

1500

2000 t(°C)

2500

3000

Figure 2.23. LECO curves obtained in the system NaF–AlF3 –Al2 O3 at the cryolite ratio CR = 4.

Main Features of Molten Salt Systems

63

atoms linked by two bridging oxygen atoms. This assumption was confirmed by analyzing the synthesized Na2Al2 O2 F4 compound. On the LECO curve, only a single peak occurred at the temperature identical with the second peak in the system NaF–AlF3 –Al2 O3 (see Figure 2.23). It was therefore concluded that this peak was due to the Al2 O2 F2− 4 complex anion. The content of oxygen present in the individual oxofluoroaluminate species was calculated as the fraction of the total alumina dissolved with respect to the area of the peak corresponding to the given oxygen-containing species. In Figures 2.24–2.26, the distributions of individual oxofluoroaluminate anions in the systems LiF–AlF3 –Al2 O3 , NaF–AlF3 –Al2 O3 at CR = 3, and KF–AlF3 –Al2 O3 at CR = 3, respectively, are shown. The obtained LECO data, described in the figures with open symbols, were compared with the results of the Raman work by Robert et al. (1997a) (described by full symbols). These two essentially different methods for determining the content of oxyfluoroaluminate species in melts gave a very similar variation of the oxygen-containing complexes with increasing contents of Al2 O3 . In the LiF–AlF3 –Al2 O3 system, only a moderate quantity of Al2 OF2− 6 is present, while Al2 O2 F2− is the dominating species over the entire composition range. The distribution 4 of species does not depend to any significant extent on the cryolite ratio, obviously due

5 Al2O2F42−

Mole % of species

4

3

2 Al2OF62−

1

0

0

1

2 Mole % Al2O3

3

4

Figure 2.24. Distribution of species in LiF–AlF3 –Al2 O3 melts at various CR. ◦, • – CR = 1.2; , – CR = 2; , – CR = 3.

64

Physico-chemical Analysis of Molten Electrolytes

12 Al2O2F42−

Mole % of species

10 8 6

Al2OF62−

4 2 Al3O3F63−

0

0

2

4 Mole % Al2O3

6

8

2− Figure 2.25. Distribution of species in NaF–AlF3 –Al2 O3 melts at CR = 3. ◦, • – Al2 OF2− 6 ; , – Al2 O2 F4 ; 3− – Al3 O3 F6 .

60 Al2O2F42-

Mole % of species

50 40 30 20

Al2OF62–

10

Al3O3F63–

0

Al3O4F43–

0

4

8 12 Mole % Al2O3

16

20

2− Figure 2.26. Distribution of species in KF–AlF3 –Al2 O3 melts at CR = 3. ◦, • – Al2 OF2− 6 ; , – Al2 O2 F4 , 3− 3− – Al3 O3 F6 ; ✩, – Al3 O4 F4 .

Main Features of Molten Salt Systems

65

to the very low solubility of alumina and the large surplus of available LiF and AlF3 to form oxofluoroaluminates. In the NaF–AlF3 –Al2 O3 melts, it was found that the first peak is split and two species having aluminum atoms linked with one oxygen bridge are obviously present. Daneˇ k et al. (2000) attributed the lower temperature shoulder to the Al2 OF2− 6 species, while the shoulder at the higher temperature was attributed to the formation of the Al3 O3 F3− 6 species. Up to 2 mole % of Al2 O3 , no other species was found in the melt. The Al2 O2 F2− 4 species are present already above the mentioned Al2 O3 content and at high Al2 O3 contents, they become dominant. The observed oxofluoroaluminate species distribution in NaF–AlF3 –Al2 O3 melts is in accordance with the results of Julsrud (1979), Sterten (1980), Kvande (1980, 1986), and Robert et al. (1997) obtained by quite different methods. As in the previous systems, in the KF–AlF3 –Al2 O3 also two main oxygen peaks were detected. The first peak shows a shoulder at the lower temperature, which means that two different oxofluoroaluminate species with single oxygen bonded aluminum atoms are present. The first shoulder was attributed to the Al2 OF2− 6 species, while the first main peak belongs most probably to the more stable Al3 O3 F3− 6 species. The second main peak has a shoulder at higher temperatures as well. The main second peak is 3 attributed to Al2 O2 F2− 4 , while the high temperature shoulder is attributed to Al3 O4 F4 . 2− Similarly as in the NaF–AlF3 –Al2 O3 melts, at low Al2 O3 content, only the Al2 OF6 and Al3 O3 F3− 6 species are observed in measurable amounts. Above a certain Al2 O3 concentration, Al2 O2 F2− 4 species arises, its content steeply increases, becoming the dominating species at high Al2 O3 concentrations. The shoulder of the second peak at a higher temperature belongs to a minor amount of Al3 O4 F3− 4 . The presence of this species is necessary in order to explain the very high alumina solubility in this system. Brooker et al. (2000) measured the Raman spectra of the ternary eutectic mixture LiF–NaF–KF (FLINAK), to which they added small amounts of AlF3 (or NaAlF3 ) and Na2 O. The temperature varied in the range 25–500◦ C. Na2 O served as the source of oxygen, since aluminum oxide does not dissolve in FLINAK. When Na2 O was added to the mixture of FLINAK + 5 mole % AlF3 (or Na3AlF6 ) at 500◦ C, new bands occurred in the Raman spectrum, the intensity of which increases with increasing amounts of added −1 −1 Na2 O, while the intensity of bands assigned to AlF3− 6 (ν1 ≈ 540cm ), (ν2 ≈ 326cm ) −1 decreases. The new bands, especially the very intensive one at 494 cm , were assigned 2− to the Al2 OF2− 6 anion. Presence of the Al2 O2 F4 anion was regarded as little probable by the authors. The authors found that the 494 cm−1 band (at 500◦ C) shifts with decreasing temperature to 509 cm−1 (at 25◦ C). The same holds also for the other Al2 OF2− 6 bands. 2− This indicates marked temperature instability of the Al2 OF6 anion. Lacasagne et al. (2002) studied the structure of the NaF–AlF3 –Al2 O3 melts up to 1025◦ C by in situ NMR of 27Al, 23 Na, 19 F, and 17 O, using a laser-heated experimental set-up. The Al2 O3 additions to cryolite varied in the range of 0.6 to 8.2 mole %. The observed changes in 27Al and 17 O chemical shifts gave direct experimental evidence

66

Physico-chemical Analysis of Molten Electrolytes

on the existence of at least two different oxofluoro-aluminate species in the cryolite– alumina melts. Zhang et al. (2002, 2003) measured the solubility of alumina in neutral and basic cryolite melts in the composition range 3 ≤ CR ≤ 12.5 at 1300 K. Using thermodynamic activity probes for Al and Na, they monitored the activity of NaF and AlF3 in the melt at different alumina concentrations. From the obtained solubility data as well as of those published by Skybakmoen et al. (1997), the distribution of the solutes in dependence of the alumina concentration was calculated for the composition range 1.5 ≤ CR ≤ 12.5 and compared with the literature data. The present model very well describes the experimentally determined alumina solubility data. The authors found that in the acidic melt at CR ≈ 1.5, Na2 Al2 OF6 is the dominant solute. In less acidic melts, Na2 Al2 O2 F4 becomes more dominant, while in the basic melts, Na4 Al2 O2 F6 provides the significant solubility of alumina. The 3D geometry for the three solutes was proposed. 2.1.7.3. Oxofluoro-complexes in niobium electrolytes

The crucial problem in niobium deposition in molten salts is the presence of oxygen in the electrolyte, because it is extremely difficult to prepare a melt free of O2− ions, especially in the case of industrial applications. It was formerly assumed that the presence of O2− ions might decrease the quality of Nb coatings or prevent the formation of Nb coatings completely. Therefore a great part of the research efforts was concentrated on the influence of O2− ions on the reduction mechanism of Nb and the formation of niobium oxofluoro-complexes in the melt. Though electrolyte impurities, such as oxides, hydroxides, chlorides, bromides, and iodides, were formerly considered to be undesirable, later it was found that the presence of small amounts of oxides in the melt increases the current efficiency of the electrolysis. Christensen et al. (1994) obtained the highest current efficiency in melts with O/Nb molar ratios in the range 1 < n0 /nNb < 0.5. The presence of oxide in the melt causes the formation of various niobium oxofluorocomplexes. It was shown that relatively pure Nb coatings can be obtained from FLINAK melts if n0 /nNb is less than 1. However, it has been shown by Konstantinov et al. (1981) and Khalidi et al. (1991) that, even a small amount of O2− ions can entirely change the mechanism of Nb deposition depending on the types of the niobium oxofluorides formed in the melt. Daneˇ k et al. (2000a) and Bocˇ a et al. (2005) showed that from the number of systems used as electrolytes for niobium deposition, the most promising seems to be the system LiF–KF–K2 NbF7 or LiF–NaF–K2 NbF7 . From the physico-chemical and spectroscopic measurements, it follows that the formation of the [NbF8 ]3− anion is the general feature of the investigated systems. However, this obviously leads to an increase in the symmetry of the C2 v local structure of the [NbF7 ]2− anion to the D4 h of the [NbF8 ]3− , and thus does not facilitate the deposition of niobium.

Main Features of Molten Salt Systems

67

When oxide ions are present in the melt, the formation of the oxofluoro-complex [NbOF5 ]2− according to the reaction [NbF7 ]2− +O2− = [NbOF5 ]2− +2F−

(2.71)

was observed. At a lower concentration of oxide ions in the melt, the ligand displacement in the coordination sphere of the [NbF7 ]2− anion by oxygen and the formation of an oxofluoro-complex lowers the local structure symmetry due to the formation of the C4 v local structure of the [NbOF5 ]2− anion. With increasing O2− contents and due to the presence of free oxide ions in the melt, [NbOF5 ]2− ions are transformed to the [NbO2 F]− according to the reaction [NbOF5 ]2− +O2− +e− = [NbO2 F]− +4F−

(2.72)

and they coexist in the melt with mono-oxofluoro-complexes despite the low concentration of oxide ions. Van et al. (1999a,b, 2000) used voltammetry, calculation of the Gibbs energy of formation of K2 NbF7 , and IR spectroscopy in order to clarify the complex formation in the high oxidation states of the metal. In Figure 2.27, the cyclic voltammogram obtained in the LiF–NaF–K2 NbF7 melt with an initial Nb(V) concentration of 166.46 mol · cm−3

0.06

Ox2 0.04

i (A)

0.02

Ox1 0.00

R1 –0.02

R3

–0.04 –2.0

R2 –1.5

–1.0

–0.5

0.0

0.5

1.0

E (V) Figure 2.27. Cyclic voltammogram taken in the LiF–NaF–K2 NbF7 melt at 750◦ C.

68

Physico-chemical Analysis of Molten Electrolytes

at 750◦ C and a scan rate of 0.36 V · s−1 is shown. Three reduction waves may be observed. The first wave, R1 , at −0.24 V is followed by a sharp peak, R2 , at −0.87 V, and then a hump, R3 , at −1.15 V. In the positive potential scan direction, the reduced species are re-oxidized in two steps, Ox2 at −0.83 V and Ox1 at −0.25 V. The third reduction wave, R3 , is caused by presence of the [NbOF5 ]2− complex, which is formed in the melt when the oxide ions are added. [NbOF5 ]2− is then reduced in a one-step-reaction at a potential of about 100 mV lower than Nb(IV). Characteristic vibrations of niobium oxofluoro-complexes were found in the 1000–400 cm−1 region. Figure 2.28 shows the infrared spectra of K2 NbF7 dissolved in equimolar LiF–KF melt with c(Nb) = 570 mol · m−3 and various n0 /nNb ratios at 700◦ C. The IR spectrum of the sample with n0 /nNb = 0.2 shows a sharp band at 926 cm−1 , which is characteristic for the Nb=O stretching vibration in [NbOF5 ]2− (Figure 2.28a). The absorption band at 738 cm−1 is supposed to correspond to the vibrations of bridging

(d)

878 (c) 807 495

Transmitance

918 875 804 (b)

1083

498

738

478 615

930

(a)

549

1200

1000

800

600

400

Wavenumber (cm−1) Figure 2.28. Infrared spectra of the LiF–NaF quenched melts with different contents of K2 NbF7 .

Main Features of Molten Salt Systems

69

oxygen–niobium bonds. The broad complex band near 550 cm−1 consists of several overlapping components due to Nb–F vibrations of both [NbOF5 ]2− and [NbF7 ]2− complexes. With the addition of the oxide up to the ratio n0 /nNb = 1, a pronounced decrease in intensity and a shift of the Nb=O band to 918 cm−1 are observed. This indicates the presence of [NbOF6 ]3− in the melt (Figure 2.28b). Moreover, two new bands at 875 cm−1 and 804 cm−1 , assigned to O–Nb–O stretching vibrations in di-oxofluoro-complexes, are observed. The Nb–F band is shifted to 498 cm−1 , the position found for [NbO2 Fy ](y−1)− . No Nb=O band near 920 cm−1 in the spectrum of the melt corresponding to the ratio n0 /nNb = 1.5 indicates no or a negligible amount of double bonds (Figure 2.28c). Absorptions at 878, 807, and 495 cm−1 are consistent with O–Nb–O and Nb–F stretching vibrations in di-oxofluoro-complex [NbO2 Fy ](y−1)− . The IR spectra indicates a transformation of [NbF7 ]2− through [NbOFx ](x−3)− to [Nb(V)O2 Fy ](y−1)− with increasing the oxide content in the melts. The ligand displacement reactions can be described as [NbF7 ]2− +O2− → [NbOFx ](x−3)− +(7−x)F−

(2.73)

[NbOFx ](x−3)− +O2− → [NbO2 Fy ](y−1)− +(x −y)F−

(2.74)

where, according to Pausewang and Rudorf (1969) and von Barner et al. (1991), x could be 5 and 6, and y may attain the value of 2, 3, or 4. It may be concluded that the successive ligand displacement in the [NbF7 ]2− anion by oxygen and the formation of oxofluoro-complexes can be described by the following general scheme 2− NbF2− → NbOF(x−3)− +xO2− → NbO2 F(x−1)− x x 7 +xO

(2.75)

The first step taking place at n0 /nNb < 0.7 lowers the local structure symmetry due to the formation of the C4 v local structure of the [NbOF5 ]2− anion and facilitates the deposition of niobium. Good experimental conditions for the deposition process can be expected in such a case. However, at n0 /nNb > 1, inhomogeneous niobium deposits containing probably potassium–niobium phase and niobium oxide solid solution were obtained. However, a non-metal deposit could be obtained as well. Infrared spectra of solid K2 NbF7 and spectra of Nb(V) in molten KF–LiF were measured by Fordyce and Baum (1966). The spectra showed bands characteristic of the 2− NbF2− 7 ion. Solid K2 NbF7 has been shown to contain discrete NbF7 ions with C2v symmetry. According to Keller (1963), the Raman spectrum of this salt has bands at 388, 630, and 782 cm−1 . Contrary to this, solid CsNbF6 , which contains octahedral NbF− 6, shows Raman bands at 280, 562, and 683 cm−1 .

70

Physico-chemical Analysis of Molten Electrolytes

According to Pausewang and Rudorf (1969), solid dioxofluorides of the type Alk3 (NbO2 F4 ) (Alk = Na, K, Rb) are also known. The (NbO2 F4 )3− ion has a sixcoordinate arrangement of the ligands with the two oxygen atoms in cis position to each other (C2v symmetry). von Barner (1991) investigated the formation of Nb(V) fluoro- and oxofluorocomplexes in eutectic LiF–NaF–KF melts (FLINAK) as a function of the oxide contents. On the basis of the Raman spectra of the melts and infrared spectra of solidified melts, the possible structures for the complexes were discussed. Raman spectra of the molten system LiF–NaF–KF–K2 NbF7 at 650◦ C indicates the −1 −1 presence of NbF2− 7 ions with vibration frequencies at 626 cm (p), 371 cm (dp), and 2− 290 cm−1 (dp). The NbF7 ions have C2v symmetry, as previously suggested by Hoard (1939) for solid K2 NbF7 . Depending on the value of the oxygen to niobium(V) ratio, at least four different species are formed in the LiF–NaF–KF melts. In the range O < mol of O2− /mol (n−3)− of Nb(V) < 1, vibration frequencies due to the NbOFn species are observed at −1 −1 −1 921 cm (p), 583 cm (p), and 307 cm (dp) for oxide to niobium(V) mole ratios up to approximately 2. (n−3)− When an oxide is added to the melt, complexes with the general formula NbOFn are formed. The most probable value of n is 5, giving rise to a monomeric NbOF2− 5 . Infrared spectra of solidified melts support the existence of a niobium–oxygen double bond in this complex. The vibration spectrum is in accord with the band pattern of monomeric NbOF2− 5 with a C4v symmetry. The nature of the species that are formed at high oxide to niobium ratios are more uncertain, but it seems likely that the first of them has a niobium–oxygen ratio of 1/2 (n−1)− (i.e. [NbO2 Fn ]n ). Bands in the Raman spectrum of the melt at 878 and 815 cm−1 and in the infrared spectrum of the solidified melt at 879 and 809 cm−1 are ascribed to the stretching vibrations of the NbO2 entity. The vibration spectra (Raman of the melt and infrared of the solidified melt) are consistent with the formation of NbO2 F3− 4 ions of C2v symmetry. In melts saturated with the oxide, structures of the [NbO3 Fn ](1+n)− type probably exist. Furthermore, some kind of polymerization is likely to occur in this melt, since frequencies typical of vibrations of distorted NbO6 octahedrons with edge shearing were observed by McConnell et al. (1976). By means of computer programs, it has been possible to obtain the Raman spectra of the three pure complexes, believed to be NbF2− 7 , 3− NbOF2− , and NbO F . 2 5 4 2.1.7.4. Alkali metal fluoride melts containing boron and/or titanium oxide

Among the systems of alkali metal fluorides containing boron oxide, the melts of the system LiF–KF–B2 O3 –TiO2 were tested as possible electrolytes in the electro-chemical synthesis of titanium diboride, especially when well-dispersed powders should be

Main Features of Molten Salt Systems

71

prepared. The use of such electrolytes was motivated by the effort to exclude special and expensive boron and titanium sources, such as potassium fluoroborate and potassium fluorotitanate, which moreover introduced a surplus of potassium fluoride. Moreover, the systems of alkali metal halides containing boron oxide are technologically important too, since they may serve as potential electrolytes for boriding of steels and other metallic surfaces. The system LiF–KF–B2 O3 –TiO2 is a considerably complicated subsystem of the quinary reciprocal system Li+ , K+ , B3+ , Ti4+ // F− , O2− , in which a number of compounds are formed. However, the phase equilibria in the system LiF–KF–B2 O3 –TiO2 have been investigated and are still unsatisfactory. The existing phase equilibrium studies are as follows. The phase diagram of the binary system LiF–B2 O3 was studied by Berul and Nikonova (1966) and by Chrenková and Daneˇ k (1992a), especially in the range of high concentrations of LiF. This system is a part of the ternary reciprocal system Li+ , B3+ // F− , O2− and according to the Gibbs energy of reaction of the metathetical reaction, 6LiF(l) + B2 O3 (l) = 3Li2 O(l) + 2BF3 (g) r G01200K = 609.6kJ·mol−1

(2.76)

the system LiF–B2 O3 is the stable diagonal of the ternary reciprocal system. Berul and Nikonova (1966) studied the phase diagram of this system up to 55 mole % B2 O3 . According to this source, the congruently melting compound LiF·B2 O3 with the temperature of fusion of 840◦ C is formed in this system. In the composition interval of 5–23 mole % B2 O3 , the liquidus curve is parallel to the composition axis at 835◦ C, probably due to the existence of immiscibility in the liquid state. According to Daneˇ k and Chrenková (1992a), a miscibility gap in the region 5–23 mole % B2 O3 was found with the monotectic temperature of 836◦ C and the upper consolute temperature of 862◦ C at approximately 14 mole % B2 O3 . The part of the phase diagram of the system LiF–B2 O3 investigated by Daneˇ k and Chrenková (1992a) is shown in Figure 2.29. Among the possible chemical reactions in the ternary reciprocal system Li+ , B3+ // − F , O2− leading to the compound formation, the following four are considered to be probable: 6LiF(l) + 4B2 O3 (l) = 6LiBO2 (l) + 2BF3 (g) r G01200 K = 38.02 kJ·mol−1 8LiF(l) + 4B2 O3 (l) = 6LiBO2 (l) + 2LiBF4 (g) r G01200 K = 432.9 kJ·mol−1

(2.77)

(2.78)

72

Physico-chemical Analysis of Molten Electrolytes

880

t(°C)

860

840

820

800

LiF

0.1

x(B2O3)

0.2

0.3

Figure 2.29. Investigated part of the phase diagram of the system LiF–B2 O3 . • – Berul and Nikonova (1966); – Danˇek and Chrenková (1992a); —– ⇒ aLiF = xLiF .

6LiF(l) + 7B2 O3 (l) = 3Li2 B4 O7 (l) + 2BF3 (g) r G01200 K = −9.23 kJ·mol−1 8LiF(l) + 7B2 O3 (l) = 3Li2 B4 O7 (l) + 2LiBF4 (g) r G01200 K = 385.7 kJ·mol−1

(2.79)

(2.80)

From the values of the Gibbs energies of reaction, it follows that the equilibrium of reactions (2.78) and (2.80) is shifted considerably to the left side and the reactions do not take place. On the other hand, reactions (2.77) and (2.79) can take place with respect to the values of the Gibbs energies of reaction especially when BF3 escapes from the system. In order to explain the structure of the LiF–B2 O3 melts, the composition of the quenched samples was determined by means of X-ray powder diffraction analysis and IR spectroscopy. Both methods eliminated the presence of LiBF4 in the mixtures. On the other hand, the presence of LiBO2 up to 5 mole % B2 O3 and of Li2 B4 O7 at higher

Main Features of Molten Salt Systems

73

contents of boron trioxide was confirmed. In dilute solutions of B2 O3 in molten LiF, reaction (2.77) obviously take place, while at higher B2 O3 contents, lithium metaborate polymerizes into more condensed polyanions. The repulsive effect of both the present anions, F− and the [B4 O7 ]2− leads at a certain content of B2 O3 to liquid separation into the LiF-like melt with a limited solubility of Li2 B4 O7 and the B2 O3 -like one, creating the immiscibility region. It should be concluded that LiF–B2 O3 is a quasi-binary system. In fact, the system is a projection of the LiF–LiBO2 –B2 O3 non-linear cross-section into the LiF–B2 O3 diagonal. The congruently melting compound LiF·B2 O3 mentioned by Berul and Nikonova (1966) is probably only the dystectic point on this cross-section. The density of the system LiF–B2 O3 was measured by Chrenková and Daneˇ k (1992c). The dependence of the molar volume on composition at 850◦ C was described by the equation V cm3 ·mol−1 = 14.365 + 21.573xB2 O3

(2.81)

and for the partial molar volume of B2 O3 in the infinitely diluted solution of LiF, the value VB2 O3 = 35.84 cm3 ·mol−1 was found, which is lower than the molar volume of pure B2 O3 at this temperature, VB02 O3 = 44.78 cm3 ·mol−1 . This volume contraction could be ascribed to the formation of lithium metaborate and the volatile BF3 according to reaction (2.74). BF3 escapes from the melt causing the volume contraction. In the ternary system LiF–Li2 O–B2 O3 , the phase diagram of the pseudo-binary system LiF–LiBO2 is known. The ternary congruently melting compound 2LiF·3LiBO2 with the temperature of fusion of 755◦ C, is formed in this system. The phase diagram of the system NaF–B2 O3 up to 65 mole % B2 O3 was studied by Bergman and Nagornyi (1943) and the cryoscopy in the system NaF–B2 O3 by Makyta (1993). It was found that in this system, sodium metaborate, sodium tetraborate, and gaseous boron trifluoride are created in the melts. The phase diagram of the ternary system LiF–NaF–B2 O3 studied Nikonova and Berul (1967). No ternary compound was found in this system. The liquidus curve of KF in the system KF–B2 O3 up to 20 mole % B2 O3 was determined by Chrenková and Daneˇ k (1992b) and Patarák et al. (1993), Patarák (1995). According to the value of the Gibbs energy of the metathetical reaction, 6KF(l) + B2 O3 (l) = 3K2 O(s) + 2BF3 (g) G01100K = 940kJ·mol−1

(2.82)

the system KF–B2 O3 should be the stable diagonal of the ternary reciprocal system K+ , B3+ // F− , O2− . However, a number of compounds are formed in this ternary reciprocal

74

Physico-chemical Analysis of Molten Electrolytes

system. From the thermodynamic analysis of the liquidus curve of KF, it follows that the reaction 8KF(l) + 7B2 O3 (l) = 3K2 B4 O7 (l) + 2KBF4 (l) G01200 K = −201.7 kJ·mol−1

(2.83)

yielding two complex compounds, KBF4 and K2 B4 O7 , takes place in the melts. The presence of both the compounds was also confirmed by means of the X-ray powder diffraction analysis and IR spectroscopy in the quenched samples. The positive deviation of the real liquidus curve from the theoretical one is most probably due to further polymerization of the borate species. Cryoscopic measurements of B2 O3 in molten sodium and potassium fluorides were performed by Makyta (1993). He obtained the best fit of the experimental and theoretical liquidus curve for the reaction 32KF(l) + 25B2 O3 (l) = 6KBO2 (l) + 9K2 B4 O7 (l) + 8KBF4 (l)

(2.84)

The last two compounds were confirmed by both the X-ray analysis and IR spectroscopy of quenched melts. K2 B4 O7 is created preferentially with an increasing content of B2 O3 in the mixture. The presence of metaborate anions, however, was not detected, most probably due to its low concentration in the melt. The density of the system KF–B2 O3 was measured by Chrenková and Daneˇ k (1992c). The density course in this system shows a minimum at 20 mole % B2 O3 . The concentration dependence of the molar volume at 1100 K was described by the second order polynomial V cm3 ·mol−1 = 30.14 + 23.33xB2 O3 −47.23xB2 2 O3

(2.85)

Differentiating Eq. (2.82) according to xB2 O3 , and inserting into the equation VB2 O3 = V + xKF

∂V ∂xB2 O3

(2.86)

we obtain the equation for the partial molar volume of B2 O3 at the temperature of 1100 K 2 VB2 O3 cm3 ·mol−1 = 6.384 + 47.225xKF

(2.87)

As it follows from this equation, for xKF → 1, the partial molar volume of B2 O3 at 1100 K attains the value VB2 O3 = 53.61cm3 ·mol−1 . This value is substantially higher than the molar volume of pure B2 O3 , VB2 O3 = 44.62 cm3 ·mol−1 , which indicates the

Main Features of Molten Salt Systems

75

formation of larger ions. The formation of K2 B4 O7 and KBF4 according to equation (2.84) was confirmed by Makyta (1993) and Chrenková and Daneˇ k (1992b) using the X-ray diffraction and IR spectroscopic analyses of quenched melts. The IR spectroscopic and X-ray diffraction investigation of the interaction of components in the system KF–B2 O3 was carried out by Babushkina et al. (2000). Samples of the quenched melts were taken for measurement. The wave numbers of the observed bands in the IR spectra of investigated samples are given in Table 2.8. As it can be shown from the table, up to 15 mole % KF, the IR spectra of samples contain only bands of pure B2 O3 . The addition of KF obviously does not cause any substantial change in the structure. In the IR spectra of samples with 30 mole % KF and more, new well-defined bands can be observed, which may be attributed to the vibration modes of different species. The addition of KF to the B2 O3 melt and the successive increase in its concentration should lead to the formation of distorted tetrahedral oxofluoroborates linked in boroxol rings and chain fragments composing a 3D network. On the other hand, the addition of KF to the B2 O3 melt initiates the depolymerization of the 3D network and the disintegration of the boroxol rings under formation of smaller fragments. The structure of possible species present in the KF–B2 O3 melts are demonstrated schematically in Figure 2.30. The formation of the [BO3 F]4− tetrahedrons in the concentration range up to 70 mole % NaF in the system NaF–B2 O3 was suggested by Maya (1977), about what indicates the bands at 770 and 1000 cm−1 in the IR spectra. However, these bands may also be attributed to the formation of other tetrahedral species, e.g. BF− 4 or [BOn F4−n ]. From the IR spectra of the system KF–B2 O3 , it can be seen that by the addition of more than 30 mole % KF to molten B2 O3 the intensity of all the bands increased and attained the maximum at the molar ratio n(KF):n(B2 O3 ) = 2:1. At this concentration, for every Table 2.8. Assignment of the wave numbers of observed bands in the IR spectra of the quenched samples of the system KF–B2 O3 to the individual structural species

xKF xB 2 O 3 Wave number cm−1 [BO3 ]3− in boroxol rings 0.00 0.05 0.10 0.15 0.30 0.50 0.70 0.85 0.90 0.95 1.00

1.00 0.95 0.90 0.85 0.70 0.50 0.30 0.15 0.10 0.05 0.00

700, 1450 700, 1450 700, 1450 700, 1370 710, 1350 550, 750, 1330 550, 730, 1340 550, 730, 1330 1350

[BO3 F]4− and [BO2 F2 ]3− distorted tetrahedrons

920, 1050 950, 1100 900, 1050 900, 1050 900, 1050 850, 1050 without bands

–O– bridges 1250 1250 1250 1250 1230 1270 1260 1260 1250 1250

76

Physico-chemical Analysis of Molten Electrolytes

  F− O − O −   \ − O − B〈O − B〈 O  + KF → − O − B 〈O − B〈 O  + KF →     O − B〈 O − B〈  O −  O −  ∞ ∞ − 1  [B3O 6 ]3nn−

[BO 3 ]3nn− , [BO 3 F] 4−

D 3h

→

D 3h

C 3v

F− − F− | F \ O − B〈 −O− B〈 O O − B〈 O− [BO 3 ]3nn− , [BO 3 F] 4− , [BO 2 F2 ]3− D 3h

C 3v

−

− O O − − 〉 〈 B O B − O O−

pyroborate, [B 2 O 5 ]4−

C 2v

−

O− O − B〈 − O

orthoborate, [BO 3 ]3−

F F F〉 B〈 F  

−

[BF4 ]− , Td

Figure 2.30. Disintegration of the 3D network of B2 O3 by addition of KF and structure of the possible species in the KF–B2 O3 melt.

atom of boron, one supplementary atom of fluorine is introduced, which theoretically enables boron to be fully in four-fold coordination. However, at this molar ratio of the components, the diffraction lines of potassium fluoride could be already observed, which would indicate the formation of all the possible structures containing boron and fluorine. At the next KF addition to B2 O3 (>70 mole % KF), no new bands or vanishing of the existing ones in the IR spectra could be observed, which refers to the steadiness of the ensemble of oxofluoroborate species in this concentration region. The phase diagram of the system LiF–KF–B2 O3 up to 30 mole % B2 O3 was determined by Chrenková and Daneˇ k (1992b). In the region of the primary crystallization of LiF, a liquid miscibility gap was found to exist at compositions from 5 to 23 mole % B2 O3 in

500

77

0 80

700

600

500

0 60

0 70

0.1

0 80

6

0.9

0.2

83

0.8

) O3 B2 x(

x (K F)

Main Features of Molten Salt Systems

KF

0.1

0.2

0.3

0.4

0.5 x (LiF)

0.6

0.7

0.8

0.9

LiF

Figure 2.31. Investigated part of the phase diagram of the system LiF–KF–B2 O3 .

the LiF–B2 O3 boundary system and up to 12 mole % KF. The reaction of components according to the equation 2LiF(l) + 6KF(l) + 7B2 O3 (l) = Li2 B4 O7 (l) + 2K2 B4 O7 (l) + 2BF3 (g) r G01200 K = −136.5 kJ·mol−1

(2.88)

was supposed to take place in the ternary system LiF–KF–B2 O3 . The phase diagram of this system is shown in Figure 2.31. The phase equilibrium study in the system LiF–KF–B2 O3 –TiO2 was performed by Chrenková and Daneˇ k (1992b). Owing to the low solubility of titanium oxide in alkali metal halides, the phase equilibrium study was performed in the cross-section with a constant content of 5 mole % TiO2 only, since the addition of TiO2 into molten LiF or the molten LiF–KF mixtures, insoluble Li2 TiO3 precipitates immediately from the melt. The solubility of TiO2 increases significantly when B2 O3 is present. However, at concentrations of LiF higher than 50 mole %, precipitation of Li2 TiO2 occurs anyway. The phase equilibrium in the ternary reciprocal system Li+ , K+ //F− , [TiO3 ]2− was studied by Nikonova and Berul (1967). Also this system was studied up to approximately 40 mole % LiF only, since above this, the LiF content Li2 TiO3 precipitates from the melt. It was found that the system KF–Li2 TiO3 is the stable diagonal of the above-mentioned reciprocal system. This could also be confirmed by the calculation of the standard Gibbs energy of the metathetical reaction 2KF(l) + Li2 TiO3 (l) = 2LiF(l) + K2 TiO3 (l) r G01200K = 32 kJ·mol−1

(2.89)

78

Physico-chemical Analysis of Molten Electrolytes

Reactions of titanates with alkali metal fluorides and the phase diagrams of the ternary 2− + + − reciprocal systems Li+ , Na+ // F− , TiO2− 3 and Li , K // F , TiO3 were investigated by Sigida and Belyaev (1957) and Belyaev and Sigida (1957), respectively. From the results of these works, it follows that Li2 TiO3 is soluble in NaF, KF, Na2 TiO3 and K2 TiO3 , but insoluble in LiF. However, in systems based on LiF, the addition of Na2 TiO3 or K2 TiO3 may cause precipitation of Li2 TiO3 from the melt. The phase diagram of the ternary reciprocal system Na+ , K+ // F− , TiO2− 3 was studied by Sholokhovich (1955). Two binary compounds, Na2 TiO3 ·K2 TiO3 and (Na2 TiO3 )3 · 2NaF, are formed in this system. The authors qualify this ternary reciprocal system as irreversible. The dissolution of M2 TiO3 (M = Li, Na, K) in molten NaF and KF and the ionic structure of the melts were studied using the cryoscopic method by Makyta and Zatko (1993). From the measurement, it follows that the number of new particles created in the melt by introducing one molecule of alkali metal titanate is 1 for systems with a common cation, and 3 for reciprocal systems. The proposed mechanism of dissolution explains the dissolution of Li2 TiO3 in NaF and KF and its precipitation from systems containing LiF as a solvent. The explanation was based on the comparison of the calculated (α calc ) and experimental (α exp ) values of the degree of conversion of the exchange reaction α 2MF + N2 TiO3 −→ 2NF + M2 TiO3

(2.90)

where α is the degree of conversion. The calculation confirmed the presumption that reaction (2.1.7.4.–15) runs in the region of the diluted solution quantitatively. The system (LiF–NaF–KF)eut –KBF4 –B2 O3 is a part of the (LiF–NaF–KF)eut – K2 TaF7 –KBF4 –B2 O3 –Ta2 O5 system, where melts were proposed as electrolytes in the electrochemical synthesis of tantalum diboride by Polyakova et al (1998, 1999). The phase equilibrium in the ternary systems NaF–NaBF4 –B2 O3 and KF–KBF4 – B2 O3 were studied by Selivanov (1960). Both are simple eutectic systems. However, Maya (1977) later found that in the system NaF–NaBF4 –B2 O3 , at low oxide contents the compound Na3 B3 O3 F6 is formed. In the mixture containing up to 33.3 mole % B2 O3 , the ternary compound Na2 B3 O3 F5 is formed. In the ternary system, KF–KBF4 – B2 O3 , the cross-section KF–B3 O3 F3 up to 60 mole % B3 O3 F3 was studied by Andriiko et al. (1988). In this system, the ternary compound K3 B3 O3 F6 , melting congruently at 560◦ C, is formed. The formation of oxofluoroborate species as a function of oxygen content in fluoride solvents was investigated using Raman and IR spectroscopy by von Barner et al. (1999). For n0 /nB mole ratios up to 0.4, it is suggested that [B2 OF6 ]2− with the B–O–B bridge is formed in equilibrium with the BF− 4 complex. In the samples with oxide to boron mole ratios near one, the ion B3 O3 F3− with a six–membered non-planar B3 O3 6 ring of alternating B and O atoms and with two fluorine atoms coordinated to each boron atom in a tetrahedral configuration, could be identified.

Main Features of Molten Salt Systems

79

The density, viscosity, X-ray diffraction analysis, and IR spectroscopy of quenched melts of the system (KF–LiF–NaF)eut –KBF4 –B2 O3 were measured by Chrenková et al. (2003b). From the results, it follows that in the binary system FLINAK–KBF4 , no new compounds are formed. However, the BF− 4 anion partially decomposes due to the high polarization ability of the Li+ cation. In the system FLINAK–B2 O3 , the components react under the formation of potassium tetraborate (K2 B4 O7 ), and volatile boron trifluoride (BF3 ), or potassium tetrafluoroborate (KBF4 ). The last compound was, however, not observed in the IR spectra. According to Maya (1977), Andriiko et al. (1988), and von Barner et al. (1999), there are two principal compounds originating in the ternary system FLINAK–KBF4 – B2 O3 . The compound K3 B3 O3 F6 originates at the molar ratio nB /n0 ≈ 1, while the compound K2 B2 OF6 is formed at nB /n0 ≥ 2. The presence of these compounds in the system FLINAK–KBF4 –B2 O3 was also confirmed by Chrenková et al. (2001). This is evident both from the X-ray diffraction pattern and the IR spectra of the investigated samples. In Figure 2.32 (according to Chrenková et al. (2001)), the IR spectra of samples with different nB /n0 ratio are compared with those of the pure compounds K2 B2 OF6 and K3 B3 O3 F6 . From Figure 2.31, it follows that at the molar ratio nB /n0 = 1, predominantly 2− the anion B3 O3 F3− 6 is formed, while at the ratio nB /n0 = 3, only the anion B2 OF6 is present. At the molar ratio nB /n0 = 2, both the species are present in the melt. 2.1.8. Systems containing jumping electrons

Electronic conduction in inorganic melts can occur when atoms of the same kind in different oxidation states are present. Such systems exhibit an increased electrical conductivity with an exponential character of its temperature dependence, caused by a diffusion-like motion called hopping mechanism. The hopping mechanism is characterized by low mobility at elevated temperatures and the charge carrier is termed as a small-polaron. The mobility of the small-polaron is much lower compared to that of the carrier in a broad semi-conductor band. At low temperatures, the small-polaron moves by Bloch-type band motion, while at elevated temperatures it moves by thermally activated hopping mechanism. Holstein (1959), Friedman and Holstein (1963), Friedman (1964) performed the theoretical calculations of small-polaron motion and showed that the temperature dependencies of the small-polaron mobility in the two regimes are different. In the high-temperature hopping regime, the electrical conductivity is thermally activated and it increases with increasing temperature. As shown by Naik and Tien (1978), its temperature dependence is characterized by the following equation κ0 Em κ(T ) = 3/2 exp − kT T

(2.91)

80

Physico-chemical Analysis of Molten Electrolytes

Relative transmitance

B2OF6–

B2OF6–

1

B2OF6–

B2OF6–

2 KF

BF4–

3 4 KF KF

5 KF

B3O3F63– B3O3F63– B3O3F63– B3O3F63–

1600

1400

1200

1000

800

600

400

Wavenumber(cm–1) Figure 2.32. Infrared spectra of quenched samples in the system (LiF–NaF–KF)eut –KBF4 –B2 O3 . 1 – K2 B2 OF6 ; 2 – nB /n0 = 3; 3 – nB /n0 = 2; 4 – nB /n0 = 1; 5 – K3 B3 O3 F6 .

where κ0 is a constant, Em is the activation energy for mobility, and T is the thermodynamic temperature. In this case, the charge of the carrier is localized at a specific site, for example on a cation, and its movement under the influence of electric field occurs in discrete leaps from one site to another. The hopping mechanism is expected at temperatures exceeding 500◦ C. In inorganic melts, the hopping mechanism is caused by the presence of “free” electrons, which are responsible for the partial electronic conductivity. “Free” electrons originate in the melt, when cations of the same kind in two oxidation states are present and the electrons jump between the two ions, being in different oxidation states. Such a

Main Features of Molten Salt Systems

81

“localized” group, i.e. the electron and the entrapping atomic displacement, denoted as the small-polaron, has only a low mobility and it can be easily described on the basis of the diffusion theory. This phenomenon was observed e.g. in metal–metal halide systems and in the system CuCl–CuCl2 . The above mechanism for the charge transport in melts was supposed by Rice (1961) and elaborated by Raleigh (1963) to explain the exponential conductivity increase with increasing metal content in the system Bi–BiCl3 . In this case the electron is jumping between Bi+ and Bi3+ ions. Rice (1961) and Raleigh (1963) supposed that the concentration of electrons is proportional to the concentration of cations in the lower oxidation state. Such a condition is well fulfilled in metal–metal halide systems in the range of high concentrations of metal halide (when the metal is a minor component). However, in systems with comparable concentrations of both the cations, the situation is somewhat different. An electron can jump only when an electron donor has an electron acceptor in its neighborhood. The probability that such an acceptor is available is equal to the product x(Mex+ )·x(Me(x+1)+ ). The exponential character of the temperature dependence of electrical conductivity is due to the fact that the concentration of cations in lower oxidation state increases with increasing temperature, which consequently increases the jump probability of the electron. Some examples of electronic conductivity in the melt will be described in the following chapters. 2.1.8.1. Metal–metal halide systems

Properties of the metal–metal halide systems have been studied since the first Davy’s observations of colored melts near the cathode at the electrolysis of alkali metal hydroxides. However, already the knowledge of these molten systems was substantially improved 50 years ago by the pioneering work of Bredig et al. (1958), who investigated the phase equilibria in a number of alkali metal–alkali metal halide and earth alkali metal–earth alkali metal halide systems. Further interest was also aroused in the case of •

• • •

rare metal–rare metal halide systems investigated by Mellors and Senderoff (1959), Eastman et al. (1950), Druding and Corbett (1961), Polyachenok and Novikov (1963), and McCollum et al. (1973), the bismuth–bismuth chloride system, whose phase diagram was investigated by Yosim et al. (1959, 1962) and Hoshino et al. (1979), aluminum–aluminum chloride, zinc–zinc chloride, and cadmium–cadmium chloride systems studied by Elagina and Palkin (1956) and Palkin and Belousov (1957), galium–galium chloride and indium–indium chloride systems investigated by Palkin and Ostrikova (1964), Chadwick et al. (1966), Chernykh and Safonov (1979), and Fedorov and Fadeev (1964).

The investigation of these systems was initiated by their importance in the electrolytic metal production and material science, their potential use in fuel cells, nuclear power

82

Physico-chemical Analysis of Molten Electrolytes

plants, etc. A short review of the metal–molten salt systems with the emphasis on the electrochemical properties of the dissolved metal was given by Haarberg and Thonstad (1989). In some systems there is a total miscibility between metal and salt, in others, the metal solubility. All the phase diagrams are characterized by the lowering of the melting point of the salt when the metal is added. This phenomenon is indicative of true solutions. Several systems exhibit a region with two immiscible liquid phases, i.e. “the miscibility gap”. Systems with miscibility gap show positive deviation from Raoult’s law, i.e. the activity coefficient of the salt is larger than unity. Above a certain temperature, which is called “the critical temperature of miscibility” or above the “consolute temperature”, salt and metal are completely miscible at all compositions. In Figure 2.33, the phase diagrams of the KX–K (X = F, Cl, Br, I) and RF–R (R = Li, Na, K, Rb, Cs) systems according to Bredig et al. (1958) are shown. The melting points of the metals are always lower than those of the respective salts. The characteristic feature of these systems is the formation of a more or less extended miscibility gap. In the potassium–potassium halide systems, the concentration extent of immiscibility increases

904°

900 LiF – Li

KF – K

850

1200

849°

NaF – Na

800

790°

1000

KCI – K 751.5°

750

728° K Br – K 717° 708°

700

KF – K

800

KI – K

RbF – Rb

658.5°

650 CsF – Cs

0

20

40 60 Mol. % K

80

100

600 RF

25

50 75 Mol. % Metal

R

Figure 2.33. Phase diagrams of the systems KX–K (X = F, Cl, Br, I) and RF–R (R = Li, Na, K, Rb, Cs) according to Bredig et al. (1958).

Main Features of Molten Salt Systems

83

with the increasing ionic radius of the anion, while in the alkali metal fluoride–alkali metal systems, it decreases with the increasing ionic radius of the cation, vanishing in the CsF–Cs system. In principle, the occurrence of the miscibility gap is caused by the polarization ability of ions and the magnitude of the repulsing forces between ions of the same kind. In the KX–K systems the repulsive forces between K+ ions are, however, shielded by anions of different size, which influences the width of the miscibility gap. The phase diagrams of the MeCl2 –Me (Me = Ca, Sr, Ba) systems according to Bredig et al. (1958) are shown in Figure 2.34. While the immiscibility in these systems extends almost over the whole concentration range, the upper critical temperature decreases from calcium to barium. This effect is coherent again with the magnitude of the repulsion forces between Me2+ cations. Figure 2.35 shows the phase diagrams of the systems MeF2 –Me (Me = Ca, Ba) according to Bredig et al. (1958). There is no miscibility gap in these systems; however, the tendency to admixture is evident. Obviously the repulsion forces between Me2+ cations are not so strong as to create a miscibility gap, since they are shielded by the F− anions. Finally, in Figure 2.36, the phase diagrams of the system BiCl3 –Bi according to Yosim et al. (1959, 1962) and of the system CeCl3 –Ce according to Mellors and Senderoff (1959) are shown. Wide regions of immiscibility are characteristic for these systems, which are due to the very strong repulsive forces between Bi3+ and Ce3+ cations, respectively. When a metal dissolves in a pure molten salt, a color change is often observed. The intensity of the color often increases with an increasing concentration of the dissolved metal. Solidified melts with dissolved metal are usually gray due to finely

1200 1200

Liquid

1200

Liquid

1000 1100

1000

Two liquids Liq.

Two liquids Two liquids

1000 800 900 825°

800 773°

800

828°

767°

600

700 CaCI2 20 40 60 Mol. %

80

Ca

Sr CI2 20 40 60 Mol. %

80 Sr

600 BaCI2 20

40 60 Mol. %

80 Ba

Figure 2.34. Phase diagrams of the systems MeCl2 –Me (Me = Ca, Sr, Ba) according to Bredig et al. (1958).

84

Physico-chemical Analysis of Molten Electrolytes

1400

1400

Liquid

Liquid

1200

1300 CaF2 ss + Liquid

1200

1000

1100 1000

800

900

730°

800

CaF2 + Ca

CaF2

20

40 60 Mol %

80

Ca

BaF2 20

40

60

80

Ba

Figure 2.35. Phase diagrams of the systems MeF2 –Me (Me = Ca, Ba) according to Bredig et al. (1958).

800 Liq.2

700 Liq.1

600

850

Liq.1 + Liq.2

500

Liq.

800

400

Two liquids

802°

BiCl3 + Liq.1

300

Liq.1 + BiCl

750

Liq.2 + Bi

200 100

Liq.2 + BiCl

BiCl + Bi

BiCl3 + BiCl

700

0 0 BiCl3

20

40

60 Mol. %

80

100 Bi

CeCI3

20

40 60 Mol. %

80

Ce

Figure 2.36. Phase diagrams of the systems BiCl3 –Bi according to Yosim et al. (1959, 1962) and CeCl3 –Ce according to Mellors and Senderoff (1959).

Main Features of Molten Salt Systems

85

dispersed metal. Corbett and von Winbush (1955), Corbett et al. (1957, 1961) studied several systems and observed color changes after the addition of metal to the pure molten salt. Transport of subhalides from the melt into the gas phase was also examined. The next characteristic feature of the metal–metal halide systems is the partial electronic conductivity caused by the presence of the same metallic atoms in two oxidation states. In general, the conductivity of the melt is given by the equation κ= zi F u i c i (2.92) i

where ci is the molar concentration of the conducting particles (ions and electrons) with charge zi , ui is their mobility, and F is the Faraday constant. In the low concentration region of the metals of the MX–M melts, the total conductivity can be divided into the ionic and electronic conductivities, respectively. The conducting particles are cations and anions, forming the ionic part of the total conductivity, and electrons, forming the electronic conductivity. Equation (2.92) can thus be written in the form

κ = κion + κel = F zM+ uM+ cM+ + zX− uX− cX− + ze− ue− ce−

(2.93)

According to Bredig (1964), the electron can jump only when a donor (M) has an acceptor (M+ ) in its neighborhood. The probability that such an arrangement is available is in due proportion to xM0 xM+ . The molar concentration of electrons then should be equal to the expression ce − =

x M 0 xM + Vmelt

(2.94)

Experimental conductivity data in binary systems can be presented in different ways. (1)

The measured conductivity of the mixture is treated as the sum of the conductivity of the molten salt and that of the dissolved metal κMX−M = κMX + κM

(2)

(2.95)

The electrical conductivity is expressed in terms of equivalent conductivities λMX−M = xMX λMX + xM λM

(2.96)

where xMX and xM are equivalent fractions of molten salt and dissolved metal, respectively. The equivalent conductivity of the dissolved metal is then given by λM =

λMX−M −(1−xM )λMX xM

(2.97)

86

(3)

Physico-chemical Analysis of Molten Electrolytes

An alternative expression is

κMX−M VMX−M = xM λM + (1−xM )κMX VMX

(2.98)

where Vi are the molar volumes. The chosen presentation of experimental data depends on the purpose for which the data will be used. 2.1.8.2. Electronic conduction in aluminum electrolytes

The current efficiency in modern cells of aluminum electrolysis may exceed 95%. It is generally accepted that the major part of loss in current efficiency is due to the reaction between dissolved metal and electrolyte. Model studies by Ødegard et al. (1988) indicates that sodium dissolves in the electrolyte in the form of free Na, while dissolved Al is predominantly present as the monovalent species AlF− 2 . Any electronic conductivity is most likely associated with the Na species, which may form trapped electrons and electrons in the conduction band. Morris (1975) ascribed the loss in current efficiency during Al production to electronic conduction. In a theoretical and experimental study, Dewing and Yoshida (1976) subsequently maintained that the electronic conductivity was too low to account for the loss in current efficiency in industrial aluminum cells. However, the existence of electronic conduction in NaF–AlF3 melts was demonstrated later by Borisoglebskii et al. (1978) also. Several authors studied the solubility of Al in melts of the system NaF–AlF3 –Al2 O3 . In addition to dissolved aluminum, sodium also has to be considered, and the solubility depends on the composition of the melt, as shown in Table 2.9. Most authors agree that the solubility decreases with decreasing NaF/AlF3 ratio and increasing alumina content. Ødegard et al. (1988) performed an extensive study of the effect of several additives on the total aluminum solubility. The results were summarized by the equation

log(cAl ) = −1.825−0.5919/CR + 3429/T −3.39×10−2 cAl2 O3 /cAl2 O3 (sat) −2.41×10−2 cMgF2 −2.03×10−2 cCaF2 −2.49×10−2 cLiF

(2.99)

where CR is the molar NaF/AlF3 ratio and all concentrations are in mass %, T is the temperature in K, and cAl2 O3 (sat) is the saturation concentration of alumina. Reaction models were fitted to experimental data using the available activity data and the Temkin’s

Main Features of Molten Salt Systems

87

Table 2.9. Solubility of Al and Na in NaF–AlF3 –Al2 O3 melts NaF/AlF3

t (◦ C)

molar ratio 1 3 3 6 3 3 3 3 2.25 2.25 3 4

1000 1000 1000 1000 980 980 980 980 962 1000 1000 1000

Al2 O3

Solubility

References

(mass %)

Al (mass %)

Na (mass %)

Total as Al (mass %)

sat. sat. sat. sat. 2 4 6 8 sat. 0 sat. sat.

– 0.065 – – 0.06 0.05 0.04 0.03 – – – –

– 0.090 – – 0.12 0.09 0.08 0.06 – – – –

0.075 0.10 0.14 0.33 0.11 0.09 0.07 0.05 0.054 0.076 0.083 0.090

Thonstad (1965); Ødegard et al. (1988); Yoshida et al. (1986).

model. Near the cryolite composition, two most probable dissolution reactions were considered Al(l) + 3NaF(l) = AlF3 (diss) + 3Na(diss)

(2.100)

+ 2Al(l) + AlF3 (diss) + 3NaF = 3AlF− 2 + 3Na

(2.101)

+ The presence of monovalent aluminum in the form of AlF− 2 rather than Al or AlF has also been suggested by Saget et al. (1975) and Yoshida et al. (1986) to explain the concentration dependence on the solubility. Thonstad and Oblakowski (1980) demonstrated that the migration of the dissolved metal species such as AlF− 2 may take place in cryolite-alumina melts. Hence a certain activity of sodium is established at the metal/electrolyte interface. The excess electrons associated with dissolved sodium likely cause electronic conduction, which may limit the obtainable current efficiency. Earlier results of electronic conductivity in the molten system Na3AlF6 –Al2 O3 (sat)–Al at 1000◦ C obtained by Haarberg et al. (1993) were also reproduced with high accuracy later by Haarberg et al. (2002). The total conductivity was determined to be 3.11 S · cm−1 , while the ionic conductivity is 2.22 S · cm−1 . Assuming that the increase in conductivity is due to the electronic conduction, the electronic conductivity is 0.89 S · cm−1 . This corresponds to a transport number of electrons of 0.29 in the aluminum-saturated electrolyte. However, a gradient with respect to the dissolved metal near the cathode during industrial electrolysis cuts down the effect of electronic conduction. Haarberg et al. (2002) found that the electronic conductivity increases with an increasing NaF content and increasing temperature. The variation of the Na activity shows the same trend. This supports the

88

Physico-chemical Analysis of Molten Electrolytes

theory that localized electrons with relatively high mobility are formed in the melt due to the interaction between Al and the electrolyte to form dissolved Na. It is commonly accepted that due to the mixing of the electrolyte in industrial cells, the concentration gradients of dissolved metal are present only in the boundary layers at the cathode and the anode, as illustrated in Figure 2.37. Therefore, during electrolysis in cryolite–alumina melts there is a gradient in the transport number of electrons, te , across the cell te =

κe κe +κion

(2.102)

In a cell where ions and electrons migrate simultaneously, the current density of electrons, ie , is proportional to the electrochemical potential of electrons ηe ie =

κe F

∂ηe ∂x

(2.103)

The ionic current density, iion , is given by iion = −

κion F

∂µNa ∂ηe − ∂x ∂x

(2.104)

Anode 0 xg

Bath

xm L n0(e) nm (e)

Cathode nL(e)

Figure 2.37. Profile of electron concentration across the bath during Al electrolysis.

Main Features of Molten Salt Systems

89

where µNa is the chemical potential of Na in the electrolyte. The total current density, i, through the cell is i = ie +iion

(2.105)

Combining and rearranging Eqs. (2.102) – (2.105) yield the following expression for ionic current density,

iion =

i L

L

L tion dx −

µNa te dµNa

κion LF

0

(2.106)

µ0Na

where L is the inter-electrode distance (see Figure 2.36). The ionic current density during electrolysis can be determined from Eq. (2.106) using the obtained data for electronic conductivity in dependence on the Na activity. The current efficiency is given by the equation CE =

iion 100 i

(2.107)

This treatment is based on the assumption that there is no convection in the electrolyte. This is far from the situation during industrial electrolysis. However, the major contribution from electronic conduction originates in the diffusion layer near the cathode, which can be assumed to be stagnant. A detailed description of the mathematical approach and the results of measurement can be found in the literature (Thonstad and Oblakowski, 1980; Haarberg et al. 1991, 1993, 1998, 2002). 2.1.8.3. The sodium oxide–sodium vanadate system

In the combustion of fuels containing vanadium and alkali in gas turbines originate ionic melts as undesirable side products. These melts consisting mostly vanadium and sodium oxides lead to the formation of corrosive sediments on construction materials of turbines and steam pre-heaters, and form corrosion products on the heating surfaces. The layer of molten ash by its chemical and transport properties accelerates the corrosion process due to the transport of oxygen to the metal–melt interface, where the anodic oxidation of the metal proceeds. The electrons and corrosion products proceed in the opposite direction through the melt to the melt–gas interface, where the cathodic depolarization reaction takes place. Regarding the study of chemistry of this corrosion, the knowledge of the physico-chemical properties of the basic system V2 O5 –Na2 O appears to be of prime importance.

90

Physico-chemical Analysis of Molten Electrolytes

Vanadium pentoxide decomposes partially at melting according to the reaction 1 α V2 O5 ←→ V2 O4 + O2 2

(2.108)

The equilibrium of this reaction depends on the partial pressure of oxygen and temperature. On the basis of the experimentally determined temperature dependence of the equilibrium constant, Pantony and Vasu (1968) calculated the heat of reaction (2.108) and obtained the value 6.12 kJ·mol−1 . The theoretical value, however, calculated on the basis of the thermodynamic data is 1.86 kJ·mol−1 . The difference, 4.26 kJ·mol−1 , represents the partial molar heat of solution of V2 O4 in V2 O5 . The high value of the heat of solution indicates the formation of vanadyl-vanadates in the melt. This is obviously responsible for the properties of molten V2 O5 , which appears to be a semi-conductor of the type n. The electronic conductivity is stipulated by the increased concentration of V2 O4 in the quasi-crystalline structure of V2 O5 and its existence is also confirmed by a small change in the conductivity at the solid→liquid phase transition of V2 O5 . The addition of Na2 O or of other alkali and alkaline earth oxides to V2 O5 favors the dissociation of V2 O5 and consequently, also the formation of defects in its quasicrystalline structure, thus facilitating the transport of oxygen across the layer of molten ash. The compounds formed by V2 O5 and alkali metal oxides are known as “vanadium bronzes” and their crystal structure is close to that of V2 O5 . Vanadium bronzes, Nax V2 O5 , where x varies in the range 0.13 < x > 0.31, represent a series of non-stoichiometric compounds with univalent metal M atoms localized in the distorted structure of V2 O5 . The valence electrons of M metals are trapped by the vanadium atoms of the V2 O5 matrix so that V(IV) atoms are in tetragonal distortion with a moderate spin-orbital coupling. The trapped electron is equivalent to the small polaron of Holstein (1959). The electrical conductivity of such a system is exponentially dependent on temperature. During solidification, vanadium bronzes release oxygen, whereas they bind it during melting. According to Flood and Sørum (1946), besides the non-stoichiometric compound Nax V2 O5 , another compound, Na0.9 V3 O8 , should be formed. On the other hand, according to Ozerov (1957) and Illarionov et al. (1957), in the above system there exists a stable compound Na0.33 V2 O5 and yet another compound, which could not be unambiguously identified. Reisman and Mineo (1962), in the study of interaction of components of the system V2 O5 –Li2 O determined that three compounds are formed in this system: the incongruently melting compounds 2Li2 O·17V2 O5 and 2Li2 O·5V2 O5 and the congruently melting lithium metavanadate Li2 O·V2 O5 . The existence of alkali metal metavanadates of the type MeVO3 was also determined in systems containing Na2 O and K2 O. Reisman and Mineo (1962) assumed that also in the system V2 O5 –Na2 O, most probably the

Main Features of Molten Salt Systems

91

incongruently melting compound 2Na2 O·5V2 O5 would be formed. The existence of an analogous incongruently melting compound 2K2 O·5V2 O5 was determined by Illarionov et al. (1956). Figure 2.38 shows the phase diagram of the system V2 O5 –NaVO3 measured by Daneˇ k et al. (1973). It was confirmed that two compounds are formed in this system: the congruently melting 2Na2 O·17V2 O5 and the incongruently melting 2Na2 O·5V2 O5 . At freezing, close to the phase transition temperature of 724◦ C the compound 2Na2 O·17V2 O5 releases oxygen, most probably according to the scheme 2Na2 O·17V2 O5 = 17Na0.235 V2 O5 + O2

(2.109)

The originating compound belongs to the group of vanadium bronzes as stated by Flood and Sørum (1946). At melting, reaction (2.109) proceeds in the opposite direction, since below the melting point the partial pressure of oxygen is evidently higher than above the melting point. The compound 2Na2 O·17V2 O5 is analogous to the compound 2Li2 O·17V2 O5 identified by Reisman and Mineo (1962). There is a narrow area of solid solutions of V2 O5 in the vanadium bronze Na0.235 V2 O5 , the existence of which was also assumed by Illarionov et al. (1957). The identification of the compound 2Na2 O·5V2 O5 confirmed the assumption of its existence presented by

800

750

650

550

2Na2O.5V2O5

600

2Na2O.17V2O5

t(°C)

700

500

V2O5

0.2

0.4

0.6 x(NaVO3)

0.8

NaVO3

Figure 2.38. Phase diagram of the system V2 O5 –NaVO3 according to Danˇek et al. (1973).

92

Physico-chemical Analysis of Molten Electrolytes

Reisman and Mineo (1962), who determined the analogous compound 2Li2 O·5V2 O5 . On the other hand, no compound of this type was found by Illarionov et al. (1957). The physico-chemical properties, i.e. density, viscosity, and electrical conductivity, of the system V2 O5 –NaVO3 were measured by Daneˇ k et al. (1974). In Figure 2.39, the density isotherms of the system V2 O5 –NaVO3 mixtures are presented. From the course of the isotherms it is evident that the density changes only slightly with increasing concentrations of Na2 O in the melt. On the other hand, the temperature dependence of the density of mixtures close to the composition corresponding to vanadium bronze Na0.235 VO3 , seems to be of definite interest. The coefficient of thermal expansion of this melt appears to be markedly lower than of any other mixtures over the entire investigated concentration range. This indicates a qualitative difference in the structure of the melt. It can be assumed that the combination of V2 O5 groups in vanadium bronze is more pronounced, which leads to a relative decrease in the free volume of the melt and is indicated by the decrease in the coefficient of thermal expansion. This assumption was also confirmed by the course of the viscosity isotherms of the molten system V2 O5 –NaVO3 , which are shown in Figure 2.40. The maximum on the viscosity isotherms corresponds to the formation of vanadium bronze Na0.235 VO3 . Such a maximum on the viscosity isotherms can be explained only by the formation of big structural entities in the melt. The isotherms of the electrical conductivity are presented in Figure 2.41. Also in this case, the maximum on the isotherms corresponds to the composition of vanadium bronze. The coincidence of these maximums on the isotherms of both the viscosity and the

2.6

ρ(g.cm−3)

2.5

2.4

2.3

2.2

V2O5

0.2

0.4 0.6 x(NaVO3)

0.8

NaVO3

Figure 2.39. Isotherms of the density of the molten V2 O5 –NaVO3 mixtures. – 750◦ C; – 800◦ C; – 850◦ C; – 900◦ C.

◦ – 650◦ C; • – 700◦ C;

Main Features of Molten Salt Systems

93

100

η(mPa.s)

80

60

40

20

0

V2O5

0.2

0.4 0.6 x(NaVO3)

0.8

NaVO3

Figure 2.40. Isotherms of the viscosity of the molten V2 O5 –NaVO3 mixtures. – 750◦ C; – 800◦ C; – 850◦ C; – 900◦ C.

◦ – 650◦ C; • – 700◦ C;

2.5

κ(S.cm−1)

2.0

1.5

1.0

0.5

0.0

V2O5

0.2

0.4 0.6 x(NaVO3)

0.8

NaVO3

Figure 2.41. Isotherms of the electrical conductivity of the molten V2 O5 –NaVO3 mixtures. • – 700◦ C; – 750◦ C; – 800◦ C; – 850◦ C; – 900◦ C.

◦ – 650◦ C;

electrical conductivity can be explained only by the existence of “free” electrons, which are responsible for the partial electronic conductivity of these melts. As the existence of neutral sodium atoms in the presence of V(V) atoms is highly improbable, the partial electronic conductivity can be explained by transfer of the valence electron from the alkali metal to the free d-orbital of the vanadium atom. A direct evidence of the existence

94

Physico-chemical Analysis of Molten Electrolytes

2.0

κ (S.cm−1)

1.5

1.0

0.5

0.0 650

700

750

800 t(°C)

850

900

950

Figure 2.42. Polytherms of the electrical conductivity of the molten V2 O5 –NaVO3 mixtures. – V2 O5 ; ◦ – 10% NaVO3 ; – 18% NaVO3 ; – 21% NaVO3 ; • – 33% NaVO3 ; – 50% NaVO3 .

of V(IV) atoms, and thus of the formation of the small polaron inside the array of V2 O5 was delivered by the EPR measurements carried out by Gendell et al. (1962) in lithium bronzes. The temperature dependencies of the electrical conductivity of the molten V2 O5 –NaVO3 mixtures are shown in Figure 2.42. The exponential character of electrical conductivity polytherms of mixtures in the concentration range 10–50 mole % NaVO3 indicates the presence of the small polaron in these melts. 2.1.8.4. Alkali metal sulfates containing cobalt sulfate

In the combustion of fuels containing sulfur and alkali in gas turbines, molten ash deposits originate, which contains alkali sulfates. Sulfur trioxide, which has been identified by Cutler (1971) as the main corrosive agent in molten alkali sulfates, is easily soluble in sulfate melts, acts as an oxidizing agent, and increases the solubility of the oxidation layers in the sulfate melts. Umland and Voigt (1970) followed the influence of cobalt content in the metal on the mechanism and rate of corrosion in molten sulfates. CoSO4 produced by the corrosion of Co-containing steels and alloys, gives rise to a complex anion [Co(SO4 )2 ]2− , which shows an increased stability in a certain temperature region. Since Co can exist in more than one oxidation state, a stabile redox system originates on the respective phase interfaces according to the following reaction scheme Co ↔ Co2+ + 2e−

(2.110)

Main Features of Molten Salt Systems

95

Co2+ + O2− ↔ CoO

(2.111)

2− CoO+SO2− 4 + SO3 ↔ [Co(SO4 )2 ]

(2.112)

The originating complex anion enables the transfer of electrons from the metallic surface to the melt/gaseous phase interface according to the following scheme [Co(SO4 )2 ]2− ↔ [Co(SO4 )2 ]− +e−

(2.113)

According to Umland and Voigt (1970), the system Co(II)/Co(III) attains maximum stability at a temperature of approximately 750◦ C. Analogical sulfate–metallic anions originate also with iron; however, these are more stable at temperatures in the range 500–600◦ C. Matiašovský et al. (1973) measured the density, viscosity, and electrical conductivity of the ternary eutectic mixture of alkali metal sulfates, Li2 SO4 –Na2 SO4 –K2 SO4 , with the addition of NiSO4 and CoSO4 as the products of the sulfate corrosion. The viscosity versus temperature plots of the melts of the alkali metal sulfate ternary eutectic mixture with the addition of CoSO4 is shown in Figure 2.43. The addition of CoSO4 increases the viscosity of the ternary eutectics. The course of the polytherms of the viscosity with a higher content of CoSO4 is not monotonous. At the temperature 700◦ C anomalies on the polytherms could be seen, beginning at the concentration of 5 mole % CoSO4 and at 10 mole % the anomaly is strongly expressed. It can be expected that in the concentration range 2–5 mole % CoSO4 , besides the ionic the electronic conductivity also starts to

60 50

η(mPa.s)

40 30 20 10 0 550

600

650

700

750

t(°C) Figure 2.43. Viscosity polytherms of the alkali metal sulfate ternary eutectics with the addition of CoSO4 . eutectics with the addition of CoSO4 . ◦ – ternary eutectics; • – 2% CoSO4 ; – 5% CoSO4 ; – 10% CoSO4 .

96

Physico-chemical Analysis of Molten Electrolytes 2.8 2.7 2.6 κ (S.cm−1)

2.5 2.4 2.3 2.2 2.1 2.0 1.9 550

600

650

t(°C)

700

750

Figure 2.44. Conductivity polytherms of the alkali metal sulfate ternary ◦ – ternary eutectics; • – 2% CoSO4 ; – 5% CoSO4 ; – 10% CoSO4

participate in the charge transfer. Its share increases with increasing the concentration of CoSO4 . This explanation is in accordance with the exponential character of the course of the electrical conductivity polytherms of the ternary sulfate eutectics with the CoSO4 addition, which are shown in Figure 2.44. It could be accepted that the anomaly on the viscosity polytherms at 700◦ C and the exponential character of the polytherms of the electrical conductivity of the melts of the system Li2 SO4 –Na2 SO4 –K2 SO4 –CoSO4 is coherent with the formation of the complex [Co(SO4 )2 ]2− in the melt. The transport of electrons from the metal/melt interface to the melt gas interface in the sulfate corrosion is possible due to the presence of the stable redox system described by Eq. (2.113) in the melt. 2.1.8.5. Silicate systems containing iron oxides

The system CaO–FeO–Fe2 O3 –SiO2 is the basic system of metallurgical slags. The ratio of di- and trivalent iron in the melt depends on the oxygen partial pressure, temperature, and composition of the melt. When the melt is held at equilibrium with metallic iron in an inert atmosphere, only Fe2+ cations are present in the melt. Toropov and Bryantsev (1965) used these conditions in the measurement of the electrical conductivity of the MgO–FeO– SiO2 melts. Air is another atmosphere with defined oxygen partial pressure. In this case, the content of di- and trivalent iron depends on temperature and the melt composition. The electrical conductivity of the FeOy –CaO–SiO2 melts in air was measured by Dancy and Derge (1966) and Engell and Vygen (1968). The conductivity of melts in the atmosphere

Main Features of Molten Salt Systems

97

of pure oxygen, when only tri-valent iron was present in the melt was measured by Hirashima and Yoshida (1972), Lopatin et al. (1973), and Morinaga et al. (1975). In the investigation of properties of the CaO–Fex Oy –SiO2 melts in air, it is necessary to know the equilibrium composition of the melt as the function of the temperature and total melt composition. Larson and Chipman (1953) and Timucin and Morris (1970) studied the phase diagram of this system, where the equilibrium contents of di- and trivalent iron in the melt were determined in quenched samples. The presence of ions of the same kind in two oxidation states results in an increased electrical conductivity of the melt due to the contribution of the electronic conductivity caused by the electrons jumping between the two ions in different oxidation states. Daneˇ k et al. (1986) measured the electrical conductivity of the melts of the system CaO–FeO–Fe2 O3 –SiO2 in air atmosphere and in the temperature range of 1530–1920 K. The composition of samples lying in two cross-sections with constant molar ratios k1 = x(CaO)/x(SiO2 ) = 1 and k1 = x(CaO)/x(Fe2 O3 ) = 4 were chosen. The equilibrium compositions of the melts at the given experimental temperature were calculated according to the equation r = 1.7273−6.592×10−4 T /K + 0.223k1 + 0.116x (Fe2 O3 )

(2.114)

where r = x(Fe2 O3 )/[x(FeO)+x(Fe2 O3 )], T is the thermodynamic temperature, and x (Fe2 O3 ) is the molar fraction of Fe2 O3 in the weighed-in mixtures. Equation (2.114) was derived by Daneˇ k et al. (1986) using the data published by Larson and Chipman (1953) and Timucin and Morris (1970). The composition dependence of the conductivity of the investigated melts at 1723 K is shown in Figure 2.45. The conductivity of the investigated melts increases with increasing concentrations of calcium oxide, iron oxides, and temperature. The exponential increase of the conductivity with the increasing content of iron oxides indicates the presence of other conducting particles in the melt. Free electrons jumping from Fe(II) to Fe(III) atoms are suggested to contribute to the conductivity. Licˇ ko and Daneˇ k (1983) showed that the electrical conductivity of ionic silicate melts is proportional to the sum of the product of the mobility and the concentration of conductive particles. κ=

zi F ui (ci −ci0 )

(2.115)

i

where ci is the molar concentration of the conductive particle i with the charge number zi , ci0 is the molar concentration of that part of the cations which do not participate in the charge transfer, F is the Faraday’s constant, and ui is the mobility of the conductive

98

Physico-chemical Analysis of Molten Electrolytes

0.00 1.2

0.05

x(FexOy) 0.10

0.15

κ(S.cm−1)

1.0

0.8

1

0.6

2

0.4

0.2 0.3

0.4

0.5 x(SiO2)

0.6

Figure 2.45. The dependence of conductivity on composition of the investigated melts of the system CaO– FeO–Fe2 O3 –SiO2 at 1723 K. 1 – section k1 = x(CaO)/x(SiO2 ) = 1; 2 – section k1 = x(CaO)/x(Fe2 O3 ) = 4.

particle i. Licˇ ko and Daneˇ k (1983) also showed that in silicate melts with high SiO2 content (minimum 40 mole %) the charge is transported exclusively by cations. In the system CaO–FeO–Fe2 O3 –SiO2 , the contribution of electrons also has to be included. Equation (2.115) then can be written in the form 0 0 κ = F [2uCa2+ (cCa2+ −cCa 2+ ) + 2uFe2+ (cFe2+ −cFe2+ )

(2.116) 0 + 3uFe3+ (cFe3+ −cFe 3+ ) + ue− ce− ]

Equation (2.116) was solved using the multiple linear regression analysis. Concentrations of Ca2+ and Fe2+ were calculated from melt densities measured by Licˇ ko et al. (1985). In the calculation of the liquidus surface of CaSiO3 in the system CaO–Fe2 O3 –SiO2 , Daneˇ k (1984) stated that half of the Fe(III) atoms is coordinated tetrahedrally, i.e. they behave as network formers. This means that only the other half of the Fe(III) atoms, which are highly coordinated, behaving as a network modifier,

Main Features of Molten Salt Systems

99

can participate in the charge transport. The concentration of the conductive Fe3+ cations is then given by the equation cFe3+ =

xFe2 O3 Vmelt

(2.117)

where Vmelt is the molar volume of the melt and xFe2 O3 is the molar fraction of Fe2 O3 in the melt. The concentration of electrons in the melt was determined as follows. In the system with comparable concentrations of both Fe2+ and Fe3+ cations, the electron can jump only when a donor (Fe2+ ) has an acceptor (Fe3+ ) in its neighborhood. The probability that such an acceptor is available is equal to 2xFe2 O3 . The concentration of electrons is then equal to cFe3+ =

2xFe2 O3 xFeO Vmelt

(2.118)

The dependence of the electrical conductivity on the concentration of conductive particles in the system CaO–FeO–Fe2 O3 –SiO2 was calculated using the multiple linear regression analysis. The dependence takes the form κ = −A+B1 cCa2+ +B2 cFe2+ +B3 cFe3+ +B4 ce−

(2.119)

where Bi = zi F ui . The standard deviation of approximation was sd = 3×10−2 S · cm−1 . The calculated constants A and Bi at the temperatures of 1723 K and 1823 K are given in Table 2.10. Comparing Eqs. (2.116) and (2.119) it follows for A 0 0 0 A = B1 cCa 2+ +B2 cFe2+ +B3 cFe3+

(2.120)

Licˇ ko and Daneˇ k (1983) regarded the limiting concentration ci0 as the cations, which do not participate in the charge transport. Those cations were supposed to be fixed in Table 2.10. Constants A, Bi , mobility, and diffusion coefficients of the conductive particles in the CaO–FeO–Fe2 O3 –SiO2 melts at 1723 and 1823 K Parameter

T (K)

Ca2+

Fe2+

Fe3+

e−

Bi (S · cm2 · mol−1 )

1723 1823 1723 1823 1723 1823 1723 1823

38.6 41.9 2.0 2.2 1.5 1.7

35.1 38.4 1.8 2.0 1.4 1.6 0.61 0.53

32.7 35.6 1.1 1.2 0.56 0.61

2170 2050 225 212 334 315

µi · 104 (cm2 · s−1 · V−1 ) Di · 105 (cm2 · s−1 ) A(S · cm−1 )

100

Physico-chemical Analysis of Molten Electrolytes

larger structural units, clusters, which could be composed of silicate polyanions linked to cations by polar covalent bonds. In the system CaO–FeO–Fe2 O3 –SiO2 such bonding occurs especially in the case of Fe(III) atoms. As regards the other conductive particles, the tendency towards such behavior can be expected at Fe(II) and less at Ca(II) atoms. The limiting concentrations of Ca(II) and Fe(II) atoms in the investigated system were estimated from the value of the absolute term A. Provided that c0 2+ /c0 2+ = rFe2+ /rCa2+ , Fe Ca ∼ 6×10−3 mol· where ri are the ionic radii, the following values were obtained: c0 = Ca2+

0 −3 mol·cm −3 . These values correspond to approximately 15 ∼ cm−3 and cFe 2+ = 7.5×10 mole % CaO and 18 mole % FeO in the systems CaO–SiO2 and FeO–SiO2 , respectively. The mobility of the conductive particles, i.e. Ca2+ , Fe2+ , and Fe3+ cations and electrons, were calculated from the Bi coefficients in Eqs. (2.119) and (2.120). The calculation was carried out for 1723 and 1823 K. From the obtained mobility values, the corresponding values of the diffusion coefficients were calculated using the Nernst–Einstein equation. The calculated values are given in Table 2.10. The mobility and diffusion coefficients of the cations were found to decrease in the order uCa2+ > uFe2+ > uFe3+ , which can be explained by different forces acting between the cations and the non-bridging oxygen atoms of the SiO4 tetrahedrons. Cations with a higher value of the z/r ratio (z is the charge number and r is the radius) form stronger bonds between them and the non-bridging oxygen atom. Due to steric reasons, the coordination number of the cation becomes lower. Such a cation is stronger bound in its position and the Me–O bond becomes more covalent, which results in the decrease of cation mobility. The mobility of the electrons was found to be two orders higher than that of the cations. For electron jump, some fluctuation in the composition or a different arrangement of the ionic atmosphere of both the exchange cations is needed. Fe(II) and Fe(III) atoms present in silicate melts are differently coordinated and the distances from the nearest neighbors (i.e. the oxygen atoms) are also different. The distances between the exchange sites are relatively great and this is why for the jump frequency, values close to the mean value of the vibration frequencies of the thermal movement in the melt can be expected. The mobility of such electrons was defined by Raleigh (1963)

ue − =

e νR R 2 6kT

(2.121)

where e is the electron charge, νR is the jump frequency, and R is the mean distance between the exchange sites given by the relation R∼ =

V NA

1/3 (2.122)

Main Features of Molten Salt Systems

101

where V is the molar volume and NA is the Avogadro’s constant. From Eq. (2.121) the mobility and jump frequency of the electron can be calculated. For 1723 K and V ≈ 23cm3 ·mol−1 , one obtains νR ≈ 1.7×1013 s−1 , which is a feasible value comparable to the vibration frequency of thermal movement. The reciprocal frequency value is the mean lifetime of the given arrangement τ ≈ 6×10−14 s. The high degree of the symmetry of oxygen covers of both the exchange cations is indicated by the νR and τ values. Energy levels of the excited states of the electron in both oxidation states of Fe atoms are probably close to each other. 2.1.9. Systems of silicate melts

The structure of glass-forming oxides SiO2 , GeO2 , P2 O5 , B2 O3 , etc., is based on the linked AO4 tetrahedrons (SiO2 , GeO2 , P2 O5 ) or AO3 triangles (B2 O3 ). Førland (1955) concluded that the structure of molten SiO2 differs only little from that of cristobalite. The plausibility of this conclusion follows from the fact that the Si–O bonds are extremely strong. However, in spite of this, the melting enthalpy of SiO2 is only 9.58 kJ · mol−1 and the melting entropy is 4.8 J · mol−1 ·K −1 , which means that molten SiO2 has an arrangement very similar to that of cristobalite. The densities of molten SiO2 and cristobalite are nearly equal; the viscosity of molten SiO2 is high. SiO2 belongs to tectosilicates, its structure consists of SiO4 tetrahedrons linked by their apexes into a three-dimensional network. The structure contains only bridging oxygen atoms, i.e. those, bound to two neighboring central silicon atoms by covalent Si–O–Si bonds. Let us consider what happens with the structure when adding another oxide (e.g. MeO) to SiO2 . The situation can be expressed by the scheme |

|

|

|

|

|

|

|

− Si −O− Si −+MeO → − Si −O− Me2+ − O− Si −

(2.123)

Now, not all of the oxygen atoms are linked to the two central silicon atoms. This is due to the addition of oxygen atoms further, so that the n(O)/n(Si) ratio exceeds 2. In principle, the process can be characterized as the decrease in the number of bridging oxygen atoms accompanied by the simultaneous formation of the two-fold number of nonbridging oxygen atoms. With increasing the MeO content, the tectosilicate structural type changes to inosilicates (e.g. pseudo-wollastonite) through sorosilicates (e.g. akermanite) to the structural type of nezosilicates (e.g. dicalcium silicate). The dependence of the number of bridging oxygen atoms on the content of MeO is linear from tectosilicates to nezosilicates. Let us further consider the glass-forming ability of the MeO–SiO2 melt. Here “glass” means the product formed by cooling the melt into the solid without crystallization. The glass-forming ability of systems depends on the energy of the bonds in the cooling melt, which disrupt in the course of crystallization. For the two limiting cases

102

Physico-chemical Analysis of Molten Electrolytes

(tectosilicate–nezosilicate), therefore there are quite different conditions for glass formation. Crystallization of SiO2 requires disruption of the very strong Si–O bonds and a new regular arrangement corresponding to the crystalline state. The crystallization process is a reconstructive one, while in the case of nezosilicates, the disruption of Si–O bonds is not necessary (the crystallization mechanism is not a reconstructive one). Practice shows that crystallization of SiO2 does not virtually take place, while nezosilicates, on the other hand, crystallize very rapidly, so that one can assume a similar, even though irregular state of arrangement of the tetrahedrons in the melt when compared with the crystalline phase. Generally speaking, the ability to crystallize is indirectly proportional to the degree of polymerization, which in turn depends on the content and properties of the MeO oxides. In the silicate chemistry, the oxides are divided into three principal groups: (1) (2) (3)

Network forming oxides, which participate in the formation of the polyanionic network, i.e. GeO2 , P2 O5 , B2 O3 . Network modifying oxides, which do not participate in the formation of the polyanionic network, i.e. alkali metal oxides and earth alkali metal oxides. Amphoteric oxides, which may act both as network forming or network modifying oxides depending on the actual composition of the melt. Typical amphoteric oxides are Al2 O3 and Fe2 O3 .

There is, of course, no sharp boundary between individual groups. For instance, magnesium oxide can act also as a network forming oxide in basic melts with a relatively high content of MgO, when sufficient non-bridging oxide atoms are available. Magnesium oxide then forms MgO4 tetrahedrons, which may link two SiO4 tetrahedrons together. Such melts exhibit relatively high viscosity. Another characteristic feature of the silicate melts is the formation of two immiscible liquids in the region of high concentration of SiO2 . Such a behavior is observed, for example, in the systems of alkali metal silicates and earth alkali metal silicates. The chemistry of silicate melts is rather complicated and special monographs devoted to this problem can be found in the literature. Therefore, only the main characterization is presented here.

2.1.10. Systems of molten alkali metal borates

Using X-ray diffraction techniques, Zarzycki (1956) showed that vitreous and molten boron oxide are composed of BO3 triangles linked by their apexes into an irregular threedimensional network. Grjotheim and Krogh-Moe (1954) pointed out that the structure of boron oxide glass is similar to that of its hexagonal crystalline form and that it consists of two types of irregular BO4 tetrahedrons. The first one is a hybrid between triangular

Main Features of Molten Salt Systems

103

and tetrahedral configuration with the boron located much closer to the three oxygen atoms. The other is a distorted tetrahedron with various B–O bond lengths. The mean coordination number of boron in the B2 O3 melt, 3.1, as determined by Biscoe and Waren (1938), does not provide explicit evidence of any change in the boron coordination. In the binary glass-forming systems of alkali metal borates, it is possible to observe change in the trend of a number of physico-chemical properties in the concentration range of approximately 20 mole % of alkali metal oxide. This phenomenon is known in the literature as “boric acid anomaly”, and it is due to the change in the structure of the B2 O3 melt caused by the alkali metal oxide addition and is related to the ability of boron to change its coordination number. Krogh-Moe (1958, 1960) concluded that the change in the coordination number of boron from 3 to 4 takes place up to a content of 33 mole % of alkali metal oxide, which corresponds to the maximum concentration of 50% of four-coordinated boron. This assumption has been explicitly experimentally confirmed using the measurements of nuclear magnetic resonance carried out by Silver and Bray (1958) and Bray and O’Keefe (1963). These authors found out that within the concentration range of x = 0–30 mole % of alkali metal oxide the concentration of four-coordinated boron, N4 , may be quite accurately expressed by the relation N4 =

x 100−x

(2.124)

Equation (2.124) can be interpreted so that each oxygen atom added will change the coordination of two boron atoms from the triangular to the tetrahedral. As a result of this, there is no non-bridging oxygen atoms present within this concentration range. This fact has also been confirmed by the X-ray structural analyses of various crystalline borates, performed by Krogh-Moe (1958, 1960). The comparison of results of NMR studies, made by Silver and Bray (1958) and Bray and O’Keefe (1963), on the glass-forming ability of alkali metal borates, indicates that the degree of polymerization in alkali metal borate systems is much higher than in the silicate systems. On the basis of the above mentioned facts, one may reasonably expect the analogy between structures of the crystalline phase and liquid phase in the systems of alkali metal borates also. The structure of alkali metal borate melt can be imagined as a three-dimensional network of apex-joined BO3 triangles and BO4 tetrahedrons, where the cations are located in the free spaces. The properties of such melts will obviously depend on the number of boron atoms in the individual co-ordination, on the amount of alkali metal cations and bridging oxygen atoms, and also on the amount of nonbridging oxygen atoms at higher concentrations of alkali metal oxide. In the region of low content of the alkali metal oxide, each added oxygen atom of the alkali metal oxide changes the coordination of two boron atoms from 3 to 4. For instance, in the compound

104

Physico-chemical Analysis of Molten Electrolytes

Na2 O · 4B2 O3 there are two boron atoms in tetrahedral coordination, six boron atoms in triangular coordination, and all oxygen atoms are bridging. The above-described trend is maintained up to approximately 30 mole % of alkali metal oxide. Above this concentration, non-bridging oxygen atoms arise as the result of the reverse transition of some boron atoms from the tetrahedral into the triangular coordination. The number of boron atoms in the tetrahedral coordination decreases, approaching zero at 70 mole % of the alkali metal oxide. It should be, however, noted that the picture of the structure of the melt is only approximate and differences between the liquid and crystalline phase may occur. The chemistry of borate melts is described in detail in special monographs devoted to this problem. Thus the above mentioned short characteristics are given here only.

2.1.11. Systems of metallurgical slags

Pyrometallurgical metal production involves as a by-product, a slag, the amount of which often exceeds several times the produced metal. Slag is a poly-component system of metallic and non-metallic oxides forming compounds and molten mixtures, and containing some amounts of metals, sulfides, and gases. The main role of slag is to act as a collector of unwanted components of the charge and of impurities formed during the melting or refining of the metal. The correct choice of slag composition affects the total loss of the metal, its quality, energy demands, consumption of refractory lining, and thus in general, the overall economy and environmental aspects of the pyrometalurgical process. In the pyrometallurgical production of iron, steel, and non-ferrous metals, the so-called combined slags, or silicate slags with a high content of calcium oxide are frequently used. The base of these slags is the system, CaO–FeO–Fe2 O3 −SiO2 . Magnesium oxide is introduced into these slags by dissolution of the basic refractory lining. Aluminum oxide is present after the concentration melting of raw materials. Zinc oxide is introduced into the pyrometallurgical slags mainly in the processing of secondary raw materials. The properties of some of these slags were discussed in Section 2.1.9. In the middle of the 1970s, the process of continuous copper production was started in Japan. This process, using calcium–ferritic slags, was developed on the basis of the theoretical work of Yazawa (1977). The base of the calcium–ferritic slags is the system CaO–FeO–Fe2 O3 that in a real pyrometallurgical processes contains as admixtures SiO2 , Al2 O3 , MgO, ZnO, and in the pyrometallurgical production of copper as Cu2 O. Takeda et al. (1980) studied the thermodynamics of calcium–ferritic slags at 1473 and 1573 K. Their work includes a formula for calculating the concentrations of Fe2+ and Fe3+ in dependence on concentration, partial pressure of oxygen, and temperature. An extensive study of physicochemical properties of the systems CaO–FeO– Fe2 O3 −Mx Oy (Mx Oy = SiO2 , Al2 O3 , MgO, ZnO) and of the system CaO–Fe2 O3 −Cu2 O was made by Vadász and Haulík (1995, 1996, 1998), Vadász et al. (1993, 2000, 2005),

Main Features of Molten Salt Systems

105

Fedor (1990), and Fedor et al. (1991). The density, surface tension, electrical conductivity, and viscosity have been measured at the temperature of 1573 K and in a relatively wide concentration range. The density and surface tension were measured by means of the maximum bubble pressure method using a device similar to that described in Section 6.2.2. The viscosity was measured using the rotational method, and the electrical conductivity, by means of the two-electrode method. From the theoretical point of view, on the basis of the complex physico-chemical analysis of these melts, considerations of their structure, i.e. the ionic composition was made. Using the multiple linear regression analysis, the equations describing molar volume and the surface tension on composition were obtained. From the individual interaction parameters, the formation of different structural entities in the melts was proposed. It has been suggested that the structure of the investigated calcium–ferritic melts is mostly influenced by the properties of the Fe(III) atoms. In the basic medium of calcium–ferritic slags, this amphoteric element exhibits an acidic character. Its acidity depends on the nature and concentration of the following oxides present in the melt. In the calcium–ferritic melts, Fe(III) atoms form simple complex anions. With increasing the concentration of oxygen anions, the co-ordination of the central Fe atom changes according to the following scheme 4− − + FeO5− 4 (4) → Fe2 O5 → FeO2 → FeO (6)

(2.125)

where the co-ordination number is given in parentheses. The obtained density and surface tension data, as well as the interpretation of the interactions found in the individual CaO–FeO–Fe2 O3 −Mx Oy systems suggest that cations Ca2+ , Fe2+ , Zn2+ , Cu+ may be present in the respective investigated melts. Due to 5− 5− the low concentration of network forming oxides, isolated SiO4− 4 , AlO4 and FeO4 tetrahedrons are formed in these basic melts, the donor of the oxygen atoms being either CaO, FeO, or both CaO + FeO oxides. The observed ternary interactions may be most 8− probably attributed to the formation of the anions SiFeO7− 7 and FeAlO7 , in which one Si atom is substituted by Fe and/or Al forming two tetrahedrons linked by a corner. 4− 9− However, it could not be excluded that also species, such as FeO+ 2 , Fe2 O5 , AlO6 , etc. may be present in the melts of the systems investigated. In the system CaO–Fe2 O3 −Cu2 O, due to the absence of silica and the low concentration of other network-forming oxides, only isolated FeO5− 4 tetrahedrons and CaO·FeO ionic pairs are formed in the basic melts. From the behavior of the cuprous oxide, the formation of the Ca4 Cu2 O5 compound might be assumed. In Table 2.11 the values of the molar volumes of essential pure oxides CaO, FeO, and Fe2 O3 , obtained in the calculation of the molar volume of individual systems are given. The agreement of values obtained from the measurement in independent systems is surprisingly excellent. The calculated values coincide fairly with the literature data.

106

Physico-chemical Analysis of Molten Electrolytes

Table 2.11. Molar volumes of pure oxides calculated from individual independent systems 0 (cm 3 ·mol−1 ) VCaO

0 (cm 3 ·mol−1 ) VFeO

0 VFe (cm3 ·mol−1 ) 2 O3

CaO–FeO–Fe2 O3 −SiO2 (1573 K) CaO–FeO–Fe2 O3 −Al2 O3 (1573 K) CaO–FeO–Fe2 O3 −MgO (1623 K) CaO–FeO–Fe2 O3 −ZnO (1573 K) CaO–Fe2 O3 −Cu2 O (1573 K)

16.41 ± 0.26 16.31 ± 0.60 16.50 ± 0.14 16.44 ± 0.38 16.42 ± 0.35

17.03 ± 0.28 17.00 ± 0.63 17.01 ± 0.15 17.22 ± 0.42 −

38.79 ± 0.48 38.11 ± 1.37 39.01 ± 0.34 38.06 ± 0.45 39.29 ± 0.36

Average

16.42 ± 0.35

17.07 ± 0.37

38.65 ± 0.60

System

Table 2.12. Surface tension of pure oxides calculated from individual independent systems σCaO (cm3 ·mol−1 )

σFeO (cm3 ·mol−1 )

σFe2 O3 (cm3 ·mol−1 )

CaO–FeO–Fe2 O3 −SiO2 (1573 K) CaO–FeO–Fe2 O3 −Al2 O3 (1573 K) CaO–FeO–Fe2 O3 −MgO (1623 K) CaO–FeO–Fe2 O3 −ZnO (1573 K) CaO–Fe2 O3 −Cu2 O (1573 K)

657.6 ± 12.8 661.6 ± 7.8 656.7 ± 11.8 617.5 ± 16.7 686.9 ± 25.7

588.6 ± 11.3 585.1 ± 7.0 582.8 ± 12.7 589.4 ± 25.4 −

376.2 ± 7.3 375.0 ± 4.5 374.4 ± 7.9 363.3 ± 14.8 379.5 ± 15.7

Average

656.1 ± 15.0

586.5 ± 14.1

373.7 ± 10.0

System

For illustration, Licˇ ko and Daneˇ k (1982) published a similar value for the molar volume of CaO at 1873 K, V 0 (CaO) = 18.28 cm3 ·mol−1 , and Bottinga and Weill (1970) published the following values for the molar volumes of oxides: V 0 (CaO) = 16.5 cm3 · mol−1 , V 0 (FeO) = 12.8 cm3 · mol−1 , and V 0 (Fe2 O3 ) = 52 cm3 · mol−1 , all at the temperature of 1723 K. The differences may be caused by different temperatures since the molar volume is rather sensitive to temperature. The calculated values of the surface tension of pure oxides are given in Table 2.12. Again, surprisingly excellent agreement can be observed. For the surface tension of pure oxides, the following values could be found in the literature for comparison: Daneˇ k and Licˇ ko (1982), based on the measurement in the system CaO–MgO–SiO2 , published for CaO at 1800 K the value σ (CaO) = 726 mN ·m−1 , Daneˇ k et al. (1985a) calculated from the measurements in the system CaO–FeO–Fe2 O3 –SiO2 for CaO, FeO, and Fe2 O3 at 1723 K, the values σ (CaO) = 689 mN ·m−1 , σ (FeO) = 502 mN ·m−1 , and σ (Fe2 O3 ) = 467 mN ·m−1 , respectively. For FeO, the value σ (FeO) = 585 mN ·m−1 was reported by Richardson (1974).

Chapter 3

Phase Equilibria 3.1. THERMODYNAMIC PRINCIPLES

The isobaric phase diagrams of condensed systems represent the graphical description of the phase equilibria in the temperature versus composition coordinates. In the phase diagrams, the fields of existence of individual phases can be seen. Phase diagrams are useful tools in many areas of industry, such as metallurgy, material science, glass making, aluminum production, etc. The construction of a trustworthy phase diagram is usually the first step in the development of a new technology. The basic laws and principles in the phase equilibrium theory are given in the next few chapters. 3.1.1. Gibbs’s phase law

Let us consider a closed system with f phases and k components that do not react mutually. The composition of each phase can be expressed using k − 1 molar fractions, since it holds that the sum of the mole fractions in each phase is equal to 1. For the description of f-independent phases with k components, we therefore need f (k − 1) independent data on composition. To these we still have to add data on temperature and pressure, so we have altogether f (k − 1) + 2 intensive data, if the temperature and pressure are equal in the whole system. If the system considered is in equilibrium, the intensive criterion of the thermodynamic equilibrium must be fulfilled, thus the chemical potentials of all the k components in all the f phases have to be equal. This criterion thus defines the number of binding conditions between the intensive variables. This number is k(f − 1), because the number of binding conditions is one less than the number of phases. Then the difference between both the quantities defines the number of intensive variables, which are independent in a system with k components and f phases being in equilibrium – the variance, or the number of degrees of freedom v v = f (k − 1) + 2 − k(f − 1) = k − f + 2

(3.1)

This is the mathematical expression of the Gibbs phase law. It is an explicit and simple guide in the study of phase equilibria.

107

108

Physico-chemical Analysis of Molten Electrolytes

Examples (1)

A closed vessel filled with gaseous oxygen. We have 1 component (oxygen) and 1 phase (gaseous). v =k−f +2=1−1+2=2

(2)

(3.2)

The system has two degrees of freedom, we can change the temperature and pressure without the occurrence of a new phase. A closed vessel partially filled with water. We have 1 component (water), 2 phases (water and its vapor pressure). v =k−f +2=1−2+2=1

(3)

(3.3)

The system has 1 degree of freedom, since the temperature and pressure are not independent any more. A certain temperature corresponds to a certain equilibrium pressure and vice versa. When the pressure decreases, the water evaporates completely. A closed vessel filled partially with a mixture of ethanol and water. We have 2 components (water and ethanol) and 2 phases (a liquid solution of ethanol and water and a mixture of gaseous ethanol and water vapor). v =k−f +2=2−2+2=2

(3.4)

The system has 2 degrees of freedom. We can change the pressure and composition of the liquid phase. However, once these quantities have a certain value, the temperature and the composition of the gaseous phase are already defined. 3.1.2. Lever rule

The lever rule can be derived from the mass balance as well as from the balance of the amount of substance. On a straight line we can show a mixture X lying in between two pure components A and B and their respective mole fractions and the amount of substances:

A

X

B

n1', x1'

n1, x1

n1'', x1''

For the amount of substance balance it holds then: n1 = n1 + n1

(3.5)

Phase Equilibria

109

n1 x1 = n1 x1 + n1 x1

n1 + n1 x1 = n1 x1 + n1 x1

(3.6)

n1 x1

(3.8)

+ n1 x1

n1 x1 − x1

=

n1 x1

(3.7)

+ n1 x1

= n1 x1 − x1

(3.9)

and finally: n1 BX x1 − x1 = ≡ n1 x1 − x1 AX

(3.10)

According to the lever rule, it follows that the ratio of the amount of substances equals the ratio of the lengths of sections to which the figurative point of the system divides the connection of both components. It can be easily shown that this rule holds for any three figurative points of a system lying on the straight line. Similarly as the lever rule holds in the two-component systems, in the three- and four-component systems, apply the triangle and quadrangle rules. According to the triangle rule, every phase, figurative point of which lies in the field of the triangle with the peaks A, B, and C, can be divided without residue into three phases, corresponding to the figurative points A, B, and C. Phase F divides first according to the lever rule into phases C and D and then the phase D divides again according to the lever rule into phases A and B. C

F

A

D

B

According to the quadrangle rule, every couple of phases, the figurative points of which lie on the ends of one of the diagonals of the general quadrangle (e.g. A, C), can

110

Physico-chemical Analysis of Molten Electrolytes

be changed to the other couple of phases, the figurative points of which lie on the other diagonal of this quadrangle (i.e. B, D).

D

A F C

B

According to the lever rule, phases A and C will first compose phase F and then phase F decomposes according to the lever rule into phases B and D. The quadrangle rule applies especially in phase diagrams of three-component systems with the formation of a congruently melting compound and in the phase diagrams of reciprocal systems. In the presence of a complex chemical compound Z = Aq Br in the system A–B, it is often advantageous to investigate the behavior of a component A in the subsystem A–Z. This approach requires transformation of the composition coordinates. The transformation of the composition coordinates of component A from the system A–B to the system A–Z is based on the following scheme xA + (1 − x)B = yA + (1 − y)Aq Br

(3.11)

where x is the mole fraction of component A in the original system A–B and y is the mole fraction of the same component A in the transformed system A–Z. The transformed composition coordinate y is given by the expression y=

(q + r)x − q (q + r − 1)x − q + 1

(3.12)

3.1.3. Thermodynamics of solutions

In this chapter, we focus on the fundamentals of the theory of solutions, which is needed for the understanding of the equilibrium phase diagrams. The thermodynamic analysis

Phase Equilibria

111

of the temperature dependence of the phase transition on the composition of the systems will be emphasized, since this approach forms the rational basis of the calculation for the phase diagrams. For the composition, we will use mole fractions since they do not depend on temperature and pressure. Different relations have been used in the literature to describe the dependence ai, l = f(Ti ). For instance, in the monographs by Blander (1964), Denbigh (1966), and Prigogine and Defay (1962) the activity is considered to be a function of the molar heat of fusion, while in others, e.g. in the monograph by Kogan (1968), the activity is considered to be a function of the partial molar heat of fusion. Since every phase diagram shows dependencies of the activity of components in their saturated solution on temperature (liquidus curves, liquidus areas) we will examine this dependency in a binary mixture. 3.1.3.1. Application of the relation µs, A = µl, A

Let us consider an isothermal process in which 1 mol of pure solid component in equilibrium with its saturated solution transforms into a liquid state (melting). In equilibrium, the equality of the chemical potentials of this component in both the phases must be valid (e.g. for the component A) µs, A = µl, A

(3.13)

Because it holds that µi = µ0i + RT ln ai , we get µ0s, A + RT ln as, A = µ0l, A + RT ln al, A

(3.14)

Since A is the pure solid component, as, A = 1 and after rearranging we get µ0s, A T

−

µ0l, A T

= RT ln al, A

(3.15)

The derivative of Eq. (3.15) according to the temperature at constant pressure yields d

µ0s, A T

dT

d −

µ0l, A T

dT

=R

d ln al, A dT

(3.16)

The derivatives on the left side of Eq. (3.16) are equal to d

µ0i T

dT

1 = 2 T

dµ0i T − µ0i dT

=

1 1 (−Gi − T Si ) = − 2 T2 T

dH dni

=−

Hi T2

(3.17)

112

Physico-chemical Analysis of Molten Electrolytes

Remember that the derivative of H, the enthalpy (not molar) of the system according to ni is the molar enthalpy, Hi . Then Eq. (3.16) attains the form Hl, A − Hs, A d ln al, A = dT RT 2

(3.18)

The difference Hl, A –Hs, A represents the change in the enthalpy at the transition of 1 mol of the pure component considered from the solid to the liquid state, which is thus the molar enthalpy of fusion Dfus HA and Eq. (3.18) has the final form d ln al, A fus HA = dT RT 2

(3.19)

Equation (3.19) is the differential form of the Le Chatelier–Shreder’s equation. We can obtain the integral form of this equation after integration in the range from the activity al, A at temperature T to the activity al, A = 1 (the pure solvent) at temperature Tfus, A . In the simplified case, we assume that the difference between the heat capacity in the liquid and the solid state of the component considered, s/l Cp , does not depend on temperature. We then get ln al, A =

fus HA R

1 Tfus,A

−

1 T

+

s/l Cp R

Tfus,A Tfus, A − 1 − ln T T

(3.20)

which is the so-called generalized Le Chatelier–Shreder’s equation in the integrated form. Assuming further that s/l Cp of the component equals zero (Dfus HA = constant), we obtain the Le Chatelier–Shreder’s equation in its simplified form ln a1, A

fus HA = R

1 Tfus, A

1 − T

(3.21)

However, in the calculation of the phase diagrams, i.e. the dependences of the activity of components in their saturated solution on temperature, we use the Le Chatelier–Shreder’s equation in the form explicit for temperature T =

fus HA Tfus, A fus HA = fus HA − RTfus, A ln al, A fus SA − R ln al, A

(3.22)

3.1.3.2. Application of the Planck function G/T

The equilibrium between the pure solid component and its saturated solution is given by the simultaneous validity of the following equations at constant pressure

Phase Equilibria

113

G0 Gl = s T T 0 Gl Gs d =d T T

(3.23) (3.24)

0 = f (T , x), where x is the mole fraction of the substance B in the At P = constant G l 0s = f (T ). Since Gl = G0 + RT ln al , with respect to liquid saturated solution, and G l Eq. (3.24) it holds 0 G0l + RT ln al ∂ Gs ∂ G0l + RT ln al ∂ dT (3.25) dT + dx = ∂T T ∂x T ∂T T It holds that

∂ ∂T

R

Since

∂ ∂T

G T

G0l T

x

= 0. By rearranging we get T

∂ ln al ∂T

= x,P

T

dT +

x

∂ ln al ∂x

T

∂ dx = − ∂T

G0l − G0s T

dT

(3.26)

−Hx,P , then T2 d ln al (T , x) =

fus H 0 dT RT 2

(3.27)

where fus H 0 = Hl0 − Hs0 . Equation (3.27) is obviously equal to Eq. (3.19). The application of the Planck function is called as the “differential” method. 3.1.3.3. Application of the relation a1 = f(T, x)

For the equilibrium of the given type it formally holds that a1 = f (T , x), P = constant, and also consequently ln a1 = f (T , x). Both al and ln al are functions of state. Then, using the properties of the exact differential, we can write ∂ ln al ∂ ln al d ln al = dT + dx (3.28) ∂T x ∂x T Equation (3.28) can be integrated if both the right-hand terms are known. From the derivative of the definition equation Gl = G0l + RT ln al according to T at x = constant it follows ∂ ln al 1 ∂(Gl − G0l ) = (3.29) ∂T x R ∂T x

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Physico-chemical Analysis of Molten Electrolytes

or

∂ ln al ∂T

=− x

(H l − Hl0 )x RT 2

(3.30)

Hence the first right-hand term of Eq. (3.28) has been established. The second right-hand term will be determined by means of Eq. (3.30) applied to the phase equilibrium liquidus–solidus

∂ ∂T

Gl T

x

∂ dT + ∂x

Gl T

∂ dx = ∂T

T

G0s T

dT

(3.31)

dT

(3.32)

or

(H l )x − T2

∂ dT + ∂x

(G0l + RT ln al ) T

T

Hs0 dx = − T2

and after rearrangement

∂ ln al ∂x

dx = T

(H l − Hs0 )x dT RT 2

(3.33)

Equation (3.33) is the second right-hand term of Eq. (3.28). Substituting the corresponding terms from Eqs. (3.30) and (3.33) into Eq. (3.28) we get d ln al =

(−H l + Hl0 )x (H l − Hs0 )x dT + dT RT 2 RT 2

(3.34)

fus H 0 dT RT 2

(3.35)

or d ln al =

Equation (3.35) is obviously identical with Eq. (3.27) and it is the differential form of Eq. (3.21). 3.1.3.4. Two liquid phases coexisting in equilibrium

From Eq. (3.33) it follows that

dT dx

∂ ln al ∂x T (H l − Hs0 )x

RT 2 =

(3.36)

Phase Equilibria

115

and since H l − Hs0 = mix H l + fus H 0 , Eq. (3.36) can be written in the form

dT dx

RT 2 =

∂ ln al ∂x

T

(3.37)

(mix H l + fus H 0 )x

The term fus H 0 is always greater than zero. What concerns the term (∂ ln al /∂x)T , the situation could be more complicated. When the course of the dependence ai = f(xi ) is monotonous over the whole concentration range, only one liquid phase exists in the system and (∂ ln al /∂x)T > 0. However, when the course of the function ai = f(xi ) is not monotonous, as it is demonstrated in Figure 3.1, the activities of the given component i in the coexisting liquid phases of compositions corresponding to the points Q and R are equal and it holds again that (∂ ln al, Q /∂x)T > 0 and (∂ ln al, R /∂x)T > 0. This situation is typical for systems exhibiting a miscibility gap and is associated with a sufficiently large positive deviation from the ideal behavior. In the system, two liquid phases coexist in equilibrium. These two liquids of composition corresponding

1.0

Q

R

0.8

0.6 ai 0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

xi Figure 3.1. The function ai = f (xi ) in a system exhibiting a miscibility gap.

116

Physico-chemical Analysis of Molten Electrolytes

to the points Q and R are evidently in equilibrium. Single-phase solutions with the composition coordinates situated between points Q and R are metastable with respect to the two-phase system. 3.1.3.5. Gibbs energy at mixing

In the calculation of the phase diagrams the mixing molar Gibbs energy plays an important role as well. Consider the change of the molar Gibbs energy at mixing of two pure liquid compounds A and B, which takes place at a certain temperature and pressure mix G = Gmix − nA µ0A − nB µ0B

(3.38)

The molar Gibbs energy of the mixture, Gmix , equals the sum A + nB G B = nA µA − nB µB Gmix = nA G

(3.39)

Equations (3.38) and (3.39) can be added together, which yields mix G = nA (µA − µ0A ) + nB (µB − µ0B )

(3.40)

Since µi − µ0i = RT ln ai , then it holds mix G = nA RT ln aA + nB RT ln aB

(3.41)

or, because of ai = xi · γi mix G = nA RT ln xA + nB RT ln xB + nA RT ln γA + nB RT ln γB

(3.42)

3.1.3.6. Ideal solutions

According to Lewis, the fugacity of an arbitrary component in the ideal solution is proportional to its mole fraction through the whole concentration region and at all temperatures and pressures fi = xi fi0

(3.43)

where fi and f 0i are the fugacities of the component in the solution and in the pure state, respectively. From this definition, two important consequences follow. (1)

The origin of solution is connected neither with volume contraction nor with volume dilatation.

Phase Equilibria

117

Proof: From Eq. (3.43) we get ln fi = ln xi + ln fi0

(3.44)

The derivative according to pressure at constant temperature and composition gives d ln fi0 d ln fi = dP dP

(3.45)

Since Vi d ln fi = dP RT

d ln fi0 V0 = i dP RT

and

(3.46)

from which it follows that Vi = Vi0

(3.47)

and finally, for the binary system it holds Vmix = nA V A + nB V B − nA VA0 − nB VB0 = 0 (2)

(3.48)

The origin of solution is connected neither with the release nor with the absorption of heat. Proof: Differentiating Eq. (3.43) according to temperature at constant pressure and composition we get d ln fi0 d ln fi = dT dT

(3.49)

Since ∗

d ln fi H − Hi = i 2 dT RT

(3.50)

and thus ∗

∗

H i − H i = H i − Hi0

(3.51)

H i = Hi0

(3.52)

from which it follows that

118

Physico-chemical Analysis of Molten Electrolytes

The change in enthalpy at the formation of the binary solution equals Hmix = nA H A + nB H B − nA HA0 − nB HB0 = 0

(3.53)

For an ideal solution γ i = 1 and Eq. (3.42) simplifies to the form mix G∗ = nA RT ln xA + nB RT ln xB

(3.54)

For every solution it holds that mix G∗ = mix H ∗ − T mix S ∗

(3.55)

and since the solution is ideal, mix H ∗ = 0 and mix G∗ = −T mix S ∗

(3.56)

mix S ∗ = −R (nA ln xA + nB ln xB )

(3.57)

from which it follows that

The molar entropy of mixing of the ideal solution is thus not equal to zero. This is due to the fact that the mixing of pure components proceeds spontaneously and is connected with rearrangement of the melt and thus with an increase in entropy. According to the value of the individual members on the right side of Eq. (3.55) we divide the solutions into the following categories (the asterisk denotes the ideal mixing) ideal solutions: mix G = mix G∗ ,

mix H = 0,

mix S = mix S ∗

(3.58)

mix H = 0,

mix S = mix S ∗

(3.59)

mix H = 0,

mix S = mix S ∗

(3.60)

regular solutions: mix G = mix G∗ , Temkin’s ideal solution: mix G = mix G∗ ,

Phase Equilibria

119

3.1.3.7. Real solutions

The difference mix G – mix G∗ in real solutions, i.e. the difference of Eq. (3.42) and Eq. (3.54), is called the excess molar Gibbs energy of mixing and is denoted as ex G ex G = mix G − mix G∗ = nA RT ln γA + nB RT ln γB

(3.61)

Differentiating partially the excess molar Gibbs energy of mixing according to the amount of substance (e.g. A) we get

∂ex G ∂nA

= RT ln γA + RT

nB

nA

∂ ln γB ∂ ln γA + nB ∂nA ∂nA

= RT ln γA

(3.62)

since the expression in parentheses is according to the Gibbs–Duhem’s equation equal to zero. A similar relation can be derived also for the component B. Equation (3.62) is frequently used in the calculation of the activity coefficients of the components. In order to describe the behavior of the real solutions, i.e. the deviation of the solution from the ideal behavior, we use different thermodynamic models. 3.1.4. Thermodynamic models of molten salts

In order to compare the theoretical models with the experimental results, we need to know the form of the functional dependence ai = f(xi ). This dependence can be obtained either on the basis of a certain structural model or empirically. The dependence of such a kind must, however, fulfill certain limiting conditions. According to the character of the functional dependence ai = f(xi ) it is convenient to distinguish molten salt systems of type I and II. The systems of type I must fulfill two conditions: dal =1 dxl

(3.63)

dal = H1∗ = 0 x1 →0 dxl

(3.64)

lim

x1 →1

lim

where H1∗ is proportional to the Henry’s constant (H1∗ = f0 H ) and is also called the Henry’s constant. The first condition expresses the validity of Raoult’s law and the second one is equivalent to Henry’s law. As soon as at least one of the above-mentioned conditions is not fulfilled, the systems are those of type II. If γ i > 1 (and thus ai > xi ), we speak about the positive deviation from the ideal behavior, if γ i < 1 (and thus ai < xi ) it deals with negative deviation. For type I solutions we postulate the validity of the Raoult’s and Henry’s laws. Then the dependence ai = f(xi ) will have the form shown in Figure 3.2. In the concentration region < 1;

120

Physico-chemical Analysis of Molten Electrolytes 1

Activity

γ<1

a=x γ<1

0 1

x(R)

Molar fraction

x(H)

0

Figure 3.2. The dependence ai = f (xi ) for systems of type II.

x(R) > it holds ai ∼ = xi . This region is called the Raoult’s region. In the concentration region < x(H); 0 > it holds ai ∼ = Hi∗ xi and this region is called the Henry’s region. 3.1.4.1. Concept of activity and activity coefficients in molten salts

In the literature on molten salts, the single ion activity coefficient is often used erroneously. The use of single ion activity coefficients depends on the chosen standard state and may thus be confusing. The ion activity and the ion activity coefficient concepts have been used for a long time in the thermodynamic treatment of dilute aqueous solutions of electrolytes and have also been extended to non-aqueous mixtures, molten salts, and slags. It was, however, generally agreed, that single ion activity cannot be experimentally determined and it cannot be calculated without non-thermodynamic assumptions. It should thus be emphasized that when the single ion activity is applied, the standard state and the convention used should be clearly defined. For simplicity, only solutions of symmetrical binary univalent electrolytes will be considered here. The measurable thermodynamic quantity of interest in this connection is the change in chemical potential, µMA , of the component MA in the mixture. The activity of MA,

Definition of single ion activity coefficient.

Phase Equilibria

121

aMA , is defined by the equation µMA = µMA − µ0MA = RT ln aMA

(3.65)

where µMA and µ0MA is the chemical potential in the given mixture and in the standard 0 , is by definition equal to one. state, respectively. In the standard state, the activity, aMA Equation (3.65) emphasizes that the choice of the standard state usually corresponds to the selection of a state, in which we want the activity to be equal to one. For a dissociating electrolyte MA, the activity coefficient, γMA may be defined by the equation µMA = RT ln(cM+ cA− γ MA )

(3.66)

where cM+ and cA− are concentrations of the ions M+ and A− , respectively. In this equation, different concentration units may be used, e.g. molarity, molality, etc. The most rational concentration units are considered to be the mole (ionic) fractions based on the species actually present in the system. In a similar way as in Eq. (3.65), the single ion activities aM+ and aA− are usually defined by the general equations µM+ = µ0M+ + RT ln aM+

(3.67)

µA− = µ0A− + RT ln aA−

(3.68)

where µM+ and µA− are the chemical potentials of the ions M+ and A− in the given solution, respectively. The single ion activity coefficient, γM+ , is defined by the equation aM+ = cM+ γM+

(3.69)

and correspondingly also for γA− . From Eqs. (3.67) and (3.68), it can be seen that the single ion activity coefficients will depend on the chosen concentration units and the chosen standard states for the ions. From the above equations it follows that the activity coefficient of the salt, γMA , is related to the single activity coefficients by the equation γMA = γM+ γA− = γ±2

(3.70)

where γ± is called the mean activity coefficient. It should be, however, stressed that none of the quantities connected with single ions, i.e. µM+ , aM+ , γM+ , or µM+ , aA− , γA− , can be obtained from thermodynamic measurements.

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Physico-chemical Analysis of Molten Electrolytes

It should be stressed that the single ion activities and activity coefficients can be introduced only formally. The reason is that due to the electro-neutrality we cannot differentiate the Gibbs energy of the system according to the amount of cations keeping the amount of the anions constant. The concept of the chemical potential of a single ion seems to be thus somewhat dubious. The chosen standard state is normally based upon either Raoult’s or Henry’s law. The particular choice depends on the convenience and experimental accessibility.

Standard states.

(1)

(2)

Henrian activities (solute standard state) When dealing with dilute solutions, it is often convenient to refer to the infinitely dilute solution when defining the standard states. In this case, the activity coefficient of the solute is taken to approach one as the solution is infinitely diluted, and the standard state is defined as the (hypothetical) state with unit concentration (ci = 1) and with unit activity coefficient (γ i = i). Mathematically, this standard state is defined by ai /ci = 1 as ci → 0. The choice of infinite solution as the basis for defining this hypothetical standard state effectively assigns to the environment of each solute species and the chemical properties of the pure solvent. This means that the variations in Gibbs energy accompanying concentration change in the Henry’s law region (γ i = 1) are ascribed solely to the change in the partial molal entropy of the solute, which is associated with expanding or contracting the volume of solution available to each solute species. This standard state is most useful when activities of different compounds are compared in dilute solutions of the same solvent, and is commonly used, for example, when dealing with aqueous solutions of electrolytes or dilute solutions in molten salts (cryoscopy). When defining a Henrian standard state, it is imperative to consider those species actually existing in the infinitely dilute solution, or behavior according to Henry’s law will not be observed. In the case of electrolytes, this means that the individual ions must be selected as the species of interest. The standard state of each ionic species is chosen so that the ratio of its activity to its concentration becomes one at infinite dilution, at 101.325 kPa pressure, and the actual temperature. It should be noted that with the state of infinite dilution as the standard state, such dilution is applied to all ions in solution and not just to the particular ion under consideration. This stipulation with respect to all ions is important because it is the ionic environment, which is mostly responsible for departures from ideality and not just the characteristics of the particular ion under consideration. Raoultian activity (solvent standard state) When dealing with liquid mixtures over a rather wide composition range, or at higher concentrations, it is most convenient to choose the pure components,

Phase Equilibria

123

supercooled if convenient, as the standard states with unit activity. With this choice, the activities of a compound in different solvents can be directly compared, since the activities are expressed relative to the same standard state, the pure component. This standard state is usually preferred in the thermodynamic treatment of fused salts and slags. The fused salt system of the ions Na+ , K+ , Cl− , and has the following restriction due to electro-neutrality

Systems of reciprocal salt mixtures.

Br−

nNa+ + nK+ = nCl− + nBr−

(3.71)

where ni is the amount of individual ions. In terms of the phase rule, the system has three components. To some extent they can be chosen arbitrarily, e.g. a mixture containing one mole of positive and negative charge can be described as xNa+ NaCl + xBr− KBr + (xCl− − xNa+ )KCl

(3.72)

with the Gibbs energy per one mole of the mixture G = xNa+ µNaCl + xBr− µKBr + (xCl− − xNa+ )µKCl

(3.73)

The same system can also be described as (seen from simple stoichiometric calculation) xNa+ xCl− NaCl + xK+ xCl− KCl + xNa+ xBr− NaBr + xK+ xBr− KBr

(3.74)

with the Gibbs energy per one mole of the mixture G = xNa+ xCl− µNaCl + xK+ xCl− µKCl + xNa+ xBr− µNaBr + xK+ xBr− µKBr

(3.75)

Since the Gibbs energy and the chemical potentials in Eqs. (3.73) and (3.75) are identical, one obtains by combining the two equations µKBr = µKCl + µNaBr − µNaCl

(3.76)

which means that by describing the system by one more components than necessary, one automatically obtains an additional equation connecting the chemical potentials. If we further introduce the activity from the equations µNaCl = µ0NaCl + RT ln aNaCl , and so forth

(3.77)

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Physico-chemical Analysis of Molten Electrolytes

we obtain the chemical equilibrium expression RT ln

aNaBr aKCl = µ0NaBr − µ0KCl + µ0NaCl + µ0KBr = −ex G0 aNaCl aKBr

(3.78)

where ex G0 is the Gibbs energy change for the metathetical reaction NaCl + KBr = NaBr + KCl

(3.79)

when all the components are in their standard state, which is here chosen to be the pure components (Raoultian standard states). According to Temkin (1945), the partial molar entropy of a salt component, e.g. NaCl, in an ideal fused salt mixture is given by S NaCl = −R ln(xNa+ xCl− )

(3.80)

where xNa+ =

nNa+ nNa+ + nK+

and

xCl− =

nCl− nCl− + nBr−

(3.81)

in accordance with this, the activity coefficient of a component, γNaCl , is therefore defined by the equation aNaCl = xNa+ xCl− γNaCl , and so forth

(3.82)

Each salt activity, aNaCl , may formally be split into separate ionic activities aNaCl = aNa+ aCl−

(3.83)

with corresponding ionic activity coefficients γNaCl = γNa+ γCl−

(3.84)

Equation (3.83) corresponds to the formal division of the standard chemical potential µ0NaCl µ0NaCl = µ0Na+ + µ0Cl−

(3.85)

However, in this case µ0NaCl refers to a state where Na+ has Cl− as a close neighbor and vice versa, and can be expressed as the Gibbs energy of formation of NaCl. In Eq. (3.85), this has arbitrarily been divided into ionic standard chemical potentials.

Phase Equilibria

125

If we further make the non-thermodynamic assumption that µ0Cl− from the component NaCl is the same as µ0Cl− from the component KCl, and so forth, one ends up with a contradiction, which can be demonstrated in the following way. For the metathetical reaction (3.79) we have G0 = µ0KCl + µ0NaBr − µ0NaCl − µ0KBr = 0

(3.86)

By introducing the standard chemical potentials of the single ions as given in Eq. (3.86) we end up with the contradiction that G 0 equals zero. The same conclusion follows, when the activity terms, Eq. (3.82), are introduced into the expression for the equilibrium constant in Eq. (3.78) and then introducing the single activity coefficients by Eq. (3.84), we get (γNa+ γBr− )(γK+ γCl− ) −ex G0 = exp (3.87) (γNa+ γCl− )(γK+ γBr− ) RT If γCl− from the component NaCl is assumed to be the same as γCl− from the component KCl and so forth, one ends up with the contradiction −ex G0 1 = exp (3.88) RT This demonstrates that it is not advisable to split up salt activities in this way into ion activities when using the Raoultian standard state. Conclusions.

(1) (2) (3)

(4)

As concluding remarks the following points should be emphasized.

It is generally agreed that single ion activities cannot be determined by thermodynamic measurements. In any work applying single ion activities, the convention used should be clearly stated and carefully analyzed. Single ion activities should be used for components where the activities are expressed in terms of the ideal dilute solution as the standard state (Henrian activities). In the treatment of thermodynamic properties of mixed systems of molten salts or slags used as standard state pure components (Raoultian standard states), the use of single ion activity is to be avoided, as it leads to contradictions.

3.1.4.2. Model of regular solutions

The model of regular solutions is very frequently used for which the conditions expressed in Eq. (3.59) are valid. This means that the deviation from the ideal behavior is ascribed to the change in enthalpy at the interaction of the components on mixing. For the excess

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Physico-chemical Analysis of Molten Electrolytes

Gibbs energy of regular solutions different relations have been postulated, for example according to Redlich and Kister ex G0 = xA xB A + BxB + CxB2 + · · ·

(3.89)

ex G0 = xA xB A + B (xA − xB ) + C (xA − xB )2 + · · ·

(3.90)

ex G0 = xA xB (xA A + xB B)

(3.91)

or Guggenheim

or Margules

or van Laar ex G0 = xA xB

AB AxA + BxB

(3.92)

as also other relations. For simple regular solutions it holds ex G0 = xA xB A

(3.93)

where A is the coefficient of interaction of the components in the simple regular solutions. From conditions (3.59) it follows that in these solutions the excess entropy of mixing is equal to zero and it thus holds ex G0 = ex H 0

(3.94)

The model of regular solutions thus ascribes deviations from the ideal solution to the non-zero enthalpy of mixing. From the excess Gibbs energy of mixing, we can calculate the activity coefficients of the components, however, the mole fractions must be calculated from the amounts of substances xA =

nA , nA + n B

xB =

nB nA + n B

(3.95)

Then it holds nB nA A nA + n B nA + n B nA nB ex G = ex G0 (nA + nB ) = A nA + n B

ex G0 = xA xB A =

(3.96) (3.97)

Phase Equilibria

127

Note that ex G is the extensive property while ex G0 is the intensive one. The partial derivative of ex G according to the amount of substance of the respective component at T, P, and ni = constant yields

∂ex G ∂nA ∂ex G ∂nB

= RT ln γA =

AxB2

= A (1 − xA ) = 2

T , P , nB

= RT ln γB =

AxA2

= A (1 − xB ) = 2

T , P , nA

∂ex H ∂nA ∂ex H ∂nB

(3.98)

T , P , nB

(3.99) T , P , nA

For the activity coefficients we then obtain the relation

A γi = exp (1 − xi )2 RT

(3.100)

If the liquid phase behaves like a simple regular solution, for the dependence of the liquidus temperature on composition (the Le Chatelier–Shreder’s equation (3.22)) we get

T =

fus Hi + A(1 − xi )2 fus Hi + RT ln γi = fus Si − R ln xl, i fus Si − R ln xl, i

(3.101)

3.1.4.3. Temkin’s model of ideal ionic solutions

Both the above-mentioned models could be applied only to the systems of type I. For the description of the dependence ai = f (xi ) of inorganic molten systems of type II, the Temkin’s model of ideal ionic solution is used most frequently. The Temkin’s model is based on three postulates. (1)

(2)

(3)

The solution as a whole as well as its components are composed solely from ions 3− (simple, e.g. Na+ , Ca2+ or complex, e.g. SO2− 4 , AlF6 ). Electrically neutral particles (molecules) are not present. With regard to the high concentration of electrically charged particles, substantial electrostatic forces originate among them. Consequently, every ion is surrounded by ions of the opposite charge. This excludes the interchange of the cation in its position by an anion and vice versa. Thus, as a whole, the solution is composed of two independent, mutually interlocked solutions, i.e. of the solution of cations and the solution of anions. In its “own” solution, all ions are equivalent regardless of their dimensions and charge and they are randomly (statistically) distributed.

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Physico-chemical Analysis of Molten Electrolytes

From the third postulate two important conclusions follow. (a) (b)

The enthalpies of mixing of the cation and anion solutions equal zero, thus also the enthalpy of mixing of the solution as a whole equals zero, mix H = 0. The entropy of mixing of the cation and anion solutions has solely the configurational character. The entropy of mixing as a whole is equal to the sum of the entropies of mixing of both the ionic solutions, thus it is not equal to the entropy of mixing of the ideal solution, mix S = mix S∗ .

Deviations from the classical ideal solution in the Temkin’s model are thus ascribed to the non-ideal entropy of mixing. Let us consider a general type of molten ionic binary mixture AX–BY (e.g. NaCl–LiI). In such a mixture cations A+ and B+ and anions X− and Y− are present. The entropy of mixing of such a mixing is equal to the sum of the entropies of mixing of both the independent solutions of cations and anions mix S = mix SA+ + B+ + mix SX− + Y−

(3.102)

Since it was postulated that in every solution the ions are mixing randomly it holds

mix SA+ + B+ = mix SA∗ + + B+ = −R nA+ ln xA+ + nB+ ln xB+

mix SX− + Y− = mix SX∗ − + Y− = −R nX− ln xX− + nY− ln xY−

(3.103) (3.104)

or

mix S = −R nA+ ln xA+ + nB+ ln xB+ + nX− ln xX− + nY− ln xY−

(3.105)

where nA+ , nB+ , nX− , and nY− are the amounts of ions and xA+ , xB+ , xX− , and xY− are their molar fractions, respectively, defined by the equations (according to the postulate) xA+ =

nA+ nB+ nX − nY − , x += ,x − = , x −= nA+ + nB+ B nA+ + nB+ X nX − + n Y − Y nX− + nY− (3.106)

If mix H = 0, then mix G = −T mix S. For mix G in general it holds mix G =

ni (µ − µi )

(3.107)

i

In order to obtain the relation for the chemical potential of the component e.g. AX we must partially differentiate Eq. (3.107) according to nAX . But when we exchange nA+

Phase Equilibria

129

for dnA+ , we automatically exchange also nX− for dnX− (the solution must remain on the whole electrically neutral). We must thus make the derivative according to nA+ and nX− simultaneous. For the chemical potential of the component AX we thus get

µAX − µ0AX

∂mix G = ∂nAX

nB , nY

∂mix G = ∂nA+

ni=A

∂mix G + ∂nX−

(3.108) ni=X

and

µAX − µ0AX = RT ln xA+ + RT ln xX− = RT ln xA+ xX− = RT ln aAX

(3.109)

For the activity of the component AX it thus holds aAX = xA+ xX−

(3.110)

Similar relations can be obtained also for the other components. Examples LiF(1)–NaF(2): x 1 x1 + x 2 = xLiF x1 + x 2 x1 + x 2 x 2 x1 + x 2 = = xNaF x1 + x 2 x1 + x 2

aLiF = xLi+ xF− = aNaF = xNa+ xF −

daLiF dxLiF = = 1; dxLiF dxLiF

daNaF =1 dxNaF

The system is thus of type I. One molecule of LiF introduces into the molten NaF one new particle, the cation Li+ . We also obtain the same result for NaF, where one molecule of NaF introduces into the molten LiF one new particle, the cation Na+ . LiF(1)–BaCl2 (2): x12 x1 x1 = x1 + x2 x1 + 2x2 2 − x1 2 4x23 2x2 x2 = = x1 + x2 x1 + 2x2 (1 + x2 )2

aLiF = xLi+ xF− = 2 aBaCl2 = xBa2+ xCl −

2x1 (2 − x1 ) + x12 4x1 − x12 daLiF = = 2 dxLiF (2 − x1 ) (2 − x1 )2 lim

x1 → 1

daLiF = kst,Ba Cl2 /LiF = 3 dxLiF

130

Physico-chemical Analysis of Molten Electrolytes

This system thus belongs to those of type II. One molecule of BaCl2 introduces into the molten LiF three new particles, Ba2+ and 2Cl− . 12x22 (1 + x2 ) − 4x23 · 2 daBaCl2 = dxBaCl2 (1 + x2 )3 daBaCl2 = kSt, LiF/BaCl2 = 2 x2 →1 dxBaCl2 lim

One molecule of LiF introduces into the molten BaCl2 two new particles, Li+ and F− . The Temkin’s relation for the ai = f(xi ) dependence represents the limiting relation and thus enables the rational introduction of the Stortenbeker’s correction factor. The dependence ai = f(xi ) for the systems of type II is shown in Figure 3.3. The slope of the tangent to the dependence ai = f(xi ) intersects on the x-axis a section, which is the reciprocal value of the Stortenbeker’s correction factor. According to the Boltzmann statistics, the probability of a given arrangement of N particles is proportional to the factor e−E/kT divided by the total sum of all possible arrangements of N particles. E is the energy of the given arrangement. The configurational

Activity

1

1/k(St) = 1/2

0 1

Molar fraction

Figure 3.3. The dependence ai = f (xi ) for systems of type II.

0

Phase Equilibria

131

partition function Q for a binary mixture composed of A and B particles can be written in the form 1 . . . e−E/kT dr1 . . . drN (3.111) Q= NA !NB ! where NA and NB are the number of particles A and B, respectively and N = NA + NB . Since E does not change when A and B do change their configurations, all configurations must have the same statistical weight. Thus A and B are randomly distributed in all the configurations of the N particles. Such a mixture forms an ideal solution. The enthalpy of mixing is zero and the entropy of mixing is given by the equation mix S = −N k(xA ln xA + xB ln xB )

(3.112)

where Nk = R. The simplest molten salts mixture must be composed of at least three kinds of particles, e.g. A+ , B+ , and C− , where A+ and B+ are cations, C− is the common anion, and NA+ + NB+ = NC− . The Gibbs energy of mixing is given by the equation mix G = −RT (xA+ ln xA+ + xB+ ln xB+ )

(3.113)

where xA+ and xB+ are the molar fraction of cations A+ and B+ , respectively, defined by the equations xA+ =

nA+ , nA+ + nB+

xB+ =

nB+ nA+ + nB+

(3.114)

Equation (3.113) together with Eq. (3.114) define the Temkin’s ideal solutions. In a binary system there are three kinds of ions, e.g. A+ , B+ /X− . However, only the amount of two of these three constituents can be independently changed because of the constraint given by the condition of electro-neutrality

Reciprocal salt mixtures.

nA+ + nB+ = nX−

(3.115)

Ternary ionic systems contain four kinds of ions, which can be constituted in three different ways: (A+ , B+ , C+ /X− ), (A+ /X− , Y− , Z− ), or (A+ , B+ /X− , Y− ). The first two arrangements are additive ternary systems (the former one is a system with a common anion, the latter one is a system with a common cation), while the last one is a ternary reciprocal system. Having in mind the restriction of electro-neutrality, there are only three independent salt components, from which the solution could be built up.

132

Physico-chemical Analysis of Molten Electrolytes

Let us now pay attention to ternary reciprocal systems. They contain two different cations and two different anions. Due to such a composition, they differ somewhat from the additive ternary systems. There are four constituents, AX, AY, BX, BY, but only three of them are independent components. The solution can be made up in four ways, i.e. AX–AY–BX, AX–AY–BY, AX–BX–BY, and AY–BX–BY. In each of these cases, the composition of the final solution will differ. Let us consider one mole of a solution in which the composition is xA+ = 0.3 and xX− = 0.5. Using three salts only the solution can be composed in two different ways. From these two figures it follows that xB+ = 0.7 and xY− = 0.5. The composition of the solution built up in the two ways is given in Table 3.1. Both of these ways will lead to the same final solution, but the Gibbs energy of mixing in both ways will differ. While the composition of an ordinary ternary system is represented by a triangle, the ternary reciprocal system is represented by a square, corners of which represent the pure constituents AX, AY, BX, BY. The final solution that we are considering lies inside the two triangles AX–BX–BY and AY–BX–BY. There are two peculiarities of the ternary reciprocal system. The first one concerns the activities and the chemical potentials. Regardless of the way the solution was made up, the activities and chemical potentials of all the four constituents are defined and are the same no matter which of the three components were chosen. In our example for the Temkin’s ideal solution aAX = xA+ xX− = 0.15, aAY = 0.15, aBX = 0.35, and aBY = 0.35. The second peculiarity results from the reciprocity of composition and thus from the equilibrium between AX, BY and AY, BX. The Gibbs energy of this equilibrium equals the sum of the standard Gibbs energies of formation of the components for the metathetical reaction AX + BY ↔ AY + BX

(3.116)

Remembering that xA+ = 0.3, xX− = 0.5, xB+ = 0.7, and xY− = 0.5, it is obvious that in the final solution there is always one salt missing, one on the left side or one on the right side. Thus the Gibbs energy of mixing must contain the term ±xi xj r G0 , where i and j are the cation and anion of the missing salt. The sign of the term is positive if i and Table 3.1. Amounts of AX, AY, BX, and BY salts to be in a ternary melt Amount of salt/mol

Salt AX AY BX BY

Case 1

Case 2

0.3 – 0.3 0.4

– 0.3 0.2 0.5

Phase Equilibria

133

j are for a salt on the right side of Eq. (3.116) and negative if it is on the left side. Thus mix G r G 0 = xA+ ln xA+ + xB+ ln xB+ + xX− ln xX− + xY− ln xY− ± xi xj + ... RT RT (3.117) This equation says that the term mix G/RT must be different from the ideal term xi ln xi if r G0 for the reaction (3.116) differs from zero. If the mixture is made up by AX, BY, and AY, reaction (3.116) is shifted to the right. Therefore mix G must increase by xB xX r G0 , since BX is formed in the melt. If the mixture is made up by AX, AY, and BX, reaction (3.116) is shifted in the opposite direction and BY is formed. This result was first obtained by Flood et al. (1954) using a thermodynamic approach and was later expanded by Blander and Yosim (1963) and Førland (1964). The activity coefficients in reciprocal salt systems can be calculated knowing the components that form the system. From the above we can deduce that one of the terms for the activity coefficient of the component, e.g. AY is RT ln γAY = ± xB xX r G0 · · ·

(3.118)

where r G0 is the Gibbs energy of the metathetical reaction (3.116) for which the “−” sign is valid. For the metathetical reaction running in the opposite direction, the “+” sign is valid. This is easily comprehensible in a qualitative way. If r G0 for reaction (3.116) is negative, the reaction will be shifted to the right and the amount of AY and BX neighbors in the mixture will be larger than predicted by random mixing. Therefore positive deviations from the ideality for AY and BX will occur. There is, however, a problem in dilute solutions, when values of xB or xX approach zero, since the activity coefficients of AY should then approach those in the binaries AX–AY or AX–BX. This problem overcomes the quasi-lattice theory. The quasi-lattice theory is based on a quasi-lattice, which consists of two interlocking sub-lattices of cations and anions. Only the nearest neighbor interactions are taken into account and the energy of any nearest neighbor pair is assumed to be independent of its environment (additivity of pair bond interactions). Nearest neighbor interactions are ignored, which means that all the binary systems are ideal. In the dilute solution of AX in BY, there is always some probability that A+ will meet − X and a new pair bond will originate. The fundamental energy change when this occurs is E. The Gibbs energy change depends also on the coordination of the A+ environment Z. The following equation must then hold for the metathetical reaction (3.116) −r G0 = ZE

(3.119)

134

Physico-chemical Analysis of Molten Electrolytes

where Z is the coordination number. Coordination numbers in molten salts appear to be about 4–5. If the quasi-lattice equations for the activity coefficients will be expanded up to the second order E, we obtain RT ln γAY = xB xX ZE − xB xX (xB xY + xA xX − xA xY )

ZE 2 + ... 2RT

(3.120)

where the second term on the right-hand side is the first correction for the non-random mixing. In order to overcome the problem with activity coefficients in dilute solutions, Førland (1964) suggested the addition of four additive binary terms ex ex ex xA Gex A + xB GB + xX GX + xY GY

= xA xX xY λA + xB xX xY λB + xX xA xB λX + xY xA xB λY

(3.121)

where Gex i is the excess Gibbs energy of mixing and λi is the interaction parameter for the binary system with i as a common ion. Adding the contribution from non-random mixing we obtain RT ln γAY = ± xB xX r G0 − xB xX (xB xY + xA xX − xA xY )

(r G0 )2 2ZRT

+ xB xX (xX − xY )λB + xX (xB xY + xA xX )λA + xB (xB xY + xA xX )λY + xB xX (xB − xA )λX

(3.122)

Equation (3.122) gives the activity coefficient and the corresponding Eq. (3.123) shows the total excess Gibbs energy of mixing when the components AX, BX, and AY are mixed ex mix Gex = ± xB xY r G0 + xA Gex A + xB GB ex + xX Gex X + xY GY + xA xB xX xY

(r G0 )2 2ZRT

(3.123)

In these equations, the “−” sign has to be chosen when r G0 is for the metathetical reaction AX + BY → AY + BX, and the sign “+” when r G0 is for the reaction AY + BX → AX + BY. Equations (3.122) and (3.123) are not necessarily valid for ionic systems. These equations have also been derived from the conformal ionic solution theory, thus providing a fundamental underpinning for this equation with a theory, which properly includes coulomb interactions. The form of the “non-random mixing” term has only been shown to be proportional to (r G0 )2 from the conformal solution theory. The term holds for a large scale of salts. The proportionality constant (2ZRT)−1 was chosen by analogy

Phase Equilibria

135

with the quasi-lattice theory. To compensate for the fact that the higher order term is not included, a large value of the parameter Z (= 6) appears to be necessary. If we look at the properties of Eqs. (3.122) and (3.123) we can see that they have right limiting values. When xB+ = xY− = 1 we obtain mix Gex = 0. If any one of the ion fraction approaches zero, then the equation reduces properly to that for the binary systems. The equation can be improved somewhat, if the additive binary terms are more generalized to read like the left-hand side of Eq. (3.121). The non-random mixing term containing (r G0 )2 can cause an S-shaped character of the ln γAY versus xB+ xX− plot. For instance, when xB+ + xX− , the ln γAY curve will have a positive contribution from this term up to xB+ = 0.333 and negative at xB+ > 0.333. An example for this behavior is the LiF–KCl system.

3.1.4.4. Molecular model

Contrary to the concept of the random mixing of ions, Fellner (1984), Fellner and Chrenková (1987) proposed the molecular model for molten salt mixtures in which it is assumed that in an ideal molten mixture, molecules (ionic pairs) mix randomly. The model composition of the melt, i.e. the molar fractions of ionic pairs in the molten mixture, is calculated on the basis of simultaneous chemical equilibrium among the components of the mixture. For instance, in the melt of the system M1 X–M2 X–M2Y one can assume + – + – + – + – 2– random mixing of the ionic pairs M+ 1 ·X , M2 ·X , M2 ·Y , M1 ·X , and 2M2 ·XY . The applicability of the model for the description of the behavior of thermodynamic properties of molten mixtures is tested by comparing the experimental and calculated solid–liquid phase equilibria in the system studied. It should be noted that this model is formally similar to the approach wherein the calculation of the thermodynamic equilibrium molecules of constituents instead of ionic pairs are considered.

3.1.4.5. Thermodynamic model of silicate melts

Owing to their polymeric character, silicate melts belong to the solutions of type II, which do not follow Raoult’s law. The classic regular solution approach is not applicable, since the limiting laws are not obeyed. The Temkin’s model of ideal ionic solution, which has been widely applied in molten salt systems, cannot be used, since the real anionic composition, owing to a broad polyanionic distribution, is not known a priori. Any structural model of silicate melts should be in agreement with certain experimentally determined physico-chemical properties of the given silicate system: •

high equivalent conductivity, increasing with a decreasing cation size, the conductivity being of ionic character, at least for the cations of I A and II A groups;

136

• • • •

Physico-chemical Analysis of Molten Electrolytes

the activation energy of viscous flow is roughly constant within the range of 10–60 mole % M2 O (M = alkali metal); the volume expansion is up to 10 mole % M2 O almost zero, approaching asymptotically for higher M2 O contents the value at about 50 mole % M2 O; the compressibility increases with increasing M2 O content up to 10–15 mole %, remaining further on constant values up to the concentration of about 50 mole %; conductivity measurements have proved the presence of anions, studies of transport phenomena have shown that the O2− anions do not contribute to the charge transfer.

It can be assumed that the course of the above relationships will also be similar in the MeO–SiO2 systems. On the basis of the above facts, the structure of the MeO–SiO2 melts can be imagined as a lattice of SiO4 tetrahedrons polymerized to a certain degree, where cations are situated in the free spaces between the tetrahedrons. This concept was used by Pánek and Daneˇ k (1977) to formulate the thermodynamic model of silicate melts. In the thermodynamic model of silicate melts, the chemical potential of component is expressed as the sum of chemical potentials of all energetically distinguishable atoms constituting the given component. Depending on the composition, oxygen atoms can be present as free oxygen anions, bridging, and non-bridging oxygen atoms. All oxygen atoms are thus not equivalent, i.e. their chemical potential is not equal. The material balance of the available oxygen atoms and the Si–O bonds then gives the amounts of individual oxygen atoms. Silicon atoms are present exclusively in fourfold coordination, while other network forming atoms, e.g. B, Al, Fe, etc., may be present both as cations and/or central atoms in tetrahedral structural units. In the latter case, these atoms participate at least partially on the formation of the polyanionic network, which has to be taken into account in the calculation of the activity of the component as well. Activities of individual components are calculated on the basis of the theory of conformal solutions (Reiss et al., 1962). This theory was derived for ionic systems without the formation of complex ions, in which both anions and cations have identical charges. This theory was later applied to systems containing ions of various valences (Saboungi and Blander, 1975). It should be emphasized that the application of this theory to silicate melts has only a formal character. Let us define the polymerization degree of silicate melts as the fraction of bridging oxygen atoms in a single formula unit. The assumptions, on the basis of which activities of components in the mixture are calculated, are as follows: (a) (b)

linear dependence of the polymerization degree on MeO content throughout the tectosilicate–nezosilicate series is assumed, polymerization degree of SiO4 tetrahedrons does not change during melting,

Phase Equilibria

(c)

137

arrangement of polyanions in the melt in the vicinity of the melting point or the temperature of primary crystallization is identical at close distance to that in the crystalline state.

Let us now consider an arbitrary melt in the MeO–SiO2 system (Me = Mg, Ca, Fe, . . .). The melt is composed of the following kinds of atoms: • • • •

the Me2+ cations, silicon atoms in tetrahedral coordination surrounded by two kinds of oxygen atoms, bridging oxygen atoms, linking two neighboring SiO4 tetrahedrons by means of Si–O–Si covalent bonds, and non-bridging oxygen atoms, bound to the silicon atom by one covalent bond and creating the coordination sphere of Me2+ cations.

The two kinds of oxygen atoms have evidently different energetic states. Their mutual molar ratio defines the structure of the melt, i.e. its polymerization degree as well as the chemical potentials of the components. With regard to this structural aspect of silicate systems, the chemical potential of an arbitrary component may be defined as the sum of chemical potentials of all atoms forming the component considered, when their particular energetic states are taken into account. The chemical potential of the ith component in an arbitrary solution is defined by the relation µi =

ni, j µj

(3.124)

j

where ni, j is the amount of atoms of the jth kind in the ith component and µj is the chemical potential of atoms of the jth kind in the solution. For instance, the chemical potential of CaSiO3 , which is formed in the CaO–SiO2 system, equals the sum of the chemical potentials of calcium atoms, silicon atoms, and the bridging and non-bridging oxygen atoms. The activity of the ith component in the solution is also defined by the equation µi = µ0i + RT ln ai

(3.125)

where µ0i is the chemical potential of the pure ith component, defined similarly to the chemical potential of this component in the solution µ0i =

j

ni, j µ0i, j

(3.126)

138

Physico-chemical Analysis of Molten Electrolytes

where µ0i,j is the chemical potential of the pure jth atoms in the pure ith component. Substituting Eqs. (3.124) and (3.126) into Eq. (3.125) we obtain

ni, j µi, j =

j

ni, j µ0i, j + RT ln ai

(3.127)

j

The real mole fraction of the jth atoms in the ith component and in the solution are given by the relations ni, j yi,0 j = ni, j

j

ni, j xi i yj = xi ni, j i

(3.128)

(3.129)

j

where xi is the mole fraction of ith component in the solution. The chemical potentials of the jth atoms in the pure ith component and in the solution may also be expressed using Eqs. (3.128) and (3.129) in the following way 0 µ0i, j = µ+ j + RT ln yi, j

(3.130)

µj = µ+ j + RT ln yj

(3.131)

where µ+ j is the chemical potential of a hypothetical liquid composed exclusively of the jth atoms. Inserting Eqs. (3.130) and (3.131) into Eq. (3.127) we get for the activity of the ith component in the solution the relation ln ai =

ni, j ln yj −

j

ni, j ln yi,0 j

(3.132)

j

or after rearrangement ai =

j

yj 0 yi,j

ni, j (3.133)

Equation (3.133) for the activity of a component derived in this way is completely universal and may be used in any system. The calculation of the activity of a component in an actual system using this equation thus respects the characteristic features of silicate melts, e.g. the different energetic state of individual atoms of the same kind. Such cases may

Phase Equilibria

139

occur in the presence of three-valence atoms like B3+ , Al3+ , Fe3+ , etc., or four-valence atoms like Ti4+ . In order to describe the structural aspect of the given silicate melt, it is necessary to perform correctly the material balance of individual oxygen atoms as well as of the above-mentioned double-acting atoms. The material balance is based on the following principles. Let us assume that in the system consisting of m oxides, the polymeric network is formed by three- and fourfold coordinated atoms jA (j = 1, 2, . . . m) linked to the bridging (−O−) and non-bridging (−O− ) oxygen atoms. Denoting the fraction of the k-fold coordinated jA atoms as αk,j , the following inequality must hold α3,j + α4,j ≤ 1

(3.134)

The distribution of the atoms according to their coordination, or to their participation in a covalent network, is then determined by the following material balance n(jA) = n0 (jA) + n3 (jA) + n4 (jA)

(3.135)

n0 (jA) = (1 − α3,j − α4,j )n(jA)

(3.136)

n3 (jA) = α3,j n(jA)

(3.137)

n4 (jA) = α4,j n(jA)

(3.138)

where

where nk (jA) is the amount of the k-fold coordinated jA atoms and n0 (jA) is the amount of those jA atoms, which are not built into the polyanionic network. Then the amount of n(jA − O) bonds is given by the relation n(jA − O) =

m

[(3α3, j + 4α4, j )n(jA)]

(3.139)

j =1

Assuming that the total amount of oxygen atoms, n(O) equals the sum of the bridging, n(−O−), and non-bridging, n(−O− ), oxygen atoms, we can calculate their amounts from the material balances of the total amounts of oxygen atoms and the jA−O bonds according to the equations n(O) = n(−O − ) + n(−O− )

(3.140)

n(−O−) = n(jA − O) − n(O)

(3.141)

140

Physico-chemical Analysis of Molten Electrolytes

If the solution does not have a physical reason (i.e. the values of n(i) are negative), it is necessary to assume that non-bridging oxygen atoms and oxide ions O2− are present. The material balance is then given by the equations n(O) = n(−O− ) + n(O2− )

(3.142)

n(−O− ) = n(jA − O)

(3.143)

For the typical modifying atoms such as alkali metals and earth alkali metals it is assumed that α3,j = α4,j = 0

(3.144)

while for the typical network-forming elements of the IV group of the periodic system like silicon and germanium α4,j = 1. In other cases, the value of αk,j can be chosen in order to fit the calculated and experimentally determined phase diagrams. The formal polymerization degree P of the melt with a given composition can be calculated from Eqs. (3.140) and (3.141). Since the polymerization degree was defined as the fraction of the bridging oxygen atoms in the total oxygen atoms, then it can be written m

n(jA − O) − n(O) n(−O−) = = P = n(O) n(O)

j =1

[n(jA)(3α3,j + 4α4,j )] n(O)

−1

(3.145)

For instance, for the MeO–SiO2 system with the composition (1−x) MeO + x SiO2 it holds that n(jA) = x, m = 2, 1A = Me, 2A = Si, α3,1 = α3,2 = α4,1 = 0, and α4,2 = 1. The polymerization degree is then P =

4x −1 1+x

(3.146)

In such a melt for x = 0.333, i.e. for the composition of orthosilicate, is P = 0, and for the pure SiO2 melt, (x = 1), P = 1. The thermodynamic model of silicate melts has been applied in the calculation of various types of binary, ternary, and pseudo-ternary phase diagrams.

Application of the model to different systems.

(i) (ii)

simple binary and ternary eutectic systems, binary and ternary systems with the formation of several congruently and incongruently melting compounds,

Phase Equilibria

141

Table 3.2. Temperatures and enthalpies of fusion of different compounds Compound Al2 O3 (A) 3Al2 O3 ·2SiO2 (A3 S2 ) B2 O3 (B) CaO (C) CaO·Al2 O3 (CA) CaO·2Al2 O3 (CA2 ) 12CaO·7Al2 O3 (C12A7 ) 2CaO·Al2 O3 ·SiO2 (C2AS) CaO·Al2 O3 ·2SiO2 (CAS2 ) CaO·MgO·2SiO2 (CMS2 ) 2CaO·MgO·2SiO2 (C2 MS2 ) CaO·SiO2 (CS) 2CaO·SiO2 (C2 S) 3CaO·2SiO2 (C3 S2 ) CaO·TiO2 (CT) CaO·TiO2 ·SiO2 (CTS) MgO·Al2 O3 (MA) MgO·SiO2 (MS) 2MgO·SiO2 (M2 S) MnO·SiO2 (MS) Na2 O (N) Na2 O·B2 O3 (NB) Na2 O·2B2 O3 (NB2 ) Na2 O·3B2 O3 (NB3 ) Na2 O·4B2 O3 (NB4 ) K2 O (K) K2 O·B2 O3 (KB) K2 O·2B2 O3 (KB2 ) Li2 O·B2 O3 (LB) Li2 O·2B2 O3 (LB2 ) SiO2 (S) TiO2 (T)

(iii)

Tfus (K)

fus H (kJmol−1 )

2293 2123 723 2843 1878 2033 1728 1868 1826 1665 1727 1817 2403 1718 2243 1656 2408 1850 2171 1564 1405 1239 1016 1045 1088 1154 1223 1088 1117 1190 1996 2103

111.4 188.3 22.2 52.0 102.5 200.0 209.3 155.9 166.8 128.3 85.7 56.0 55.4 146.5 127.3 139.0 200.0 75.2 71.1 66.9 47.6 72.4 81.1 105.7 133.4 32.7 64.8 104.1 67.7 120.4 9.6 66.9

References Barin et al. (1973, 1977) Barin et al. (1973, 1977) Barin et al.(1973, 1977) Barin et al. (1973, 1977) Barin et al. (1973, 1977) Barin et al. (1973, 1977) Barin et al. (1973, 1977) Barin et al. (1973, 1977) Barin et al. (1973, 1977) Ferrier (1971) Barin et al. (1973, 1977) Barin et al. (1973, 1977) Barin et al. (1973, 1977) Barin et al. (1973, 1977) Estimated Nerád et al. (2000) Barin et al. (1973, 1977) Barin et al. (1973, 1977) Barin et al. (1973, 1977) Barin et al. (1973, 1977) Barin et al. (1973, 1977) Barin et al. (1973, 1977) Barin et al.(1973, 1977) Estimated Estimated Therm. Propert. (1965) Estimated Barin et al. (1973, 1977) Barin et al. (1973,1977) Barin et al. (1973, 1977) Barin et al. (1973,1977) Barin et al. (1973, 1977)

ternary systems with four crystallization areas, where the figurative point of the fourth crystalline phase lies beyond the pseudo-ternary diagram.

Silicate, alumino-silicate, ferro- and ferri-silicate, calcium–titanium–silicate, and alkali metal borate systems were chosen. The basic thermodynamic data of the individual components, i.e. the temperatures and enthalpies of fusion, were mainly taken from the literature and are summarized in Table 3.2. Data published by Bottinga and Richet (1978) were taken into account as well. However, for some components, the values of the enthalpy of fusion were not known. In such cases, these data were estimated on the basis of thermodynamic analogy. Besides, the full chemical formula and the acronym of the component is given in the table.

142

Physico-chemical Analysis of Molten Electrolytes

In the application of the model to calculate the phase diagrams, the liquidus temperature of component i, Ti,liq , is calculated using the values of the enthalpy and temperature of fusion according to the simplified and adapted Le Chatelier–Shreder equation Ti,liq =

fus Hi Tfus,i fus Hi − RTfus,i ln ai

(3.147)

where Tfus, i and fus Hi are the temperature and enthalpy of fusion of the ith component, respectively, and ai is its activity calculated according to Eq. (3.133). In most cases it could be assumed that fus Hi = constant. However, when necessary, i.e. at great differences between the melting and eutectic temperatures, the change of fus Hi with temperature could be expressed in the form fus Hi (T ) = fus Hi (Tfus ) − CP , s/l (Tfus − T )

(3.148)

The first crystallizing phase at the given composition is determined according to the condition Tpc = max(Ti, liq ) i

(3.149)

Pánek and Daneˇ k (1977) calculated various binary or pseudo-binary phase diagrams of the ternary system CaO–MgO–SiO2 . The systems, SiO2 –CaO·SiO2 , CaO·MgO·2SiO2 –SiO2 , CaO·SiO2 –CaO·MgO·2SiO2 , CaO·SiO2 – 2CaO·MgO·2SiO2 , CaO·MgO·2SiO2 –2CaO·MgO·2SiO2 , CaO·SiO2 –2CaO·SiO2 , and CaO·MgO·2SiO2 –2MgO·SiO2 were chosen. The experimental phase diagrams of these systems were taken from Levin et al. (1964, 1969, 1975) for comparison. The authors neglected in all cases the change in the enthalpy of fusion with temperature. In Figures 3.4 and 3.5, the comparison of the calculated and experimentally determined phase diagrams of the systems, CaSiO3 –Ca2 MgSi2 O7 and CaSiO3 –Ca2 SiO4 , respectively, are shown as examples. In the CS–C2 S system, the incongruently melting compound C3 S2 is formed, which affects the activity of CS or C2 S in the melt. For this reason, this fact should be considered in the calculation of the partial systems CS–C3 S2 and C3 S2 –C2 S.

The CaO–MgO–SiO2 system.

The comparison of liquidus surfaces of CaSiO3 calculated using the thermodynamic model of silicate melts with the experimental ones was used by Daneˇ k (1984) to study the real structure of melts of the CaO–“FeO”–SiO2 system in equilibrium with metallic iron and that of the CaO– “Fe2 O3 ”–SiO2 system in equilibrium with air (P (O2 ) = 21kPa), i.e. the participation of individual Fe(II) and Fe(III) atoms in the polyanionic network. The phase equilibrium

The CaO–“FeO”–SiO2 and CaO–“Fe2 O3 ”–SiO2 systems.

Phase Equilibria

143

1800

T(K)

1750

1700

1650

CS

0.2

0.4 0.6 x(C2MS2)

0.8

C2MS2

Figure 3.4. Phase diagram of the CaSiO3 –Ca2 MgSi2 O7 system.

2400

T(K)

2200

2000

C3S2

1800

1600 CS

0.2

0.4

x(C2S)

0.6

0.8

C2S

Figure 3.5. Phase diagram of the CaSiO3 –Ca2 SiO4 system.

144

Physico-chemical Analysis of Molten Electrolytes

in the four-component CaO–FeO–Fe2 O3 –SiO2 system has been studied by Muan and Osborne (1965) and Timucin and Morris (1970). Relatively unambiguous is the situation in the CaO–“FeO”–SiO2 system, where Fe(II) atoms are placed in the interstitial sites with higher than fourfold coordination. However, according to the results of density measurement performed by Lee and Gaskell (1974), these melts tend to micro-segregate into regions richer in calcium oxide and regions richer in iron oxide. On the other hand, Fe(III) atoms in the CaO–“Fe2 O3 ”–SiO2 system can enter the polyanionic network being in the fourfold coordination (network former) or, similar to the Fe(II) atoms, being in higher coordination and behave as the network modifier. The dependence of interatomic distance Fe–O on composition is similar to that of Si–Si, which corresponds to the transition of Fe(III) atoms from the octahedral sites to the tetrahedral ones. The calculated part of the phase diagram of the CaO–“FeO”–SiO2 system is shown in Figure 3.6. A similar part of the experimental phase diagram according to Muan and Osborn (1965) is shown in Figure 3.7 for comparison. The boundary curves of the CaSiO3 liquidus surface were also obtained by the calculation of the liquidus surfaces of Ca2 SiO4 and Fe2 SiO4 . The boundary curve with tridymite could not be determined because of a large immiscibility gap in the region of high SiO2 concentration, which the thermodynamic model does not take into account.

70

00

1100

°C

°C

°C 1200

CS

1300 °C

°C 1400

1500

CS 50

°C

xm ole %

SiO

2

60

10

C3S2 40 C2S

0 °C

100

C2S

F2S

30 0

10

20 30 x mole % FeO

40

50

Figure 3.6. Calculated part of the phase diagram of the CaO–“FeO”–SiO2 system.

Phase Equilibria

145

70

Rridymite

Pseudowl

°C

Wollastonite

°C 00

13

Ro

C2S

°C

nk

°C

00

14

C3S2 40

00

00

14

11

CS 50

1200 °C

xm

ole

%S

iO

2

60

Oilvine

C2S 30 0

10

20 30 x mole % FeO

40

50

Figure 3.7. Part of the phase diagram of the CaO–“FeO”–SiO2 system taken from Muan and Osborn (1965).

As it follows from Figures 3.6 and 3.7, a relatively good agreement between the experimental and calculated liquidus surface was obtained when all Fe(II) atoms present are in higher coordination and behave as the network modifier. The dotted lines in Figure 3.8 represent isotherms in the CaSiO3 liquidus surface calculated for the case, when all Fe(II) atoms behave as a network former. It is evident that such an assumption is not correct. However, some of the Fe(II) atoms in tetrahedral coordination cannot be excluded. The calculation of the CaSiO3 liquidus surface in the CaO–“Fe2 O3 ”–SiO2 system was carried out for two cases: (i) (ii)

all Fe(III) atoms are in the tetrahedral coordination and together with the SiO4 tetrahedrons participate in the polyanionic network formation, only half of the Fe(III) atoms is in tetrahedral coordination, while the other half behaves as the network modifier being in higher coordination.

The third possibility, that all Fe(III) atoms behave as network modifiers was, as improbable, not considered. The calculated part of the phase diagram of the CaO–“Fe2 O3 ”–SiO2 system is shown in Figure 3.8. The similar part of the experimental phase diagram according to Muan and Osborn (1965) is shown in Figure 3.9 for comparison. The boundary curves of the CaSiO3 liquidus surface were also obtained by the calculation of the liquidus surfaces

146

Physico-chemical Analysis of Molten Electrolytes

C

CS 50

°C

CS

C3S2 40

13 00 °C

00 14

13 00 °C

15 00 °

xm ole

%

SiO

2

60

Fe2O3

C2S

C2S 30 0

10

20 x mole % Fe2O3

30

Figure 3.8. Calculated part of the phase diagram of the CaO–“Fe2 O3 ”–SiO2 system.

60

Tridymite

C2S

Ra

nk

1300

13

°C

00

°C

00

°C

°C 14

xm

C3S2 40

00

15

CS 50

ole

%

SiO

2

Pseudowol

Hematite

C2S

P

30 0

10

20 x mole % Fe2O3

30

Figure 3.9. Part of the phase diagram of the CaO–“Fe2 O3 ”–SiO2 system taken from Muan and Osborn (1965).

of Ca2 SiO4 and Fe2 O3 . The other limitations concerning the boundary limits and the immiscibility gap are similar to the CaO–“FeO”–SiO2 system. The dotted lines in Figure 3.8 represent isotherms of the CaSiO3 liquidus surface for the first case (i). This assumption is obviously not fulfilled. Very good agreement with the experimental phase diagram was attained in the second case. It may therefore be

Phase Equilibria

147

stated that in the melts of the CaO–“Fe2 O3 ”–SiO2 system having the x(CaO)/x(Si O2 ) modulus in the range 0.6–1.5, roughly half of the Fe(III) atoms present is coordinated tetrahedrally and behaves as the network former, while the other one contributes to the polyanion destruction and behaves as the network modifier. Such an arrangement does 4− not exclude the possibility of the anion formation of the Fe2 O2− 4 and Fe2 O5 type as it is mentioned in the article of Mori and Suzuki (1968). The coordination of Al(III) atoms in aluminosilicate melts is one of the most important research directions in the structure of the melts containing alumina. This interest is associated with an extensive utilization of these melts in different sections of the silicate industry, such as glass, cement, porcelain production, etc. Conclusions on the structure of aluminosilicate melts are mainly based on the interpretation of physico-chemical properties, such as density and viscosity, in terms of composition, particularly of the x(CaO)/x(Al2 O3 ) ratio. The basic idea was the concept that in melts with the ratio x(CaO)/x(Al2 O3 ) ≥ 1, all the Al(III) atoms are in tetrahedral coordination and that the octahedral coordination of Al(III) atoms occurs only at ratios x(Ca)/x(Al2 O3 ) < 1, i.e. at the excess of alumina with respect to CaO. The coordination of Al(III) atoms in the CaO–Al2 O3 –SiO2 system melts was studied using the thermodynamic model of silicate melts by Liška and Daneˇ k (1990). This system is rather complicated because of the formation of a number of binary and ternary compounds. The calculation was performed on the assumption that one half of the Al(III) atoms are in tetrahedral coordination over the entire concentration range in question, and that the other half of the Al(III) atoms are in higher coordination, obviously in the octahedral one. The latter part thus does not participate in the formation of the polyanionic network. The calculated phase diagram of this system is shown in Figure 3.10, while the experimentally determined phase diagram according to Muan and Osborn (1965) is shown in Figure 3.11. The immiscibility region near the SiO2 apex was neglected in the calculation as such behavior is not considered in the thermodynamic model. Furthermore, because of the lack of thermodynamic data, the crystallization of rankinite, tricalcium silicate, tricalcium aluminate, and calcium hexaaluminate was not included in the calculation. From the comparison of the calculated and experimentally determined phase diagram of the CaO–Al2 O3 –SiO2 system, it follows that the Al(III) atoms are in partial tetrahedral coordination also in the composition range for x(CaO)/x(Al2 O3 ) ≥ 1. This finding is relatively surprising considering the available information on the behavior of Al(III) atoms in silicate melts. However, it is in agreement with a very similar behavior of Fe(III) atoms in the melts of the CaO–Fe2 O3 –SiO2 system. It can be concluded that the thermodynamic model of silicate melts is very suitable for describing phase equilibrium also in aluminosilicate systems.

The CaO–Al2 O3 –SiO2 system.

148

Physico-chemical Analysis of Molten Electrolytes

SiO2

CS

CAS2

C2S A3S2

C2AS

20

CaO

40 C12A7 60CA CA1280 Mass %

Al2O3

Figure 3.10. Calculated phase diagram of the CaO–Al2 O3 –SiO2 system.

SiO2

CS C3S2

CAS2

C2S C3S

C2AS

20 CaO

A3S2

C3A40 C12A7 60CA CA280 CA6 Al2O3 Mass (%)

Figure 3.11. Experimental phase diagram of the CaO–Al2 O3 –SiO2 system according to Muan and Osborn (1965).

Phase Equilibria

149

Silicate systems containing TiO2 are of considerable technological and geochemical interest. Titanium dioxide is a common component of industrial glasses, enamels, pyro-ceramics, and of some metallurgical slags. The structural role of Ti(IV) in silicate melts has been studied in many spectroscopic investigations, e.g. by Yarker et al. (1986), Abdrashitova (1980), Schneider et al. (1991), Mysen and Neuville (1995), and Liška et al. (1995). It is a complex function of several variables, namely TiO2 and SiO2 concentration, type, and content of the modifying cations and temperature. Despite the number of investigations, a consensus has been reached neither regarding the coordination state of Ti(IV) atoms, nor on how the structure of the melts is modified by their presence. The results obtained by various methods are often contradictory. In situ high-temperature Raman spectroscopy of melts along the Na2 Si2 O5 −Na2 Ti2 O5 join performed by Mysen and Neuville (1995) has shown that the Raman spectra of Ti-bearing glasses and melts are consistent with Ti(IV) in at least three different structural positions: The CaO–TiO2 –SiO2 system.

(a) (b) (c)

Ti(IV) substitutes for Si(IV) in tetrahedral coordination in structural units of the melt (acts as network former), Ti(IV) forms TiO2 -like clusters with Ti(IV) in tetrahedral coordination, Ti(IV) as network modifier, possibly occurring in octahedral or fivefold coordination.

The structural behavior of Ti(IV) was also determined by Nerád and Daneˇ k (2002) by the calculation of phase diagrams of the pseudo-binary systems, CaSiO3 –CaTiSiO5 , CaSiO3 –CaTiO3 , Ca2 SiO4 –CaTiO3 , CaTiSiO5 –CaTiO3 , and CaTiSiO5 –TiO2 , in the binary system CaO–TiO2 , as well as in the whole ternary system CaO–TiO2 –SiO2 . The comparison of the calculated phase diagram of the ternary system CaO–TiO2 – SiO2 (Figure 3.12) and that of the experimentally determined (De Vries et al., 1955), Figure 3.13) is shown for demonstration. In the CaO–TiO2 –SiO2 system the following phases are present: CaO, TiO2 , SiO2 , Ca3 SiO5 , Ca2 SiO4 , Ca3 Si2 O7 , CaSiO3 , CaTiO3 , Ca3 Ti2 O7 , and CaTiSiO5 . Some existing phases, namely rankinite and tricalcium silicate, were not included in the calculation because of lack of relevant thermodynamic data. The calculation was performed taking into account that all the Ti(IV) atoms present are in tetrahedral coordination, i.e. they act as network formers. In the region of high content of SiO2 , the calculation of the phase equilibrium fails as the formation of two liquids is not considered in the thermodynamic model of silicate melts. This is also the reason for the enlarged liquidus surface of CaTiSiO5 up to the high content of silica.

150

Physico-chemical Analysis of Molten Electrolytes

SiO2 0.8

0.8

2

x(S iO )

0.6

0.6

CaSiO3

0.4

0.4 CaTiSiO5

Ca2SiO4

0.2

0.2

Ca3Ti2O7 CaTiO3

CaO

0.2

0.4

x(TiO2)

0.6

0.8

TiO 2

Figure 3.12. Calculated phase diagram of the CaO–TiO2 –SiO2 system.

SiC2 1713˚

1698˚

1550˚

1780˚ Silico

1650

1698˚

1600

00

16 00

15

CaSiO3 1318° ° 98 13

16

00

° 25 14

° 03 14

1353° 1348°

1373 Rutile

Sphere

1800

CT5 1382°

1365° 1375°

1375°

00 °

1335°

19

Ca3SiO3

00 20

2130˚-Ca3SiO4 2065˚

1700 1600 1535

00

14

CaSiO3 1544˚ 1463˚ 1467˚ Ca3Si2O7

1670° ? Ca0

1670° ?

Perovskite

16

1365° Rutile

1650°

16

00

00

2000

2200

2400

1800

00

15

18

CaO 2570˚

00

1695˚1780˚ Ca3Ti2O7 Ca3Ti2O3 1970˚

1460˚

1760˚

Ti2O3 1830˚

Figure 3.13. Experimental phase diagram of the CaO–TiO2 –SiO2 system according to De Vries et al. (1955).

Phase Equilibria

151

In the TiO2 -rich region, the experimental liquidus surface of TiO2 fits best the simple model ai = xi . The reason may be the basic nature of the TiO2 -rich melts. More probable is that not all the Ti(IV) atoms are in tetrahedral coordination. From the comparison of the calculated and experimental phase diagrams, it follows that the thermodynamic model of silicate melts is suitable for the description of the phase equilibrium also in titania-bearing silicate systems and provides deeper information on the behavior of Ti(IV) atoms. It was, however, shown that Ti(IV) atoms behave in silicate melts as network formers, except in the region of its high concentration, and in highly basic melts. Liška and Daneˇ k (1990) also calculated the phase diagrams of the following ternary and pseudo-ternary systems:

Other silicate systems.

• • • •

MgO·SiO2 –CaO·MgO·2SiO2 –CaO·Al2 O3 ·SiO2 , CaO·MgO·2SiO2 –MnO·SiO2 –CaO·Al2 O3 ·2SiO2 , MgO·Al2 O3 –2CaO·SiO2 –2CaO·Al2 O3 ·SiO2 , 2CaO·Al2 O3 ·SiO2 –MgO·Al2 O3 –CaO·Al2 O3 ·2SiO2 .

The experimentally determined phase diagrams were taken from Levin et al. (1964, 1969, 1975). It was again assumed that one half of the Al(III) atoms are in tetrahedral coordination and the other half are in higher coordination and behave as network modifiers. The last system with four crystallization areas is interesting where the figurative point of the fourth crystalline phase lies beyond the pseudo-ternary diagram. The calculated and experimentally determined phase diagrams of this system are shown in Figures 3.14 and 3.15, respectively.

Cor u

1500

ndu 15 m 50

CAS2

00

1550

16

C2AS

20

40 60 Mass (%)

80

MA

Figure 3.14. Calculated phase C2AS–MA–CAS2 system.

152

Physico-chemical Analysis of Molten Electrolytes CAS2

Corundum

00

00

15

15

16

00

1600

C2AS

20

40 60 Mass (%)

80

MA

Figure 3.15. Experimental phase diagram of the diagram of the C2AS–MA–CAS2 system according to Levin et al. (1964, 1969, 1975).

As mentioned in Section 2.1.10, in the binary glass-forming systems of alkali metal borates there is the ability of boron to change from the threefold to fourfold coordination. This observation was verified by Daneˇ k and Pánek (1979) on calculating the liquidus curves in alkali metal borate systems using the thermodynamic model of silicate melts adapted according to the characteristic feature of borate melts. These authors calculated the liquidus curves in binary systems, B2 O3 –Na2 O·4B2 O3 , Na2 O·4B2 O3 –Na2 O·2B2 O3 , Na2 O·2B2 O3 –Na2 O·B2 O3 , Li2 O·2B2 O3 –Li2 O·B2 O3 , and K2 O·2B2 O3 –K2 O·B2 O3 . The values of the enthalpy and temperature of fusion were taken from Barin et al. (1973, 1977). In the calculation of the activity of components, the numbers of individual kinds of atoms, i.e. the M+ cations, the boron atoms in the threefold and fourfold coordination, and the bridging and non-bridging oxygen atoms, in the pure components were taken into account. The numbers of individual kinds of atoms in the formula unit of pure components are listed in Table 3.3. Systems of alkali metal borates.

Table 3.3. Numbers of individual kinds of atoms in pure alkali metal borates Component* B2 O3 (B) M2 O·4B2 O3 (MB4 ) M2 O·3B2 O3 (MB3 ) M2 O·2B2 O3 (MB2 ) M2 O·B2 O3 (MB) * M = Li, Na, K.

M+

B(3)

B(4)

–O–

–O–

– 2 2 2 2

2 6 4 2 1

– 2 2 2 1

3 13 10 6.5 2.5

– – – 0.5 1.5

Phase Equilibria

153

The number of three- and four-coordinated boron atoms in the formula unit was calculated according to Eq. (2.124). The calculation of the non-bridging oxygen atoms was based on the experimental data measured by Bray and O’Keefe (1963). For illustration, some of the calculated phase diagrams are shown. The experimental and calculated phase diagrams of the Na2 O·4B2 O3 –B2 O3 systems and Na2 O·4B2 O3 – Na2 O·2B2 O3 are shown in Figures 3.16 and 3.17, respectively. In the calculation of the liquidus curve of the Na2 O·4B2 O3 –Na2 O·2B2 O3 system the presence of the incongruently melting compound Na2 O·3B2 O3 has been taken into account. There is slightly less agreement with experimental liquidus curves as observed in the Li2 O·2B2 O3 –Li2 O·B2 O3 and K2 O·2B2 O3 –K2 O·B2 O3 systems, where the assumption of the similarity of the structure of solid and liquid phases holds to a lesser degree. The experimental and calculated phase diagrams of the Li2 O·2B2 O3 –Li2 O·B2 O3 and K2 O·2B2 O3 –K2 O·B2 O3 systems are shown in Figures 3.18 and 3.19, respectively. The effect of different polarizing ability of the individual alkali metal cations plays evidently a more substantial role. From the comparison of the experimental and calculated liquidus curves, it follows that the thermodynamic model of silicate melts describes satisfactorily the courses of liquidus curves in these complex glass-forming systems. It is expected that this model can also be applied in other inorganic glass-forming systems, like germanates, phosphates, etc.

1100 1050

T(K)

1000 950

850 800 NB4

0.2

0.4

NB9

NB5

900

0.6

x(B)

0.8

B

Figure 3.16. Phase diagram of the Na2 O·4B2 O3 –B2 O3 system. Dashed line – calculated.

154

Physico-chemical Analysis of Molten Electrolytes

1125 1100

T(K)

1075 1050

NB3

1025 1000 975 NB2

0.2

0.4 0.6 x(NB4)

0.8

NB4

Figure 3.17. Phase diagram of the Na2 O.4B2 O3 –Na2 O·2B2 O3 system-Dashed line – calculated.

1225

1200

T(K)

1175

1150

1125

1100

1075 LB

0.2

0.4 0.6 x(LB2)

0.8

LB2

Figure 3.18. Phase diagram of the Li2 O·2B2 O3 –Li2 O·B2 O3 system. Dashed line – calculated.

Phase Equilibria

155

1250

1200

T(K)

1150

1100

1050

1000 KB

0.2

0.4

0.6

0.8

KB2

x(KB2) Figure 3.19. Phase diagram of the system K2 O·2B2 O3 –K2 O·B2 O3 . Dashed line – calculated.

3.2.

PHASE DIAGRAMS OF CONDENSED SYSTEMS

In the following chapters, examples of phase diagrams of two-, three-, and multicomponent systems are described with the characterization of their topology, crystallization paths, transformation processes, and the equilibrium of the coexisting phases in agreement with the Gibbs phase law. Emphasis will be put first to phase diagrams frequently used in the molten salts technology, i.e. those with simple eutectics, with solid solutions, with the formation of complex compounds melting congruently and incongruently, and with the polymorphic transformation of components. 3.2.1. Binary systems

The phase diagrams of two-component systems are represented in the two-dimensional space, where the composition is shown on the x axis (in molar or in mass fractions) in agreement with the lever rule, and the temperature is given on the y axis (in ◦ C or in Kelvin). They are the so-called isobaric diagrams, since the constant pressure, mostly the atmospheric one, is assumed. The Gibbs phase law attains thus the form v =k−f +1=2−f +1=3−f

(3.150)

156

Physico-chemical Analysis of Molten Electrolytes

The individual curves or straight lines in the phase diagrams, which are denoted as boundary lines, represent the equilibrium between two phases. In a phase diagram of the binary system, the plane above both the boundary lines has the highest degree of freedom possible, i.e. v = 2. On the boundary lines, the degrees of freedom decreases to v = 1 and in the eutectic point, where the boundary lines meet, the degree of freedom is equal to zero. In the following text, the composition in all cases will be expressed in the coordinates of the molar fraction of the component B, thus x(B). 3.2.1.1. Simple eutectic systems

The phase diagram of a simple two-component eutectic system contains four planes (Figure 3.20). The plane L represents the region of presence of the homogeneous solution of the components A and B. The plane A + L is the region of coexistence of the crystals of the solid phase A and the melt saturated with the component A, the plane B + L is the region of coexistence of the crystals of the solid phase B and the melt saturated with the component B. Finally, the plane A + B is the region of coexistence of both the solid phases A and B, it is thus their solid, mechanical mixture.

Tfus, A

X

L Tfus, B Tpc, A

A+L

T nL

nA B+L

T3 A+B A

x(B)

x1(B)

xe

Figure 3.20. Phase diagram of the simple binary eutectic system.

B

Phase Equilibria

157

Points Tfus, A and Tfus, B represent the melting points of components A and B. The lines starting from the individual melting points represent curves of primary crystallization, i.e. their liquidus curves. Both the liquidus curves meet at the eutectic temperature Te in the eutectic point with the composition xe . The eutectic temperature is the lowest temperature at which the liquid phase is present in the system. At cooling of the melt with composition x(B), from the temperature representing to the figurative point X in the plane L, the composition of the melt does not change until the point Tpc, A is attained. In the plane L, the system has two degrees of freedom (k = 2, f = 1, v = 2), thus we can arbitrarily change the temperature and composition of the melt without the appearance of a new phase. At the point Tpc, A , first crystals of component A appear, and the melt starts to coexist with the solid phase A. The cooling of the melt slows down owing to the evolution of the crystallization heat of component A. The system has one degree of freedom (k = 2, f = 2, v = 1), we can change the temperature, and the composition is already defined. This situation is shown at the temperature T, where the solid component A coexists with the melt with composition x1 (B). The amount of the solid phase and the melt is given by the lever rule, the system is composed of nA mol of component A and nL mol of melt with composition x1 (B). When the systems cool down to the eutectic temperature also the solid phase B starts to crystallize. At the eutectic temperature the system has no degree of freedom (k = 2, f = 3, ν = 0), which means that its cooling stops due to the evolution of the crystallization heat of component B, even when the surroundings cool down further. The system stays at the eutectic temperature until its full solidification. A wholly analogical situation is also in the plane of the primary crystallization of component B. Below the eutectic temperature the system has again one degree of freedom (k = 2, f = 2, v = 1), at the given composition the temperature can be arbitrarily changed.

3.2.1.2. Systems with the formation of solid solutions

Solid solutions can be substitutional or interstitial. In the case of substitutional solid solutions, the foreign atoms substitute the atoms of the host structure. In the case of interstitial solid solutions, the foreign atoms are dislocated in free spaces of the host structure. The phase diagram of the two-component eutectic system with the formation of a solid solution of one component has five planes (Figure 3.21). The plane L represents the region of homogeneous solution of components A and B. The plane Ass + L is the region of coexistence of crystals of the saturated solid solution of the component B in component A and the melt saturated with the component A. The plane B+L is the region of coexistence of the crystals of the pure solid phase B and the melt saturated with the component B. The plane Ass is the region of non-saturated solid solutions of the component B in component

158

Physico-chemical Analysis of Molten Electrolytes

Tfus, A

L

X

Tfus, B Tpc, Ass Ass + L NL nAss

B+L

Ass

Te

Ass + B A

x(Ass)

x(B) x1(B)

B

Figure 3.21. Phase diagram of the two-component eutectic system with formation of solid solution Ass.

A and, finally, the plane Ass + B is the region of coexistence of both solid phases, i.e. the crystals of the saturated solid solutions of the component in component A and the crystals of the component B. It deals thus again with the mechanical mixture of both the solid phases. Points Tfus, A and Tfus, B represent the melting points of components A and B. The upper line, which originates from the melting point of the component A, represents the curve of its primary crystallization, the line exiting from the melting point B represents the curve of the primary crystallization of the component B. Both the curves of primary crystallization meet at the eutectic temperature Te in the eutectic point with the composition xe . The lower line originating in the melting point of the component A, breaks at the eutectic temperature and at lower temperatures approaches the y axis and represents the line of saturated solid solutions of the component B in component A. From the course of this curve, it follows that the composition of the saturated solid solutions changes with temperature. At cooling of the melt with the composition x(B) from the temperature representing the figurative point X, the composition of the melt does not change in the plane L until

Phase Equilibria

159

the point Tpc, Ass is reached. In the plane L the system has two degrees of freedom (k = 2, f = 1, v = 2); we can arbitrarily change the temperature and composition without occurrence of a new phase. At further cooling at the point Tpc, Ass , the first crystals of the saturated solid solution Ass arise, the melt starts to coexist with the solid solution Ass. Due to the evolution of the crystallizing heat of the solid solution Ass, the cooling of the system slows down. The system has one degree of freedom (k = 2, f = 2, v = 1), we can change the temperature, and the composition of the melt is already defined. At further cooling, the solid solution Ass coexists with the melt of the composition x1 (B). The amount of the solid phase and of the melt is given by the lever rule, the system is composed of the nAss mol of the solid solution Ass and n1 mol of the melt with the composition x1 (B). When the system cools down to the eutectic temperature the solid phase B also starts to crystallize. At the eutectic temperature, the system has no degree of freedom (k = 2, f = 3, v = 0), which means that due to the evolution of the crystallization heat of the component B the cooling stops, even though the surroundings further cools down. The system stays at the eutectic temperature until it totally freezes. Below the eutectic temperature, the system again has one degree of freedom (k = 2, f = 2, v = 1), at the given composition we can arbitrarily change the temperature.

3.2.1.3. Systems with limited solubility in the liquid phase

As a consequence of impossibility to arrange particles in the liquid phase in one solution only, the formation of two liquids, which are not miscible, may occur. Such a situation can be frequently observed in glass-forming silicate melts in the region of high concentrations of SiO2 . Partial miscibility in the liquid phase is quite uncommon in binaries of molten salts. Partial miscibility is more frequent in reciprocal salt systems and probably as a rule, in the high metal concentrations of the metal–metal halide systems. The phase diagram of the two-component eutectic system with limited miscibility in the liquid phase has five planes (Figure 3.22). The plane L represents the region of existence of the homogeneous solution of the components A and B. The plane A + L is the region of coexistence of the crystals of component A and the melt saturated with component A. The plane B + L represents the region of coexistence of crystals of the pure component B and the melt saturated with component B, and the plane A + B is the region of coexistence of both the solid phases A and B. The plane L1 + L2 is the region of coexistence of two so-called conjugate liquids L1 and L2 . The phase admixture in the melt begins at the so-called critical temperature Tcrit . On cooling the melt of the composition x(B) from the temperature represented by the figurative point X, the first drops of the conjugate liquid having the composition x1 (B) can be observed on the admixture curve at the point Tad . The system has only one degree of freedom (k = 2, f = 2, ν = 1), we can change only the temperature, and

160

Physico-chemical Analysis of Molten Electrolytes

X Tfus, A Tcrit Tad nL 1

L1

L

nL L2 2

Tfus, B

L1 + L2

Tm

nA

nL

Tpc, A

A+L

B+L

Te

e A+B A

xL 1

x 1(B ) x(B )

xL x'(B) 2

B

Figure 3.22. Phase diagram of the two-component eutectic system with limited miscibility in the liquid phase.

the composition of both conjugate liquids is given by the admixture curve. On further cooling, the composition of both the conjugate liquids is given by the points L1 and L2 . The amounts of both the conjugate liquids are given again by the lever rule. On further cooling, at the monotectic temperature Tm , an invariant equilibrium exists between the solid phase A and the two liquid phases L1 and L2 having the composition xL1 and xL2 . At this temperature the monotectic reaction L1 = A + L2

(3.151)

occurs. The cooling stops since k = 2, f = 3, ν = 0 until all L2 disappears. On continued cooling, the system behaves similar to the simple eutectic system. 3.2.1.4. Systems with unlimited solubility in solid and liquid phases

The common feature of these phase diagrams is that the melt is in univariant equilibrium with one crystalline phase of a variable composition. A complete series of solid solutions are formed.

Phase Equilibria

161

L

Tfus, A L+

(A

X +B

) ss

Tpc nL n (A + B)ss

Tfus, B

(A + B)ss

A

x'(A)

x(A)

B

Figure 3.23. Phase diagram of the system of continuous solid solubility in the solid and liquid phases.

Systems with one two-phase region. The phase diagram of the system with unlimited solubility in the solid and liquid phases having one two-phase region is shown in Figure 3.23. There are three planes in this phase diagram. The plane L represents the region of existence of homogeneous liquid solution of compounds A and B. The plane (A + B)ss + L is the region of the coexistence of the crystals of the solid solutions Ass and the melt saturated with component B. The plane (A + B)ss represents the region of continuous solid solutions of phases A and B. During cooling the mixture of the composition x(A), at Tpc solid solution Ass with the composition x (A) starts to crystallize. On further cooling, the composition of the saturated liquid follows the upper curve, while the composition of the solid solutions follows the lower one. The amounts of the saturated liquid L and the solid solution Ass change according to the lever rule. At the lower curve, the liquid phase disappears and the mixture solidifies. The phase diagram of the system with unlimited solubility in the solid and liquid phases having two two-phase regions is shown in Figure 3.24. Such phase diagrams have four planes. The plane L is the region of homogeneous liquid solutions of A and B. The plane L + (A + B)ss is the region of coexistence of crystals of the solid solution (A + B)ss and the melt saturated with Systems with continuous solid solutions with a minimum.

162

Physico-chemical Analysis of Molten Electrolytes

Tfus, A L

Tfus, B

L

L+ + (A

(A

+B

ss B)

)s

s+

X

Tm

m (A + B) ss

A

xm

B

Figure 3.24. Phase diagram of the system with continuous solid solutions with a minimum.

components A. The plane (A + B)ss + L is the region of coexistence of crystals of the solid solution (A + B)ss and the melt saturated with component B. Finally, the plane (A + B)ss represents the region of the continuous solid solutions of phases A and B. The crystallization path of a mixture of any composition except of xm , is similar as in the previous case of the systems with one two-phase region. When cooling the mixture of composition xm , at temperature Tm the liquid and solid phases are in equilibrium and they have identical composition. The system seems to be univariant, since there are two components, two phases, and thus one degree of freedom. However, in the case of an extreme on the boundary line, the degree of freedom decreases by one. The system in the minimum is thus non-variant and the cooling of the system stops until all the melt completely freezes. 3.2.1.5. Systems with the formation of a congruently melting compound

Even when these systems have formally three components, only two of them are independent, since the number of components decreases by the number of reactions that take place among them. In this case, it is the reaction A + B = AB. Sometimes the phases A and B are called components and the phase AB is called a constituent.

Phase Equilibria

163

L

Tfus, A T fus, AB

X1

A+L

X2 Tfus, B

T pc, AB

T pc, A AB + L

nL AB + L

nA

Tel

nL

n AB

e1

B+L

Te2

e2 A + AB AB + B A

x1(B)x'1(B)

AB x2(B)x'2(B)

B

Figure 3.25. Phase diagram of the two-component eutectic system with the formation of the congruently melting compound.

The phase diagram of the two-component eutectic system, in which one congruently (explicitly) melting compound is formed, has seven planes (Figure 3.25). The compound AB divides the phase diagram into two simple eutectic systems. The plane L represents the region of existence of the homogeneous solution of compounds A and B. The plane A + L is the region of coexistence of the crystals of the solid phase A and the melt saturated by the component A. The plane B + L is the region of coexistence of the crystals of the solid phase B and the melt saturated by the component B. Both the divided planes AB + L are the regions of coexistence of the component (constituent) AB with the respective melts saturated with the component (constituent) AB. The planes A + AB and AB + B are the regions of coexistence of the two solid phases A and AB, and AB and B, respectively. Points Tfus, A , Tfus, B and Tfus, AB represent melting points of the components A and B, and AB that of the compound AB. Lines originating from the individual melting points represent the curves of their primary crystallization. The curves of primary crystallization meet at the eutectic temperatures Te1 and Te2 at the eutectic points e1 and e2 . The crystallization processes of components in the systems of this type are identical as in the simple eutectic systems.

164

Physico-chemical Analysis of Molten Electrolytes

3.2.1.6. Systems with the formation of an incongruently melting compound

The phase diagram of the two-component eutectic system in which the incongruently (hidden) melting compound is formed has six planes (Figure 3.26). But in this case the compound formed does not divide the phase diagram into two simple eutectic systems. The plane L represents the region of existence of the homogeneous solution of components A and B. The plane A + L is the region of the coexistence of crystals of the solid phase A and the melt saturated with component A. The plane A4 B + L is the region of the coexistence of crystals of the solid A4 B and the melt saturated with component A4 B. The plane B + L is the region of coexistence of crystals of the solid phase B and the melt saturated with component B. The planes A + A4 B and A4 B + B are the regions of coexistence of two solid phases A and A4 B and A4 B and B, respectively. The points Tfus, A and Tfus, B are again the melting points of components A and B. The lines originating from the individual melting points represent the curves of their primary crystallization. The curve of primary crystallization of the component A stops at the peritectic temperature at the peritectic point P. Below the peritectic temperature, the curve

X1 Tfus, A

L Tpc, A X2 nL

Tp

nA

Tfus, B

A+L

P

A4B+ L nL

A+A4B

Tpc, A4B

B+L

nA4B

Te

e

A4B+B A x1(B) A4B x'1(B) x2(B)x'2(B)

B

Figure 3.26. Phase diagram of the two-component eutectic system with the formation of the incongruently melting compound.

Phase Equilibria

165

of the primary crystallization of A4 B continues. The curves of primary crystallization of the components A4 B and B meet at the eutectic temperature Te in the eutectic point e. At cooling of the melt with the composition x1 (B) from the temperature of the figurative point X1 the first crystals of component A arise at the temperature Tpc, A , the melt starts to coexist with the solid phase A. The cooling of the system slows down due to the evolution of the crystallization heat of the component A. The system has one degree of freedom (k = 2, f = 2, v = 1). Below this temperature the solid component A coexists with the melt saturated with the component A. For the case shown in Figure 3.26 the saturated melt has the composition x1 (B). The amount of the solid phase and of the melt is given by the lever rule, the system is composed of nA mol A and nL mol of melt with the composition x1 (B). When the system attains the peritectic temperature the peritectic reaction A + L = A4 B starts and crystals of the compound A4 B appear. Since now there are three phases in the system (component A, compound A4 B, and melt L), the system has no degree of freedom (k = 3 – 1 = 2, f = 3, v = 0), which means that its cooling due to the evolution of the reaction heat of the peritectic reaction must halt, even when the surrounding cools further. The system stays at the peritectic temperature until the melt disappear and the system solidifies. Below the peritectic temperature, there is again a mechanical mixture of the crystals of component A and the crystals of compound A4 B. At cooling of the melt with composition x2 (B) from the temperature representing the figurative point X2 , the first crystals of the compound A4 B arise at the temperature Tpk,A4B and the melt coexists with the solid compound A4 B. The cooling of the system slows down due to the evolution of the crystallization heat of the compound A4 B. The system has one degree of freedom (k = 2, f = 2, v = 1). Below this temperature, the solid compound A4 B coexists with the melt having the composition x2 (B). The amount of the solid phase and that of the melt is given by the lever rule. The system is composed of nA4 B mol A4 B and nL mol melt with the composition x2 (B). When the system attains the eutectic temperature the solid phase B also starts to crystallize. At the eutectic temperature, the system has no degree of freedom (k = 2, f = 3, v = 0) and its cooling will stop. The system maintains the eutectic temperature until its total solidification. In the field of primary crystallization of the component B, the crystallization proceeds like in the simple eutectic system. Below the eutectic temperature, we have a mechanical mixture of crystals A4 B and B, the system has again one degree of freedom (k = 2, f = 2, v = 1).

3.2.1.7. Systems with polymorphic transformation of one component

In this case one of the components exists in two or more crystallographic modifications and the temperature of the highest polymorphic transformation lies above the eutectic temperature of the system.

166

Physico-chemical Analysis of Molten Electrolytes

X1

L

Tfus, A

A+L nL Tpt

Tpc, A

X2

Tfus, B

nAα

A+L

nA

Tpc, A

B+L

nL nL Te

nAβ e

A+ B A

x1(B)

x'1(B)

x2(B) x'2(B) x''1(B)

B

Figure 3.27. Phase diagram of the two-component eutectic system with polymorphic transformation of one component.

The phase diagram of the two-component eutectic system with the polymorphic transformation of one component has five planes (Figure 3.27). The plane L represents the region of the existence of a homogeneous solution of the components A and B. The plane Aα + L is the region of coexistence of the crystals of the α-modification of the solid phase A and the melt saturated with the component Aα . The plane Aβ + L is the region of coexistence of the crystals of the β-modification of the solid phase A and the melt saturated with the component Aβ , and the plane B + L is the region of coexistence of the crystals of the solid phase B and the melt saturated with the component B. Finally, the plane Aβ + B is the region of coexistence of two solid phases Aβ and B. The points Tfus, Aα and Tfus, B represent the melting points of the components Aα and B. The lines originating in the individual melting points represent the curves of their primary crystallization. The curve of the primary crystallization of component Aα ends at the temperature of the polymorphic transformation Tpt . Below this temperature, this curve continues as the curve of the primary crystallization of component Aβ . The curves of primary crystallization of components Aβ and B meet at the eutectic temperature Te at the eutectic point e.

Phase Equilibria

167

Cooling the melt with composition x1 (B) from the temperature of the figurative point X1 , the first crystals of the component Aα arise at temperature Tpc, Aα , where the melt coexists with the solid phase Aα . Below this temperature, the solid component Aα coexists with the melt of the composition x1 (B). The amount of the solid phase and the melt is given by the lever rule. The system is composed of nAα mol Aα and of nL mol melt of the composition x1 (B). When the system attains the temperature of the polymorphic transformation of the component A, Tpt , the polymorphic transformation Aα → Aβ

(3.152)

starts to take place and the crystals of the β-modification of the solid phase A arise. Since three phases (Aα , Aβ , and the melt L) are present, the system has no degree of freedom (k = 3 − 1 = 2, f = 3, v = 0), which means that its cooling must stop due to the evolution of the transformation heat. The system keeps at the polymorphic transformation temperature until all crystals of the α-modification of the component A transform to the β-modification. Below the temperature of the polymorphic transformation, the crystals of the β-modification coexist with the melt of the composition x1 (B). The amount of the solid phase and of the melt is given by the lever rule and the system is composed of nAβ mol Aβ and nL mol melt with the composition x1 (B). Cooling the melt with the composition x2 (B) from the temperature of the figurative point X2 , the first crystals of Aβ arise at the temperature Tpc, Aβ , the melt starts to coexist with Aβ . Below this temperature, Aβ coexists with the melt of composition x2 (B). The amount of the solid phase and of the melt is given by the lever rule, the system is composed of nAβ mol Aβ and nL mol melt with the composition x2 (B). When the system cools down to the eutectic temperature, the solid phase B also starts to crystallize, the system has no degree of freedom (k = 2, f = 3, v = 0) and its cooling stops. Below the eutectic temperature we have the mechanical mixture of crystals Aβ and B, the system has again one degree of freedom (k = 2, f = 2, v = 1). 3.2.2. Ternary systems

The isobaric phase diagrams of three-component systems are determined by four variables: three concentration coordinates and the temperature. With regard to the condition that the sum of mole or mass fractions equals one, it is possible to show these phase diagrams in three-dimensional space, where the concentration coordinates are shown in the x–y plane and the temperature on the z axis. The representation of the figurative point of a three-component system in the equilateral triangle using three concentration coordinates is shown in Figure 3.28. The parallels with

168

Physico-chemical Analysis of Molten Electrolytes

M

xB(M)

A

xC(M) x(B)

) x(B

x(C )

C

xA(M)

B

Figure 3.28. Representation of the figurative point of a three-component system in a triangle diagram.

the individual axes passing the figurative point M cut off on each axis sections which are equal to the concentration coordinates of the individual components. The composition of the figurative point M in the A–B–C system can be read, for example on the A–B axis, where the first section represents the molar (mass) fraction of the component B, the second one represents the molar (mass) fraction of the component C, and the third one is the molar (mass) fraction of the component A. 3.2.2.1. Simple eutectic systems

The spatial view of the phase diagram of the simple ternary eutectic system is shown in Figure 3.29. The three-sided prism represents the shape of this phase diagram. On its sides lie the individual binary subsystems. Points Tfus, A , Tfus, B , and Tfus, C represent the melting points of the components A, B, and C. Lines originating from the melting points of the components and lying on the sides of the prism represent the curves of their primary crystallization. Each component has thus two curves of primary crystallization belonging to two binary systems. Curves of primary crystallization meet at the respective eutectic temperatures Te1 , Te2 , and Te3 at the eutectic points e1 , e2 , and e3 .

Phase Equilibria

169

Figure 3.29. Spatial view of the simple ternary eutectic system.

The boundary curves, coming out from the individual binary eutectic points, represent curves of the common crystallization of both components of the respective binary subsystems. All boundary curves meet at the ternary eutectic point, which is the lowest temperature where there is the liquid phase in this ternary system. With respect to the fact that in the ternary system the temperature of primary crystallization of each component is affected by the presence of two other components, the phase diagram of the ternary system has three planes of primary crystallization belonging to the individual components. On the plane Tfus, A −e1 −et −e2 component A crystallizes primarily, the plane Tfus, B −e1 −et −e3 is the primary crystallization plane of the component B, and the plane Tfus, C −e2 −et −e3 belongs to the primary crystallization of the component C. The space above the three planes mentioned represents the region of homogeneous solution of all the three components. The space between the plane Tfus, A −e1 −et −e2 and the temperature of the ternary eutectics is the region of coexistence of the crystals of the solid phase A and the melt saturated with the component A. Between the plane

170

Physico-chemical Analysis of Molten Electrolytes

Tfus, B −e1 −et −e3 and the temperature of the ternary eutectics is the region of coexistence of the crystals of the solid phase B and the melt saturated with the component B, and, finally, between the plane Tfus, C −e2 −et −e3 and the temperature of the ternary eutectics is the region of coexistence of crystals of the solid phase C and the melt saturated with the component C. Below the temperature of the ternary eutectics lies the region of the mechanical mixture of three solid phases A, B, and C. As the graphical presentation of the ternary phase diagrams using the triple-sided prism is inconvenient, the vertical projection of the diagram into the x–y plane is used as it is shown in Figure 3.30. Projections of the binary eutectics e1 , e2 , and e3 in the x–y plane (Figure 3.30) are e1 , e2 , and e3 points in the Figure 3.29. The projection of the ternary eutectic point et (Figure 3.30) is the point et (Figure 3.29). Projections of the boundary lines e1 −et , e2 −et , and e3 −et in the x–y plane (Figure 3.30) are curves e1 −et , e2 −et , and e3 −et in Figure 3.29. In the triangle diagram thus three planes originate. The plane pc(A) is the projection of the plane of primary crystallization of the component A, the plane pc(B) is the projection of the plane of primary crystallization of the component B,

C

pc(C ) e1

e2

> >

pc(A)

A

<

> et

>

X

M

e3

pc(B)

B

Figure 3.30. Vertical projection of the phase diagram of the simple ternary eutectic system into the x–y plane.

Phase Equilibria

171

and the plane pc(C) is the projection of the plane of primary crystallization of component C. In every projection of a ternary phase diagram (like altitude curves on geographic maps) also isotherms, i.e. curves having identical temperature of primary crystallization (not shown in Figure 3.30) are shown. In a mixture A–B–C, the composition of which is on Figure 3.30 shown by the figurative point X lying in the field of primary crystallization of the component A. This mixture is at a temperature above the temperature of primary crystallization of A and is thus a homogeneous solution of all the three components. The melt has three degrees of freedom (k = 3, f = 1, v = 3), we can arbitrarily change its temperature, and the content of two components without the appearance of a new phase. At its cooling from the given temperature the composition of the solution does not change until the temperature of primary crystallization of component A is attained. Therefore any shift of the point X cannot be seen. At the temperature of primary crystallization, the first crystals of component A appear, the solid phase A begins to coexist with the melt saturated with component A. The cooling rate of the system slows down due to the evolution of the crystallization heat of component A. At further cooling, the solid component A crystallizes from the saturated melt which becomes depleted of this component. The composition of the melt thus moves in the course of the arrow on the straight line A–X facing the point M. Between the temperature of primary crystallization and the boundary line (on the X–M straight line) the system has two degrees of freedom (k = 3, f = 2, v = 2), for example we can change two concentration coordinates of the melt and the temperature will be already determined, or the temperature and one concentration coordinate of the melt and the second concentration coordinate will be already determined. In point M also the component C begins to crystallize from the melt. The cooling rate slows down still more due to the evolution of the further crystallization heat of component C. On the boundary line e1 −et the system has only one degree of freedom (k = 3, f = 3, v = 1), only one variable can be changed, for example one concentration coordinate of the melt or the temperature, and the other variables will be determined by the line e1 −et . On continued cooling of the system both the components, A and C, will crystallize simultaneously and the composition of the melt will move on the boundary line e1 −et until the ternary eutectic point et is attained, where also the component B begins to crystallize. At the eutectic temperature the system has no degree of freedom (k = 3, f = 4, v = 0), which means that its cooling will stop. The system maintains the eutectic temperature until its whole solidification. Completely analogical is the crystallization process of mixtures lying in the crystallization fields of the components B and C. Below the eutectic temperature the system has again one degree of freedom (k = 3, f = 3, v = 1), at the given composition we can change arbitrarily the temperature.

172

Physico-chemical Analysis of Molten Electrolytes

3.2.2.2. Systems with formation of a congruently melting binary compound

In the ternary system, in which a congruently melting binary compound is formed, the figurative point of the binary compound AB lies in the boundary binary system A–B. The vertical projection of the phase diagram of such a ternary eutectic system is shown in Figure 3.31. The phase diagram of such a system has four planes. The plane pc(A) is the vertical projection of the plane of primary crystallization of the compound A, plane pc(B) represents the plane of primary crystallization of the component B, and the plane pc(C) refers to the primary crystallization of compound C. Finally, the plane pc(AB) is the projection of the plane of primary crystallization of compound AB. In the case of the congruently melting compound AB its figurative point lies inside the plane of the primary crystallization of this compound. In comparison with the simple eutectic ternary system a new boundary line, et1 − et2 , which represents the common crystallization of compounds C and AB, will arise. The joint AB–C divides the ternary system A–B–C into two simple

C

X3 pc (C)

e4 >

e4 <

>

et1 <

et2

<

M1

>

>

pc (A) X1

M2 >

M3

>

p c(B) X2 pc (AB)

A

e1

AB

e2

Figure 3.31. Phase diagram of the ternary eutectic system with the formation of the congruently melting binary compound.

Phase Equilibria

173

eutectic ternary systems A–AB–C and AB–B–C. The point where the joint AB–C crosses the boundary line et1 − et2 represents the highest temperature at the boundary line et1 − et2 . Let us have a mixture A–B–C, the composition of which is on Figure 3.31 represented by the figurative point X1 . The crystallization process at cooling from temperatures higher than the temperature of primary crystallization is equal to the case of a simple ternary eutectic system. The compound will crystallize first, the composition of the melt will move to the point M in which it begins to crystallize the compound AB also. At the next cooling, compounds A and AB crystallize simultaneously and the composition of the melt moves on the boundary line from the point M up to the eutectic point et1 , where also the component C will crystallize and where also the whole system solidifies (according to the triangle rule the point X1 lies in the triangle A–AB–C). An analogical situation happens also in the case of the mixture, the composition of which is given by the figurative point X3 . However, it depends on the position of X3 , in which of the two ternary eutectic points, the system will solidify. Since the summit of the boundary line et1−et2 is the eutectic point of the binary system AB–C, the boundary line falls down from its summit towards both ternary eutectic points. Thus mixtures, the composition of which lies in the triangle A–AB–C will solidify in the eutectic point et1 , while mixtures with composition lying in the triangle AB–B–C will solidify in the eutectic point et2 . If we have a mixture, the composition of which is shown by the figurative point X2 , then on cooling, the compound AB begins to crystallize first and this compound will coexist with the melt saturated by it. At the following cooling from the saturated solution even further amounts of the solid compound AB and the melt will by impoverished by this compound. The composition of the melt will thus move along the course of the arrow on the straight line AB–X2 towards the point M2 . In this point also the compound C begins to crystallize from the melt. At the ensuing cooling of the system the composition of the melt moves along the boundary line et1 − et2 in the direction of the arrow up to the ternary eutectic point et2 , where also the solid phase B starts to crystallize until the total solidification of the system (according to the triangle rule the figurative point X2 lies in the triangle AB–B–C).

3.2.2.3. Systems with formation of an incongruently melting binary compound

Similarly as in the previous case, incongruently melting binary compounds can originate in the ternary system as well. The vertical projection of the phase diagram of such a system is shown in Figure 3.32. The phase diagram of the ternary system with the formation of an incongruently melting compound has four planes. Planes pc(A), pc(B), pc(C), and pc(A4 B) are the projections of the planes of primary crystallization of the compounds A, B, C, and A4 B, respectively. The figurative point of the binary compound A4 B lies in the boundary of the binary

174

Physico-chemical Analysis of Molten Electrolytes

C

X3 pc(C) e3

" X1

>

pc(A)

Pt

X1 A

A4B

X1"

'

>

et λ

X2

>

>

<

>

e2

pc(B)

pc(A4B) Pb

e1

B

Figure 3.32. Phase diagram of the ternary eutectic system with the formation of an incongruently melting binary compound.

system A–B, however, it lies outside the plane of its primary crystallization. In Figure 3.32 it is shown on the left side of the binary peritectic point Pb . Contrary to the previous case, the phase diagram of this type has only one ternary eutectic point et . The second singular point is the ternary peritectic point Pt . The boundary line Pt −et represents the simultaneous crystallization of the component C and the compound A4 B. The crystallization path of mixtures, the figurative points of which lie in the region of the primary crystallization of the component A, pc(A), depend on the part of this crystallization field, in which the figurative point of the system lies. Three cases may occur. If point X lies in the area of the quadrangle A–A4 B–α–e2 (point X1 on Figure 3.32), according to the triangle rule its crystallization path must end in the mechanical mixture of the solid A, A4 B, and C (apexes of the triangle A–A4 B–C) at the point Pt . Since we are in the crystallization area of component A, at the cooling of the melt crystals, A will appear

Phase Equilibria

175

first and these will coexist with the melt saturated by component A. At the following cooling, crystals of A crystallize from the saturated melt next and the composition of melt will move on the straight line A−X1 in the direction of the arrow up to the point β. In point β also compound A4 B begins to crystallize, the system has one degree of freedom and the composition of the melt will move from point β on the boundary line Pb −Pt up to the point Pt , where also component C begins to crystallize until the whole system solidifies. If the mixture has the composition placed in the triangle A4 B–Pt –α (point X1 in Figure 3.32) or A4 B–Pb –Pt (point X1 ), their crystallization paths, according to the triangle rule, must end in the mechanical mixture of solid A4 B, B, and C (apexes of the triangle A4 B–B–C) in the ternary eutectic point et . The crystallization paths, however, will in both cases proceed in a different way. At cooling of mixture X1 primarily component A begins to crystallize and after the composition of the melt attains on the boundary line e2 −Pt point γ , also component C begins to crystallize moving the composition of melt into point Pt . At point Pt , however, always four phases coexist: A, A4 B, C, and the melt. The system has thus no degree of freedom. At the temperature of the ternary peritectic point the cooling of the system will stop and according to the quadrangle rule the peritectic reaction A + L ⇒ A4 B + C

(3.153)

takes place until the component A disappears. The system thus attains one degree of freedom and at its ensuing cooling, the composition of the melt moves from point Pt to the point et , where the whole system solidifies into the mechanical mixture of crystals A4 B, B, and C. From the mixture X1 component A crystallizes first as well moving the composition of the melt from point X1 towards point β . After the composition of melt attains point β on the boundary line Pb −Pt , also compound A4 B crystallizes, since with regard to the quadrangle rule the melt starts to dissolve compound A according to the reaction A + L1 ⇒ A4 B + L2

(3.154)

The composition of the melt moves then on the boundary line Pb −Pt (one degree of freedom) towards point Pt . However, the composition of the melt will never attain point Pt , as if the composition of the originating melt L2 attains point β on the straight line X1 −A4 B, the original composition X1 changes according to the lever rule to the mixture of the melt and A4 B, i.e. the whole component A has been dissolved. The system attains one degree of freedom and the primary crystallization of A4 B proceeds until point β , followed by the secondary crystallization of C and the composition of melt moves on the

176

Physico-chemical Analysis of Molten Electrolytes

boundary line Pt −et into point et , where the whole system solidifies under the formation of a mechanical mixture of crystals A4 B, B, and C. On cooling the mixture with the composition X2 , which lies in the crystallization field of the compound A4 B, the primary crystallization of A4 B proceeds. The composition of the melt moves on the straight line A4 B−X2 from point X2 up to the point λ, where also component B starts to crystallize. The system has only one degree of freedom and the composition of the melt thus moves on the boundary line e1 − et to the ternary eutectic point et , where the whole system solidifies under the formation of a mechanical mixture of crystals A4 B, B, and C. In the case of a mixture shown on Figure 3.32 by the figurative point X3 , the situation is similar as in a simple eutectic ternary system. Starting with the crystallization of component C, the composition of the melt moves up to the point δ, where the component A4 B begins to crystallize. At the ensuing cooling, the composition of the melt moves on the boundary line Pt −et up to the ternary eutectic point, where also component B crystallizes until the whole system solidifies. Below the temperature of the ternary eutectic point the system formed is a mechanical mixture of the solid phases A, A4 B, and C if the composition of the system lies in the triangle A–A4 B–C, or A4 B, B, and C if the composition of the system lies in the other part of the concentration triangle. 3.2.2.4. Systems with formation of a congruently melting ternary compound

The vertical projection of the phase diagram of this ternary system is shown in Figure 3.33. The phase diagram has four planes pc(A), pc(B), pc(C), and pc(ABC), representing the vertical projections of the planes of the primary crystallization of the individual compounds. The figurative point of the ternary compound ABC lies inside the ternary system A–B–C. There are three boundary lines et1 −et2 , et2 −et3 , and et3 −et1 representing the common crystallization of compounds B, C, and A with AB, respectively. The joints of the figurative point ABC with the apexes of the concentration triangle divide the phase diagram into three simple eutectic ternary systems A–ABC–B, B–ABC–C, and A–ABC– C. The points, where the individual joins of ABC with the apexes cross the boundary lines, form the summits of the boundary lines. The crystallization path of mixtures lying in the planes pc(A), pc(B), and pc(C) is completely equal to a simple ternary eutectic system. When the figurative point of the mixture lies inside the plane pc(ABC) (e.g. of that represented by the figurative point X) the compound ABC will crystallize first, the composition of the melt will move to the point M in which compound A also begins to crystallize. At the following cooling, compounds A and ABC crystallize simultaneously and the composition of the melt moves on the boundary line from point M to the eutectic point et1 , where also the component B will crystallize and where also the whole system solidifies (according to the triangle rule the point X lies in the triangle A–ABC–B).

Phase Equilibria

177

Figure 3.33. Phase diagram of the system with the formation of the congruently melting ternary compound.

3.2.2.5. Systems with formation of an incongruently melting ternary compound

The vertical projection of the phase diagram of this system is shown in Figure 3.34. The phase diagram has four planes pc(A), pc(B), pc(C), and pc(ABC), which are the projections of the planes of primary crystallization of the compounds A, B, C, and ABC, respectively. The figurative point of the ternary compound ABC lies outside the plane of its primary crystallization. In contrary to the previous case, the phase diagram of this type has only one ternary eutectic point et and two singular points, which are the ternary peritectic points Pt1 and Pt2 . The boundary lines Pt1 −et and Pt2 −et represent the simultaneous crystallization of components B and ABC, and C and ABC, respectively. The crystallization path of mixtures, the figurative points of which lie in the region of the primary crystallization of the component A, pc(A), depend on the part of this crystallization field, in which the figurative point of the system lies. The crystallization paths are rather complicated and therefore they will not be discussed here. However,

178

Physico-chemical Analysis of Molten Electrolytes

C

pc(C)

e3

Pt

2

<

ABC

<

> > pc(ABC)

>

Pt

pc(A)

et

>

e2

pc(B)

>

1

AB

e1

Figure 3.34. Phase diagram of the system with the formation of the incongruently melting ternary compound.

three final compositions are reached below the temperature of the ternary eutectic point. The system is formed from the mechanical mixture of the solid phases • • •

A, ABC, and C if the composition lies in the triangle A–ABC–C, A, ABC, and B, if the composition lies in the triangle A–ABC–B, B, ABC, and C, if the composition lies in the triangle B–ABC–C.

3.2.2.6. Ternary reciprocal systems

Ternary reciprocal systems are those of the general type Ap Xq −Br Ys , i.e. not having a common ion. For example, the systems LiF–NaCl, KF–Na2 SO4 , KCl–Na3AlF6 , etc. belong to this category. Like in the other ternary systems, in the ternary reciprocal systems several congruently and incongruently melting binary and ternary compounds can originate. Phase diagrams of the ternary reciprocal systems are represented in the vertical projection as rectangles, whereas on the opposite apexes of the diagonals, figurative points of the compounds without the common ion are placed.

Phase Equilibria

179

e3

AY pc(AY)

BY pc(BY)

>

et

1

e4

X2

>

e2

>

x(AY)

X1

M1 S

>

et

pc(AX) x(BX)

AX

<

2

pc(BX)

>

M2

e1

BX

Figure 3.35. Phase diagram of the simple ternary reciprocal system.

The phase diagram of a simple ternary reciprocal system is schematically shown in Figure 3.35. In the systems of this type the exchange in the metathetical reaction between components takes place, for example p = q = r = s = 1,

Simple ternary reciprocal systems.

AX + BY = AY + BX

G0

(3.155)

Although these systems contain four chemical compounds, in fact they belong to the three-component systems, since the number of components is lowered by the metathetical reaction (3.34). The exchange reaction is characterized by the change in the Gibbs energy G0 , the value of which determines the stabile pair of compounds. If for the above exchange reaction, the change of the Gibbs energy is negative, the reaction runs from the left to the right side and the stabile pair is the system AX–BY. As a next rule, the stabile pair of compounds is formed by the small–small and big–big ionic compounds.

180

Physico-chemical Analysis of Molten Electrolytes

The phase diagram of the AX–BY system has four planes, pc(AX), pc(BX), pc(AY), and pc(BY), which are the projections of the crystallization planes of the components AX, AY, BX, and BY, respectively. There are five boundary lines, representing the simultaneous crystallization of two compounds. The dashed line marks the stabile system AX–BY, which divides the reciprocal system into two simple eutectic ones. The value of G0 defines the equilibrium composition of the system. Any figurative point of the system, which lies inside the given stabile triangle, can, according to the triangle rule, decompose into the respective three components. This means that in the triangle AX−BX−BY the component AY cannot be present while the component BX cannot be present in the triangle AX−AY−BY. Of course, on the diagonal AX−BY, according to the quadruple rule all four components AX, BX, AY, and BY are present. Their equilibrium concentration is determined by the value of the equilibrium constant of the exchange in the metathetical reaction, i.e. by the value of the reaction Gibbs energy. If we have a mixture AX–AY–BY, the composition of which is in Figure 3.35 shown by the figurative point X1 , the crystallization path at its cooling is completely similar as in the case of a simple ternary eutectic system. The component BY begins to crystallize first, the composition of the melt moves towards point M1 , where also component AX starts to crystallize. At the ensuing cooling, both the components fall out from the melt simultaneously and the composition of the melt moves on the boundary line et1 − et2 from point M1 up to the eutectic point et1 , where also component AY starts to crystallize and where also the whole system will solidify. Similar situation will happen also in the case of the mixture X2 , lying inside the triangle AX−BX−BY. The component BY begins to crystallize first, the composition of the melt moves up to the point M2 , in which the component AX also starts to crystallize. At further cooling both components, BY and AX, crystallize from the melt simultaneously and the composition of the melt moves on the boundary curve et1−et2 from point M1 up to the eutectic point et2 , where also the phase BY starts to fall out and where also the whole system solidifies. It depends thus on the composition of the mixture in which both the ternary eutectic points of the system will solidify. The boundary line et1−et2 falls down from its summit S towards both the eutectic points. This summit is simultaneously the eutectic point of the pseudo-binary system AX–BY. The phase diagram of the ternary reciprocal system A+ , B2+ // X2− , Y2− is shown in Figure 3.36. Since the bivalent cation B2+ is smaller than the univalent A+ one, the stable diagonal is formed by the BX–A2Y pair. Two binary compounds, A2 BX2 and A2 B2Y3 , are formed. These two binary compounds divide the concentration rectangle into three simple ternary eutectic systems A2 BX2 –A2Y–BX, A2 X–BX–A2 B2Y3 , and BX–A2 B2Y3 –BY, and the ternary system A2 X–A2Y–A2 BX2 with continuous solid solutions in the binary boundary system

Systems with formation of two congruently melting binary compounds.

Phase Equilibria

A2X

181

A2Y

A2(X,Y)ss β-A2X e1

A2 Y e2

A2BX2

e3 A2B2Y3

BX e4

BY BX

BY

Figure 3.36. Phase diagram of the ternary reciprocal system A+ , B2+ // X2− , Y2− .

A2 X–A2Y. The compound A2 X undergoes a solid–solid phase transformation prompted in the phase diagram by the dashed line. The phase diagram of the A+ , B2+ // X2− , Y2− system has six crystallization areas belonging to primary crystallization of the individual phases. The solidification of any mixture may end in four ternary eutectic points, depending on its starting composition. The phase diagram of this type forms, for example the K2 SO4 –PbSO4 –K2 WO4 – PbWO4 system, which was measured by Belyaev and Nesterova (1952). This system has marginal importance in the production of tungsten. It is the irreversible transition-type ternary reciprocal system with stable diagonal PbWO4 –K2 SO4 . The phase diagram of the ternary reciprocal system A+ , B2+ // X– , Y2−− is shown in Figure 3.37. There are three incongruently melting binary compounds: ABX3 and A2 BX4 , originating in the AX–BX2 system, and A2 B3Y4 , originating in the A2Y–BY system. The AX–BY system System with formation of three binary incongruently melting compounds.

182

Physico-chemical Analysis of Molten Electrolytes

A2Y

AX

E2

ABX3 P3 P2 E1 P1

A2BX4

A2B3Y4

BX2

BY

Figure 3.37. Phase diagram of the ternary reciprocal system A+ , B2+ // X– , Y2−− .

forms the stable diagonal of the ternary reciprocal system. In the ternary subsystem AX–BX2 –BY, there are two ternary peritectic and one ternary eutectic point. In the ternary subsystem AX–A2Y–BY there is one ternary peritectic point, where three solid and one liquid phases meet. The variance of the system equals zero and thus the following reaction takes place BY + L1 → AX + L2

(3.156)

After all the BY disappears, the system moves up to the ternary eutectic point, where the whole system solidifies. The phase diagram of this type forms the NaCl–MgCl2 –Na2 SO4 –MgSO4 system and was measured by Speranskaya (1938). The melts of this system find its utilization in the electrochemical production of magnesium.

Phase Equilibria

183

A2BY2

A2Y

BY

E2 E3

A3B2X 3Y2 E1

E5 P E4

AX

ABX3

BX2

Figure 3.38. Phase diagram of the ternary reciprocal system A+ , B2+ // X– , Y2−− .

The phase diagram of the ternary reciprocal system A+ , B2+ // X– , Y2−− is shown in Figure 3.38. Two binary compounds ABX3 and A2 BY2 , and one ternary compound A3 B2 X3Y2 , all melting congruently, originate in this system. The joins between the figurative point of the compound A3 B2 X3Y2 and the individual simple and binary compounds divide the ternary reciprocal system into four simple ternary eutectic phase diagrams, one phase diagram with the crystallization field of a compound, the figurative point of which lies outside its concentration triangle, and one phase diagram without a ternary eutectic point. There are five ternary eutectic and one peritectic points. The crystallization path of all mixtures, figurative points of which lay inside the triangles BY–A3 B2 X3Y2 –BX2 and A3 B2 X3Y2 –P–BX2 , starts with the primary crystallization of BY, A3 B2 X3Y2 , or BX2 , followed by the crystallization of the neighboring salt till the peritectic point P is reached. System with formation of two binary and one ternary congruently melting compounds.

184

Physico-chemical Analysis of Molten Electrolytes

Here the system contains three solid and one liquid phases and the variance equals zero. Two different reactions take place BY + L → BX2 + L

(3.157)

BY + L → A3 B2 X3 Y2 + L

(3.158)

or

After all the BY disappears, the system moves towards the ternary eutectic point E4 , where the whole system solidifies. The phase diagram of this type forms the KCl–K2 SO4 –MgCl2 –MgSO4 system and was measured by Jänecke (1912). The melts of this system are technologically important in the electrochemical production of magnesium. 3.2.3. Quaternary systems

Quaternary systems are composed of four components with a common ion and form four ternary and six binary systems. The concentration diagram of the quaternary system can be represented by a tetrahedron. The vertical projection of every ternary system is placed on one face of the tetrahedron. The phase diagram of the quaternary system A+ , B+ , C2+ , D2+ // X– with the formation of the congruently melting binary compound BCX3 is shown in Figure 3.39. The binary compound BCX3 divides the AX–BX–CX2 and DX2 –BX–CX2 ternary systems into four simple ternary subsystems. The crystallization paths of ternary mixtures end in one of the six ternary eutectic points Ei , where the ternary mixtures solidify. The crystallization path of any quaternary mixture follows the dotted boundary lines inside the concentration tetrahedron and ends in one of the two quaternary eutectic points Eqi . Phase diagrams of quaternary systems are represented in the literature often also in the form of unfolded tetrahedron surface. Such representation of the phase diagram of the quaternary system A+ , B+ , C2+ , D2+ // X– is shown in Figure 3.40. This picture serves a better understanding of the phase equilibria within the ternary systems and on the boundary between them, but does not enable to study the phase equilibria inside the quaternary system. This phase diagram looks like that of the NaF–KF–CaF2 –BaF2 system, measured by Bukhalova and Sementsova (1967). 3.2.3.1. Quaternary reciprocal systems

Quaternary reciprocal systems are composed of six components, which form three ternary reciprocal systems and two ternary systems with a common ion. The concentration diagram of this system can be represented by a three-sided prism, on each side is the vertical

Phase Equilibria

185

AX

E5 E1 E6 E2

E q1

CX 2

E q2

E4

BCX3

DX2

E3

BX Figure 3.39. Phase diagram of the quaternary system A+ , B+ , C2+ , D2+ // X– with the formation of a congruently melting binary compound.

projection of one of the ternary reciprocal systems and on both the bases are the vertical projections of the ternary system with a common ion. The three-sided prism of the quaternary reciprocal system A+ , B+ , C2+ // X– , Y– is shown in Figure 3.41. As it can be seen from the figure, in the CX2 –CY2 system the incongruently melting compound CX2 ·CY2 originates and in the systems AX–CX2 and AY–CY2 , the congruently melting compounds ACX3 and ACY3 originate, respectively. These compounds give rise to several eutectic points and two peritectic points. Such a representation of the phase diagram, however, does not tell anything about the phase relation inside the prism of the quaternary reciprocal system. This phase diagram resembles the phase diagram of the Na+ , K+ , Ca2+ // F− , Cl− system, which was measured by Bukhalova and Maslennikova (1962). As it can be seen, in Figure 3.41 are depicted only phase equilibria on the surface of the three-sided

186

Physico-chemical Analysis of Molten Electrolytes

AX

E5

CX 2

DX2 E4

BCX3

E1

E3 E6

E2

AX

BX

AX

Figure 3.40. The unfolded tetrahedron surface of the phase diagram of the quaternary system A+ , B+ , C2+ , D2+ // X– with the formation of a congruently melting binary compound.

prism, i.e. those of the ternary and ternary reciprocal systems. The phase equilibria of the quaternary reciprocal system, i.e. those inside the three-sided prism could not be drawn, since they have not been studied. Phase diagrams of quaternary reciprocal systems are represented in the literature only in the form of the unfolded surface of a three-sided prism. Such presentation of the phase diagram of the quaternary reciprocal system A+ , B+ , C2+ // X– , Y– is shown in Figure 3.42. 3.2.4. The CaO–Al2 O3 –SiO2 system

The phase diagram of the CaO–Al2 O3 –SiO2 system is an indispensable tool in metallurgy, production of refractory ceramics, special glasses, and in cement production. In Figure 3.43, the phase diagram of this system according to Osborn and Muan (1960), revised in the CaO·6Al2 O3 field by Gentile and Foster (1963), is shown.

Phase Equilibria

187

BY

AY

ACY 3 CY 2

CXY solid soln.

AX

BX

ACX 3

CX 2 Figure 3.41. Phase diagram of the quaternary reciprocal system A+ , B+ , C2+ // X– , Y– with the formation of two congruently and one incongruently melting compound.

The CaO–Al2 O3 –SiO2 system forms a rather complicated phase diagram, in which a number of binary and two ternary compounds are formed. In the binary system CaO–SiO2 two incongruently melting compounds, 3CaO·SiO2 and 3CaO·2SiO2 , and two congruently melting compounds, 2CaO·SiO2 and CaO·SiO2 , are formed. While the two congruently melting compounds show a broad field of primary crystallization, the two incongruently melting compounds show only a narrow field of primary crystallization, which is located on both sides of the crystallization field of dicalcium silicate. Tricalcium and dicalcium silicates are the main hydraulic components of the cement clinker. In the SiO2 -rich side of the CaO–SiO2 phase diagram, the region of the two liquids may be seen, extending also into the ternary system. On the SiO2

188

Physico-chemical Analysis of Molten Electrolytes

CY 2

ACY 3 CY 2

ACY3

e e

AY

CY 2

e

CXY

e

e

p

e

e

e

solid soln.

ACX3

CXY

e

p

CX 2

BY

BX

AX

CX 2

e ACX3

e

CX 2 Figure 3.42. The unfolded surface of the three-sided prism of the phase diagram of the quaternary reciprocal system A+ , B+ , C2+ // X– , Y– with the formation of two congruently and one incongruently melting compound.

corner of the ternary system, the crystallization fields of the two modifications of SiO2 , the cristobalite and tridymite, can be seen. Very complicated and many times revised, due to the formation of a number of binary compounds, is the CaO–Al2 O3 system. From the CaO side it is at first the incongruently melting compound 3CaO·Al2 O3 , with the peritectic decomposition temperature of 1539◦ C. The next compound 5CaO·3Al2 O3 melts congruently at 1392◦ C. However, Nurse et al. (1965) later found out that it is the compound 12CaO·7Al2 O3 that crystallizes congruently at the temperature 1455◦ C. According to Rolin and Thanh (1965), the next two compounds, CaO·Al2 O3 and CaO·2Al2 O3 , melt congruently at 1605◦ C and 1750◦ C, respectively. The last compound in this system is CaO·6Al2 O3 , which melts incongruently at the peritectic decomposition temperature of 1850◦ C. Only one compound forms in the Al2 O3 –SiO2 system, the mullite, 3Al2 O3 ·2SiO2 , which melts according to some authors congruently and according to the others incongruently. The enigma of the fusion of mullite is discussed in detail by Davis and Pask (1972) and Risbud and Pask (1978).

Phase Equilibria

189

80

C

ris to b 15 ali 00 te

20

16 00

2L iqu ids

170 0

SiO2

00 14 1600

Anorthite

20

00

00

14

1500

2CaO.SiO2

Gehelnite

1700

19 00

20

00 15

3 3CaO AI2O3

20 CaO

00 CaO.2AI2O3 CaO.AI2O3

CaO.6AI2O3

1900

15

16 1800

2000

2200

2400

Lime

1800

00

17

00

18

80

3AI2O3.2SiO2

Corundum

2CaO.AI2O3-SiO2

00

3CaO.2SiO2

40 1600

13 00

CaO.AI2O32SiO2

1400

2CaO.2SiO2

Mullite

00 15

CaO.2SiO2

60

1800

140 0

1500

CaO2.SiO2

Pseudo– woilastonite

2000

40

1700

e ymit 0 Tnd 130

3CaO.AI2O3 5CaO.3AI2O3 CaO.AI2O3 CaO.2AI2O3 CaO.6AI2O3 AI2O3

Figure 3.43. Phase diagram of the CaO–Al2 O3 –SiO2 system according to Osborn and Muan (1960) and Gentile and Foster (1963).

The two congruently melting ternary compounds, gehlenite – 2CaO·Al2 O3 ·SiO2 and anorthite – CaO·Al2 O3 ·2SiO2 , melt at temperatures 1593◦ C and 1553◦ C, respectively, and show broad fields of primary crystallization.

3.3.

EXPERIMENTAL METHODS

3.3.1. Thermal analysis

Experimental determination of phase diagrams is convenient by using the thermal analysis method at which the temperature of the investigated sample is registered at its cooling by a constant rate of 2–5◦ C/min. Due to the thermal effects connected with the phase transformations (crystallization, polymorphic transformation), breaks appear on the cooling

190

Physico-chemical Analysis of Molten Electrolytes

T fus

temperature

T pc

Te

Te

1

2

3

time Figure 3.44. Cooling curves of samples with different composition.1 – pure component, 2 – arbitrary mixture, 3 – eutectic mixture.

curve of the sample as well as delays pertinent to the respective phase transformation. Cooling curves of three different types of samples in a simple binary eutectic system are schematically shown in Figure 3.44. Curve No. 1 shows freezing of the pure component. On the curve, only one delay can be seen, which is caused by the evolution of the crystallization heat of the component. At melting temperature, this one-component system has no degree of freedom, since two phases coexist: the solid compound and its melt (k = 1, f = 2, v = 0). The temperature of the system therefore stays constant until its whole solidification. In practice we can, however, observe at the end of the delay a temperature decrease caused by the transport of bigger heat amounts to the surroundings, than it could be evolved at crystallization of the compound. Curve No. 2 shows the cooling of the binary mixture with an arbitrary composition except of the eutectic one. On the curve, one break and one delay can be seen. The break is due to the start of the primary crystallization of one of the components. At primary crystallization, the binary system has one degree of freedom, as the solid component and the melt saturated with the component coexist (k = 2, f = 2, v = 1). Thus the cooling

Phase Equilibria

191

due to the evolution of the crystallization heat only slows down until the eutectic point is attained. At the eutectic temperature a cooling delay happens since the system does not have any degree of freedom (k = 2, f = 3, v = 0). The temperature of the system again keeps constant theoretically up to the complete solidification of the mixture. Again in practice a decrease in the temperature can be seen, caused by the transport of more heat into the surroundings than it evolves by crystallization. Curve No. 3 shows freezing of the eutectic mixture. Only one delay can be seen on the curve, caused by evolution of the crystallization heat of both the components. At the eutectic point the binary system has no degree of freedom, since two solid phases coexist: two solid phases and the melt saturated by both the components (k = 2, f = 3, v = 0). The temperature of the system thus stays constant again until its total solidification. In practice, however, we can see again at the end of the delay a decrease in temperature caused by the evolution of more heat to the surroundings, than it can be evolved by the crystallization of components. By thermal analysis of a sufficient number of samples, it is possible to construct a phase diagram of the investigated system. Figure 3.45 shows the cooling curve of the mixture 5% NaF + 95 mole % NaBF4 measured by Chrenková (2001). The figurative point of this mixture in the phase diagram of the NaF–NaBF4 system lies in between the eutectic point and pure NaBF4 . The enthalpy of fusion of NaBF4 is fus HNaBF4 = 13.5kJ/mol at 408◦ C. The first break on the cooling curve occurs at 400◦ C, where NaBF4 crystallizes as a primary phase. The break at the cooling curve is not very distinct due to the small amount of heat evolved at the crystallization of NaBF4 . The shape of the break depends also on the cooling rate and at fast cooling, the break could not be detected at all. The temperature halt at 380◦ C corresponds to the eutectic temperature with the simultaneous crystallization of NaF and NaBF4 . A small under-cooling by approximately 6◦ C could be observed at the eutectic crystallization followed by a short temperature halt. Both the little-distinct break and the small under-cooling indicate the rather fast cooling of the melt. The duration of the temperature halt at the eutectic temperature depends on the amount of heat evolved at the eutectic crystallization. It is evident from the phase rule that the amount of eutectic melt increases from the pure component to the eutectic composition and in the same manner also the amount of heat evolved at the eutectic crystallization increases. Plotting the time duration of the eutectic crystallization versus the composition, we get the so-called “Tamman triangles” (Figure 3.46). This procedure helps to detect the position of the eutectic point.

3.3.2. Cryoscopy

Cryoscopy is an analytical method measuring the melting point depression of the solvent A caused by the addition of a small amount of the substance B. The classical cryoscopy

420

410

400°C

t(°C)

400

390

380°C

380

370 4000

4500

5000 τ (sec)

5500

6000

Figure 3.45. Cooling curve of the mixture 5% NaF + 95 mole % NaBF4 .

T fus, A T fus, B

Te eutectic halts

A

x(B)

B

Figure 3.46. The halts of temperature at eutectic crystallization form the Tamman triangles.

Phase Equilibria

193

in molecular solvents is usually used for the determination of either the heat of fusion of the solvent or the molecular mass of the solvent. As will be shown in the next section, cryoscopy in ionic melts is used either for the determination of the heat of fusion of the solvent or the number of new (foreign) ionic particles, which occur in the solvent when a very small amount of solute is added to the solvent. However, cryoscopy in ionic melts may sometimes provide ambiguous results. These problems occur in such cases, when association reactions take place in the solvent and the molecular mass is not definitely known. Also the uncertainty in the value of the heat of fusion can result in incorrect results. On the other hand, the interpretation of the cryoscopic result could be sometimes uncertain, especially then, when the number of new particles can be caused by two or more chemical reactions. A special case is cryoscopy performed in solvents with dystectic melting, i.e. in those, which undergo at melting a thermal dissociation. This case will be discussed in Section 3.3.2.2. 3.3.2.1. Theoretical background

The equilibrium between the activity of the saturated solution of the solvent A and temperature in a simple binary eutectic system A–B is described by Le Chatelier–Schreder’s equation ln aA =

fus HA R

1 Tfus, A

−

1 T

(3.159)

where aA is the activity, Tfus, A is the temperature of fusion, and fus HA is the enthalpy of fusion of the component A. Let us now examine the course of the liquidus curve upon the melting point of the component A. Since we are working with dilute solutions, for the solvent A the following limiting relations are valid xA → 1, aA → xA , T → Tfus, A

(3.160)

ln xA = ln(1 − xB ) ∼ = −xB

(3.161)

For ln xA we can write

and for the right side of Eq. (3.159) fus HA R

1 Tfus, A

−

1 T

=

fus HA T − Tfus, A fus HA −Tfus, A = 2 R T Tfus, A R Tfus, A

(3.162)

194

Physico-chemical Analysis of Molten Electrolytes

After substitution and rearrangement we get Tfus, A =

2 RTfus, A

fus HA

xB = Ktd, A xB

(3.163)

where Tfus, A is the melting point depression and Ktd, A is the constant of thermal depression of the solvent A. Ktd, A is the analogy of the cryoscopic constant and depends solely on the properties of the solvent A. Equation (3.163) is the classical equation valid for cryoscopic measurements in solutions, which fulfills Raoult‘s law. For solutions being ionic in character, the processing of the Le Chatelier–Shreder’s equation in the vicinity of the melting point of the component A must be slightly different T − Tfus, A R ln aA = fus HA (3.164) T Tfus, A Rearranging the right side we get RT ln aA = T

fus HA − fus HA = T fus SA − fus HA Tfus, A

(3.165)

and further T (fus SA − R ln aA ) = fus HA

(3.166)

and finally T =

fus HA fus SA − R ln aA

(3.167)

Equation (3.167) is the simplified and modified Le Chatelier–Shreder’s equation expressed explicitly for T. Differentiating Eq. (3.167) according to temperature we get dT 1 daA Rfus HA = 2 dxA (fus SA − R ln aA ) aA dxA

(3.168)

For the tangent to the liquidus curve in the point xA = 1 it holds 2 RTfus, dT daA daA A = lim = Ktd, A lim xA →1 dxA xA →1 dxA fus HA xA →1 dxA

k0 = lim

(3.169)

For systems being ionic in character is daA = kSt, A xA →1 dxA lim

(3.170)

Phase Equilibria

195

where kSt,A is the semi-empirical correction factor introduced by Stortenbeker (1892), which is numerically equal to the number of new (foreign) particles of molecular dimensions, which one molecule of the solute B introduces into the pure solvent A. In practice, we lay the polynomial best fit through the dependence of the experimentally determined temperatures of primary crystallization, T, on molar fractions of the solvent, xA , and the limit of its first derivative for xA → 1 gives the slope k0 . Knowing the enthalpy of fusion of the solvent and thus also the value of the constant of thermal depression, Ktd, A , we can easily calculate the Stortenbeker’s correction factor. The cryoscopic measurement is frequently used in the literature. Examples of such measurement can be found also in this book. 3.3.2.2. Cryoscopy in solvents with dystectic melting

In binary systems of alkali metal halides and transition metal halides, sulfates, molybdates, tungstates, etc., like NaF–AlF3 , KF–NbF5 , KF–K2 SO4 , additive complex compounds originate, which undergo at melting a more or less extended thermal dissociation according to the general scheme α

Ap Bq −−→ pA + qB

(3.171)

Evidences of such behavior for p = q = 1 were discussed, for example by Daneˇ k and Cekovský (1992) and Daneˇ k and Proks (1999). Due to the thermal dissociation of the complex compound AB, its liquidus curve exhibits a curvature at the temperature of fusion with a slope at the composition of AB equal to zero. Such a phenomenon is called the dystectic mode of melting (Figure 3.47). Thus at the temperature of fusion the following equation is fulfilled

dT dxw (AB)

=0

(3.172)

xw (AB)=1

where xw (AB) is the weighed-in mole fraction of solvent AB. The curvature radius of the liquidus curve depends on the degree of dissociation α of the complex compound AB. The flatter the liquidus curve, the higher is the dissociation degree. Very interesting from the theoretical point of view is the evaluation of cryoscopic measurements made in these solvents, when a chemical reaction takes place between the solvent and solute. In such a case, there arises the problem of defining the number of foreign particles in order to determine the nature of the probable chemical reaction. A typical example of such systems with great technological importance is cryolite, Na3AlF6 , with dystectic mode of melting. On dissolving some compounds, like metal oxides in cryolite, besides the formation of new compounds due to the chemical reactions between cryolite and the solute, also the formation of compounds identical with cryolite dissociation products

196

Physico-chemical Analysis of Molten Electrolytes

T fus(A)

T fus(B)

T fus(AB)

T eut, 1

T eut, 2

A

x w(B)

B

Figure 3.47. Phase diagram of the A–B system with dystectic melting of component AB.

may occur, regardless of the accepted dissociation scheme of cryolite. The theoretical derivation of relations valid for cryoscopic measurements in such solvents was given by Proks et al. (2002). Let us consider the dissolution of the admixture X in the dystectically melting solvent AB. The course of the liquidus curve of substance AB in the AB–X system (Figure 3.48) is described by the Le Chatelier–Shreder equation, which in the limit for xr (AB) → 1 we can express in the differential form lim

xr (AB)→1

dxr (AB) 1 fus H (AB, T ) = lim xr (A)→1 xr (AB) dT RT 2 =

fus H (AB, Tfus (AB)) = Ktd 2 (AB) RTfus

(3.173)

where xr (AB) is the real mole fraction of solvent AB (i.e. the equilibrium activity) and Tfus (AB) and fus H(AB, Tfus (AB)) is the temperature and enthalpy of fusion of AB, respectively. Turning over and extending Eq. (3.173) we get

dT dT dxw (AB) 1 xr (AB) = lim = xr (AB)→1 dxr (AB) xr (AB)→1 dxw (AB) dxr (AB) Ktd lim

(3.174)

Phase Equilibria

197

Tfus(AB)

kSt = 1

kSt = 2

Teut kSt = 3

AB

x(X)

X

Figure 3.48. Phase diagram of the AB–X system with dystectic melting of component AB and a chemical reaction between solvent AB and solute X.

Now we will examine the behavior of xr (AB) with regard to xw (AB) in the limiting region xw (AB) → 1, when X reacts with AB under the formation of new, foreign compounds and compounds identical with the products of thermal dissociation of AB, for example according to the scheme AB + X = AX + B. The increase in the amount of substance AB caused by the reaction of 1 mol of admixture X with substance AB is denoted by l (l is non-zero only when AB melts dystectically) and the decrease in the amount of substance AB caused by the reaction of 1 mol of admixture X with substance AB is denoted as m. For xr (AB) we get lim

xw (AB)→1

= =

lim

xr (AB)

xw (AB)→1

xw (AB) + l[1 − xw (AB)] − m[1 − xw (AB)] xw (AB) + l[1 − xw (AB)] − m[1 − xw (AB)] + kSt [1 − xw (AB)]

xw (AB) + n[1 − xw (AB)] xw (AB) = =1 xw (AB)→1 xw (AB) + n[1 − xw (AB)] + kSt [1 − xw (AB)] xw (AB) lim

(3.175)

198

Physico-chemical Analysis of Molten Electrolytes

where l−m = n. It is thus possible to substitute in the limiting region xw (AB) for xr (AB). Equation (3.174) then transforms to the form lim

dT dxr (AB) 1 = lim dxw (AB) xw (AB)→1 dxw (AB) Ktd

(3.176)

lim

dT dxr (AB) .Ktd = lim xw (AB)→1 dxw (AB) dxw (AB)

(3.177)

xw (AB)→1

or

xw (AB)→1

The right side of Eq. (3.177) can be expressed in the form xw (AB) + n[1 − xw (AB)] xw (AB) + n[1 − xw (AB)] + kSt [1 − xw (AB)]

xr (AB) =

xw (AB) + n[1 − xw (AB)] F

=

(3.178)

Differentiating Eq. (3.178) according to xw (AB) we get dxr (AB) 1 xw (AB)F n n[1 − xw (AB)]F = − − − dxw (AB) F F F2 F2 =

F − F xw (AB) − nF − F n[1 − xw (AB)] F2

(3.179)

and in the limiting region dxr (AB) F (1 − n) − F {xw (AB) + n[1 − xw (AB)]} = lim xw (AB)→1 dxw (AB) xw (AB)→1 F2 lim

(3.180)

For F and its first derivative F in the limiting region we get lim

xw (AB)→1

F =

lim

xw (AB)→1

{xw (AB) + n[1 − xw (AB)] + kSt [1 − xw (AB)]} = 1 (3.181)

lim

xw (AB)→1

F = 1 − n − kSt

(3.182)

and finally for Eq. (3.180) dxr (AB) 1 − n − (1 − n − kSt ) = = kSt xw (AB)→1 dxw (AB) 1 lim

(3.183)

Phase Equilibria

199

The derivative on the left side of Eq. (3.183) is not necessary to calculate separately. Its value is always equal to the number of new substances (kSt ) originating in the reaction between the solvent AB and the admixture X, distinguishable from the dissociation products of the solvent AB. However, it is necessary to choose such reaction schemes between AB and X for which kSt equals the left side of the experimentally determined value of Eq. (3.177). In the following application of the above-presented thermodynamic approach to the Na3AlF6 –Al2 O3 system, we confront the common approach with the new one with regard to the general dissociation scheme of cryolite, which transforms into the form α

A3 B −−−→ AB + 2B

(3.184)

According to the recent Raman spectroscopic investigation performed by Robert et al. (1997a) and direct oxygen LECO analysis carried out by Daneˇ k et al. (2000b), aluminum oxide dissolves in cryolite under the formation of two main oxofluoroaluminates, Na2Al2 O2 F4 and Na2Al2 OF6 , according to the general scheme 2Na3 AlF6 + Al2 O3 = Na2 Al2 OF6 + Na2 Al2 O2 F4 + 2NaF

(3.185)

which indicate the introduction of two new substances when 1 mol Al2 O3 is dissolved in an infinite amount of cryolite, i.e. kSt = 2. The phase diagram of this system according to Chin and Hollingshead (1966) and the calculated slopes of the liquidus curves for three chosen kSt are shown in Figure 3.49. The material balance regarding the above reaction is as follows: Let us consider 1 mol of mixture with the composition x1 (=xw (Na3AIF6 ) mol Na3AlF6 + x2 (=xw (X)) mol Al2 O3 , where x2 << x1 . Cryolite dissociates according to reaction (3.184) with a dissociation degree α. The dissolution of Al2 O3 in cryolite is accompanied by the reaction (3.185). At equilibrium, we get the following amounts of the individual substances n(Na3 AlF6 ) = [(1 − α)x1 − 2x2 ] mol n(NaAlF4 ) = αx1 mol n(NaF) = (2αx1 + 2x2 ) mol n(Na2 Al2 OF6 ) = x2 mol n(Na2 Al2 O2 F4 ) = x2 mol The total amount of all substances is ni = [x1 (1 + 2α) + 2x2 ]mol. As we are in the region of dilute solutions, the mole fractions can be set equal to the activities, i.e. the

200

Physico-chemical Analysis of Molten Electrolytes

1340

1320

T(K)

1300

1280 kSt = 1

1260

1240 kSt = 3

1220

Na3AlF6 0.05

0.10

kSt = 2

0.15 0.20 x(Al2O3)

0.25

0.30

Figure 3.49. Phase diagram of the Na3AlF6 –Al2 O3 system according to Chin and Hollingshead (1966).

real mole fractions of substances. For the real mole fraction of cryolite, we then get n(Na3 AlF6 ) + n(NaAlF4 ) + n(NaF) x1 (1 + 2α) = ni x1 (1 + 2α) + 2x2

xr (Na3 AlF6 ) =

(3.186)

Differentiating Eq. (3.186) according to x1 and inserting the limiting conditions (x1 = 1, x2 = 0) we get lim

xr (Na3 AlF6 )→1

∂xr (Na3 AlF6 ) ∂x1

=

(1 + 2α)(x1 + 2x1 α + 2x2 ) − (x1 + 2x1 α)(1 + 2α − 2) (x1 + 2x1 α + 2x2 )2

=

(1 + 2α)(1 + 2α) − (1 + 2α)(2α − 1) 2 = kSt = 2 (1 + 2α) (1 + 2α)

(3.187)

Phase Equilibria

201

as according to Grjotheim et al. (1982) α, attains at 1000◦ C the value of 0.3, the Stortenbecker’s correction factor equals 1.25, which is in contrast to the expected value. Such a procedure obviously does not yield the correct result. However, as it follows from the approach presented by Proks et al. (2002), it is not necessary to take into account the dissociation of cryolite in the calculation of kSt , since the solvent does not consider the originating dissociation product of cryolite as a foreign species. The material balance is thus as follows. The equilibrium amounts of individual substances are n(Na3 AlF6 ) = (x1 − 2x2 ) mol n(NaF) = 2x2 mol n(Na2 Al2 OF6 ) = x2 mol n(Na2 Al2 O2 F4 ) = x2 mol The total amount of all substances is cryolite we then get the relation xr (Na3 AlF6 ) =

ni = 1 + x2 mol. For the real molar fraction of

n(Na3 AlF6 ) + n(NaF) x1 = ni 1 + x2

(3.188)

Differentiating Eq. (3.188) according to x1 and inserting the limiting conditions (x1 = 1, x2 = 0) we get lim

xr (Na3 AlF6 )→1

∂xr (Na3 AlF6 ) ∂x1

=

(1 + x2 ) + x1 = 2 = kSt (1 + x2 )2

(3.189)

which is in very good agreement with the experimentally determined value kSt = 1.99. This means that 1 mol Al2 O3 introduces two new species into an infinite amount of cryolite. This is in accordance with the assumed reaction (3.185). The new species are the compounds Na2Al2 OF6 and Na2Al2 O2 F4 , since NaF is already present in cryolite due to its thermal dissociation. 3.3.2.3. Cryoscopy in eutectic mixtures

For the equilibrium solidus–liquidus of the component i in a simple eutectic system in which no compounds are formed and the solubility of the components in the solid state does not exist, the differential form of the Le Chatelier–Shreder’s equation holds d ln ai =

fus Hi dT RT 2

(3.190)

202

Physico-chemical Analysis of Molten Electrolytes

where ai and fus Hi are the activity and the enthalpy of fusion of the component i, respectively, T is the temperature, and R is the gas constant. For simplification, it is assumed that fus Hi = f (T ) and the solution is ideal, i.e. ai = xi . Then by integration and rearrangement of Eq. (3.190), for the melting point depression, T, (in fact it is the depression of the temperature of primary crystallization) of the solvent A that was effected by minor additions of the solute B in the A–B system, the well-known relation is obtained T =

2 RTfus, A

fus HA

(3.191)

T xB

The problem is whether it is possible to carry out analogous measurements also in the eutectic mixtures A–B, i.e. to determine the depression of the eutectic temperature affected by minor additions of the substance C. For this case, Førland (1964) derived an equation similar to Eq. (3.191) substituting Tfus, E (eutectic temperature) in Eq. (3.191) for Tfus, A , fus HE (enthalpy of fusion of the eutectic mixture, which is independent of the concentration of the component C, xC ) for fus HA , and xC for xB . However, it could be a matter of discussion, whether the equation proposed by Førland has a general validity. Fellner and Matiašovský (1974) derived an equation for cryoscopy in the eutectic mixtures, which was not found to be analogous with the Le Chatelier–Shreder’s equation. Let us consider an ideal ternary system A–B–C. In this case Eq. (3.190) must be valid for all the three components. From the integral form of Eq. (3.190) for the components A and B, the equation of a projection of the monovariant line of the simultaneous crystallization of these components, E–P in Figure (3.50), can be derived ln xA ln xB 1 − = fus HA fus HB R

1 Tfus, A

−

1 Tfus, B

(3.192)

The concentrations of components A and B at the eutectic point will be denoted as xA∗ and xB∗ . In Eq. (3.192), the temperature is not expressed explicitly; it can be calculated by introducing Eq. (3.192) into the integral form of Eq. (3.190) for the components A and B, respectively. As mentioned above, Førland’s equation for the depression of the eutectic temperature is based on the assumption that fus HE = constant. Thus assuming the relation fus HE = xA∗ fus HA +xB∗ fus HB , the ratio of xA and xB on the monovariant line of the simultaneous crystallization of A and B should be also constant, i.e. it must not be a function of xC . However, this condition is fulfilled only on the assumption that the monovariant line E–P is, at least in the proximity of the binary eutectic point E, identical with the E–C. The diagram presented by Førland for illustration of the proposed relation was drawn in such a way that it might fulfill the above condition. However, in a common case, the slope of the tangent to the line of the simultaneous crystallization of the components A and B can

Phase Equilibria

203

C

P

E

A

B

Figure 3.50. Phase diagram of the ternary system A–B–C in which no solid solutions are formed.

be calculated, after substituting for A in Eq. (3.192) the expression xA = 1–xB –xC , by differentiating Eq. (3.192) as an implicit function dxC xA fus HA = −1 − dxB xB fus HB

(3.193)

Equation (3.193) can be derived also directly from the differential form of Eq. (3.190). The slope of the connection E–C in oblique-angled coordinates equal –1/x+ B , whereas according to Eq. (3.193), for this slope, we obtain x ∗ fus HA d xC = −1 − A∗ xC →1 d xB xB fus HB lim

(3.194)

This limit equals to −1/xB∗ solely in the case when fus HA = fus HB . Only in this case the ratio xA /xB in an ideal system is constant and equal to the ratio xA∗ /xB∗ . Consequently, the statement that the ratio xA /xB is constant implies the condition that the enthalpies of fusion of both components, which compose the eutectic mixture, are equal.

204

Physico-chemical Analysis of Molten Electrolytes

Setting i = B from Eq. (3.193) into Eq. (3.190), the following relation is obtained dT = −

RT 2 d xC xA fus HA + xB fus HB

(3.195)

which actually appears to be the differential form of the equation of the monovariant line of the simultaneous crystallization of the components A and B. This equation may be derived also directly from Eq. (3.190). From a comparison of Eqs. (3.195) and (3.190) it follows that in Eq. (3.195) the increase in the concentration of component C is presented as dxC and not as d ln xC . Only if fus HA = fus HB , by rearranging Eq. (3.195) the following relation is obtained dT =

RT 2 RT 2 d(xA + xB ) = d ln(xA + xB ) fus HA (xA + xB ) fus HA

(3.196)

In the limiting case for xC → 0, from Eq. (3.195) it follows RTE2 dT =− + xC →0 dxC xA fus HA + xB+ fus HB lim

(3.197)

Thus the cryoscopic measurements may be performed also in eutectic mixtures, since the dT value of the lim dx can be determined. However, it is to be taken into account that the xC →0

C

reasons, why the function T = f xC ) is not linear, are in this case somewhat different. This difference must be considered in experimental work, especially in the estimation of maximum value of xC , at which the error in the determination of temperature resulting from the non-constant value of the denominator in Eq. (3.195) still lies below the limit of experimental determinability. In a common case, for the calculation of the enthalpy of fusion of the eutectic mixture evidently also the temperature dependence of fus HA and fus HB , as well as the enthalpy of mixing of components has to be considered. This, however, does not affect the principal meaning of the above-derived dependencies. Cryoscopic measurements in the eutectic mixture used Daneˇ k et al. (1975) in order to determine the behavior of K2 TiF6 in the LiF–LiCl eutectic mixture. K2 TiF6 in pure LiCl reacts with the chloride anions according to the scheme K2 TiF6 + Cl− = 2K + + 6F− + TiCl4

(3.198)

creating nine new particles, which means that the fluoride ions in TiF4 were substituted by the chloride ions. However, the equilibrium of reaction (3.198) would be substantially shifted to the left side at sufficiently high concentrations of fluoride ions. According to the

Phase Equilibria

205

thermodynamic calculation, this situation could occur in the LiF–LiCl eutectic mixture. The experimental verification of this assumption was made by cryoscopic measurements. The experimentally measured values of the eutectic temperature depression caused by the addition of K2 TiF6 were compared with those calculated on the basis of the determined value of the cryoscopic constant of the LiF–LiCl mixture. The Stortenbeker’s correction factor was found to be kSt = 3, indicating the formation of three new particles due to the dissociation of K2 TiF6 . Hence it follows that in the LiF–LiCl eutectic mixture the dissociation of K2 TiF6 can be described by equation K2 TiF6 = 2K + + 2F− + TiF4

(3.199)

and the substitution reaction does not take place. The verification of this result anticipates the knowledge of the enthalpy of fusion of the LiF–LiCl eutectic mixture. The corresponding data were not known and they were thus determined using the cryoscopic method. In the measurement of the eutectic temperature depression in the LiF–LiCl eutectic mixture, sodium chloride was used as the solute. From the slope of the tangent to the determined monovariant line the value of the enthalpy of fusion of the eutectic mixture fus HE = 20.1 ± 1.3 kJ·mol−1 was obtained. The value of the enthalpy of fusion of the LiF–LiCl eutectic mixture was verified by calculation according to the equation fus HE = xLiF fus H LiF + xLiCl fus HLiCl + mix H

(3.200)

where fus HLiF , fus HLiCl , and fus HE are the enthalpies of fusion of pure components and of the eutectic mixture at the eutectic temperature, respectively, xLiF and xLiCl are the molar fractions of LiF and LiCl, respectively, and mix H is the enthalpy of mixing. The dependence of the enthalpy of fusion on temperature was calculated using the JANAF Thermochemical Tables (1971). The activity coefficients of LiF and LiCl in the eutectic mixture at the eutectic temperature were calculated from the phase diagram determined by Haendler et al. (1959). Since the values of the activity coefficients were found to be close to one (γ LiF = 0.951, γ LiCl = 0.991), the solution could be considered as ideal and thus the enthalpy of mixing close to zero. Assuming that the solution is regular, the value of −0.15 kJ·mol−1 was calculated, which is below the limit of the accuracy of the measurement. The calculated value of the enthalpy of fusion of the LiF–LiCl eutectic mixture fus HE ∼ = 20.9 kJ·mol−1 is in very good agreement with the experimentally measured value. 3.3.3. Differential thermal analysis

Differential thermal analysis is a powerful tool in the investigation of matter. Using this method, different phase transformations and reactions can be determined. The principle of

206

Physico-chemical Analysis of Molten Electrolytes

this method is similar to the thermal analysis, but in the differential thermal analysis, the temperature difference between the sample and the reference substance during constant heating rate is measured. A typical experimental set-up is shown schematically in Figure 3.51. Two small crucibles made preferably of platinum are placed abreast in a ceramic block, which serves as the heat absorber. The amount of sample ranges from 0.05 to 0.5 g. In the temperature range of measurement, the reference substance must not undergo any phase transformation or any reaction with the surroundings. Fine aluminum oxide powder, carefully purified and dried, is generally used as a reference substance. To the bottom of each crucible the hot joint of one thermocouple is attached. Both the thermocouples are linked outside the ceramic block by equal wires. Such a linkage enables the measurement of temperature difference of both the crucibles during heating and the difference versus time is recorded. Such a connection, however, yields zero voltage when

Heat shields

Heating tube

Ceramic block Reference substance Sample Thermocouples

µV Figure 3.51. The experimental set-up for differential thermal analysis.

Phase Equilibria

207

the sample shows no heat effect and a straight line is then registered. However, the final scan strongly depends on the configuration of the equipment used. The temperature detection, restricted to the surface of the crucibles, does not fully respond to the heat flow in the bulk sample. This results in a baseline drift of thermograms that may cause some uncertainties in the evaluation of the temperature of the thermal effect. The use of the differential thermal analysis in the phase diagram determination is illustrated in Figure 3.52. A hypothetical binary eutectic system A–B with the formation of the incongruently melting compound A4 B was chosen. There are three thermograms (a) to (c) shown as examples. In thermogram (a), the first heat effect at temperature a1

a

c

b

Temperature

a2

a

b3 a1

A4 B

b2

A

-dt c1

b1

a1

B

x(B)

a2

Temperature c

b

-dt

-dt

b1

b2

b3

Temperature

c1

Temperature

Figure 3.52. Phase diagram determination using differential thermal analysis. Thermograms (a) to (c) correspond to different compositions of the hypothetical system.

208

Physico-chemical Analysis of Molten Electrolytes

corresponds to the melting of the eutectic mixture in the subsystem A–A4 B, followed by the progressive melting until the liquidus is reached at the temperature a2 , where the complete system becomes liquid. In thermogram (b), the first heat effect at temperature b1 corresponds to the eutectic melting in the subsystem A4 B–B, followed by the progressive melting of A4 B up to the temperature b2 , where the amount of pure A4 B, still present as a solid, undergoes isothermal peritectic transformation into solid A and a liquid. Thereafter, the next progressive melting of A takes place up to the liquidus temperature b3 . Finally, the thermogram (c) represents the fusion of the eutectic mixture in the A4 B–B system. Thermograms similar with (c) can also be obtained for the fusion of pure components A, B, as well as for the intermediate compound A4 B.

3.4.

CALCULATION OF PHASE DIAGRAMS

The theoretical calculation of phase diagrams is an important tool both in forecasting the properties of materials, as well as in the study of the structure of the melts, and became a self-sustaining field of science. Several characteristics of the phase diagrams can be related to the calculations and to particular terms in the theory. Extension of the deviation from ideal behavior is an important feature, which can explain the probability of the formation of compounds. The calculation of phase diagram can prove to be useful for many practical reasons. One can (i) (ii) (iii)

predict ternary phase diagrams a priori from the data of pure materials and on the binary systems, utilize the thermodynamic self-consistency of the theory to check phase diagrams for inconsistency or phenomena, which are not likely to occur, utilize the theory fitted to a very few experimental points in order to extrapolate from those few measurements. Thus, the use of the theory can minimize the number of measurements one needs to characterize any given system.

3.4.1. Coupled analysis of thermodynamic and phase diagram data

For the calculation of the phase diagrams using coupled analysis of thermodynamic and phase diagram data, the thermodynamic data represent enthalpies of fusion, enthalpies of mixing, heat capacities, and all other data that are available from the literature. The phase diagram data are the measured temperatures of primary crystallization, temperatures of secondary crystallization, etc. as well as the temperatures of the eutectic temperatures. The calculation of the phase diagrams of condensed systems using the coupled analysis of the thermodynamic and phase diagram data is based on the solution of a set of equations

Phase Equilibria

209

of the following type fus G0i (T ) + RT ln

al, i (T ) =0 as, i (T )

(3.201)

where fus G0i is the standard molar Gibbs energy of fusion of the component i at the temperature T, R is the gas constant, and as, i (T) and al, i (T) are the activities of component i in the solid and liquid phase, respectively. Assuming immiscibility of components in the solid phase (as, i = 1) and that the enthalpy of fusion of the components does not change with temperature, for the thermodynamic temperature of primary crystallization of the component i, Tpc, i , we get Tpc, i =

fus Hi0 + RTpc, i ln γl, i fus Si0 − R ln xl, i

(3.202)

where fus H0i and fus S0i are the standard enthalpy and standard entropy of fusion, respectively, xl, i and γ l, i is the mole fraction and the activity coefficient of component i, respectively. The activity coefficients can be calculated from the molar excess Gibbs energy of mixing

RTpc, i ln γl. i

∂ nGEter = ∂ ni

(3.203)

T , p, nj =i

where ni is the amount of component i and n is the total amount of all the components. In the ternary system A−B−C the molar excess Gibbs energy of mixing in the liquid phase, GEter can be described by the following general equation GEter =

k(j ) l(j ) m(j )

xA xB xC

Gj

(3.204)

j

where xi are the mole fractions of the components, Gj are the empirical coefficients in the composition dependence of the molar excess Gibbs energy of mixing and k(j), l(j), m(j) are adjustable integers. For the boundary binary systems one of the integers equals zero. Using Eq. (3.202), the following mathematical model for the coupled thermodynamic analysis was used Tpc, i = F0, i +

j

Fj, i Gj

(3.205)

210

Physico-chemical Analysis of Molten Electrolytes

where Tpc, i was obtained from the phase diagram measurement. The first term on the right side represents the ideal behavior and the second one, the deviation from the ideal behavior. For the auxiliary functions F0, i and Fj, i , with respect to the Gibbs–Duhem relation, the following equations hold fus Hi0

F0,i =

(3.206)

fus Si0 − R ln xl, i   k(j ) l(j ) m(j ) ∂ nxA xB xC   ∂ ni nj =i

Fj, i =

(3.207)

fus Si0 − R ln xl, i

If an intermediate compound Z = Ap Bq Cr (p + q + r = 1) is formed in the ternary system, Eqs. (3.204), (3.206), and (3.207) for this compound must be modified GEter =

k(j ) l(j ) m(j )

xA xB xC

− p k(j ) q l(j ) r m(j ) Gj

(3.208)

j

F0, Z =

p Fj, Z =

∂ G ∂ nA

fus HZ0

+q nB , nC

(3.209)

p q

fus SZ0 − R ln KxA xB xCr ∂ G ∂ nB

+r

nA , nC p q

∂ G ∂ nC

nA , nB

fus SZ0 − R ln KxA xB xCr

(3.210)

where

−1 K = pp q q r r

(3.211)

k(j ) l(j ) m(j ) G = n xA xB xC − p k(j ) q l(j ) r m(j )

(3.212)

and

If the molar enthalpy of mixing of the system is known, the molar Gibbs energy of mixing can be expressed as follows Gmix = Hmix + T Smix

(3.213)

Phase Equilibria

211

The enthalpy of mixing can be expressed as a function of composition in the form Hmix =

α

β

i i xAX xBX Hi

(3.214)

i

where Hi are empirical coefficients determined, for example using the least-squares method from experimental data, and α i and βi are integers. A similar equation can be written for the entropy of mixing Smix =

α

β

i i xAX xBX Si

(3.215)

i

In the three-component systems, two different procedures may be used for the calculation of the molar excess Gibbs energy of mixing. In the first one, the molar excess Gibbs energy of mixing is calculated separately for the individual binary boundary systems using the binary phase diagram data. The resulting coefficients for the binary systems are then used for the calculation of the molar excess Gibbs energy of mixing in the ternary system using the ternary phase diagram data only. This approach is frequently used in the so-called “symmetrical” ternary systems, in which the excess Gibbs energy of mixing of all three binary systems is approximately of the same order. In the second procedure, the calculation of the molar excess Gibbs energy of mixing in the ternary system is performed in one step using Eq. (3.204). The use of this approach is advantageous in the “asymmetrical” ternary systems, in which the excess Gibbs energy of mixing of one binary system substantially differs from those of the other two binaries. The application of the first approach to such ternary systems describes well the binaries, but does not yield a reliable ternary diagram. For such a case, the latter approach is more convenient, however, on account of the lower precision of the calculated binary diagrams. Qiao et al. (1996) defined the thermodynamic criterion for judging the symmetry of the ternary systems and its application. Based upon the previous significant studies and the enlightening analysis made by works of Toop (1965), Ansara (1979), Hillert (1980), Lukas (1982), and taking into account interactions among components, the thermodynamic criterion for defining the symmetry of ternary systems from the energetic point of view, was explicitly proposed as follows: “If the excess thermodynamic properties of the three binary subsystems of the A–B–C ternary system are similar to each other, the ternary system is symmetric. If the deviation of the binary system A–B and A–C from the ideal behavior are similar, but differ markedly from that of the binary system B–C, then the A–B–C ternary system is asymmetric. In the asymmetric system the component A in two binary subsystems with thermodynamic similarity should be chosen as the thermodynamic asymmetric component.”

212

Physico-chemical Analysis of Molten Electrolytes

As an example of the asymmetric ternary system, Qiao et al. (1996) presented the system PrCl3 –CaCl2 –MgCl2 . The excess molar Gibbs energy of the binary systems CaCl2 –MgCl2 and PrCl3 –MgCl2 show positive deviations from ideal solutions, while the PrCl3 –CaCl2 system shows negative deviations from ideality. Hence, according to the above-mentioned thermodynamic criterion, MgCl2 is the asymmetric component. For most LnCl3 –CaCl2 –MgCl2 (Ln = rare earth metals) systems, MgCl2 can be reasonably chosen as the asymmetric component. According to Papatheodorou et al. (1967), in pure MgCl2 there are Mg–Cl–Mg bridging bonds. When CaCl2 or LnCl3 is added to MgCl2 , the original bridging bonds are broken. This additional energy is absorbed and the total energy of the system increases, which shows in the thermodynamic properties of the system. As an example of the asymmetric system, NdCl3 –CaCl2 –LiCl could be mentioned, where apparent negative deviations from ideal solutions have been observed in all binary subsystems. However, in the binary system NdCl3 –LiCl, the negative deviations are much bigger than in the NdCl3 –CaCl2 and CaCl2 –LiCl systems. From the view of the asymmetric distribution of energy in the ternary system, it is reasonable to choose CaCl2 as the asymmetric component. The asymmetric model was successfully applied also in molten slag systems. As it was shown by Pelton and Blander (1986), in the CaO–FeO–SiO2 system there is one acidic (SiO2 ) and two basic (CaO and FeO) components. Thus the most appropriate seems to be SiO2 as the asymmetric component. However, according to the above-mentioned criterion, FeO should be chosen as the thermodynamic asymmetric component, since the CaO–SiO2 system shows much more negative deviations from the ideal behavior than the systems, CaO–FeO and FeO–SiO2 . After a systematic comparison of the calculated and measured results of excess thermodynamic properties and phase diagrams, it can be concluded that the key point in the choice of the suitable geometric model to predict ternary thermodynamic properties from the binary ones is to choose reasonably between the symmetric or asymmetric models. In most cases, the main source of error in the calculation arises from the incorrect choice of the asymmetric component in the asymmetric model. The coupled thermodynamic analysis, i.e. the calculation of the coefficients Gj in Eq. (3.204) is performed using the multiple linear regression analysis omitting the statistically non-important terms according to the Student test on the chosen confidence level. As the optimizing criterion for the best fit between the experimental and calculated temperatures of primary crystallization, the following condition was used for all the p measured points p

Tpc,exp,i − Tpc,calc,i

i=1

2

= min

(3.216)

Phase Equilibria

213

Beside the condition (3.216) for the calculation of the molar excess Gibbs energy of mixing, the minimum necessary Gj coefficients for attaining thermodynamically consistent phase diagram and a reasonable standard deviation of approximation are required. E = f (T ). The calculation is mostly performed assuming fus Hi = f (T ) and Gter Using a coupled analysis of thermodynamic and phase diagram data, phase diagrams of the Fe–Cr–O, Fe–Ni–O, Cr–Ni–O systems (Pelton et al., 1979a), Mn3 O4 –Co3 O4 and Fe3 O4 –Co3 O4 spinelitic systems (Pelton et al., 1979b), binary and ternary systems of the NaCl–Na2 SO4 –Na2 CrO4 –Na2 MoO4 –Na2 WO4 system (Liang et al., 1980), AlF3 –LiF– NaF system (Saboungi et al., 1980), Na+ , K+ // F− , SO2− 4 system (Hatem et al., 1982), etc., were calculated. Thompson et al. (1987) has developed a special program F*A*C*T (Facility for the Analysis of Chemical Thermodynamics) to calculate the phase diagrams of various systems, for example 70 binary alkali metal halide systems with common ions (Sangster and Pelton, 1987) and a number of binary and ternary alkali metal salt systems (Sangster and Pelton, 1987). In the following section, an example of the coupled analysis of the thermodynamic and phase diagram data will be presented. 3.4.2. Calculation of the phase diagram of the quaternary system KF−KCl−KBF4 −K2 TiF6

As an example, the coupled analysis of the thermodynamic and phase diagram data of the KF−KCl−KBF4 −K2 TiF6 system performed by Chrenková et al. (2001) is presented. This system is important because of its potential use as an electrolyte for electro-deposition of titanium diboride. The molar excess Gibbs energy of mixing in the liquid phase in the quaternary system KF−KCl−KBF4 −K2 TiF6 was calculated as the sum of the molar excess Gibbs energies of mixing in the binary systems, of the molar excess Gibbs energies of mixing in the ternary systems, and of the quaternary interaction terms. The molar excess Gibbs energy of mixing in the boundary binary systems was descibed using the following general equation b(k) c(k) GEi,bin = x 1 x 2 Gk (3.217) k

and in the ternary systems GEj,ter

=

3

GEi,bin +

b (k) c (k) d (k) x 2 x 3 Gk

x1

(3.218)

k

i=1

For the molar excess Gibbs energy of mixing in the quaternary system, the final equation is then 4 b (k) c (k) d (k) e (k) GEquat = GEj,ter + x1 x2 x3 x4 Gk (3.219) j =1

k

214

Physico-chemical Analysis of Molten Electrolytes

where the second term represents the quaternary interactions. Coefficients b (k), c (k), d (k), and e (k) are integers in the range 1−3. Two additive binary compounds, K3 TiF7 and K3 TiF6 Cl, are formed in the system. In the calculation this fact was taken into account using Eqs. (3.208) to (3.212). The coupled thermodynamic analysis, i.e. the calculation of coefficients Gk , Gk , and Gk in Eqs. (3.204), (3.205), and (3.206), respectively, has been performed using the multiple linear regression analysis omitting the statistically non-important terms in the molar excess Gibbs energy of mixing at 0.99 confidence level according to the Student’s test. The values of the enthalpy of fusion of individual components used in the calculation are summarized in Table 3.4. The experimentally determined temperatures of primary crystallization in the ternary KF−KCl−KBF4 system were measured by Patarák and Daneˇ k (1992), those of the KF−KBF4 −K2 TiF6 system by Chrenková et al. (1996), and those of the KCl−KBF4 −K2 TiF6 system by Chrenková et al. (1995). The temperatures of primary and eutectic crystallization in the KF-KCl−KBF4 −K2 TiF6 system were measured by Chrenková et al. (2001). For the excess molar Gibbs energy of mixing in the boundary binary systems the general equation GEi, bin = x1 x2 G1 + G2 x2 + G3 x22

(3.220)

was found to be valid. The values of the coefficients Gi together with the standard deviations of the fits are given in Table 3.5. For the excess molar Gibbs energy of mixing in the individual ternary systems the following equations were obtained

GEKCl−KF−KBF4 =

3

GEi, bin + G1 x1 x22 x3 + G2 x1 x22 x32

(3.221)

i=1

Table 3.4. Temperatures and enthalpies of fusion of compounds used for the phase diagram calculation Compound

fus (KJ · mol−1 )

Tfus (K)

Reference

KF KCl KBF4 K2 TiF6 K3 TiF7 K3 TiF6 Cl

27.196 26.154 17.656 21.000 57.000 47.000

1131 1045 843 1172 1048 969

Knacke et al. (1991) Knacke et al. (1991) Knacke et al. (1991) Adamkoviˇcová et al. (1995a) Adamkoviˇcová et al. (1995b) Adamkoviˇcová et al. (1996)

Phase Equilibria

215

Table 3.5. Coefficients Ci of the concentration dependence of the molar excess Gibbs energy of mixing and the standard deviations of the temperature of primary crystallization in the binary subsystems of the quaternary system KF−KCl−KBF4 −K2 TiF6 System KCl−KF KF−KBF4 KF−K2 TiF6 KCl−KBF4 KCl−K2 TiF7 K2 TiF7 −KBF4

G1 (J · mol−1 )

G2 (J · mol−1 )

G3 (J · mol−1 )

σ (◦ C)

2144 ± 547 3836 ± 233 −11507 ± 1743 50 ± 22 −7531 ± 2458 8475 ± 830

−7379 ± 284 −14434 ± 790 −19918 ± 6050 3725 ± 736 25700 ± 6335 −25810 ± 1426

6111 ± 829 6625 ± 833 33344 ± 5830 −7175 ± 2367 −31125 ± 8457 12905 ± 713

1.2 1.9 6.5 5.6 5.0 6.1

GEKF−K2 TiF6 −KBF4 =

3

GEi, bin + G1 x13 x2 x3 + G2 x13 x2 x32

(3.222)

i=1

GEKCl−K2 TiF6 −KBF4 =

3

GEi,bin + G1 x1 x23 x32 + G2 x1 x2 x33 + G3 x12 x22 x33

i=1

GEKCl−KF−K2 TiF6 =

3

+ G4 x13 x2 x3

(3.223)

GEi, bin + G1 x1 x2 x3 + G2 x1 x22 x3 + G3 x13 x22 x33

(3.224)

i=1

The calculated coefficients of the concentration dependence of the excess molar Gibbs energy of mixing together with the standard deviations of the fit temperature in the ternary systems are given in Table 3.6. The ternary system KCl−KF−KBF4 is a simple eutectic with the coordinates of the eutectic point 21 mole % KF, 19 mole % KCl, 60 mole % KBF4 , te = 409◦ C. Table 3.6. Coefficients Ci of the concentration dependence of the molar excess Gibbs energy of mixing and the standard deviations of the temperature of primary crystallization in the ternary subsystems of the quaternary system KF−KCl−KBF4 −K2 TiF6 System KF–K2 TiF6 –KBF4

Coefficient

KCl–KF–KBF4

KCl–KF–K2 TiF6

G1 (J · mol−1 ) G2 (J · mol−1 ) G3 (J · mol−1 ) G4 (J · mol−1 ) σ (◦ C)

−22709 ± 1429 −36041 ± 1843 – – 6.8

36975 ± 2173 −31585 ± 2067 −1241825 ± 89126 – –

−14718 ± 5725 −198846 ± 16972 – – 15.2

KCl–K2 TiF6 –KBF4 −263005 ± 16623 92415 ± 8284 796972 ± 72485 41055 ± 5182 17.6

216

Physico-chemical Analysis of Molten Electrolytes

The inaccuracy in the calculated ternary phase diagram is ± 6.8◦ C. In the ternary system KF−K2 TiF6 −KBF4 , the intermediate compound K3 TiF7 divides the system into two simple eutectic systems. The calculated coordinates of the two ternary eutectic points are: e1 : 26 mole % KF, 68 mole % KBF4 , 6 mole % K2 TiF6 , te1 = 450◦ C, e2 : 3 mole % KF, 69 mole % KBF4 , 28 mole % K2 TiF6 , te2 = 435◦ C. The inaccuracy in the calculated ternary phase diagram is ± 15.2◦ C. In the ternary system KCl−K2 TiF6 −KBF4 , the intermediate compound K3 TiF6 Cl divides the ternary system into two simple eutectic systems. The calculated coordinates of the two ternary eutectic points are: e1 : 18 mole % KCl, 66 mole % KBF4 , 16 mole % K2 TiF6 , te1 = 449◦ C, e2 : 6 mole % KCl, 63 mole % KBF4 , 31.0 mole % K2 TiF6 , te2 = 417◦ C. The inaccuracy in the calculated phase diagram is ± 17.6◦ C. In the ternary system KF−KCl−K2 TiF6 two intermediate compounds, K3 TiF7 and K3 TiF6 Cl, are formed. The calculated coordinates of the three ternary eutectic points are: e1 : 13 mole % KF, 25 mole % KCl; 62 mole % K2 TiF6 , te1 = 635◦ C, e2 : 21 mole % KF, 50 mole % KCl, 29 mole % K2 TiF6 , te2 = 620◦ C, e3 : 46 mole % KF, 44 mole % KCl; 10 mole % K2 TiF6 , te3 = 586◦ C. The phase diagram was calculated from the binary boundaries and from the quaternary mixtures measured, therefore the standard deviation of the fit is not given. The calculated phase diagram of this system is shown in Figure 3.53. The experimentally determined phase diagram measured by Chernov and Ermolenko (1973) is probably not quite correct because of little experimental data in the region of the presented peritectic point. Finally, for the excess molar Gibbs energy of mixing in the quaternary system KCl(1)–KF(2)–K2 TiF6 (3)–KBF4 (4), the following equation was obtained: GEquat =

4

GEj, ter + G1 x1 x2 x3 x43 + G2 x13 x2 x32 x42 + G3 x1 x2 x33 x4

(3.225)

j =1

The calculated coefficients of the excess molar Gibbs energy of mixing and the standard deviation of the measured and calculated temperature of primary crystallization for the quaternary system are listed in Table 3.7.

Phase Equilibria

217

K2TiF6

80

0 .8

75

70 65

0 .6

0

0 .8

0

0

0

0 .6 750

650

65 0

70 0

e1

0 .4

0 .4

e2 0 .2

0 .2

0.2

0.4

0 80

75

0 65

0

60 0

e3

70

0

750

KCl

0

0

65

70

0.6

0.8

KF

Figure 3.53. The calculated ternary phase diagram of the KCl−KF−K2 TiF6 system.

Three quaternary eutectic points were found in the quaternary phase diagram KF−KCl−KBF4 −K2 TiF6 . Their coordinates are: e1 : 2.8 mole % KF, 5.8 mole % KCl, 64.5 mole % KBF4 , 26.9 mole % K2 TiF6 , te1 = 413◦ C, e2 : 4.9 mole % KF, 13.7 mole % KCl, 73.9 mole % KBF4 , 7.5 mole % K2 TiF6 , te2 = 389◦ C,

Table 3.7. Coefficients Ci of the concentration dependence of the molar excess Gibbs energy of mixing and the standard deviation of the temperature of primary crystallization in the quaternary system KF−KCl−KBF4 −K2 TiF6 G1 (J · mol−1 ) G3 (J · mol−1 ) G2 (J · mol−1 )

2009367 ± 145628 249546 ± 20491 15367354 ± 1123575

80% K2TiF6 + 20% KBF4 800 0.6

750 700

0.6

0.4

0.4 0 70

65 0

0 60

0.2

0.2

0.4

0.6

75 0

70 0

60 0

600

0 65

80% KCl + 20% KBF4 0.2

65 0

60 0

80% KF + 20% KBF4

Figure 3.54. Cross section of the quaternary phase diagram of the KF−KCl−K2 TiF6 −KBF4 system at 20 mole % KBF4 .

60% K2TiF6 + 40% KBF4 700 65 0 60 0 0.4

0.4

60 0

0.2

0.2

550

0.2

0 50 50 5

0.4

0 60

0 65

70 0

500 500

60% KCl + 40% KBF4

0 55

60 0

60% KF + 40% KBF4

Figure 3.55. Cross section of the quaternary phase diagram of the KF−KCl−K2 TiF6 −KBF4 system at 40 mole % KBF4 .

Phase Equilibria

219

40% K2TiF6 + 60% KBF4 50 0 450 0.3

0 45

450

0.3

0.2

0.2

50 0

0.1

0.1

0 50

450 450

40% KCl + 60% KBF4

0.1

0 50

0 45

0.2

0.3

0 55

40% KF + 60% KBF4

Figure 3.56. Cross section of the quaternary phase diagram of the system KF−KCl−K2 TiF6 −KBF4 at 60 mole % KBF4 .

e3 : 14.0 mole % KF, 11.2 mole % KCl, 74.0 mole % KBF4 , 0.8 mole % K2 TiF6 , te2 = 336◦ C. The inaccuracy in the calculated phase diagram is ± 20.7◦ C. The cross sections of the quaternary phase diagram KF−KCl−KBF4 −K2 TiF6 with constant content of 20, 40, and 60 mole % KBF4 are shown in Figures 3.54–3.56.

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Chapter 4

Enthalpy The heat measurement of various reactions is the first step in entering the realm of thermodynamics. Every study of any chemical process starts with the laboratory work connected with the first law of thermodynamics. On the other hand, when the thermodynamic considerations do not lead to a reasonable result, it is necessary to go back to the laboratory work. The basic unit of heat is joule (J). The older unit is calorie (cal). The relation between them is 1 J = 4.314 cal.

4.1. THERMODYNAMIC PRINCIPLES

The measurement of enthalpy, or heat, of different chemical processes is the objective of the first thermodynamic principle – the law of energy conservation. Every chemical process is connected with a certain amount of enthalpy, which the system receives from or delivers to the surroundings. Consider a closed system in which a chemical reaction takes place and that can exchange only heat with the surroundings. During the reaction, an amount of heat is liberated to the surroundings. We can define the function U – internal energy, as the energy, which the system has exchanged with the surroundings at transition from the start to the final state. The change in the internal energy for this process is given by the sum of the heat delivered to the system and the exerted work dU = dQ + dW

(4.1)

Equation (4.1) is the mathematical expression of the first law of thermodynamics. In the adiabatic process, the system does not exchange heat with the surroundings, thus dQ = 0

and

dU = dW

(4.2)

and for the adiabatic expansion, it holds dU = P dV 221

(4.3)

222

Physico-chemical Analysis of Molten Electrolytes

Similarly, when the system does not exchange work with the surroundings, then dW = 0

and

dU = dQ

(4.4)

In both the cases, the change in the internal energy depends only on the starting and final state of the system. From experience, we know that work can be changed to heat (e.g. heating of the liquid at mixing). There is thus an equivalence of work W and heat Q:W = J · Q, where J is a physical quantity called the mechanical equivalent of heat and its value and the dimensions are: J = 4.184 J/cal (J is the unit of heat and work called Joule). This is the principle of equivalency of heat and work. The first thermodynamic principle includes two basic laws enabling us to measure heat of any reaction at the optional temperature: the Hess’s and the Kirchoff’s laws. 4.1.1. Hess’ law

Most of the processes take place at constant pressure. In 1840, Hess found that the total heat, which the system releases or consumes at a chemical reaction, is equal regardless of whether the reaction runs in one step or stepwise. This is the principle of Hess’s law. The Hess’s law can be demonstrated on the oxidation of sulfur S + 32 O2 = SO3

QP = −395.68 kJ

S + O2 = SO2

QP = −296.75 kJ

SO2 + 12 O2 = SO3

QP = −98.93 kJ

Total

QP = −395.68 kJ

Now we introduce a new function, the enthalpy H = U + PV

(4.5)

Enthalpy is the function of state and its value depends only on the starting and final state of the system (QP = H2 – H1 ). In the case of a chemical reaction, its value depends only on the state of the chemical reactants and the reaction products. The molar heat capacities at constant volume and constant pressure are defined by the relations dQV dQP ∂U ∂H CV = ; CP = (4.6) = = dT ∂T V dT ∂T P and their units are [CV ] = [CP ] = J/(mol · K). The unit of specific capacity is [CV ] = [CP ] = J/(kg · K).

Enthalpy

223

Heat capacities depend on temperature and pressure, or on temperature and volume. For practical use, the temperature dependence of CP is given in tables in the form of empirical equations of the type CP = a + bT + cT 2 + dT 3 + · · ·

(4.7)

CP = a + bT + eT −2

(4.8)

These equations are valid in the temperature interval in which the heat capacity is measured. The total amount of heat needed for heating 1 mol of a substance is T2 QP =

CP dT

(4.9)

T1

4.1.2. Kirchoff’s law

Denoted by H1 , the heat of reaction, which the system exchanges with the surroundings at the temperature T1 , what is the heat of reaction, if the reaction takes place at temperature T1 ? In other words, we are looking for the dependence of enthalpy on temperature at constant pressure

∂H ∂T

= P

=

∂Hproducts

∂T

−

P

nj CP , products −

∂Hreactants

∂T

P

ni CP , reactants = CP

(4.10)

Equation (4.10) is the differential form of Kirchoff’s law. In the integral form, it becomes T2 H2 = H1 +

CP dT

(4.11)

T1

4.1.3. Enthalpy of reaction

If an exothermic reaction takes place at constant pressure adiabatically (the system does not exchange heat with the surroundings), the released heat causes an increase in temperature of the reaction products from temperature T1 to temperature T2 . Then it holds: HT +

T2 T1

CP + iCP , inert dT = 0

(4.12)

224

Physico-chemical Analysis of Molten Electrolytes

where i denotes the amount of inert substance, which does not participate directly in the reaction. Equation (4.12) enables us to calculate the final temperature T2 , of the reaction products at the end of the adiabatic reaction. Most of the chemical processes are accompanied by energetic changes. Since most reactions proceed at constant pressure, it is convenient to characterize the reaction by enthalpy change. The stoichiometric equation is thus completed by the respective enthalpy change, H

HT0 =

bB(S) + cC(g) = rR(1) + sS(S) nj Hj0 − ni Hi0 = rHR0 + sHS0 − bHB0 − cHC0

(4.13)

As it can be seen from the reaction (4.13), the enthalpy change depends also on the coefficients r, s, b, and c, thus on the given stoichiometry. For instance 1 H2(g) + O2(g) = H2 O(1) 2

(4.14)

0 = −285.24 kJ · mol−1 H298

The index 0 means that the starting as well as the final substances, are considered to be in a standard state. Since we do not know the absolute values of the enthalpies or the internal energies, we have chosen for practical purposes a certain exactly defined standard state, which we compare with the given quantity. The following standard states are mostly used. Gases: The gas in the ideal state at a pressure of 101 325 Pa and the given temperature (e.g. at 298.16 K, or at a temperature of the system). Liquids: The liquid at a pressure of 101 325 Pa and the given temperature. Solids: The most stable crystallographic modification at a pressure of 101 325 Pa and the given temperature. The enthalpy of elements is considered to be zero. A very important type of reaction enthalpy is the enthalpy of formation, which is the amount of heat the system obtains from the surroundings at the origination of 1 mol of the given compound directly from the elements in the standard state. There are different kinds of reaction enthalpy that we can observe in molten salt chemistry. Bonding energy is the mean value of the dissociation energy of the given bond (in a reaction where atoms of elements in the gaseous state originate from the molecules); Enthalpy of mixing is the enthalpy connected with the formation of 1 mol of solution of the given concentration from pure liquid components;

Enthalpy

225

Enthalpy of dissolution is the enthalpy connected with the dissolution of the given solution; it depends on the composition and can be calculated from the integral dissolution enthalpies of solutions in the final and starting states; Integral enthalpy of dissolution is the same as in the previous case, when one of the components is in the solid or gaseous state; it depends on the composition of the solution and for dilute solutions, it approaches a certain limiting value; Enthalpy of fusion is the heat connected with the transition of the compound from the solid to the liquid state; Enthalpy of polymorphic transformation is the heat connected with the transition of the compound from one crystallographic modification to the other. 4.1.4. Estimation of enthalpy of fusion

For thermodynamic calculations and analyses of phase diagrams of binary systems whose components form a binary compound, partially or totally dissociating at melting, it is necessary to know the enthalpies of fusion of the components of the binary compounds, and of both eutectic mixtures, as input quantities. When these data cannot be found in the literature, it is possible to estimate them using entropy or enthalpy balances. 4.1.4.1. Enthalpy of fusion of binary compounds (Aq Br )

The enthalpy of fusion of a binary compound Aq Br can be estimated in two quasi-similar ways. In principle, the calculation is based on the summation of entropies of substances at the temperature of fusion of the additive compound Aq Br . (a)

If the temperatures and enthalpies of fusion of both the substances A and B are known, the enthalpy of fusion of the additive compound Aq Br can be estimated from the entropies of fusion of both substances A and B according to the relation fus H (Aq Br ) = [qfus S(A) + rfus S(B)] · Tfus (Aq Br )

(b)

(4.15)

In this estimation, the entropy of mixing of both the compounds A and B to form the additive compound Aq Br is neglected. The entropy of the Aq Br melt at the temperature of fusion can be calculated either as the sum of the entropy of the crystalline compound Aq Br at the temperature of fusion and the entropy of fusion of Aq Br Smelt (Aq Br ) = Scr (Aq Br ) + Sfus (Aq Br )

(4.16)

or as the sum of the entropy of the mechanical mixture of the A and B melts in the corresponding stoichiometric ratio at the temperature of fusion, and the entropy of mixing

226

Physico-chemical Analysis of Molten Electrolytes

of this mixture at the same temperature Smelt (Aq Br ) = qSmelt (A) + rSmelt (B) + mix Smelt (Aq Br )

(4.17)

The entropy of mixing can be calculated from the relation mix Hmelt (Aq Br ) − R [q ln amelt (A) + r ln amelt (B)] Tfus (Aq Br )

mix Smelt (Aq Br ) =

(4.18)

The enthalpy of fusion of Aq Br can then be calculated according to the relation fus H (Aq Br )

  Tfus  (A)  CP , cr (A)  = q dT + fus S(A) +  T  0

  +r

Tfus (B)

CP , cr (B) dT + fus S(B) + T

 CP , melt (A)  dT  T

Tfus (A)

Tfus (Aq Br )

0

+

Tfus (Aq Br )

 CP , melt (B)  dT  T

Tfus (B)

mix Hmelt (Aq Br ,Tfus ) − R [q ln a(A, melt) + r ln a(B, melt)] Tfus (Aq Br )

Tfus (Aq Br )

− 0

  CP , cr (Aq Br )  · Tfus (Aq Br )  T 

(4.19)

In Eq. (4.19) a(X) = γ (X) · x(X), where a(X), γ (X), and x(X) are the activity, activity coefficient, and mole fraction of X (X ≡ A or B), respectively, and x(A) = q/(q + r), and x(B) = r/(q + r). Equation (4.19) is the complete entropy balance to form the molten additive compound Aq Br at the temperature of fusion from the crystalline substances A and B at the temperature of absolute zero. In Eq. (4.19), it is assumed that the heat capacity of the crystalline substances is known over the whole temperature range from absolute zero to the temperature of fusion. Several simplifications can, however, be made if some quantities are not available. If the heat capacity of the crystalline compound Aq Br is not known, it can be estimated in the first approximation using the Neumann–Kopp’s rule CP , cr (Aq Br ) = qCP , cr (A) + rCP , cr (B)

(4.20)

Enthalpy

227

Substituting Eq. (4.20) into Eq. (4.19), and using the relation fus CP (X) = CP ,melt (X) − CP ,cr (X)

(4.21)

Equation (4.19) is simplified to the form      fus H (Aq Br ) = Tfus (Aq Br ) · q fus S(A) +     + r fus S(B) +

Tfus (Aq Br )

Tfus (Aq Br )

Tfus (A)

 fus CP (A)  dT  T

(4.22)



fus CP (B)  dT  T

Tfus (B)

   mix Hmelt (Aq Br ) + − R q ln amelt (A) + r ln amelt (B)  Tfus (Aq Br ) 

In systems with dystectic melting of the Aq Br compound, the activities and activity coefficients of constituents depend on the degree of dissociation, α, of the dissociation α reaction Aq Br −−→ qA + rB. At equilibrium, the constituents A, B, and Aq Br are present in the melt in the ratio αq:αr:(1 − α). 4.1.4.2. Enthalpy of fusion of an eutectic mixture

In the thermodynamic balance calculation as well as in the cryoscopic measurement in eutectic mixtures, the enthalpy of fusion of the eutectic mixture is necessary. In the literature, however, such data are very scarce and thus they must be estimated on the basis of the enthalpy or entropy balance. The enthalpy of fusion of the eutectic mixture can be calculated by evaluating the changes of the enthalpy or entropy accompanying the following processes: (a)

(b)

Starting from the solid state, the crystalline components A and A q Br or B and A q Br are first mixed to form the corresponding eutectic mixture at the eutectic temperature and then the solid mixture is melted, Starting from the solid state, the crystalline components A and A q Br or B and A q Br are first heated from the eutectic temperature to their melting points, consequently they are melted at this temperature, then cooled to the eutectic temperature, and finally, at this temperature, they are mixed to form the molten eutectic mixture.

228

Physico-chemical Analysis of Molten Electrolytes

The enthalpy or entropy balance (the latter being obtained using the LeChatelier– Shreder’s equation applied to both components) yields the following relation for the heat of fusion of the eutectic mixture fus H (eut) = y(eut, A or B)fus H (A or B)   + y(eut, Aq Br ) fus H (Aq Br ) +

Tfus (eut)

  fus CP (Aq Br )dT 

Tfus (Aq Br )

+ mix Hmelt (eut)

(4.23)

where y(eut, Y) is the mole fraction of Y in the A–A q Br or B–A q Br system. If the enthalpy quantities of the A q Br compound are not known, the value of f us H(A q Br ) is estimated according to Eq. (4.19) (or Eq. (4.22)), and that of f us CP (A q Br ) is calculated using the Neumann–Kopp’s rule. 4.1.4.3. Applications

For most binary systems, the thermodynamic quantities in Eqs. (4.19) (or Eq. (4.22)) and (4.23) are not known. It is also, however, possible to use simplified forms of these relations to estimate the heats of fusion of binary compounds and binary eutectics. In the estimation of the heats of fusion of binary compounds, simplifying assumptions can be divided into three limiting groups: (a)

mix Hmelt (Aq Br ) = 0 Ex (A B ) = 0 mix Smelt q r

γ (X, melt) = 1 (ideal solutions) (b)

mix Hmelt (Aq Br ) = 0 Ex (A B ) = 0 mix Smelt q r

γ (X, melt) = 1 (regular solutions) (c)

mix Hmelt (Aq Br ) = 0 Ex (A B ) = 0 mix Smelt q r

γ (X, melt) = 1 or = 1

Enthalpy

229

In the first approximation, in order to estimate the heats of fusion of binary eutectics, the most simplified form of Eq. (4.23) can be used under the following conditions mix Hmelt (eut) = 0

(4.24)

fus CP (Aq Br ) = 0

(4.25)

Kosa et al. (1993) used Eqs. (4.22) and (4.23), simplified according to the conditions (a), (b), and (c), in the estimation of heats of fusion of the binary compounds Na3 FSO4 and K3 FSO4 as well as for the heats of fusion of the eutectic mixtures in the systems NaF–Na3 FSO4 , Na3 FSO4 –Na2 SO4 , KF–K3 FSO4 , and K3 FSO4 –K2 SO4 (q = r = 1). The temperatures and heats of fusion and the temperature dependencies of the heat capacities for the crystalline and liquid phases of NaF, KF, Na2 SO4 , and K2 SO4 were taken from Barin and Knacke (1973). mix Hmelt (AB), mix Hmelt (eut), and the activity coefficients of components were calculated using the data published by Hatem et al. (1982) assuming that they are independent of temperature. The coordinates of the individual eutectic points were taken from Kleppa and Julsrud (1980). The estimated values of the heats of fusion of Na3 FSO4 and K3 FSO4 were compared with those measured by Adamkovicˇ ová et al. (1991, 1992). The estimated values of the heat of fusion of binary compounds Na3 FSO4 and K3 FSO4 are given in Table 4.1 and those of the individual eutectic mixtures in the systems NaF–Na2 SO4 and KF–K2 SO4 in Table 4.2. The estimation errors for the heats of fusion of binary compounds are related to the experimental values. The estimation errors for the heats of fusion of the eutectic mixtures are related to the values calculated using

Table 4.1. Estimated heats of fusion of Na3 FSO4 and K3 FSO4 under different simplifying conditions mix H melt (AB) (kJ · mol−1 ) 1. (a) (b) (c) 2. (a) (b)

γ

E (AB) mix Smelt (J · mol−1 · K −1 )

estim (AB) mix Hmelt (kJ · mol−1 )

0 0 4.08

60 60 64

−13 −13 −7

0 0

75 75

−13 −13

(%)

Na3 FSO4 :1060 K, fus Hexp = (69 ± 4) kJ · mol−1 0 2.78 2.78

1 =1 γ NaF = 0.837 γ Na2 SO4 = 1.003

K3 FSO4 :1148 K, fus Hexp = (86 ± 3) kJ · mol−1 0 −2.58

1 = 1

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Physico-chemical Analysis of Molten Electrolytes

Table 4.2. Estimated heats of fusion of eutectic mixtures in subsystems of systems NaF–Na2 SO4 and KF–K2 SO4 , for various values of input parameters fus H(AB) (kJ · mol−1 ) 1.

57 57 58 52 54

0 2 −9 −2

0.36 0 0 0 0

= 0 = 0 0 0 0

41 40 40 36 38

−2 −2 −12 −5

−0.61 0 0 0

= 0 = 0 0 0

39 39 40 38

0 3 −3

= 0 = 0 0 0

73 73 72 64

0 −1 −12

0.42 0 0 0 0

KF–K3 FSO4 :1051 K 86 86 86 75b

4.

= 0 = 0 0 0 0

(%)

Na3 FSO4 –Na2 SO4 :1021 K 69 69 69 60b 64b

3.

fus H estim (eut) (kJ · mol−1 )

NaF–Na3 FSO4 :1054 K 69 69 69 60b 64b

2.

fus Cp (Y)a (J · mol−1 · K −1 )

mix Hmelt (eut) (kJ · mol−1 )

K3 FSO4 –K2 SO4 :1143 K 86 86 86 75b

−0.18 0 0 0

= MF, M2 SO4 , or M3 FSO4 (fus CP (M3 FSO4 ) according to Neumann–Kopp’s rule). Estimated values (c.f. Table 4.1.).

aY b

experimentally determined heats of fusion of binary compounds and mixing enthalpies of the corresponding eutectic mixtures. As it follows from Table 4.1, the estimated heats of fusion of Na3 FSO4 and K3 FSO4 are lower than the measured values and the error in the estimation lies within the range 7–13%. This difference is probably due to the limited validity of the Neumann–Kopp’s rule, which in most cases, yield higher values of heat capacities of crystalline AB compounds in Eq. (4.19). The first three estimations of the heat of fusion of individual eutectic mixtures (see Table 4.2) indicate that the zero value of the enthalpy of mixing and the difference in the heat capacity at fusion have a very low influence on the estimated value of the enthalpy of fusion of the eutectic mixtures. The estimation error is in the range 0–12%.

Enthalpy

231

In general, the reliability of estimation of the enthalpy of fusion of the binary compounds and eutectic mixtures depends on • • •

the volume of input information, the choice of simplifying conditions, the difference between the melting points of the components and binary compounds and the eutectic temperatures.

The more the experimental knowledge on the system in question is available, the more reliable is the estimation of the sought quantity. On the other hand, it is sometimes better to omit a dubious value of the experimentally determined input quantity. 4.1.5. Enthalpy balance

The enthalpy balance of significant processes can be evaluated by solving the equation ψ(H, composition, T ) = 0

(4.26)

For this reason, the aim of a complete enthalpy analysis is to ascribe an unambiguous assignment of enthalpy values (measured or calculated) to the individual figurative points of the phase diagram of the system in question, i.e. to determine the shape of the function ψ. Since the absolute values of the internal energy cannot be determined, ψ could be established merely for the increase in enthalpy from some basic values corresponding to a certain reference state of the system. This could be, for instance the enthalpy value of the elementary particles, the combination of which yields the respective system at 298 K. If we designate the increase in enthalpy due to the combination of the building blocks into the given system as form H (Tref = 298 K), then the absolute enthalpy of the system at the experimental temperature Tm is given by the equation Habs (Tm ) =

Tm νi Hi (Tref ) + form H (Tref ) +

Csyst (T )dT

(4.27)

Tref

where Hi (Tref ) are the absolute enthalpies of particles constituting the system in its reference state, ν i are the stoichiometric coefficients in the equation according to which the system in question is formed from the elementary particles, and Csyst (T ) is the isobaric molar heat of the system. 4.2.

EXPERIMENTAL METHODS

The selection of a calorimetric method suitable for the measurement of heat effect of the investigated process is affected by several factors. Almost all of these factors influence the precision of the measurement. Such factors are for example, the rate of the measured

232

Physico-chemical Analysis of Molten Electrolytes

chemical reaction, physical properties of the sample, reaction of the sample with the crucible, temperature, etc. Since no criterion exists on the basis of which it would be possible to evaluate individual calorimetric methods, the aim of this chapter is to give an informative survey about them. However, the emphasis will be on calorimetry at medium and high temperatures. 4.2.1. Calorimetry

Calorimetry constitutes a powerful tool to investigate materials. It is a measurement technique that enables us to obtain values of the thermodynamic quantities of substances. The methods used for the characterization of thermodynamic properties of molten salts include temperature, enthalpy, and heat capacity measurements as mixing enthalpy and phase diagram determinations for their mixtures. 4.2.1.1. Principle of calorimetry

Calorimeter is a device for the measurement of heat evolved or consumed at the change of state of the system. This change can evoke change of the phase composition, temperature, volume, or chemical composition. The main components of calorimeter are schematically shown in Figure 4.1. If we denote the temperature of the calorimetric vessel or cover by Tc , the temperature of the calorimetric cover by Ts , then the amount of heat Q, which the calorimeter gains during a time unit from the investigated system at the given process, can be calculated from the calorimetric equation proposed by Tian (1923) dQ dTc (4.28) =W + K (Tc − Ts ) dt dt where W is the heat capacity of the calorimetric vessel and K is the coefficient of the heat transfer. The second member on the right-hand side of Eq. (4.28) represents the amount of heat transferred from the calorimetric vessel to the calorimetric cover. Based on Tian’s calorimetric equation, the calorimeters are conveniently classified according to the following scheme: 1. Isoperibolic calorimeter 2. Adiabatic calorimeter 3. Isothermal calorimeter 4. Calorimeter with constant heat flow

Ts = constant, Tc changes Tc = Ts , both temperatures change Tc = Ts = constant dQ/dt = constant, Tc = Ts = constant

There are also other criteria for the classification of calorimeters. They can also be divided according to the aim for which they are used, i.e. (a) (b)

calorimeters for reacting and non-reacting systems, low-, middle-, and high-temperature calorimeters,

Enthalpy

233

A B C D E

Figure 4.1. Main parts of the calorimeter. A – calorimetric vessel, B – calorimetric cover, C – thermometer, D – shield, E – stirrer.

(c) (d) (e)

simple and twin calorimeters, static and dynamic calorimeters, calorimeters and micro-calorimeters.

One of the characteristics for the isoperibolic calorimeter is to maintain the temperature of the cover at a constant value. The heat effect of the process taking place in the calorimeter is evaluated from the time course of the temperature curve. The typical course of this curve at an exothermic process is shown in Figure 4.2. The temperature difference between the calorimetric vessel and the calorimeter cover read-off from the temperature versus time curve has to be corrected for the heat introduced by stirring as well as for the heat exchange between the calorimetric vessel and the cover. Isoperibolic calorimeters also comprise calorimeters based on the measurement of heat flow, since they fulfill the condition Ts = constant, Tc changes. With these calorimeters, the temperature difference (Tc – Ts ) is not measured but directly the heat flow between the calorimetric vessel and the cover. Isoperibolic calorimeter.

Physico-chemical Analysis of Molten Electrolytes

Temperature

234

Ts

Time Figure 4.2. Temperature versus time plot of an exothermic process measured in an isoperibolic calorimeter.

With the adiabatic calorimeter, exchange of heat between the calorimetric vessel and the cover is suppressed. This happens so that the temperatures of the vessel and the cover are maintained at almost the same temperature. The condition (Tc – Ts ) = 0 can be attained at constant cover temperature by heating or cooling the calorimetric vessel using an internal heater or heat sink placed inside the calorimetric vessel. This “compensation” method is suitable for endothermic processes. For the adiabatic method, the characteristic feature is not only the equality of temperatures of the calorimetric vessel and of the cover, but also their changing value – the measurement proceeds at dynamic conditions, where the temperature of the calorimetric cover follows the temperature of the calorimetric vessel.

Adiabatic calorimeter.

With the isothermal calorimeter, the condition (Tc – Ts ) = constant, is attained by a total transfer of heat of the process taking place in the calorimetric vessel to the heat sink, where it will cause a partial phase transformation of the substance in the heat sink. The thermal effect of the investigated process is then determined from the volume change of the calorimetric substance. According to the phase transformation, isothermal calorimeters using transformation of the solid phase to the liquid or of the liquid to the vapor phase are known.

Isothermal calorimeter.

Enthalpy

235

Constant heat flow calorimeters are characterized by a constant temperature difference between the calorimetric vessel and the cover. To this group of calorimeters also belong the high-speed calorimeters for the measurement of heat capacities and the heats of modification transformation of substances, which are electrical conductors or semiconductors, where the heating is provided by their electrical resistance.

Calorimeters with constant heat flow.

4.2.1.2. Measurement of temperature

In order to calculate the heat evolved or absorbed in the calorimeter, it is necessary to measure (except for the isothermal calorimeter) the change if temperature of the calorimetric vessel and cover, as well as the heat capacity of the calorimeter. For calorimetry performed at medium and high temperatures, thermocouples are used most frequently. For example, they are made of iron and constantan for medium temperatures or of platinum, rhodium, and iridium for high temperatures. Very frequently, a number of thermocouples linked in series in order to obtain a voltage measurable with sufficient precision, are used. Presently, resistance thermometers are the most suitable temperature meters because of their high precision and stability. Mainly, they are used when resistance elements are wound directly on the surface of the calorimetric vessel and cover. Change of resistance with temperature can be in the current range of the temperature change of the calorimeter (less than 3 K) regarded as linear. Thermistors are resistance thermometers, where the temperature-sensible element is the semiconductor, and are made of a mixture of different metal oxides. The large resistance of the thermistor enables us to lower substantially its dimensions in comparison to the resistance thermometer. Thermistors are very sensible and give a fast response, which is very suitable for use in small calorimeters. Quartz crystals have been used as thermometers in calorimetry relatively recently. Their use is based on the fact that the resonance frequency of the quartz crystal cut in a certain orientation to the axis of its crystal structure depends on temperature, whereby the temperature dependence is high and almost linear. 4.2.1.3. Calibration of calorimeters

Calibration of the calorimeter means the measurement of heat capacity, and depends on the type of calorimeter, the purpose of its use, as well as on the kind of the thermal effect. Reaction calorimeters are frequently calibrated using a known heat of a chemical reaction. No standard reaction is internationally accepted. For the measurement of heat capacities, drop calorimeters are frequently used and the calibration is made using a substance, the temperature dependence of which on heat capacity is known. As substances, metals like Cu, Ag, Au, and aluminum oxide in the form of sapphire are used. Calorimeters

236

Physico-chemical Analysis of Molten Electrolytes

for the measurement of exothermic heat effects are calibrated most effectively directly using electrical energy. For calibration, the dependence of voltage on the calibration resistance, the current intensity, and the time have to be measured with utmost accuracy.

4.2.1.4. Calorimetry of reacting systems

Changes in enthalpy of chemical reactions are determined by calorimetry of reacting systems. The reaction heat is calculated from measured heats of formation, solution, and mixing, using the Hess’ law. One important factor, which influences the selection of the measurement method, is the duration of the chemical reaction investigated. For fast reactions, i.e. when the reaction is over at least within 30 min, the most suitable calorimeter is the isoperibolic calorimeter, since the heat effect can be determined quite precisely. The reaction proceeds in the calorimetric vessel, provided with a thermocouple, the drop-device of one component into the other, and the stirrer. The calorimetric cover is heated to a constant temperature. At the measurement of heats of mixing, respectively of heats of dissolution, both liquid components must be thoroughly heated before mixing to the same temperature before mixing. This can be made in such a way that the one component is placed just above the other one in a second crucible, which is then overturned or immersed and both components are mixed. On the other hand, for slow reactions, adiabatic and isothermal calorimeters are used and in the case of very small heat effects, heat-flow micro-calorimeters are suitable. Heat effects of thermodynamic processes lower than 1 J are advantageously measured by the micro-calorimeter proposed by Tian (1923) or its modifications. For temperature measurement of the calorimetric vessel and the cover, thermoelectric batteries of thermocouples are used. At exothermic processes, the electromotive force of one battery is proportional to the heat flow between the vessel and the cover. The second battery enables us to compensate the heat evolved in the calorimetric vessel using the Peltier’s effect. The endothermic heat effect is compensated using Joule heat. Calvet and Prat (1955, 1958) then improved the Tian’s calorimeter, introducing the differential method of measurement using two calorimetric cells, which enabled direct determination of the reaction heat. In the following list, the Calvet-type twin micro-calorimeter, working up to temperature of 1200 K, is shortly described. The micro-calorimeter consists of the following main parts: • • •

innerside of the external steel jacket is provided with a cylindrical furnace G; the calorimetric block, composed of three sections: top, middle, and bottom, aluminum oxide parts D, F, H; the alumina block has two holes, inside which are the thermopiles E, consisting of several hundreds of thermocouples and providing maximum thermal flux;

Enthalpy

• • •

237

each thermopile surrounds the calorimeter cell B, made of thin-walled alumina, 17 mm in diameter and 80 mm in height; the calorimetric block is surrounded by two mantles C, serving as thermal and electric shielding; the thermocouple A, placed in the center of the calorimeter measures the total temperature.

In order to provide good time and temperature stability of the calorimeter, the two thermopiles are connected in opposite direction, which eliminates most of the problems with external thermal disturbances. The computer, processing all the input temperature signals, controls the calorimeter. The isoperibolic Calvet’s twin micro-calorimeter is schematically shown in Figure 4.3. The most important conditions to be taken into account in the calorimetric measurement are the elimination of all the effects arising from the interaction between the material

A B C

D

E F G H

Figure 4.3. Schematic representation of the isoperibolic Calvet’s twin micro-calorimeter. A – thermocouple, B – calorimetric cell, C – thermal and electric shielding, D, F, H – three parts of the calorimetric block, E – thermopile, G – furnace.

238

Physico-chemical Analysis of Molten Electrolytes

and atmosphere (e.g. oxidation, water attack, etc.), sample and crucible, solvent–solute mixing effects, mixing of components with non-equal temperature, etc. All these effects can substantially lower the precision and accuracy of the calorimetric measurement.

4.2.1.5. Calorimetry of non-reacting systems

Calorimetry of non-reacting systems involves the measurement of heat capacity dependencies on temperature, which enables us to calculate the enthalpies of phase transformations. Based on the prevailing mode of the heat exchange between their individual parts, calorimeters for this purpose can be classified as low-, medium-, and high-temperature calorimeters. In the measurement of thermodynamic parameters of molten electrolytes, mostly the last two types of calorimeters are used. In the mid-temperature region for the measurement of heat capacities, the adiabatic calorimeters with direct heating of the sample and the drop calorimeters with heating of samples outside the calorimeter – in a furnace, are used. For measurement of reaction heat during heating of the sample, differential scanning calorimeters (DSC) are suitable. In the DSC calorimeter produced by the concern Perkin-Elmer, the temperature of the sample during heating is maintained based on temperature of the standards. In the temperature region of the reaction, additional heating of the sample, and respectively, of the standards, compensates the heat effect in order to attain a zero temperature difference between them. The reaction heat is then calculated from the delivered amount of energy. DSC calorimeters are also produced by the concerns Dupont, Setaram, and Rigaku Denki. In the high-temperature region, the main method of measurement is the drop calorimetry, where the sample is heated to the chosen temperature outside the calorimeter in a furnace and the heat capacity is calculated from the temperature dependence of the enthalpy changes measured after dropping the sample into the calorimeter. The application of this technique affects, however, the behavior of the sample heated in the furnace (decomposition, reaction with the crucible, etc. should be avoided) as well as at the cooling from the furnace temperature to that of the calorimeter. Sometimes the sample does not complete its phase transition at cooling (e.g. at the temperature of fusion, a part of the sample crystallizes while the other part becomes glassy). In such a case, the drop calorimeter must be supplemented by a solution calorimeter in order to get the enthalpy differences of all the samples to a defined reference state. For the drop technique, the isoperibolic calorimeters are most frequently used. The calorimetric device consists of two main parts: a furnace and a heated block. Between the calorimetric block and the furnace, there is a system of shields controlled by a mechanic, hydraulic or electromagnetic device, which prevents the heat transfer from the furnace to the calorimetric block. The calorimeter is made of copper with a cavity closed by a shield. A resistance thermometer wound on the block measures its temperature. Such a calorimeter can work up to 1700°C, especially when the furnace

Enthalpy

239

is heated using a Pt–Rh wire wound directly on the sintered alumina (Degussite) tube. However, for high-temperature measurements some problems arise, such as finding a suitable material for the sample carrier when very reactive samples are measured, the vapor expansion when a volatile sample is sealed in a Pt–Rh tube, etc.

4.2.1.6. Heat capacity measurement

An exhaustive survey of different experimental calorimetric techniques used for heat capacity determination was given very recently by Gaune-Escard (2002) and is excerpted here with her kind permission. The heat capacity at constant pressure, CP , is the derivative with respect to temperature of the enthalpy change induced by temperature variation (c.f. Eq. (4.6)). At high temperature, the methods used for CP determination are based on the simultaneous measurement of the enthalpy temperature variation versus time at a programmed rate of heating. In indirect methods, heat content measurements are performed over a large temperature range, for instance by drop calorimetry, and CP is derived by the analytic derivation of heat content plots versus temperature. In direct methods, the sample is heated over a large temperature range either continuously, or by successive small temperature increments with a linear dependence of temperature on time. Heat content determinations at high temperatures are generally easier than direct heat capacity measurements. They were widely used in early stages of measurement to obtain CP data of molten salts. The enthalpy increment of a substance between temperatures T1 and T2 , HT2 − HT1 (T2 > T1 ), is measured in general using a drop calorimeter. Two techniques are employed, depending on the way the measurements are carried out.

Measurement of the heat content.

(i)

(ii)

The sample is heated to a high temperature T2 and the actual heat content measurement is then performed in a calorimeter at the experimental temperature T1 , which is usually 25°C. The sample is at a low temperature T1 and the actual heat content measurement is performed in a high-temperature calorimeter at an experimental temperature T2 . This method is the so-called “inverse drop method”. It should be preferred in principle for melts, having the tendency to form glasses, since in this case, nonequilibrium final states could be obtained on cooling.

The dependence of the enthalpy increments HT − H298 on temperature in the temperature range 610–867°C for K3 NbF8 measured by Nerád et al. (2003) is shown in Figure 4.4. The enthalpy of fusion of K3 NbF8 was determined as the difference

240

Physico-chemical Analysis of Molten Electrolytes

360

Hrel (kJ.mol–1)

320

280

240

200

160 850

900

950

1000

1050

1100

1150

1200

T (K) Figure 4.4. Determination of the enthalpy of fusion of K3 NbF8 according to Nerád et al. (2003).

between extrapolated values of the enthalpy of melt and the enthalpy of crystalline phase at the temperature of fusion. The enthalpy of fusion fus H(K3 NbF8 ) = (60.5 ± 3.1) kJ . mol−1 at 777°C temperature has been evaluated. The following heat capacity values of the solid and liquid K3 NbF8 were obtained from the temperature dependence of HT − H298 :Cp (K3 NbF8 , sol) = (380.4 ± 18.8) J · mol−1 · K −1 for the crystalline phase and Cp (K3 NbF8 , liq) = (395.9 ± 31.9) J · mol−1 · K −1 for the melt. These values are valid in temperature intervals of the data used to evaluate the relative enthalpy equations. Similarly, Holm et al. (1973) measured the enthalpy increments HT − H298 of the congruently melting compounds of 2:1 and 1:1 alkali metal chlorides and magnesium chloride using a high-precision adiabatic drop calorimeter. From the results obtained at several experimental temperatures corresponding with solid and liquid samples, they determined the enthalpies of fusion of these compounds. Values for heat capacities for the molten salt mixtures were also derived. The heat capacities for the 2:1 and 1:1 compounds were estimated from those of the binary compounds according to the relation CP = nCP (AlkCl) + CP (MgCl2 ),

n = 1 or 2

(4.29)

The enthalpies of fusion of K2 TiF6 and K3 TiF7 were determined by Adamkovicˇ ová et al. (1995a,b) using the high-temperature calorimeter Setaram HTC 1800 K. The calorimeter was in the DSC mode at scanning rate of 1 K · min−1 . The sample was sealed in a platinum crucible and placed in the upper alumina crucible of the calorimetric cell.

Enthalpy

241

The lower alumina crucible of this cell contained a platinum crucible with small pieces of Al2 O3 as the reference substance. The data of the sample temperature and the difference between the crucibles were recorded. Na2 SO4 and KCl were used as calibration salts. The following enthalpies of fusion of K2 TiF6 and K3 TiF7 were determined: fus H (K 2 TiF6 ) = (21 ± 1) kJ · mol−1 at the temperature of fusion of 1172 K and fus H (K 3 TiF7 ) = (57 ± 2) kJ · mol−1 at the temperature of fusion of 1048 K. When a sample is subjected to a linear temperature increase, the rate of heat flow into the sample is proportional to its instantaneous heat capacity. Regarding this rate of heat flow as a function of temperature, and comparing it with that for a standard sample under the same conditions, we can obtain the heat capacity as a function of temperature. The procedure has been described in detail by O’Neil (1966). The principle of this method is shown schematically in Figure 4.5. Empty cells are placed in the sample and reference holders. An isothermal baseline is recorded at the lower temperature, and the temperature is then programmed to increase over a range. An isothermal baseline is then recorded at the higher temperature as indicated in the lower part of Figure 4.5. The two baselines are used to interpolate a baseline over the scanning section, as shown in the upper part of Figure 4.5. The procedure is repeated with a known amount of sample in the sample cell, and a dH/dT versus time trace is recorded. The deviation from the baseline is due to the absorption of heat by the sample. We may then write The ratio method.

dH dTP = mCP dT dt

(4.30)

where m is the mass of the sample, CP is the heat capacity, and dTP /dt is the programmed rate of temperature increase. While the above equation would yield values of CP directly, in order to minimize experimental errors, the procedure is repeated with a known amount of the standard sample, the heat capacity of which is known with a sufficient precision. Thus, only two ordinate deflections at the same temperature (Y and Y ) are required to yield a ratio of the CP values of the sample and the standard. This global method yields modestly accurate results. It consists of small successive temperature increments made during the linear increase in temperature with time (Figure 4.6). Each small temperature step is followed by an isothermal delay, which ensures thermal equilibration of the sample. From the difference of the thermal equilibrium aberration between the two cells during a heat pulse, the heat capacity of the sample placed in the working cell can be obtained as a function of temperature. The two crucibles contained in the cells are chosen in such a way so as to have as similar amounts as possible.

The “step” method.

242

Physico-chemical Analysis of Molten Electrolytes

dH/dT Standard material

CP C ′P

=

m′ ⋅ ′ Y m Y

Y'

Sample

Y Baseline time

T dTP /dt

time Figure 4.5. Heat capacity determination by the ratio method.

As the experimental parameters are time dependencies of enthalpy on temperature, the heat capacity can be written in the form dH dt P = dT dt P

CP =

dH dT

P

(4.31)

The temperature T varies linearly with time t, thus the integration between times t1 and t2 , which corresponds to temperatures T1 and T2 , yields an average heat capacity

Enthalpy

243

H, T

Sample Blank

Ti

Ti-1

Time Figure 4.6. Heat capacity measurement using the “step method.”

value C P in the small temperature interval T1 – T2 t2 t1

dH dt

dt = P

t1 CP t2

dT dt

T1

dt = P

T2

dH dT

T1 dt =

P

CP dt = CP (T2 − T1 ) (4.32) T2

This method thus provides nearly “true” heat capacity values of materials, except in the vicinity of phase transition temperatures, where the corresponding enthalpy increments superimpose those induced by temperature increments of the “step method” and invalidate the C P evaluation. During the experiment, the cells are maintained in a purified argon flow. In the temperature range 300–1100 K, CP measurements are carried out step by step, each heating step usually 5 K at the heating rate of 1.5 K/min, is followed by a constant temperature plateau for 400 s. The same experiment should be repeated with two empty cells identical to those used for the experimental sample run (blank experiment). The heat capacity of the sample is obtained at each temperature from the difference between the enthalpy increments obtained at each temperature step, in the two experimental series.

244

Physico-chemical Analysis of Molten Electrolytes

This method has been applied by Gaune-Escard et al. (1996a), Rycerz and GauneEscard (1999), and Gaune-Escard and Rycerz (1999) to several rare earth halides and their compounds with alkali metal halides. 4.2.1.7. Determination of enthalpy of mixing

A survey of different experimental techniques used in mixing calorimetry was given very recently by Gaune-Escard (2002). Calorimetry for the determination of mixing enthalpy can be divided into two groups: for temperatures up to 1200 K and above 1200 K. Several experimental techniques may be used for the determination of mixing enthalpies. The main problem met in all of them is the elimination of all side effects arising from: • • • • •

interaction between material and the atmosphere, like oxidation or moistening, interaction between the crucible and the melt, difficulties with the mixing of components with very different densities, stirring necessary to homogenize the mixture, mixing of components with different temperatures, etc.

The main techniques for calorimetry of liquid systems at high temperatures will be described in the following chapters. The simplest method to measure the heat at mixing a certain amount nA of a liquid salt A, considered as the solvent and kept at the experimental temperature TE , with the weighted amount nB of a solid salt B kept at the room temperature T0 , is the “drop method.” This method is very easy to operate and has been described already by Kubashewski and Evans (1964) and was applied to micro-calorimetry. The schematic description of this method is shown in Figure 4.7a. The measured enthalpy corresponds to the enthalpy of mixing according to the equation Mixing of liquids with solids.

nA A(l,TE ) + nB B(s,T0 ) = (nA + nB )AB(l,TE )

(4.33)

and includes not only the enthalpy of mixing, which should be determined, but also for the salt B, the heat capacity term of solid B in the temperature range T0 –Tfus , the enthalpy of fusion, and the heat capacity increment for liquid B from Tfus to TE . For many systems, however, there is the uncertainty in the sum of the enthalpy increments of the same order of magnitude as the enthalpy of mixing itself and can lead to unreliable results of the mixing enthalpy measurement. An improvement of the “drop method” was achieved by the “indirect drop method” (Figure 4.7b). In this variant of the drop method, the solid sample B is preheated and the mixing is carried out in two steps. First, the sample B is dropped into the

Enthalpy

245

1

1

2 2

3

3

4 5

(a)

4

6

5

7

(b)

Figure 4.7. The “drop” (a) and the “indirect drop” (b) methods. (a) 1 – drop tube; 2 – quartz liner; 3 – quartz or platinum crucible; 4 – molten salt; 5 – kaowool base; (b) 1 – drop tube; 2 – stopper; 3 – funnel; 4 – quartz liner; 5 – quartz or platinum crucible; 6 – molten salt; 7 – kaowool base.

lower end of the funnel provided with a stopper, in which it preheats. After thermal equilibration of sample B, the vertical shift of the drop tube enables the mixing of both salts. Vertically moving the drop tube in the molten mixture, or using an additional stirring device, ensures the homogeneity of the mixture. In order to enable to work in vacuum or in an inert atmosphere, the upper part of the system is provided with a set of taps and tightening rings, and the manipulation with the drop tube and with the stirrer is eventually operated electro-magnetically. The “indirect drop method” was used in various modifications, for example, in the determination of the enthalpy of mixing for the system K2 S2 O7 –K2 SO4 –V2 O5 by Fehrmann et al. (1986). The above described drop methods enable successive additions. The main advantage of the drop methods are the simplicity and readiness. This is in contrast with the more direct and accurate liquid–liquid mixing techniques, which enable measuring the mixing of liquid components directly and, in principle, with better accuracy. Mixing of two liquids. The “break-off bubble” method was developed by Gaune-Escard (1972) and has often been used in the mixing enthalpy measurements of molten salt mixtures. The schematic representation of this method is shown in Figure 4.8a. Inside

246

Physico-chemical Analysis of Molten Electrolytes

1 2 1

1

3

2

2

4

3

3 5

4

4

5

5 6 6

6

7

7 8

7 8 (a)

(b)

(c)

Figure 4.8. The “break-off bubble” (a), the “break-off ampoule” (b), and the “suspended cup” (c) methods. (a) 1 – drop tube; 2 – plug; 3 – quartz liner; 4 – quartz or platinum crucible; 5 – break-off ampoule with molten salt B; 6 – molten salt A; 7 – kaowool base. (b) 1 – manipulation tube; 2 – nichrome support; 3 – pyrex ampoule; 4 – pyrex crucible; 5 – silver tube; 6 – molten salt B; 7 – molten salt A; 8 – break-off tip; (c) 1 – crucible holder; 2 – quartz liner; 3 – silver crucible; 4 – silver tongs; 5 – silver cup; 6 – molten salt B; 7 – molten salt A; 8 – kaowool base.

the pyrex (or quartz) liner, a cylindrical crucible contains the liquid salt A, while salt B is placed in a spherical thin-walled pyrex or quartz ampoule. The ampoule has to be made of thin glass to enable being broken with a single strike. The mixing of the two liquids happens by crushing the bubble against the bottom of the crucible, or by striking the plug against the wall of the bubble. The heat effect arising from the ampoule break was checked by blank experiments and has been found to be very small and reproducible. The break-off bubble method was used recently in the determination of the enthalpy of mixing in the alkali halide – rare earth halide systems by Gaune-Escard et al. (1996). The last method is similar in its principle to that developed by Kleppa (1960) in his pioneering and extensive calorimetric investigation of molten salts. In his arrangement, salt B was placed in an ampoule made of a pyrex tube fitted with a break-off tip (Figure 4.8b). In comparison with the break-off bubble method, a slight shift in composition due to the effect of surface tension and the wetting of the crucible and the ampoule

Enthalpy

247

can happen. In the previous break-off bubble method, the homogeneity of the sample would be preserved, as the melt is contained in a single vessel instead of two. In the case, when the melt reacts with the glass, as for instance the molten alkali metal hydroxides, a “suspended cup” method (Figure 4.8c) was developed by Aghai-Khafri et al. (1976). Inside the pyrex or quartz liner, a cylindrical silver crucible contained the liquid hydroxide B while the salt or hydroxide A is contained in a small silver cup. This silver cup is held by tongs and could be released by external manipulation of the tongs. The mixing of the two liquids occurs by dropping the silver cup into the crucible. However, the thermal effect connected with the drop of the cup into the liquid A is very small and reproducible. All the mixing techniques described are only a few of those used in practice. The choice of a suitable mixing technique used for a particular system depends on the physicochemical properties of the melts under investigation. Most experimental constraints might be met when working with volatile substances like AlCl3 . In the investigation of the ternary system KCl–AlCl3 –AlCl3 NH3 performed by Hatem et al. (1988), both reactants, the liquid KCl−AlCl3 mixture and the liquid AlCl3 NH3 , were contained in closed pyrex ampoules, both kept inside the calorimeter. The inner ampoule containing AlCl3 NH3 had a very thin base, which could easily be broken by the sharp edge of the outer ampoule. Rather sophisticated calorimetric devices were used and great care was taken in order to obtain reliable enthalpy of mixing data of other very reactive melts. For instance, in the investigation of zinc halide-containing melts made by Papatheodorou and Kleppa (1973), a novel experimental arrangement and procedure was adopted in order to control the mass loss due to evaporation of the zinc halides. Inside the fused silica liner, a cylindrical crucible contained the low vapor pressure salt in the liquid state. Zinc halides were contained in an evacuated “double break-off” fused silica bubble. The mixing of the two liquids starts by crushing the break-off tip against the bottom of the crucible. Due to a slightly lower pressure in the bubble, a part of the melt sucks up into the bulb and mixes with the zinc halide. Crushing of the upper break-off tip then finishes the mixing, when the zinc halide is released into the crucible and mixes with the remainder of the melt. In some cases, a slight endothermic baseline shift might be observed after mixing. This may happen every time a volatile sample is investigated due to the mass loss of the sample. In order to correct for this baseline shift, a series of blank experiments should be performed. The heat of solution of lead oxide in oxide melts was investigated by Holm and Kleppa (1967) and Østvold and Kleppa (1969) using the following experimental arrangement. Approximately 60 g of oxide melt was contained in an Au20Pd crucible with 17 mm diameter and 75 mm height. The lead oxide, which should be dissolved in the melt of another oxide, was placed in a very shallow platinum cup with about a 10 mm diameter. This cup was attached by means of three platinum wires to a fused silica tube, which could

248

Physico-chemical Analysis of Molten Electrolytes

be manipulated from the outer side of the furnace. The solution reaction was initiated by lowering the platinum cup into the melt. Stirring was provided either by means of a platinum plunger or simply by moving the platinum cup up and down in the Au20Pd crucible. Corrections were made for the endothermic heat associated with the mass displacement due to the vertical temperature gradient of the calorimeter and the non-homogeneity of the melt. However, the additional heat due to stirring, represented 10–50% of the total heat of reaction, while without stirring, the correction was only 10–20%. Unlike most other salts, measurement of mixing enthalpies of molten alkali metal carbonates, due to their corrosive nature, cannot be performed in fused silica containers and be mixed by the usual “break-off” technique. Andersen and Kleppa (1976) showed that the Au20Pd alloy was corroded only negligibly by alkali metal carbonate melts kept under a relatively high CO2 pressure. The experimental arrangement included plunger as well as a dipper crucible that could be manipulated from the outer side of the furnace. The mass loss of the most volatile carbonate, Rb2 CO3 , was about 0.3%. However, in spite of the relatively small vaporization losses, the attack of vapors on the fused silica liner was considerable and the lower part of the device had to be discarded after 10–15 experiments. For temperatures above 1200 K, different calorimeters should be used because of construction material restrictions. Due to considerable difficulties with the “homemade” calorimeters, in the majority of cases, commercial equipment is used. One of the most widespread devices working up to 1800 K is the calorimeter Setaram. The whole calorimetric assembly comprise the following parts. The vertical cylindrical furnace consists of a graphite resistor surrounding a gas-tight alumina tube, with an inner diameter of 23 mm and a length of 600 mm, in which the calorimetric detector is placed and the experimental chamber, localized. The geometry of the resistor provides a 140-mm long constant temperature zone in the central part of the tube. The furnace has an external water-cooled jacket and can be heated up to about 2000°C. The furnace is supplied with low-voltage current. Its temperature is controlled by an electronic system monitored by a Pt–Pt13Rh thermocouple within the central part of the furnace. A set of valves and flow meters enables evacuation of the furnace and the experimental chamber, and either the gas flow or maintenance under pressure of the purified gas. The samples are introduced into the calorimeter and maintained under experimental conditions at ambient temperatures after preliminary evacuation, using a very simple charging device, similar to that previously designed for Calvet’s calorimeters. The features of the calorimetric detector and of the mixing and stirring systems, located within the experimental chamber, vary according to the nature of the experiment. The acquisition and processing of the experimental data is operated by a computer. Mixing calorimetry above 1200 K.

Enthalpy

249

F

D Cw B A Cr

E

Figure 4.9. Vertical section of the high-temperature calorimeter Setaram. See text for description.

Figure 4.9 shows the vertical section of the transducer high-temperature calorimeter Setaram. The calorimetric transducer is made up of a calorimetric transducer circuit A and a temperature-sensing thermocouple B held by alumina mountings. The transducer consist from set of thermocouples supported by capillary tubes forming a differential network and whose junctions are arranged according to two superimposed sets of ring gear housed in a cylindrical covering D. Cw and Cr are the working and reference crucibles, respectively, made of sintered alumina. The reference crucible Cr is slid into the bottom ring gear and held in place by a transverse bar E on which it rests and is supported by its ends on the wall of the covering. The covering is held by three parallel suspension wires F sealed into its wall and fitted with longitudinal canals incasing the conductors running to the calorimetric transducer and the temperature-sensing thermocouple. The tube D holds the ring supporting the thermocouples in the proper place and constitutes a thermal screen of the assembly. The whole assembly is suspended in the central

250

Physico-chemical Analysis of Molten Electrolytes

part of the furnace by three hollow alumina tubes H, which protect the junction wires of the thermocouples. A systematic study showed that to obtain reproducible and accurate values of the thermal changes of mixing, this kind of calorimeter is characterized by the following two conditions: (i) (ii)

high sensitivity of the detector, appropriate integration of the thermal flux produced within the working cell by the heat change upon mixing.

The former condition is best met using a sufficient number of thermocouples and the latter, by having a regular arrangement of the thermopile junctions around the working crucible. Therefore in their Setaram calorimeter, Gaune-Escard and Bros (1974) and Hatem et al. (1981) found it necessary to make some modifications of the commercial design. Several calorimetric detectors were built in accordance with these conditions. In one detector version, the 16 upper thermocouple junctions were located alternately on two horizontal levels about 10 mm apart. For technical reasons, and since no thermal excursion arises within the reference crucible Cr , the lower set of junctions were kept on the same horizontal level. Finally, in a number of later investigations another kind of detector having more thermocouples was permanently used. Due to easy operation, the “drop method” is usually used to obtain the enthalpy of formation of a liquid mixture A–B. However, it should be emphasized once more that the enthalpy change of the liquid–liquid mixing is calculated as the difference between two rather large quantities, thus introducing large errors. The situation could be illustrated by the molten mixture NaF–K2 SO4 , which was investigated by Hatem and Gaune-Escard (1979). The enthalpy of mixing was found to be 6.7 kJ/mol at x NaF = 0.6, and the correction term was about 250 kJ/mol. In order to improve the results, it is therefore necessary to eliminate or substantially decrease the enthalpy increment term arising from the enthalpy of fusion of the solid sample B added to the liquid bath A and its heat capacity term. This can be achieved if the sample B is introduced into the liquid bath A at a temperature equal or as close as possible to the experimental temperature TE . Also the “indirect drop method” was adapted in order to bring nearer the temperatures of both the salts. A small alumina sphere of 6 mm in diameter, closing the aperture of the funnel, attached on the upper side, an manipulation alumina rod and on the bottom, a thin alumina stirrer, was added. In the alumina rod, a Pt6Rh–Pt30Rh thermocouple is located, measuring the temperature of the solid salt B. When sample B reaches thermal equilibrium, the funnel is opened by lifting the alumina rod, enabling the mixing of both salts. The thin alumina stirrer is used to homogenize the mixture by moving up and down.

Methods of mixing.

Enthalpy

251

Many factors are to be taken into account to obtain reliable results. The powdered solid salt B may sinter before opening the funnel, causing incomplete mixing. The salt B should have suitable physico-chemical properties, such as vapor pressure, chemical reactivity with respect to the funnel and the container, etc., to remain in the funnel during thermal stabilization and to fall down completely when the funnel is open. In some experiments, a large difference between the densities of the liquids A and B does not enable to obtain a homogeneous mixture. In these cases, the addition of a stirring device is necessary. A thin alumina rod sufficiently long to immerse into the liquid is added and acts as stirrer. Such a device was used by Hatem and Gaune-Escard (1980) to investigate the system LiF–K2 SO4 . A number of ionic mixtures were investigated in this way over a temperature range between 1000 and 1500°C, such as the systems NaF–Na2 SO4 and KF–K2 SO4 (Hatem et al., 1982), NaF–Rb2 SO4 (Hatem and Gaune-Escard, 1984), ZrF4 –MF (M = Li, Na, K, Rb) (Hatem et al., 1989), AlF3 -based mixtures (Peretz et al., 1995), KF–NdF3 (Hatem and Gaune-Escard, 1993), etc. In the course of a critical analysis by Hayer et al. (1993) of all the thermodynamic data, it was shown that the difference between two sets of measurements at 1000 K obtained either with a Calvet’s micro-calorimeter or with the apparatus described above is less than 2%. At higher temperatures, such a comparison was not possible due to the scarce data available. An automatic sample charger was also designed and developed that enables a completely automated operation at very high temperatures. It allows a complete experimental run with successive addition of 30 samples. Each individual mixing experiment is computer operated and calorimetric thermograms are also automatically integrated. It should be, however, noted that no apparatus or method can be considered universal in the domain of high-temperature mixing calorimetry, and adaptations should always be made in accordance with the particular requirements of the system under investigation. 4.2.1.8. Double calorimetry of glass-forming systems

Owing to their polymeric nature, many silicate compounds and systems tend to form glasses. When cooling rapidly from the molten state, a part of the sample crystallizes, while the other part remains glassy. This is the main disadvantage while measuring their heat capacity, heat content, enthalpy of fusion, and mixing. Direct measurement of enthalpy changes taking place during physico-chemical processes in silicate systems is intrinsically inaccurate mainly because of the following reasons: (a) (b)

processes in question are frequently very slow, thermal losses due to the dissipation of heat into environment of the calorimeter during high-temperature processes in silicate systems are difficult to calculate or compensate.

252

Physico-chemical Analysis of Molten Electrolytes

However, the enthalpy balance of processes can be evaluated by solving Eqs. (4.26) and (4.27). The enthalpy balance is related to the heating and to the general reaction taking place in the system. In both cases, the initial and the final states of the system comprise the same number of elementary particles, hence the first term on the right-hand side of Eq. (4.27) is always canceled. When calculating the changes of enthalpy in the two types of processes, one can thus operate with relative enthalpy values, Hrel (Tm ) Tm Hrel (Tm ) = Hform (Tref ) +

Csyst (T )dT

(4.34)

Tref

The solution of the sample in a 2:1 mixture of concentrated hydrofluoric and nitric acids at Tref = 298 K was chosen as the reference state. The relative enthalpy, Hrel (Tm ), was measured by indirect method of double calorimetry. This procedure enables us to determine Hrel (Tm ) as the sum of enthalpy increase measured during the cooling of the system in a drop calorimeter (cool H) and during its dissolution in a solution calorimeter (sol H). Equation (4.34) can thus be written in the form Hrel = −(cool H + sol H )

(4.35)

The Hrel values determined by double calorimetry are independent of the magnitude of both cool H and sol H. In silicate systems, they often vary when one composition is repeatedly measured. This may happen because the value of cool H (and hence that of sol H also) depends on the non-reproducible state of the sample after cooling down in the drop calorimeter. The samples may often consist of a mixture of glass and crystalline phases including not only the components of the system but also the products of their decomposition. When a reaction takes place in the system, the respective reaction enthalpy is given by the following relationship when using the method of double calorimetry cool Hstart + sol Hstart − cool Hprod − sol Hprod (4.36) react H = = Hrel, prod − Hrel, start Utilization of the dissolution enthalpies in Eq. (4.36) is justified when the individual phases are adequately diluted in an amount of the chosen solvent. With regard to the error in the solution calorimetry, the enthalpy of mixing and the enthalpy of dissolution could be neglected only when the amount of the solvent in the solution formed has not changed. The isobaric enthalpy analysis of silicate systems was developed by Proks et al. (1977a) and Eliášová et al. (1978). These authors used two calorimeters to measure the enthalpy

Enthalpy

253

difference between the molten sample at high temperature and its dilute solution at 298 K, which was the chosen reference state. Samples of approximately 1 g weight are heated in sealed cylindrical Pt10Rh crucibles to the chosen temperature in the drop calorimetric furnace. The drop calorimeter was described in detail by Proks et al. (1977b). After equilibrating the temperature and keeping it for one hour, the crucible was dropped into the calorimetric block kept at 298 K. The temperature of the sample in the furnace was measured by a Pt30Rh–Pt6Rh thermocouple with an accuracy of ± 3 K. Increase in temperature in the calorimetric block after the sample was dropped into it was measured by means of a sensitive resistance bridge. The value of the enthalpy increase, cool H, was obtained by evaluating the total heat dissipation from the block to the surroundings. The total error of measurement in the drop calorimeter did not exceed 5 J at enthalpies measured up to 3000 J. After measurement in the drop calorimeter, the crucible with the sample was opened by carefully cutting off the lid. The entire sample was removed, ground to the particle size less than 0.04 mm and homogenized. A part of the sample was subjected to X-ray powder diffraction and IR spectroscopic analyses. The heat of solution, sol H, of the sample was then measured in the solution calorimeter, described in detail by Proks et al. (1967). Approximately 0.05 g of sample was dissolved in 100 ml of the 2:1 mixture of concentrated hydrofluoric and nitric acids. The measurement of the heat of dissolution was repeated 3 times on an average. The sum of the two values, the enthalpy of cooling, cool H, and the enthalpy of dissolution, sol H, is the so-called relative enthalpy, Hrel , of the sample. From the temperature dependence of the relative enthalpy, the heat capacity as well as all the following enthalpy, changes could be calculated. By this method, the heat of fusion of Akermanite, Ca2 MgSi2 O7 , (Proks et al., 1977a), the enthalpy analysis of the system 2CaO · MgO · 2SiO3 –CaO · MgO · 2SiO2 (Eliášová et al., 1978), the heat of fusion of Wollastonite, CaSiO3 (Adamkovicˇ ová et al., 1980), the heat of incongruent decomposition of Merwinite, 3CaO · MgO · 2SiO2 (Kosa et al., 1981), the enthalpy of incongruent decomposition of tricalcium aluminate, 3CaO · Al2 O3 (Adamkovicˇ ová et al., 1985), the enthalpy of crystallization of the eutectic melt in the system 2CaO · Al2 O3 · SiO2 –CaO · Al2 O3 · 2SiO2 (Kosa et al., 1987), the heat of fusion of Gehlenite, 2CaO · Al2 O3 · SiO2 (Žigo et al., 1987), etc., were determined for instance.

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Chapter 5

Density 5.1. THEORETICAL BACKGROUND

Density is defined as the mass per unit volume at constant temperature and pressure ρ=

m V

(5.1)

The unit of density is kg · m−3 . In practice, smaller units, e.g. (g · cm−3 ), are frequently used. The reciprocal value of density is the specific volume. The density of melts depends on the temperature and pressure. The dependence on temperature is expressed by the coefficient of thermal expansion 1 α= V

∂V ∂T

(5.2) P

where V is the volume of the system. The density dependence on temperature at constant pressure is usually expressed in empirical equations of the type ρ = ρ0 (1 + at + bt 2 + · · · )

(5.3)

For simple melts in which the composition does not change with temperature, density is a linear function of temperature.

5.1.1. Molar volume

The molar volume can be calculated from density according to the relation V =

M ρ

(5.4)

When the composition of the melt does not depend on temperature, the molar volume is, like the density, a linear function of temperature V = V0 (1 + αt) 255

(5.5)

256

Physico-chemical Analysis of Molten Electrolytes

For the n-component mixture Eq. (5.1) looks as follows n

xi M i

i=2

V =

(5.6)

ρexp

In Section 3.1.3.1., it was shown that the ideal mixing of components is connected neither with volume contraction nor with volume dilatation. However, in real binary mixtures, positive as well as negative deviations from the ideal behavior can be observed. The dependence of molar volume on composition is usually expressed in the polynomial form V =

n

ai x2i

(5.7)

i=0

where ai are the experimental polynomial coefficients. 5.1.2. Partial molar volume

Let us consider a two-component melt. The volume of this system can be expressed as the function of temperature, pressure, and the amount of substances of both the components dV =

∂V ∂T

dT + P , n1 , n2

∂V ∂P

dP + T , n1 , n2

∂V ∂n1

dn1 +

T , P , n1

∂V ∂n2

dn2 T , P , n2

(5.8) The partial derivatives

∂V ∂n1 T , P , n 2

and

∂V ∂n2 T , P , n 1

are the partial molar volumes of

the components and are denoted as V 1 and V 2 . In general, the partial molar volume of the ith component is defined by the relation Vi =

∂V ∂ni

(5.9) T , P , nj =i

The partial molar volume depends on temperature, pressure, and the composition of the system. The physical sense of the partial molar volume is the volume increase caused by the addition of 1 mol of component to an amount of solution such that the composition of the solution at constant temperature and pressure does not change. The partial molar volume can thus attain a negative value also.

Density

257

5.1.3. Application to binary and ternary systems 5.1.3.1. Binary systems

The evaluation of partial molar volumes from the experimentally determined dependency of the molar volume on composition in binary systems is made generally using the method of intercepts. Let us consider a binary melt at constant temperature and pressure in which just one mole of the mixture is present. Then it holds dV = V 1 dx1 + V 2 dx2 where V is the molar volume of the mixture. As it holds that dx1 = −dx2 , then ∂V = V2 −V1 ∂x2 T , P

(5.10)

(5.11)

For the system it also holds that

V = x1 V 1 + x2 V 2 = V 1 + x2 V 2 − V 1

(5.12)

Inserting Eq. (5.11) into Eq. (5.12) we get V 1 = V − x2

∂V ∂x2

(5.13) T,P

and similarly also V 2 = V + x1

∂V ∂x2

(5.14) T,P

Thus, if the dependency of the molar volume of the system on composition is known from the density measurement, it is possible to calculate the partial molar volumes of both the components according to Eqs. (5.13) and (5.14). In the graphic representation they are the intercepts, which for a given composition, cut the tangent of the molar volume versus composition plot on the y axes at x2 = 0 and x2 = 1. 5.1.3.2. Ternary systems

The partial molar volume of a component in the ternary system can be calculated in sections with constant ratio of two components. For instance, in the system A–B–C the partial molar volume of the component A can be calculated according to a relation similar to Eq. (5.14). Since (1 – xA ) = xB + xC , Eq. (5.14) transforms to ∂V V A = V + (xB + xC ) (5.15) ∂xA T , P

258

Physico-chemical Analysis of Molten Electrolytes

For the section xB /xC = N Eq. (5.15) can be written in the form

VA

∂V = V + xC (N + 1) ∂xA

(5.16) T,P

Similar equations can be derived also for component B and C. For excess molar volume in real ternary systems, the validity of the Redlich– Kister’s (1948) general equation can be assumed. For the description of the composition dependency on the molar volume, the following equation is used V =

3 i=1

x i Vi +

3 i, j =1 i=j

xi xj

n n=0

Anij xjn +

m

Bm x1a x2b x3c

(5.17)

a, b, c=1

The first term represents the additive (ideal) behavior, the second the binary interactions, and the third the interactions of all the three components. The coefficients of the regression equation (5.17) are calculated using the multiple linear regression analysis method. Omitting the statistically non-important terms on the chosen confidence level and minimizing the number of relevant terms a solution is obtained, which describes the concentration dependence of the investigated property with a standard deviation of the fit, in the same order as the experimental error. For statistically important binary and ternary interactions, appropriate chemical reactions are sought. Their thermodynamic probability is checked calculating their standard reaction Gibbs energies. The reaction products are identified using the X-ray phase analysis and IR spectroscopy of the quenched melts. Again it is assumed that the composition at high temperature is at least qualitatively conserved after quenching. In the following, two examples are presented. They differ in the complexity of available experimental data, as in the second example, only a part of the ternary system was experimentally accessible. The first example presents analysis of the volume properties of the ternary system, which is experimentally accessible through the whole concentration triangle. The density of melts of the system LiF−NaF−K2 NbF7 was measured using the Archimedean method by Chrenková et al. (2005). The values of the constants a and b of the temperature dependence on density, ρ = a − bt, obtained using the linear regression analysis together with the standard deviations of approximation are given in Table 5.1. In the first step, we examine the behavior of K2 NbF7 in its infinitely diluted solution in LiF and NaF. This requires a calculation of the concentration dependency of the partial molar volume of K2 NbF7 in the systems LiF–K2 NbF7 and NaF–K2 NbF7 . Example 1.

Density

259

Table 5.1. Coefficients a and b of the temperature dependency of density, ρ = a − bt, and the standard deviations of approximation of the investigated melts of the system LiF−KF−K2 NbF7

xLiF xNaF xK2 NbF7 a g · cm−3 sd × 104 g · cm−3 b × 104 g · cm−3 · ◦ C t (◦ C) 1.000 0.000 0.000

0.000 1.000 0.000

0.000 0.000 1.000

2.1968 2.5814 3.2791

4.6247 6.360 10.928

0.4 0.5 1.2

860−960 1010−1100 750−860

0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100

0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

2.2598 2.2857 2.3268 2.3810 2.4037 2.4269 2.4301 2.5116 2.7061

5.102 5.118 5.277 5.552 5.596 5.575 5.434 6.043 7.789

– – – – – – – – –

850−1050 850−1050 850−1050 850−1050 850−1050 850−1050 860−1050 950−1050 990−1060

0.250 0.500 0.750

0.000 0.000 0.000

0.750 0.500 0.250

3.2028 3.2572 3.2347

10.1782 11.0828 11.3694

1.7 0.3 1.7

720−820 770−850 820−910

0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.250 0.250 0.500 0.500 0.500 0.750 0.750

0.750 0.750 0.500 0.500 0.500 0.250 0.250

3.0598 2.9936 3.0677 2.9314 2.9922 2.8300 2.9170

8.150 7.4251 8.357 6.880 7.6184 6.2926 7.044

8.5 14.6 11.1 8.0 9.0 8.7 7.1

690−790 690−800 770−860 770−880 770−870 880−950 880−960

0.563 0.563 0.563 0.375 0.375 0.375 0.187

0.187 0.187 0.187 0.375 0.375 0.375 0.563

0.250 0.250 0.250 0.250 0.250 0.250 0.250

2.9668 2.8604 2.8101 2.9981 3.0499 3.0752 2.8983

7.925 6.4847 6.1368 7.834 8.5244 8.8990 7.044

17.0 16.9 20.9 8.3 36.9 24.9 12.1

750−850 750−810 740−820 730−800 730−800 740−800 770−870

0.375 0.375 0.250 0.250 0.250 0.125 0.125 0.125

0.125 0.125 0.250 0.250 0.250 0.375 0.375 0.375

0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500

2.9529 3.0489 2.9729 2.8596 2.9235 3.0352 2.8412 2.8779

7.265 8.5547 7.763 6.4971 7.3037 8.484 5.6766 6.2613

12.3 19.9 5.9 25.7 20.3 10.3 41.6 16.9

730−830 750−830 730−830 730−820 730−830 670−770 670−770 670−770

0.125 0.125

0.125 0.125

0.750 0.750

2.9332 2.8884

6.864 6.3995

13.0 17.3

700−800 700−800

260

Physico-chemical Analysis of Molten Electrolytes

The concentration dependency of the molar volume of the system LiF–K2 NbF7 at 950◦ C was described by the equation

V = 14.73 + 117.74 xK2 NbF7 + 3.25 xK2 2 NbF7 cm3 · mol−1

(5.18)

Differentiating Eq. (5.18) according to x(K2 NbF7 ) and inserting it into the equation V K2 NbF7 = V + xLiF

∂V

∂xK2 NbF7

(5.19)

we get for dependency of the partial molar volume of K2 NbF7 on composition the equation

3 2 V K2 NbF7 = 135.72 − 3.25xNaF cm · mol−1

(5.20)

For xNaF →1 we obtain for the partial molar volume of K2 NbF7 in the infinitely diluted solution in LiF the value V K2 NbF7 = 132.47 cm3 · mol−1 . This value is a little lower than the molar volume of pure K2 NbF7 , VK0 NbF7 = 135.72 cm3 · mol−1 . 2 The concentration dependency of the molar volume of the system NaF–K2 NbF7 at 950◦ C is described by the equation

V = 21.24 + 104.78 xK2 NbF7 + 9.53 xK2 2 NbF7 cm3 · mol−1

(5.21)

Differentiating Eq. (5.21) according to x(K2 NbF7 ) and inserting into an analogous equation as Eq. (5.19) we get for dependency of the partial molar volume of K2 NbF7 on composition of the equation

3 2 V K2 NbF7 = 135.60 − 9.58 xNaF cm · mol−1

(5.22)

For the partial molar volume of K2 NbF7 in the infinitely diluted solution in NaF we obtain the value V K2 NbF7 = 126.02 cm3 · mol−1 . This value is again lower than the molar volume of the pure K2 NbF7 . Remembering the physical reason of the partial molar volume it can be concluded that the small volume contraction upon addition of K2 NbF7 into molten LiF and NaF indicates the formation of the [NbF8 ]3− complex anion in both systems, since its volume would be smaller than the sum of the volumes of F− and [NbF7 ]2− anions. The calculation of the dependency of the molar volume of the ternary system LiF−NaF−K2 NbF7 on composition was performed according to Eq. (5.17). The regression coefficients were calculated using the multiple linear regression analysis omitting the statistical non-important terms on the 0.99 confidence level. The molar volume of

Density

261

the system LiF(1)−NaF(2)−K2 NbF7 (3) at a temperature of 950◦ C was described by the equation V = 14.844 x1 + 21.303 x2 + 134.102 x3 − 7.048 x2 x32 − 69.872 x1 x2 x3

+ 122.746 x1 x2 x32 cm3 · mol−1

(5.23)

The standard deviation of approximation is sd = 0.21 cm3 · mol−1 . The molar volume of the system LiF−NaF−K2 NbF7 at a temperature of 950◦ C is shown in Figure 5.1 and the excess molar volume of this system at the same temperature is shown in Figure 5.2. K2NbF7 120

0.8

0.8

x(K

2 Nb

F7

)

100

0.6

0.6

80

0.4

0.4

60

0. 2

0.2

40

20

LiF

0.2

0.4

0.6

0.8

NaF

x(NaF) Figure 5.1. Molar volume of the system LiF−NaF−K2 NbF7 (cm3 · mol−1 ) at the temperature of 950◦ C.

K2NbF7 0.8

0.6

0.6

x(K

2 Nb

F7 )

0.8

0.4

0.4

0. 2 −0.2

LiF

−0.6

0.2

−1.0

−1.4

0.4

0. 2

−1.8

0.6

0.8

NaF

x(NaF) Figure 5.2. Excess molar volume of the system LiF−NaF−K2 NbF7 (cm3 · mol−1 ) at the temperature of 950◦ C.

262

Physico-chemical Analysis of Molten Electrolytes

From Eq. (5.23), it follows that the statistically important interactions were found in the binary NaF−K2 NbF7 and in the ternary LiF−NaF−K2 NbF7 systems, which are obviously due to the formation of the more voluminous [NbF8 ]3− anions. The strong polarization ability of the Li+ cations does not allow the formation of the [NbF8 ]3− anions in the system LiF−K2 NbF7 , but when also Na+ cations are present, the effect of the Li+ cations is restricted. The second example presents the analysis of the volume properties of a ternary system, which is experimentally accessible only in part of the concentration triangle. In order to perform the analysis, the density of one component must be approximated. In addition, the ternary eutectic mixture was chosen for a following component. The density of melts of the system KF−K2 MoO4 −SiO2 was measured using the Archimedean method by Chrenková et al. (2002). The melts of this system seem to be promising electrolytes for electro-deposition of molybdenum from fused salts, especially when smooth, adherent molybdenum coatings on metallic surfaces have to be prepared. The temperature dependence of density was expressed in the form of the linear equation ρ = a −bt. The constants a and b, as well as the standard deviations of approximation for the investigated melts, obtained using the linear regression analysis, are given in Table 5.2. First, we examine the volume properties of the system KF–K2 MoO4 . In this system the congruently melting compound K3 FMoO4 is formed. The density of the system KF−K2 MoO4 was measured by Chrenková et al. (1994). The density in this system increases monotonically with an increasing content of K2 MoO4 . From the values of the excess molar volume, it follows that the system shows only small positive deviations from the ideal behavior, as the excess molar volume attains at x(KF) = 0.5 and 827◦ C the value V ex = 1.94 cm3 · mol−1 only. This can be probably ascribed to the formation of the complex anion [FMoO4 ]3− in the melt. However, the low value of the positive deviation of the molar volume may indicate a pronounced thermal dissociation of the complex anion. The obtained density data were used by Chrenková et al. (1994) in the calculation of the degree of thermal dissociation of the additive compound K3 FMoO4 . The degree of dissociation of K3 FMoO4 attains the value α0 (827◦ C) = 0.86, which agrees very well with the value determined by Daneˇ k and Chrenková (1993) from the phase diagram analysis, α0 = 0.81. The existence and structure of the complex anion [FMoO4 ]3− , however, may be a subject of discussion. Even when it cannot be identified by spectroscopic methods, obviously due to the weak Mo−F or O−F bonds and probably also a short lifetime, this complex anion can be considered as an associate. Its acceptance is well founded at least thermodynamically and serves as a useful example to understand the nature and behavior of the investigated melts. Now, we examine the behavior of SiO2 in its infinitely dilute solutions in KF and K2 MoO4 . The system KF−K2 MoO4 −SiO2 was experimentally accessible using the given Example 2.

Density

263

Table 5.2. Coefficients a, b, and the standard deviations of the temperature dependency of density in the system KF−K2 MoO4 −SiO2

xKF xK2 MoO4 xSiO2 a g · cm−3 sd × 104 g · cm−3 b × 104 g · cm−3 · ◦ C t (◦ C) 1.00 0.90 0.80 0.70

0.00 0.00 0.00 0.00

0.00 0.10 0.20 0.30

2.5579 2.5665 2.6130 2.6480

7.5523 7.1977 7.3544 7.2570

2.53 7.80 7.18 7.67

870−970 860−970 850−970 840−970

0.00a 0.00 0.00 0.00 0.00

0.00 1.00 0.90 0.80 0.80

1.00 0.00 0.10 0.20 0.20

3.2500 2.9189 3.1834 3.3196 3.0926

7.5000 6.1871 8.5258 9.6037 7.6621

− 9.63 8.69 3.40 3.41

− 940−990 940−990 940–990 940−1000

0.75b 0.50b 0.25b

0.25 0.50 0.75

0.00 0.00 0.00

2.8505 3.0120 3.0690

9.2080 9.3800 8.9000

2.90 3.10 3.50

840−960 880−990 920−1010

0.72 0.54 0.36 0.18 0.18

0.18 0.36 0.54 0.72 0.72

0.10 0.10 0.10 0.10 0.10

2.7760 2.6623 2.9539 2.9797 3.0168

7.5530 7.5348 7.4946 7.5348 7.4010

5.74 5.46 3.70 4.72 5.20

860−960 850−970 870−950 900−1020 920−1040

0.64 0.64 0.48 0.48 0.32 0.32 0.16

0.16 0.16 0.32 0.32 0.48 0.48 0.64

0.20 0.20 0.20 0.20 0.20 0.20 0.20

2.7919 2.8218 2.9040 2.8579 2.8553 2.8746 3.0239

7.2104 7.3336 7.6849 7.0653 6.4150 6.5020 7.4179

7.50 6.50 9.25 8.72 6.75 6.80 8.39

850−970 850−970 860−990 860−990 900−1020 900−1020 900−1040

0.56 0.42

0.14 0.28

0.30 0.30

3.0387 3.1598

7.9259 8.5503

8.59 9.18

980−1070 1030−1100

a b

estimated by Liˇcko and Danˇek (1982). published by Chrenková et al. (1994).

measuring device only up to approximately 30 mole % SiO2 . In the calculation of the concentration dependency of the molar volume of the system KF–SiO2 the extrapolated values of the molar volume of pure liquid SiO2 estimated by Licˇ ko and Daneˇ k (1982) were thus accepted. The concentration dependency of the molar volume of the system KF–SiO2 at 827◦ C was then described by the equation

3 2 cm · mol−1 V = 30.084 − 3.641xSiO2 − 3.591xSiO 2

(5.24)

Differentiating Eq. (5.24) according to xSiO2 and inserting it into the equation V SiO2 = V + xKF

∂V ∂xSiO2

(5.25)

264

Physico-chemical Analysis of Molten Electrolytes

we get for the partial molar volume of SiO2 the equation

3 2 cm · mol−1 V SiO2 = 26.443 − 7.181xSiO2 + 3.591xSiO 2

(5.26)

For the partial molar volume of SiO2 in its infinitely diluted solution (xSiO2 → 0) in KF the value V SiO2 = 26.44 cm3 · mol−1 was obtained. This value is a little higher than that 0 of the pure SiO2 , VSiO = 22.85 cm3 · mol−1 , which may refer to some weak chemical 2 interaction of both the components, leading to a small volume expansion. However, the nature of the chemical interaction could not be determined on the basis of the density measurement only. The concentration dependence of the molar volume of the system K2 MoO4 –SiO2 at 827◦ C can be described by the equation

3 2 V = 98.744 − 95.095xSiO2 + 19.209xSiO cm · mol−1 2

(5.27)

Differentiating Eq. (5.27) according to xSiO2 and inserting it into an analogous equation as Eq. (5.25), for the partial molar volume of SiO2 the following equation was obtained

3 2 V SiO2 = 3.649 + 38.418xSiO2 − 19.209xSiO cm · mol−1 2

(5.28)

For the partial molar volume of SiO2 in its infinitely diluted solution (xSiO2 → 0) in K2 MoO4 the value V SiO2 = 3.65 cm3 · mol−1 was obtained. This value is much lower 0 than that of the pure SiO2 , VSiO = 22.85 cm3 · mol−1 , which may refer to a strong chem2 ical interaction of both components accompanied by a substantial volume contraction. Such behavior can be explained by the formation of heteropolyanions [SiMo12 O40 ]4− in the melt according to the reaction 12K2 MoO4 + 7SiO2 + 36KF = K 4 [SiMo12 O40 ] + 6K2 SiF6 + 22K2 O

(5.29)

as it was proposed by Silný et al. (1993) and Zatko et al. (1994), who explained the positive role of SiO2 in the molybdenum electro-deposition by change in the structure of the electrolyte. The X-ray diffraction analysis of the solid deposit on the top closure and furnace wall proved that the deposit consists of pure K2 SiF6 , which supports the assumption of the formation of the above-mentioned heteropolyanions. The calculation of the molar volume of the ternary system was performed according to Eq. (5.15). The regression coefficients were calculated using the multiple linear regression analysis omitting the statistically non-important terms on the 0.99 confidence level. The dependency of the molar volume of the ternary system KF(1)−K2 MoO4 (2)−SiO2 (3) on

Density

265

composition at temperature 827◦ C was described by the equation V = 30.52 x1 + 96.22 x2 + 22.91 x3 + 10.16 x1 x2 + 256.5 x1 x22 x3 − 479.0 x1 x2 x32 (5.30) The standard deviation of the approximation of Eq. (5.30) was sd = 0.73 cm3 · mol−1 . The molar volume of the system KF−K2 MoO4 −SiO2 at 827◦ C is shown in Figure 5.3 and the excess molar volume at the same temperature in Figure 5.4. As it can be seen from Figure 5.4, two different regions are present in the investigated part of the system. In the region of low content of SiO2 , near the K2 FMoO4 corner, there is a region of the volume expansion, while at higher SiO2 concentration, a region of volume contraction is present. The volume expansion indicates the formation of more voluminous K2 FMoO4 associates. On the other hand, the large volume contraction in the region of high SiO2 concentration is most probably due to the loss of free volume caused by the formation of very voluminous species. As was suggested by Zatko et al. (1994) and

2)

0.4 0.3

iO x (S

−3 −2 −1 0

0.2

0.4 0.3 1

0.2

2 3

0.1

0.1

0.2

KF

0.4

0.6

0.8

K2MoO4

x(K2MoO4)

Figure 5.3. Molar volume of the system KF−K2 MoO4 −SiO2 (cm3 · mol−1 ) at the temperature of 827◦ C.

30

0.3

40

0.3

0.4

50

x (S

iO

2)

0.4

0.2

70

60

0. 2

0.1 90

80

0.1

KF

0.2

0.4

0.6

x(K2MoO4)

0.8

K2MoO4

Figure 5.4. Excess molar volume of the system KF−K2 MoO4 −SiO2 (cm3 · mol−1 ) at the temperature of 827◦ C.

266

Physico-chemical Analysis of Molten Electrolytes

Silný et al. (1993), this species may be the heteropolyanions [SiMo12 O40 ]4− originating in the melt according to Eq. (5.30). However, the thermodynamic probability of this reaction could not be proved due to the lack of thermodynamic data.

5.2.

EXPERIMENTAL METHODS

There are only a few suitable methods for high-temperature density measurement. The reason is the corrosive nature of molten salts and the thermal dilatation of the materials used for measurement. Most convenient for molten salts are the methods of hydrostatic weighing and the maximum bubble pressure method. For more viscous liquids, such as some silicate melts, the falling body method is suitable. These three methods will be described in detail here. For further study the reader is referred to an excellent book by Mackenzie (1959). 5.2.1. Method of hydrostatic weighing

The hydrostatic weighing is the most common and most precise method of density measurement in molten salts. It is also well known as the Archimedean method, as it is based on the Archimedean law. The principle of this method is measurement of the mass of a body (sphere in most cases) of a known volume in the air and in the liquid. The density is then calculated from the relation ρ=

m0 − (mt − δσ ) Vt

(5.31)

where mt and m0 are the mass of the body in the liquid at temperature t and in air, respectively, δσ is the correction for the surface tension effect exerted on the suspension wire, and Vt is the volume of the body at temperature t. The volume of the measuring body is determined by calibration and if the calibration is performed at different temperatures, then the density measurement and the volume of the body must be recalculated at the measurement temperature. The recalculation can be done either using the known coefficients of the volume expansion of the body, or using the weighing of the body in a liquid with a well-known dependency of the density on temperature (the indirect calibration). In case of aggressive liquids and at high temperatures, one of the most suitable materials for measuring the body is platinum or the Pt–Rh, resp. Pt–Ir alloys. The correction to the surface tension effect exerted on the wire is given by the expression δσ =

π dσ g

(5.32)

Density

267

where d is the diameter of the wire, σ is the surface tension of the liquid, and g is the gravitation constant. An automated device for density measurement of fused salts was published by Silný (1990). The simplified scheme of the device used is shown in Figure 5.5. The key part of the measuring device is the precise analytical balance, provided by automated balancing, working on the principle of zero aberrance. This principle is considered to be the most precise and reliable method of automated balancing. The maximum sensitivity of the balance was 0.1 mg and the maximum load was 200 g. On changing the balancing mass, the balance arm starts to deviate from the equilibrium position, which causes the position reader to send a signal proportional to the magnitude of the deviation. The signal is integrated and enforced in the electronic part of the device and then fed to the electromagnet coil. The resulting effect is the forced action of the magnet in the coil, which is suspended on the second balance arm, and which moves in the direction of lowering the deviation from the equilibrium position. The current in the coil is directly proportional to the equilibrating mass. The current (and thus also the mass) can be recorded using a suitable device in dependency on temperature (potential of the thermocouple) or time. It is also possible to show the recorded dependency on a TV screen directly in the chosen units and in the chosen temperature range.

Balance

Position reader

Counterbalancing electromagnet

Water-cooled plate

Electronic balance control

Furnace

PC screen

Data-processing PC

Figure 5.5. Scheme of the device for the automated density measuring of molten salts.

268

Physico-chemical Analysis of Molten Electrolytes

A water-cooled plate separates the balance and the furnace in order to avoid heating and corrosion of the balance. The hollow platinum sphere 20 mm in diameter and 46 g in mass is suspended from the balance arm using a 0.2 mm Pt20Rh wire. The program for the control PC is written in Basic 5.0/G and enables density measurement, calibration of the measuring sphere, and display of the density versus the temperature plot. The final output is the regression dependency of density on temperature. The program is written as a dialog with the operation personnel. The total precision of the measuring device does not exceed 0.1%. The Archimedean method can be used for the density measurement in not-tooviscous liquids, in order to enable the sphere to equilibrate. For liquids, the viscosity of which exceeds 0.5 Pa.s, the falling body method is suitable. This method can be used for simultaneous measurement of viscosity and density and is described in detail in Section 9.2.2. 5.2.2. Maximum bubble pressure method

Density measurement by the method of maximum bubble pressure is essentially the same as the measurement of surface tension. However, the precision of this method in the density measurement of molten salts is far below the method of hydrostatic weighing and is used only exceptionally. On the other hand, this method is used with an advantage at higher temperatures to measure simultaneously density and surface tension of the oxide systems. As this method is substantially more important in the measurement of surface tension, its detailed description is given in Section 6.2.2. Here only the aspects of the density measurement are discussed. Measuring the maximum bubble pressure, pmax, i , at two depths of immersion, hi , we have

r pmax, 1 − gρh1 2

r σ = pmax, 2 − gρh2 2 σ =

(5.33) (5.34)

where σ is the surface tension, g is the acceleration due to gravity, and ρ is the density of the measured liquid. Subtracting Eq. (5.34) from Eq. (5.33) and rearranging we get ρ=

1 g

pmax, 2 − pmax, 1 h2 − h 1

(5.35)

As can be seen from Eq. (5.35), in the density measurement the radius of the capillary does not need to be known. This also means that the capillary orifice does not need to be machined as carefully.

Density

269

This method was used for density measurement of the systems Fe2 O3 −FeO−CaO−X (X = MgO, Al2 O3 ) by Vadász et al. (1993), and for simultaneous density and surface tension measurement of the system Fe2 O3 −FeO−CaO−SiO2 by Vadász and Havlík (1995), of the system Fe2 O3 −FeO−CaO−ZnO by Vadász and Havlík (1996), and of the system Fe2 O3 –CaO–Cu2 O by Vadász and Havlík (1998). A platinum capillary of 1.285 mm inner diameter was used. The depth of immersion was measured using a micrometer with an accuracy of ± 0.005 mm. Pure nitrogen was used for the bubbles. The gas flow rate was adjusted to produce 3–5 bubbles per minute. The pressure in the bubbles was measured using a tilt-arm micro-manometer filled with distilled water. The accuracy of reading the pressure in the bubble was ± 2.5 Pa. The slag samples of about 100 g in weight were melted in a platinum crucible, placed into a protecting corundum crucible, then in the electrical resistance furnace in an air atmosphere. The steady-state composition was attained after 1–1.5 h, depending on the composition of the slag. The melt composition was allowed to equilibrate for at least the following 2 h. In some cases, the partial oxygen pressure above the melt was reduced by blowing pure nitrogen into the furnace chamber. The temperature of the slag was measured using two Pt6Rh–Pt30Rh thermocouples; one of them was immersed before and after measurement directly into the melt and the second one was attached to the bottom of the crucible. The accuracy in temperature measurement was ± 2.5 K.

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Chapter 6

Surface Tension One of the most important technological parameters in molten salt chemistry is surface tension, as the majority of important reactions take place at the interface of electrolytes or molten reacting media. In aluminum electrolysis, for instance, this parameter influences the penetration of the electrolyte into the carbon lining, the separation of carbon particles from the electrolyte, the coalescence of aluminum droplets and fog in the electrolyte, the dissolution rate of aluminum oxide in the electrolyte, etc. Similar is the effect of interface in aluminum recovery. It is well known that liquids tend to attain the shape with the least surface energy, mainly the shape of a sphere. The force acting per cm for length l of a plain film of liquid is σ =

F 2l

(6.1)

where σ is the surface tension of the liquid. The number 2 in the denominator is due to the fact that the liquid film has two surfaces. In the SI system, the unit of surface tension is mN · m−1 . The surface energy is the energy of the unit area of the surface. The unit of the surface energy in the SI system is J · m−2 , it is thus numerically equal to the surface tension. In general, the surface tension of most liquids decreases linearly with increasing temperature σ = a − bT

(6.2)

where σ is the surface tension in mN · m−1 and T is the temperature in K.

6.1. THERMODYNAMIC PRINCIPLES

The reversible work at constant pressure and temperature required to increase the liquid surface by an increment d A is σ d A and equals the increase of the Gibbs energy of the system dG = σ dA 271

(6.3)

272

Physico-chemical Analysis of Molten Electrolytes

The surface Gibbs energy per unit area is thus ∂G Gs = σ = ∂A T , p, n

(6.4)

Since the above-mentioned process is reversible, the heat associated with it represents the surface entropy dq = T dS = T S s dA

(6.5)

where S s is the surface entropy per unit area of the surface. As (∂G/∂T )P = −S, it follows that s ∂G (6.6) = −S s ∂T or with regard to Eq. (6.4), dσ = −S s dT

(6.7)

The surface tension usually decreases linearly with increasing temperature. Therefore the slope of this dependence equals the surface entropy. The surface enthalpy per unit area of the surface, H s , is H s = Gs + T S s

(6.8)

dσ dT

(6.9)

or Hs = σ − T

Consider now a spherical droplet of melt with the radius r. Its total surface Gibbs energy is 4πr 2 σ . If the radius increases by dr (e.g. due to temperature increase), the change in surface energy would be 8π rσ dr. Since the expansion of the droplet increases its surface energy, this process must be associated with a pressure difference P . The work done by this pressure difference, P 4π r 2 dr, is equal to the increase in the surface Gibbs energy. Thus, P 4π r 2 dr = 8π rσ dr

(6.10)

or, finally P =

2σ r

(6.11)

Surface Tension

273

This is a special case of the fundamental equation of capillary and is known as Laplace equation. In general, two radii of curvature should be taken into account to describe any curved surface and Eq. (6.11) then will attain the form P = σ

1 1 + r1 r2

(6.12)

It is evident that for a flat surface P = 0. The problem of capillary rise will be discussed in Section 6.2.3. Let us now examine the effect of curvature of the surface on the molar Gibbs energy of a liquid substance. From the thermodynamics it follows that the change of the molar Gibbs energy by a change of pressure at constant temperature is G =

V dP

(6.13)

At constant volume and using Eq. (6.12) for P we get G = σ V

1 1 + r1 r2

(6.14)

Expressing the Gibbs energy of the liquid in terms of its vapor pressure and assuming that the vapor behaves ideally, i.e. G = G0 + RT lnP , for a case of a spherical surface with radius r, we have RT ln

P 2σ V = r P0

(6.15)

where P 0 is the normal vapor pressure of the liquid and P is the pressure above the curved surface. Equation (6.15) is Kelvin’s equation and together with Laplace’s equation, they are the fundamental relations in surface chemistry. For liquid droplets, P is positive and there is an increased vapor pressure over the droplets. 6.1.1. Gibbs equation

The surface tension reflects the nature of chemical bonds between species of the system under question. Since molten salts are ionic in character, the surface tension is predominantly given by the chemical nature of the present ionic species. Due to different coulombic interactions between species, the ions more covalent in character are concentrated on the surface and become surface active. Therefore the concentration dependency of the surface tension of binary systems will be substantially influenced by the ionic composition. Due to the equilibrium between the bulk and the surface, the course of the

274

Physico-chemical Analysis of Molten Electrolytes

composition dependency of the surface tension will therefore reflect the actual chemical equilibrium in the bulk. Since the surface tension is a thermodynamic property, one of the main problems is to define the surface tension of the ideal solution. The first attempt to define the behavior of the ideal solution was made by introducing the simple additivity law based on the molar fraction scale. However, such a behavior was never observed in real systems and several sophisticated attempts were therefore made to describe the composition dependence of surface tension in binary systems with sufficient accuracy taking into account the properties of both the components. Most approaches are based on the substitution of molar fractions by the volume fractions. Such an approach seems not to be quite reliable because of the energetics and not because of the volume character of this quantity. The Gibbs energy of the liquid surface depends on temperature T, pressure p, surface area A, and the amounts of substances of all components in the surface layer, nsi , and is defined by the relation Gs = f (T , A, p, nsi ) = σ A +

nsi µsi

(6.16)

i

where µsi is the chemical potential of components in the surface layer. Differentiating at constant pressure we get dG = s

dGs dT

dT +

A, ni

= Adσ + σ dA +

dGs dA

dA + T , ni

µi dni +

Remembering that

dG s dT A, ns i

= −S s ,

i

i

dGs dni

dni

T , A, ni=j

ni dµi

(6.17)

i

dGs dA T , ns i

= σ, and

dGs dnsi T , A, ns i =j

= µsi , divid-

ing by the surface area A, and rearranging we immediately get the Gibbs equation for surface tension dσ = −S s dT −

i

dµsi

(6.18)

i

where σ is the surface tension, S s is the surface entropy per unit area, i is the surface adsorption and µsi is the chemical potential of the component i in the surface layer. However, one of the main problems in the treatment of the surface tension data is the possibility of the substitution of surface properties by those of the bulk. To get a reliable answer to this crucial question, we have not yet obtained enough experimental evidence. Only quite recently an attempt was made to derive the relations for the concentration

Surface Tension

275

dependency of the surface tension and surface adsorption in the ideal as well as in some real binary mixtures. Some examples are presented to demonstrate the concentration dependency of the surface tension and of the surface adsorption in different binary systems exhibiting only small deviations from the ideal behavior. At equilibrium, the chemical potentials of components in the bulk liquid, in the surface g layer, and in the gas phase must be equal: µsi = µli = µi . Thus we can replace the chemical potential of component µsi in the surface layer by that in the bulk liquid phase, µli . At constant temperature and for a binary mixture, we can then write dσ = −1 dµ1 − 2 dµ2

(6.19)

The separation of the bulk liquid and the vapor phase is not abrupt, but as shown in Figure 6.1, there is a region where the composition, density, and pressure vary. Since 1 and 2 are defined as relative to an arbitrarily chosen dividing plane, it is in principle possible to place the dividing plane between the bulk and surface phases so that 1 = 0. We then have

dσ dµ2

= −

2

(6.20)

T , p, n2

Concentration

Component 1

Component 2

Arbitrary dividing surface

Liquid Vapor Distance normal to surface Figure 6.1. Liquid–vapor phase separation.

276

Physico-chemical Analysis of Molten Electrolytes

The derivative of the experimentally determined concentration dependency of surface tension, σ = f (x2 ), according to x2 we can extend as follows

dσ dx2

=

T,p

dσ dµ2

T,p

dµ2 dx2

(6.21) T,p

For the chemical potential of the second component in the liquid phase, we can write µ2 = µ02 + RT ln a2 = µ02 + RT ln x2 γ2

(6.22)

where x2 is the molar fraction and γ2 is the activity coefficient of the second component in the bulk liquid. Differentiating Eq. (6.22) with respect to x2 we get

dµ2 dx2

= RT

T,p

1 d ln γ2 + x2 dx2

(6.23)

Inserting Eqs. (6.20) and (6.23) into Eq. (6.21) we get dσ RT = −2 dx2 x2

d ln γ2 1 + x2 dx2

(6.24)

where γ2 is the activity coefficient of the second component in the bulk liquid, which is defined by the relation

∂ (nex G) RT ln γ2 = ∂n2

(6.25) n1

ex G is the excess Gibbs energy of mixing in the bulk liquid. Inserting Eq. (6.25) into Eq. (6.24), for the surface adsorption of the second component, 2 we get, the general relation

2

x2 dσ RT dx2 =− x2 ∂ 2 (nex G) 1+ RT ∂ n2 x2 n1

(6.26)

The calculation of the surface adsorption 2 can be made using the excess Gibbs energy of mixing obtained, e.g. from the thermodynamic analysis of the phase diagram. For ideal solutions Eq. (6.24) attains the form dσ RT = −2 x2 dx2

(6.27)

Surface Tension

277

Equation (6.27) is frequently used in the calculation of surface adsorption in dilute solutions. For strictly regular solutions, the activity coefficient is defined as RT ln γ2 = B (1 − x2 )2

(6.28)

where B is the interaction parameter of the excess Gibbs energy of mixing. Inserting Eq. (6.28) into Eq. (6.24) we get

dσ dx2

=− T,p

2

x2

(RT − 2Bx1 x2 )

(6.29)

This equation was derived by Hildebrand (1936) and is based on the simple Gibbs model of a dividing surface. It should be emphasized that the interaction parameter B is related to the interaction of components in the bulk liquid and therefore, the surface tension is influenced by the behavior of the bulk. A more generalized approach to the variation of surface tension with composition, which does not require defining the dividing surface, was given by Guggenheim (1977). In the general case, considering a system represented by a liquid binary mixture denoted as (l) and the gaseous phase denoted as (g), separated by a monolayer surface phase denoted as (s), this approach leads to the equation σ = σi +

xs RT ln il A xi

(6.30)

where σ and σi are the surface tensions of the solution and that of the pure component i, respectively, A is the area of the interface, and xis and xil are the molar fractions of the component i in the surface and liquid phases, respectively. Equation (6.30) was derived by Butler (1932). Now, we calculate the surface tension of an ideal binary solution from Eq. (6.30). Since the system is in equilibrium, it holds g

µsi = µli = µi

(6.31)

l 0, s l s s µ0, i + RT ln xi = µi + RT ln xi − σ (xi )Ai

(6.32)

and for the liquid phase (l) it holds

It is known that one of the conditions for solutions approaching ideal behavior is the similar size of the component particles. We can then assume that the surface areas

278

Physico-chemical Analysis of Molten Electrolytes

occupied by the particles are the same for all compositions. For the pure component therefore we have s 0, l µ0, i − µi = σi A

(A1 = A2 = A)

(6.33)

where σi is the surface tension of the pure component i. Setting i = 1, 2, using the relation x1s + x2s = 1 (x1l ≡ x1 ; x2l ≡ x2 ), and adding Eq. (6.32) for both the components, we get exp

−σ A RT

= x1 exp

−σ1 A RT

+ x2 exp

−σ2 A RT

(6.34)

An analogical – but empirical – equation was given earlier by Szyszkowski (1908). When the surface tension of both the components does not differ much, Eq. (6.34) leads to a simple additivity law for surface tensions. If σ1 and σ2 are sufficiently close, then the exponentials can be expanded into a MacLaurin series giving σ1 A σ2 A x 1 σ1 A x 2 σ2 A σA = x1 1 + + x2 1 + =1+ + 1+ RT RT RT RT RT

(6.35)

which results in the final equation σ = x1 σ1 + x2 σ

(6.36)

The surface tension of an ideal solution should thus follow the simple additivity formula with good approximation. 6.1.2. Surface adsorption of ideal and strictly regular binary mixtures

Daneˇ k and Proks (1998) proposed a new approach to calculate the surface adsorption of the ideal and strictly regular binary systems. Taking into account the dependency of 2 on composition, Eq. (6.29) can be integrated. However, any composition dependency of the surface adsorption must fulfill the following boundary conditions: for x2 = 0 is 2 = 0, but for x2 = 1, 2 attains some non-zero value. Such properties fulfill well the general function

2

= x2

n

Ci x2i

(6.37)

i=0

For ideal solutions, the second term in parentheses on the right side of Eq. (6.29) vanishes and for such solutions, we can assume that the surface adsorption increases linearly with composition, i.e. 2 = x2 C0 . Such a behavior follows indirectly, e.g. from the results of

Surface Tension

279

molecular dynamics calculation of the surface layer at the molten salt mixture–saturated vapor phase boundary. Hence dσ = −RT

x 2 C0 dx2 = −RT C0 dx2 x2

(6.38)

and after integration σ = σ1 − RT C0 x2 = σ1 x1 + (σ1 − RT C0 ) x2 = σ1 x1 + σ2 x2

(6.39)

From Eq. (6.39), it follows that for ideal solutions the surface tension obeys the simple additivity law. For the surface tension of the second component it follows from Eq. (6.39) that σ2 = σ1 − RT C0

(6.40)

or σ1 − σ2 = RT C0 = RT

2

x2

(6.41)

and 2

=

σ1 − σ 2 x2 RT

(6.42)

From Eq. (6.42) it follows that in the ideal solutions, the surface adsorption is proportional to the difference in the surface tension of the pure components. For strictly regular solutions we may consider, e.g. the function 2

= x2 (C0 + C1 x2 )

(6.43)

Inserting this expression into Eq. (6.29) and integrating we get σ

x2 dσ = −RT

σ1

0

C0 x2 + C1 x22 dx2 + 2B x2

x2

C0 x2 + C1 x22 (1 − x2 ) dx2

(6.44)

0

and after integration σ −σ1 = x2 (−RT C0 ) + x22

2BC0 −RT C1 2B (C1 −C0 ) −2BC1 + x23 + x24 2 3 4

(6.45)

280

Physico-chemical Analysis of Molten Electrolytes

or, finally σ = σ1 + D1 x2 + D2 x22 + D3 x23 + D4 x24

(6.46)

where D1 = −RT C0 ; D2 =

2BC0 − RT C1 2B (C1 − C0 ) 2BC1 ; D3 = ; D4 = − (6.47) 2 3 4

It would be thus possible to evaluate parameters C0 and C1 for surface adsorption and the interaction parameter B of the excess Gibbs energy of mixing in the bulk liquid from the polynomial coefficients of the concentration dependency of the surface tension. It should be noted that there are three unknown variables and four equations. Hence also the third-order polynomial would be adequate to calculate the parameters. 6.1.2.1. Examples of binary systems

In order to apply the above-described theory it is necessary to know both the surface tension and the excess Gibbs energy of mixing of the system. As examples, the calculation of the surface adsorption in binary systems KF–KBF4 and LiF–K2 NbF7 measured by Daneˇ k and Proks (1998) and Nguyen and Daneˇ k (2000a), respectively, are presented. The phase diagram of the binary system KF–KBF4 was measured by Barton et al. (1971), Daneˇ k et al. (1976), and later again by Patarák and Daneˇ k (1992), who performed also the coupled analysis of thermodynamic and phase diagram data, yielding the excess Gibbs energy of mixing. It is a simple eutectic system not far from the ideal behavior. The surface tension of this system was measured by Lubyová et al. (1997) using the maximum bubble pressure method. The values of constants a and b of the temperature dependency of surface tension, σ = a − bt, obtained using the linear regression analysis, together with the values of the standard deviations of approximation, and the values of the surface tension at 823°C for the investigated KF–KBF4 melts are given in Table 6.1. Using the linear regression analysis of the experimental data, the following equation was obtained for the dependency of the surface tension on xKBF4

The system KF–KBF4 .

2 3 σ = (0.14597 − 0.28352xKBF4 + 0.26538xKBF − 0.06911xKBF ) N · m−1 4 4

(6.48)

The coefficients C0 and C1 of the concentration dependency of the surface adsorption and the interaction parameter B of the excess Gibbs energy of mixing in the bulk liquid were then calculated from the polynomial coefficients of Eq. (6.48) according to Eq. (6.47). The following values were obtained: C0 = 4.144 × 10−5 mol · m−2 ,

Surface Tension

281

Table 6.1. Coefficients a and b of the temperature dependency of the surface tension, the standard deviations of approximation, and the surface tension at 823°C of individual melts of the KF–KBF4 system xKF 1.00 0.75 0.50 0.50 0.25 0.25 0.00

xKBF4

a (N · m−1 )

b (N · m−1 .° C−1 )

sd (N · m−1 )

0.00 0.25 0.50 0.50 0.75 0.75 1.00

213.46 176.32 143.91 159.86 138.16 137.75 130.92

0.08088 0.10869 0.08373 0.11223 0.10492 0.10498 0.08661

0.72 0.57 0.54 0.54 0.84 0.59 1.16

σ (823°C) (N · m−1 ) 146.57 86.43 74.67 67.05 51.39 50.93 59.29

C1 = −4.727 × 10−5 mol · m−2 , and B = 5012 J · mol−1 . The surface adsorption of KBF4 in the system KF–KBF4 was then calculated according to Eq. (6.43). The excess molar Gibbs energy of the system KF–KBF4 was calculated by Patarák and Daneˇ k (1992) on the basis of the coupled analysis of the thermodynamic and phase diagram data. They obtained the equation 2 ex G = (3014 + 6760xKF + 394xKF ) J · mol−1

(6.49)

To compare the calculated interaction parameter B from the surface tension measurement with that from the phase diagram, the non-symmetrical course of the excess Gibbs energy of mixing in the investigated system was approximated by a simple regular behavior. For the interaction parameter B the following value was obtained: B = 6512 J · mol−1 . The surface adsorption of KBF4 was then calculated according to Eq. (6.25) inserting the excess Gibbs energy of mixing of the liquid obtained by Patarák and Daneˇ k (1992). The comparison of the surface adsorption of KBF4 in the system KF–KBF4 calculated according to Eq. (6.43) and that calculated according to Eq. (6.46) are shown in Figure 6.2. The surface tension of the system KF–KBF4 decreases with the increasing content of KBF4 , obviously due to the covalent character of the bonds in the BF− 4 complex anions, which are surface active and concentrate on the melt surface. Similar values as well as the shape of the surface adsorption curve were found when it was calculated from the polynomial coefficients and from the excess Gibbs energy of mixing in the liquid phase. Even both the calculated interaction parameters B are relatively close. The phase diagram of the system LiF–K2 NbF7 was determined and calculated using the coupled analysis of the thermodynamic and phase diagram data by Chrenková et al. (1999). The system LiF–K2 NbF7 is the stabile diagonal of the ternary reciprocal system Li+ , K+ // F− , [NbF7 ]2− and forms a simple eutectic phase diagram with coordinates of the eutectic point 72 mole% K2 NbF7 and 670°C.

The system LiF–K2 NbF7 .

160

1.5E-9

140

1.0E-9

120

5.0E-10

100

0.0E+0

Γ (KBF4) (mol.cm−2)

Physico-chemical Analysis of Molten Electrolytes

σ (mNm−1)

282

80

−5.0E-10

60

−1.0E-9

40 0.0

0.2

0.4

0.6

0.8

−1.5E-9 1.0

x(KBF4) Figure 6.2. Surface tension of the system KF–KBF4 (◦) and surface adsorption of KBF4 . 2 – Eq. (6.43), – Eq. (6.46).

The surface tension of the system LiF–K2 NbF7 was measured by Nguyen and Daneˇ k (2000b) using the maximum bubble pressure method. The coefficients a and b of the linear temperature dependency of the surface tension, ρ = a − bT , obtained using the linear regression analysis, together with the values of the standard deviations of approximation, and the values of the surface tension at 823°C for the investigated LiF–K2 NbF7 melts are given in Table 6.2. For the dependency of surface tension on xKBF4 the following equation was obtained σ = 0.2378 − 1.0355xK2 NbF7 + 2.5793xK2 2 NbF7 − 2.8480xK3 2 NbF7

+ 1.1524xK4 2 NbF7 N · m−1

(6.50)

Table 6.2. Coefficients a and b of the temperature dependency of the surface tension, the standard deviations of approximation and the temperature range of measurement of individual melts of the system LiF–K2 NbF7 xKF 1.000 0.750 0.500 0.250 0.000

xK2 NbF7

a (N · m−1 )

b (N · m−1 .◦ C−1 )

sd (N · m−1 )

0.000 0.250 0.500 0.750 1.000

346.50 303.25 259.02 261.07 226.54

0.09880 0.18463 0.16190 0.16898 0.12774

0.71 0.92 0.71 0.65 0.44

σ (823°C) (N · m−1 ) 237.82 100.16 80.93 75.19 86.03

Surface Tension

283

For the coefficients C0 and C1 of the concentration dependency of the surface adsorption and the interaction parameter B of the excess Gibbs energy of mixing, the following values are: C0 = 1.132 × 10−4 mol · m−2 , C1 = −1.339 × 10−4 mol · m−2 , and B = 17412 J · mol−1 . The surface adsorption of K2 NbF7 in the system LiF–K2 NbF7 was then calculated according to Eq. (6.43). For the excess molar Gibbs energy of the system, Chrenková et al. (1999) obtained the following equation ex G = (21437 − 61335xK2 NbF7 + 99548xK2 2 NbF7 − 75576xK3 2 NbF7 ) J · mol−1 (6.51) After approximating the non-symmetrical course of the excess Gibbs energy of mixing in the investigated system by simple regular behavior, for the interaction parameter B, the following value was obtained: B = 5519 J · mol−1 . The surface adsorption of K2 NbF7 was then calculated according to Eq. (6.25) inserting the excess Gibbs energy of mixing. The comparison of the surface adsorption of K2 NbF7 in the system KF–K2 NbF7 calculated according to Eq. (6.43) and that calculated according to Eq. (6.46) are shown in Figure 6.3. As it can be seen from Figure 6.3, the surface adsorption of K2 NbF7 has a maximum at approximately 0.5 mole% K2 NbF7 and is very similar in course regardless of the calculation procedure used. This indicates that the surface of the melt has very similar properties as the bulk liquid. The maximum in the surface adsorption course indicates

250

5E-5

σ (mNm−1)

200

3E-5 2E-5 1E-5

150

0E+0 −1E-5

100

Γ (K2NbF7) (mol/m2)

4E-5

−2E-5 50 0.0

0.2

0.4

0.6

0.8

1.0

−3E-5

x(K2NbF7) Figure 6.3. Surface tension of the system LiF–K2 NbF7 (•) and surface adsorption of K2 NbF7 . 2 – Eq. (6.43), – Eq. (6.46).

284

Physico-chemical Analysis of Molten Electrolytes

the presence of still more surface active species than the complex anions [NbF7 ]2− in the melt, which could be only the [NbF8 ]3− . The last conclusion is also in accordance with the results of the phase diagram measurements performed by Chrenková et al. (1999) concerning the structure of the LiF–K2 NbF7 melts. 6.1.3. Surface tension in ternary systems

The application of the Gibbs equation to ternary systems can be made only in crosssections with a constant ratio of the amounts of substances, e.g. in the system A−B−C the pseudo-binary system A / B−C. The Gibbs equation in the ternary system is dσ = −1 dµ1 − 2 dµ2 − 3 dµ3

(6.52)

For the dividing plane between the bulk and surface phases, we can choose the condition 1 ≈ 2 ≈ 0. For 3 then we get dσ = −3 (6.53) dµ3 T , p, n2 The next procedure and the calculation of the surface adsorption of the third component is analogical as it was shown for binary systems. To get some information on the structure of melts in the ternary systems, it is very important to define the course of surface tension in the ideal solutions. The general approach used for the variation of surface tension with composition was given by Guggenheim (1977), who stated that the surface tension of ideal solutions should follow the simple additivity formula with a good approximation. The excess surface tension in real systems could be described by the Redlich and Kister (1948) excess function. For the surface tension of real ternary systems it can then be written σ =

3 i=1

σi x i +

3 i=1 i=j

xi xj

k n=0

Anij xjn

+ x 1 x2 x3

l

Bm x1a x2b x3c

(6.54)

m=1

where σi s are the surface tension values of the pure components and xi s are their molar fractions in the mixture. Coefficients a, b, c are integers in the range 0 3. The first term on the right side of Eq. (6.54) represents the ideal behavior, the second one the interactions in binary systems, and the third one the interactions of all the three components. The calculation of the coefficients Anij and Bm is performed using the multiple linear regression analysis, excluding the statistically non-important terms on the chosen confidence level. For statistically important binary and ternary interactions, we look for appropriate chemical reactions. Calculating their standard reaction, Gibbs energy checks their thermodynamic probability.

Surface Tension

285

6.1.3.1. Example of a ternary system

As an example of calculation of the surface tension in the ternary system, KF–KCl–KBF4 is presented. The surface tension of this system was measured by Lubyová et al. (1997) using the maximum bubble pressure method. The system is a simple eutectic system experimentally attainable in the whole concentration triangle. The values of constants a and b of the temperature dependency of surface tension, σ = a − bt, were obtained using the linear regression analysis, together with the values of the standard deviations of approximation. The calculation of the dependency of the surface tension of the ternary system KF(1)−KCl(2)−KBF4 (3) on composition was performed according to Eq. (6.54). The regression coefficients were calculated using the multiple linear regression analysis excluding the statistically non-important terms on the 0.99 confidence level. The surface tension of this system at temperature of 823◦ C was described by the equation σ N · m−1 = σ1 x1 + σ2 x2 + σ3 x3 + x1 x2 A012 + x1 x3 A013 + A113 x3 + A213 x32 + x2 x3 A023 + A123 x2 + A223 x22 + x1 x2 x3 (B0 + B1 x1 x2 ) (6.55) The surface tension of the pure components, σ i , coefficients Anij and Bm , and the standard deviations of approximation for the chosen temperatures of 723, 823, and 923◦ C are given in Table 6.3.

Table 6.3. Coefficients σ i , Anij , Bm of the concentration dependency of surface tension of the system KF(1)–KCl(2)–KBF4 (3) and the standard deviations of approximation, sd, at individual temperatures Coefficient (N · m−1 ) σ1 σ2 σ3 A012 A013 A113 A213 A023 A123 A223 B0 B1 sd

Temperature 723◦ C 152.91 ± 2.23 105.67 ± 1.15 68.07 ± 1.07 −48.69 ± 7.82 −318.36 ± 28.23 668.99 ± 95.31 −586.33 ± 84.94 −106.95 ± 7.93 114.21 ± 15.73 – – 1401.58 ± 368.34 1.26

823◦ C 144.37 ± 2.55 98.41 ± 1.31 59.57 ± 1.15 −54.06 ± 8.63 −311.54 ± 28.24 594.47 ± 96.67 −515.87 ± 87.69 −118.88 ± 9.39 98.98 ± 17.61 – 238.18 ± 39.51 – 1.34

923◦ C 137.26 ± 3.33 91.10 ± 1.72 51.13 ± 1.36 −65.94 ± 11.12 −341.24 ± 35.30 638.23 ± 118.38 −541.95 ± 106.20 −114.11 ± 7.91 – 89.39 ± 23.90 390.45 ± 47.28 – 1.58

286

Physico-chemical Analysis of Molten Electrolytes

KBF4 1.00

0.80

0.80

0.60

0.60

0.40

0.40 −35 −30

−25

0.20

0.00

0.20

0.40

KF

0.20 −20

0.60

−15

−10

−5

0.80

1.00

Mole fraction

KCl

Figure 6.4. Excess surface tension of the ternary system KF–KCl–KBF4 at the temperature of 823◦ C.

The excess surface tension of the ternary system KF–KCl–KBF4 at a temperature of 823◦ C is shown in Figure 6.4. The interpretation of the observed interactions (coefficients Anij and Bm ) in the system KF–KCl–KBF4 is given in Section 2.1.3.2. Surface tension and surface adsorption in the ternary system CaO–MgO–SiO2 and in the quaternary system CaO–FeO–Fe2 O3 –SiO2 were calculated by Daneˇ k and Licˇ ko (1982) and Daneˇ k et al. (1985a), respectively. 6.1.4. Surface tension models

Several attempts were made to describe the course of the dependency of surface tension on composition in the binary and ternary systems. Grjotheim et al. (1972) used for calculation of the surface tension of binary mixtures three different modifications of the Guggenheim’s equation (6.34). A.

−σ a exp RT

−σ1 a = x1 exp RT

−σ2 a + x2 exp RT

(6.56)

Surface Tension

287

The surface area per molecule, a, is a=

x 1 M1 + x 2 M2 ρN

2/3 (6.57)

where M is the molecular mass, ρ is the density of the mixture, and N is the Avogadro’s number. B.

−σ A exp RT

−σ1 A = φ1 exp RT

−σ2 A + φ2 exp RT

(6.58)

In this equation the molar fractions were replaced by the volume fractions x 1 V1 x 2 V2 and φ2 = x1 V1 + x2 V2 x 1 V1 + x 2 V2

φ1 =

(6.59)

Vi are the molar volumes of pure components, and the average surface area per molecule, A, is given by the equation A=

C. exp

−σ A RT

φ 1 V1 + φ 2 V 2 N

= φ1 exp

−σ1 A1 RT

2/3 (6.60)

+ φ2 exp

−σ2 A2 RT

(6.61)

where Ai is the surface area per molecule for component i in the mixture A1 =

M1 ρ1 N

2/3 =

V1 N

2/3 and A2 =

V2 N

2/3 (6.62)

where Mi is the molar mass, Vi the molar volume, and ρ i the density of component i. The reason for introducing Eqs. (6.58) and (6.61) is that in Eq. (6.56) for the two components equal molar volumes and equal surface areas per molecule were originally assumed. However, these two assumptions are not valid in molten salt mixtures as can be seen from the molar volume data. In ternary systems of alkali metal–earth alkali metal halides, Grjotheim et al. (1972) calculated the surface tension values from the binary systems according to the following semi-empirical equation 3 3 RT −σ1 a σ123 (A) = − xi exp xi xj λij ln + a RT i=1

i<j =1

(6.63)

288

Physico-chemical Analysis of Molten Electrolytes

Here xi are the molar fractions of components in the ternary system and λij are the interaction parameters. The surface area per molecule is a=

3

2/3 xi Mi /Nρ

(6.64)

i=1

The interaction parameters are given as the binary excess surface tension (the difference between the experimental value and that calculated according to Eq. (6.56) divided by the molar fractions in the binary system λij (A) = σijex /xi xj

(6.65)

The surface tension σ123 (B) and σ123 (C) have been calculated in the same manner using interaction parameters derived from Eqs. (6.58) and (6.61), respectively, and on the basis of the volume fractions. The authors found that none of the calculated values differ by more than ± 3% from the experimental values. Calculating the surface tension of the systems NaCl–NaF, NaBr–NaF, and NaBr–NaCl, Grjotheim et al. (1976) used a different approach proposed originally by Eberhart (1966). In this approach, it is assumed that the surface tension is a linear function of the surface molar fraction, xiτ , defined in terms of the experimental parameters by the equations σ = x1τ σ1 + x2τ σ2

(6.66)

x1τ + x2τ = 1

(6.67)

Assuming an equilibrium between the bulk and the surface phase, a concentration independent enrichment factor, S12 , is defined by the equation S12 =

(x1τ γ1τ /x2τ γ2τ ) (x1 γ1 /x2 γ2 )

(6.68)

where γ is the activity coefficient and the superscript τ , as before, denotes the surface phase. Setting the activity coefficient ratios in Eq. (6.68) equal to one, we get S12 =

(x1τ /x2τ ) (x1 /x2 )

(6.69)

Combining Eqs. (6.67) and (6.69), it holds σ =

S12 x1 σ1 + x2 σ2 S12 x1 + x2

(6.70)

Surface Tension

or rearranged in the form S12

x2 = x1

σ2 − σ σ − σ1

289

(6.71)

The assumed temperature-independent factor S12 can be determined from Eq. (6.70) by the least squares fitting procedure or by testing the linearity of the rearranged Eq. (6.70) as it was proposed by Bratland et al. (1966) σ − σ1 x1 σ − σ 1 +1 (6.72) = −S12 σ2 − σ 1 x 2 σ2 − σ 1 Another consequence of this model is that if we have experimentally determined the enrichment factor S12 and S23 , then according to Eq. (6.69) it holds S13 = S12 S23

(6.73)

The enrichment factors Sij , together with the deviations from the experimental data of surface tension for the investigated systems are given in Table 6.4. The Eberhart’s model, which introduces the parameter Sij , seems to give a very satisfactory fit of the experimental surface tension values, the standard deviation being in the range 0.4–1% and hence of the same order of magnitude as the experimental standard deviations (0.1–0.9%). This model is able to accommodate a larger deviation from additivity than the Guggenheim’s model. The neglect of the activity coefficients, however, limits the Eberhart’s model to systems with medium chemical interactions and it may not be suitable for strongly interacting systems such as those containing cations of higher valence. In systems with strong complexing tendency, such as those containing cryolite, Utigard (1985) treated the liquid–gas interface as a separate phase. He assumed that it consisted of a single monolayer and that all the surface properties could be ascribed to it. According to Utigard, the surface tension of a binary liquid mixture may be calculated using the equation σ12 =

−1 θσ1 −1 1 + (1 − θ)σ2 2

(6.74)

−1 θ −1 1 + (1 − θ)2

Table 6.4. Enrichment factors and deviations of the fit from the experimental values of surface tension at 900◦ C System NaCl–NaF NaBr–NaF NaBr–NaCl

Sij 5.05 ± 0.25 5.84 ± 0.17 1.86 ± 0.60

sd (%) 0.97 1.03 0.36

290

Physico-chemical Analysis of Molten Electrolytes

where σ i is the surface tension of pure components, i is the surface adsorption of pure components, and θ is the fractional surface coverage of species 1. θ may be calculated from the standard Gibbs energy change, G 0 , for the exchange reaction 1(bulk) + 2(surface) = 1(surface) + 2(bulk)

(6.75)

For cryolite-containing systems the model of Utigard is, however, not suitable, since the activity of an ionic component cannot be assumed to be proportional to the surface concentration of this component. Fernandez and Østvold (1989) have modified the Utigard’s model and for the surface tension of the cryolite-based melts, they used the equation σ12 =

−1 xσ1 −1 1 + (1 − x)σ2 K 2 −1 x −1 1 + (1 − x)K 2

(6.76)

In this equation, K is an adjustable parameter, and x is the molar fraction of component 1 in the binary melt. For those systems for which the surface tension of the final member was known, this calculation may be performed.

6.2.

EXPERIMENTAL METHODS

6.2.1. Capillary method

An approximate treatment of the phenomenon of the capillary rise can be easily made in terms of Laplace’s equation. If the liquid wets the wall of the capillary, the liquid surface is forced to lie parallel to the wall, and the liquid surface has to be concave in shape. The pressure in the liquid below the surface is less than that in the gas phase above the liquid surface. If the capillary is circular in cross section and not too wide in radius, the meniscus will be approximately hemispherical, as is illustrated in Figure 6.5. Such a case is described well by Eq. (6.11). If h denotes the height of the meniscus above the flat liquid surface, then at equilibrium, P must also be equal to the hydrostatic pressure of the liquid column inside the capillary. Thus P = ρgh, where ρ is the difference in density between the liquid and gas phases and g is the acceleration due to gravity. Equation (6.11) then becomes ρgh = 2σ/r

(6.77)

a 2 = 2σ/ρg

(6.78)

or

The quantity a, defined by Eq. (6.78), is known as the capillary constant.

Surface Tension

(a)

291

(b) r

h h r ∆P = 0

Figure 6.5. Capillary rise (a) and capillary depression (b).

Similarly, for a liquid that does not completely wet the walls of the capillary, the simple treatment yields an identical equation. There, a capillary depression could be observed, since the meniscus is convex and h is now the depth of depression. For liquids that do not wet the walls of the capillary completely, i.e. the liquid meets the capillary wall at some angle θ , from a simple geometric consideration it follows that R2 = r/cos θ , and because R1 = R2 , Eq. (6.77) attains the form ρgh = 2σ cos θ/r

(6.79)

An exact solution of the capillary rise phenomenon must, however, take into account the deviation of the meniscus from sphericity, i.e. the curvature must fulfill the Laplace’s equation at each point above the flat liquid surface. The approximate solution has been obtained by Lord Rayleigh (1915) using a series of approximations. For a nearly spherical meniscus, i.e. for r h, expansion around a deviation function leads to the equation a 2 = r(h + r/3 − 0.1288r 2 / h + 0.1312r 3 / h2 . . .)

(6.80)

The first term represents the elementary Eq. (6.78) and the second one takes into account the mass of the meniscus assuming it to be spherical. The following terms provide corrections for deviation from sphericity. A different approach was developed by Bashforth and Adams (1883) and extended by Sugden (1921). When the meniscus at the bottom of the capillary rise is the figure of revolution, both the radii of curvature must be equal at the apex. Denoting this radius of

292

Physico-chemical Analysis of Molten Electrolytes

curvature by b, and the elevation of a general point on the surface by z = y − h, then the dimensionless quantity β is given by the equation β = ρgb2 /σ = 2b2 /a 2

(6.81)

The parameter β is positive for oblate figures of revolution, i.e. for the meniscus in a capillary, a sessile drop, and a bubble under a plate, and is negative for prolate figures, i.e. for a pendant drop or an adjacent bubble. Bashforth and Adams (1883) reported their results as tables. For more detailed information, see for example in Adamson (1967). 6.2.2. Maximum bubble pressure method

In this method, the pressure of a bubble that is formed at the tip of a capillary immersed into a liquid and through which a gas is slowly fed, is measured. For the gas pressure inside the bubble formed the following equation holds at any moment p=

2σ + ghρ r

(6.82)

where p is the pressure inside the bubble, σ is the surface tension, r is the radius of the bubble, h is the depth of immersion, ρ is the density of the liquid, and g is the gravity constant. The second term on the right side of Eq. (6.82) represents the pressure of the liquid column pushed out from the capillary. As it can be seen from Figure 6.6, the radius of the bubble first decreases until the radius of the bubble equals the radius of the capillary, then it increases, and the bubble escapes. When the radius of the bubble is at its minimum, the pressure inside the bubble is maximal and p = pmax . The measuring device used for measurement of the surface tension of molten salts is well described by Daneˇ k and Østvold (1995) and was applied to different molten salt systems. The apparatus consisted of a resistance furnace provided with an adjustable head fixing the position of the platinum capillary, the Pt–PtRh10 thermocouple and a platinum wire, serving as an electrical contact to adjust the exact contact of the capillary with the liquid surface. A suitable temperature controller adjusted the operational constants needed for the temperature control of the furnace using an additional Pt–PtRh10 control thermocouple placed between the working and heating shafts of the furnace. The platinum capillary with an outer diameter of 3 mm had two different inner diameters one at each end, which enabled to measure liquids with very different surface tensions. The orifice diameter on one end was approximately 1 mm, on the other end approximately 2 mm. The higher diameter was used in those melts, where the bubble exerts a high adhesion to platinum. In order to obtain precise results, the capillary tip

Surface Tension

293

h r1 > r2 > r

r1 r r2

Figure 6.6. Maximum bubble pressure measurements.

must be carefully machined. The orifice has to be as circular as possible, with a sharp conical edge. A precise inner diameter of the capillary is very important if accurate measurements have to be performed. The diameter of the orifice was measured using a microscope. The actual capillary radius at a given temperature was calculated using the thermal expansion data for platinum. A special water-cooled furnace lid was used for the capillary support. A micrometric screw, fixed on the lid, determined the position for the exact contact of the capillary with the liquid surface and gave the desired immersion depth with an accuracy of 0.01 mm. A digital micro-manometer with two measuring ranges, 200 and 1000 Pa, was used for pressure measurements. This enabled to measure the pressure with an accuracy of ± 1 Pa. Nitrogen was used to form the bubbles and to maintain an inert atmosphere over the sample. The gas was slowly fed through the capillary during the experiment to avoid condensation in the upper part of the capillary. The nitrogen flow was adjusted using a fine needle valve. The rate of bubble formation was approximately 1 bubble per 20–30 s. The surface tension can be calculated according to the equation σ =

r (pmax − ghρ) 2

(6.83)

where pmax is the maximum bubble pressure when the bubble is a hemisphere with the radius equal to the capillary radius. However, there is also the possibility to calculate the

294

Physico-chemical Analysis of Molten Electrolytes

surface tension of the liquid without knowing the density of the melt. By eliminating the density, ρ, from Eq. (6.83) for two different immersion depths we obtain the equation

σ =

r 2

pmax, 1 h2 − pmax, 2 h1 h2 − h 1

(6.84)

where pmax, i is the maximum bubble pressure at immersion depth hi . Since the density data for these melts are often not known Eq. (6.83) was frequently used. The surface tension of each sample should be measured at 5–7 different temperatures in a range of 100–120◦ C starting at approximately 20◦ C above the temperature of the primary crystallization. The measurements are usually carried out at four different depths of immersion (e.g. 2, 3, 4, and 5 mm) yielding six surface tension values for each temperature. In surface tension measurements using the maximum bubble pressure method several sources of error may occur. As mentioned above, the exact machining of the capillary orifice is very important. A deviation from a circular orifice may cause an error of ± 0.3%. The determination of the immersion depth with an accuracy of ± 0.01 mm introduces an error of ± 0.3%. The accuracy of ± 1 Pa in the pressure measurement causes an additional error of ± 0.4%. The sum of all these errors gives an estimated total error of approximately ± 1%. Using the above-described apparatus, the standard deviations of the experimental data based on the least-squares statistical analysis were in the range 0.5% < sd > 1%. The systems MF–AlF3 (M = Li, Na, K, Rb, Cs).

The surface tension of the systems NaF–AlF3 and KF–AlF3 was measured using the pin detachment method by Fernandez and Østvold (1989) and of the systems LiF–AlF3 , RbF–AlF3 , and CsF–AlF3 using the maximum bubble pressure method by Daneˇ k and Østvold (1995). In all the investigated systems, the surface tension decreases with an increasing con− centration of AlF3 . This behavior could be explained by the formation of AlF3− 6 and AlF4 anions, which are more covalent in character than the pure MF melt. Due to their covalent character, these complex anions concentrate on the surface of the melt. There is, however, a marked change in the concentration dependency of the surface tension going from LiF to CsF, which can be observed in Figure 6.7. Approximately at x(AlF3 ) = 0.25, there is an increasing tendency to the change in the slope of the σ versus x(AlF3 ) plot, when the alkali metal cation is increasing in size. This may be interpreted as an increasing tendency to stabilize the AlF3− 6 anion with the increasing size of the alkali metal cation. This tendency is, however, not observed in Raman studies of the MF–AlF3 (M = Li, Na, and K) melts. The above hypothesis is also in disagreement with the thermodynamic study of Hehua Zhou (1991). On the other hand, the phase diagrams of the RbF–AlF3 and CsF–AlF3 systems measured by Puschin and Baskow (1913),

Surface Tension

295

250 Li F NaF KF

200

RbF

σ (mN.m−1)

CsF 150

100

50

0

0.1

0.2

0.3

0.4

0.5

x(AlF3) Figure 6.7. Surface tension of the systems MF–AlF3 .

show sharp maxima at the Rb3AlF6 and Cs3AlF6 composition, respectively, which may refer to a very low dissociation degree of the AlF3− 6 anion in these systems. For a system in equilibrium, the Gibbs equation for the surface tension is valid. The Gibbs equation enables us to calculate the surface entropy from the temperature dependency of the surface tension. The surface entropy is related to the structure and the distribution of species on the surface. Hence this property will also be related to the distribution of ions in the bulk due to the equilibrium between the surface and bulk. In Figure 6.8, a plot of the surface entropy versus x(AlF3 ) is shown. The course of the surface entropy has a maximum, which is shifted to higher AlF3 concentrations with the increasing size of the alkali metal cation. This reflects the relatively simple structure of the pure MF and MAlF4 melts in comparison with melts having 0 < x(AlF3 ) < 0.5. Simultaneously this maximum decreases and becomes broader in the sequence Li → Cs. This behavior is probably due to a more ordered structure for the KF, RbF, and CsF melts than for the NaF and LiF containing MF–AlF3 melts.

296

Physico-chemical Analysis of Molten Electrolytes

0.25 Li NaF 0.20

KF RbF

Ss (J.mol−1K−1)

CsF 0.15

0.10

0.05

0.00 0.0

0.1

0.2

0.3

0.4

0.5

x(AlF3) Figure 6.8. Surface entropy of the systems MF–AlF3 .

6.2.3. Detachment methods

Measuring the force (the weight) needed to break away a body from the liquid surface is the basis for many variations of the detachment method. The body may be in the form of a rod with a circular base (Pin detachment method), a rod with a rectangular base (Wilhelmy slide method), a horizontal circular ring (Ring method), etc. The different forms of the detachment method have been used for surface tension measurement of liquids from room temperature up to approximately 1000◦ C. It is evident that the detachment methods can be used only in cases when the liquid wets the measuring body. Surface tension is defined as the energy per unit area of surface or the force per unit length. In principle, by the detachment method, the liquid adhering to the bottom of the body is raised up till the column of the liquid hanging from the body is broken. If the perimeter of the basis of the body is known, then the force necessary to detach the body

Surface Tension

297

from the liquid surface, max F, is given by the equation max F = σ · L

(6.85)

where σ is the surface tension and L is the perimeter of the body. However, as it was shown by Freud and Freud (1930), Eq. (6.85) is not valid when a ring is detached from the liquid surface, since in this case, the risen liquid has quite a different shape. 6.2.3.1. Pin methods

The maximum volume of liquid that is held up above the surface by a circular body is given by the fundamental Laplace’s equation max F = P =

2σ r

(6.86)

where r is the radius of the circular body at the point on the surface and P is the pressure at this point relative to the pressure on the undisturbed plane surface. In this case, it is the surface at an infinite distance from the body. For cylindrical bodies it thus holds σ =

max F 2π r

(6.87)

where r is the radius of the pin. The maximum force, which can hold the liquid attached to the pin, is related to its maximum volume V max F = V gρ

(6.88)

where V is the volume of the liquid above the undisturbed surface, ρ is the density of the liquid, and g is the gravitation constant. This situation is shown in Figure 6.9. The surface tension can thus be calculated when the maximum force, max F (the maximum weight) and the radius of the pin are known. It should be emphasized that the pin detachment method is an absolute method since it does not need any calibration. The mathematical analysis of the detachment of the circular (pin method) and rectangular (Wilhelmy method) rod from the liquid surface was given by Lillebuen (1970). He showed that the surface tension of the liquid is given by the equation σ =

max F π rr 2π r Vr

(6.89)

298

Physico-chemical Analysis of Molten Electrolytes

y

1

2 3

x Figure 6.9. The pin detachment method. 1 – the measuring body; 2 – the circular base; 3 – the liquid.

The calculation of the term (π rr /Vr ) is possible from knowing the detachment force max F, the pin radius r, and the density of the liquid ρ according to the equation 3 3 2 3 3 3 πrr r r r r = 0.992 + 2.564 × 10−6 / r − 6.605 r + 73.25 r − 454.0 r Vr Vr Vr Vr Vr (6.90) where the quantity

rr3 Vr

=

r3 (max F /gρ)

(6.91)

However, for rectangular bodies Eq. (6.87) can also be used if the edge effects are neglected. Grjotheim et al. (1972) described a relatively simple experimental device allowing precise determination of density and surface tension to be made in one run. An electronic, recording thermo-balance, described in detail in connection with the density measurement by Grjotheim et al. (1971), was employed. The density sinker with the pin for the surface tension measurement is shown in Figure 6.10. It was made of Pt10Rh alloy and the diameter of the pin at 25◦ C was 1.98 mm. In the calculation of

Surface Tension

299

1

2

3

Figure 6.10. Platinum body for simultaneous density and surface tension measurement. 1 – Pt10Rh wire; 2 – density sinker; 3 – pin.

the diameter of the pin at the actual temperature the thermal expansion data for platinum were used. In the experimental setup, the body suspended on a thin wire is attached to the balance, which records the weight of the rod when it is slowly lowered into the liquid surface. The measured weight shows a certain maximum just before the contact between the liquid surface and the base of the rod is broken. The measurement of surface tension is performed in the following way. The sinker is suspended on the wire from the balance and is positioned just above the melt. The furnace with the crucible containing the melt is then slowly and continuously raised. When a contact is established between the surface of the melt and the pin, there is a sudden increase in the weight of the sinker, σpin . The increase is due to the effect of the surface tension exerted on the pin, which strives to wet the pin and drag it a little into the melt. The furnace is then slowly and continuously lowered, until the column of the liquid is broken. During this procedure the electronic balance records the weight of the sinker. The weight increases and exhibits a maximum just before the column of the melt hanging from the pin is broken. The difference between the weight of the sinker before the contact is detached and its maximum weight, max F, is proportional to the

Physico-chemical Analysis of Molten Electrolytes

Force

300

∆maxF

mrem

Time Figure 6.11. Typical balance record at the surface tension measurement using the pin detachment method. max F – force needed to detach the pin from the melt; mrem – mass of the remaining melt resting on the pin.

surface tension. The weight, however, does not sink up to the original baseline of the sinker, but hangs back at a certain value mrem , since a small amount of melt remains on the bottom of the pin. The typical balance record is schematically shown in Figure 6.11. The pin detachment method was used in the measurement of the surface tension of cryolite-based melts, e.g. by Bratland et al. (1983), Fernandez et al. (1986), Fernandez and Østvold (1989), and Daneˇ k et al. (1995). 6.2.3.2. The Wilhelmy slide method

This method is attributed to Wilhelmy (1863) and is relatively simple. However, there are several variations so that no unique procedure is established. The basic observation is that when a thin plate is withdrawn from a liquid, the surface around the plate will no longer be a surface of revolution. Equation (6.87) is therefore in this case not valid. Let us assume a rectangular body, length L of which is much greater than the width B (Figure 6.12). For such case the maximum volume that can be held up by the rod is given by the function V = f (L + B). The simplification implicit in this consideration is that the edge effects have been neglected. Then with a good approximation for the surface tension

Surface Tension

301

1

2 3

Figure 6.12. Detachment Wilhelmy slide method. 1 – plate; 2 – rectangular end-face; 3 – liquid.

the following equation is valid σ =

max F 2(L + B)

(6.92)

and the denominator 2(L + B) is the perimeter of the plate. The surface tension can be calculated knowing the maximum force max F, L, and B, by means of Eq. (6.92). The density of the liquid need not to be known. Another modification of this method is a plate suspended from the balance and partially immersed in the liquid (Figure 6.13). A general equation holds

σ cos θ =

Wtot − (Wplate − b) P

(6.93)

where Wtot is the weight (i.e. the force exerted) of the plate partially immersed, Wplate is the weight of the plate suspended in the air, b is the buoyancy correction for the immersed portion, and P is the perimeter of the plate. In order to determine b, the depth of immersion h of the plate must be known. The perimeter is twice the sum of the width L and the thickness B of the plate. The buoyancy correction b is then equal to LBhgρ, i.e. to the weight of the liquid displaced by the immersed part of the plate.

302

Physico-chemical Analysis of Molten Electrolytes

h

Figure 6.13. Wilhelmy slide method using plate immersion.

If the thickness of the plate is more than 0.1 mm, the end correction must be taken into account. As a precaution, plates of several thickness and width should be used to verify the value of the end effect. From Eq. (6.93), it follows that the contact angle θ must be known. 6.2.3.3. The ring method

It is a widely used detachment method, in which the force to detach a ring from the surface of the liquid is determined. In the first approximation, the detachment force is supposed to be equal to the surface tension multiplied by the periphery of the surface to be detached. For a ring we can thus write Wtot = Wring + 4π σ R

(6.94)

However, Harkins and Jordan (1930) found that Eq. (6.94) is not precise and proposed an empirical correction factor, which depends on two dimensionless ratios f = (σ/p) = f (R 3 /V , R/r)

(6.95)

where p denotes the “ideal” surface tension calculated according to Eq. (6.94) and r is the thickness of the ring wire. The detailed theory of this method was derived by

Surface Tension

303

To balance

R r

Figure 6.14. Schematic representation of the ring method.

Freud and Freud (1930) and is rather complicated, but the calculated values of f agree with the empirical values within the experimental precision of about 0.25%. The schematic representation of the ring method is shown in Figure 6.14. The method is very precise when some experimental conditions are fulfilled. The ring is usually made of platinum wire and should be kept as horizontal as possible. A decline of 1◦ causes an error of 0.5%. Care must be taken to provide that the ring is in the horizontal position and to avoid any disturbance of the surface when the critical point of detachment is approached. The ring is usually heated to red glow before use to remove surface contaminants. A zero or near zero contact angle is necessary, otherwise wrong values of surface tension could be measured. 6.2.4. The drop methods

Since surface forces depend on the magnitude of the area, the drops tend to be as spherical as possible. Distortions due to gravitational forces depend on the volume of the drop. In principle, it is however possible to determine the surface tension by measurement of the shape of the drop, when gravitational and surface tension forces are comparable. Two principally different methods must be taken into account. There are methods based on the shape of a static drop lying on a solid surface or a bubble adhering underneath a solid plate, and dynamic methods, based on continuously forming and falling drops. It should be noted that all the principles described here for drops are valid also for bubbles.

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Physico-chemical Analysis of Molten Electrolytes

h r

Figure 6.15. Measurement of the sessile drop shape.

6.2.4.1. The sessile drop method

In general, a drop is formed and care must be taken to avoid any disturbance of its shape. Then the dimensions of the drop are measured, e.g. from a photograph. Usually a rather large drop is formed, because only one radius of curvature, that in the plane of drawing, is considered. For this very simple method, the contact angle is not required and only the distance between the equatorial plane and the apex is measured (Figure 6.15). Surface tension is calculated from the equation σ = ρgh2 /2

(6.96)

Using the Bashforth and Adams tables, Porter (1933) calculated the difference between h2 /2r2 and a2 /2r2 , where r is the equatorial radius. The variation of with h/r could be fitted accurately by means of the empirical equation = 0.3047

h3 r3

1−

4h2 r2

(6.97)

When this method is applied to liquids at room temperature with glass or metallic plates as bases, a precision of about 0.2–0.5% can be attained. However, at high temperatures, when the shape of the sessile drop is not as sharp from the photograph, the precision decreases very rapidly. A problem can also be to keep the base in horizontal position. When ceramic or graphite bases are used the precision can be also 10–20%. The reason is that ceramic and graphite materials are often porous and very hard to polish, which causes irregular wetting. 6.2.4.2. The drop weight method

Weighing drops falling from a tube into a container is fairly accurate and perhaps the most convenient laboratory method of surface or interfacial tension measurements.

Surface Tension

305

The principle of this method, proposed by Tate (1864), consists of the slow dripping of a liquid from a tube into a crucible. When enough drops are in the crucible, the average mass of the drops is calculated. The more the drops collected in the crucible, the more accurate is the result. The mass of one drop is given by the expression W = 2π rσ

(6.98)

However, when observing a falling drop, it is obvious that a part of it will remain in the tube. Thus, the actually obtained mass of the drop W is less than that when nothing were to remain on the tube. 6.3.

CONTACT ANGLE

A drop of a liquid touches the horizontal solid surface under a certain angle θ , which is called the contact angle (Figure 6.16). There are three interfacial boundaries: solidus–liquidus, solidus–gaseous, and liquidus–gaseous, which are characterized by the interfacial energies σsl , σsg , and σlg . At equilibrium, a simple relation can be derived among these three interfacial energies cos θ =

σsg − σsl σlg

(6.99)

which is the well-known Young equation. From Eq. (6.99) it can be seen that (a) when σsg > σsl , the contact angle θ < 90◦ and the liquid wets the solid surface, (b) when σsg < σsl , the contact angle θ > 90◦ and the liquid does not wet the solid surface. 6.3.1. Contact angle measurement

Sessile drop method is the most frequently used method (c.f. Section 6.2.4.1.). In the measurement at high temperatures the drop of the melt is lying on the solid base placed in a horizontal furnace tube. The contact angle is measured on photographs of the drop.

sl/g ss/g

θ

sl/s

Figure 6.16. Equilibrium between a liquid drop lying on a solid surface.

306

Physico-chemical Analysis of Molten Electrolytes

Even when the principle of the method seems to be very simple, the measurement of the contact angle at high temperatures using this method is rather difficult for several reasons. The accuracy and reliability of this method is very low, in general an error up to 20% may occur. The low reproducibility has many reasons. The value of the contact angle is affected by the inhomogeneity of the material of the solid pellets used. Solid materials are usually composed of grains of different size and it can hardly be polished as well as it would be needed. The liquid also wets the solid amongst the grains causing uneven contact angles at different positions. It is also almost impossible to keep the strictly horizontal position of the solid pellet. When the contact angle on platinum bases is measured, Pt sheets, mostly not absolutely even, are usually used. The measurement of contact angle of the cryolite melts on graphite bases is often described in the literature. However, it is not possible to polish porous graphite materials. Substantial discrepancies in the published data could be ascribed to the above reasons, as well as to the insufficient purity of the chemicals used. The measurement of the contact angle between aluminum and graphite is also affected by the reaction of aluminum with carbon during the formation of aluminum carbide, as explained by Thonstad et al. (2001). Silný (1987) has carried out very detailed contact angle measurements of cryolite melts on graphite and the measurement of surface tension using the sessile drop method. He used the Leitz microscope for photographing the sessile drop and used a sophisticated computerized approach to calculate the contact angle and the surface tension from the shape of the drop. However, the results showed a dispersion of approximately 20%.

6.4.

INTERFACIAL TENSION

The interfacial tension between two liquids plays an important role in some metallurgical processes, e.g. in the magnesium, aluminum, and copper production. It is also essential for any detailed understanding of the behavior of the cells, since the interfacial tension is a dominant factor controlling the interfacial stability. Although interfacial phenomena have no effect on the equilibrium between metals, slags, and gases, they may exert profound effects on the rates of reactions, which occur across interfaces involving these phases. Non-reacting surface-active solutes may tend to keep reactants, which are less surfaceactive, out of the interface and so slow down the reaction rate. They may also retard mass transfer and hence reactions, by impeding the surface renewal. But surface-active solutes, which enter into reactions, may speed up surface renewal and accelerate reactions by causing turbulence in the vital interfacial region. Examples of all the three effects have been discussed and their relative importance considered by Richardson (1982). It would appear that the interfacial blocking may retard rates up to one-hundred fold, whereas retardation or enhancement of surface renewal in stirred systems is only likely to decrease or increase rates by a factor of five or ten. Interfacial phenomena may also

Surface Tension

307

affect the coalescence of metal drops in molten salts and slags and influence the foaming of them and the nucleation of gases in metals. The theoretical background of the interfacial tension is identical with that of the surface tension. The only difference is that the interface liquidus–gaseous is replaced by the interface liquidus 1–liquidus 2. Currently, one of the best-known examples is the interfacial tension between aluminum and cryolite melts. However, the results of Zhemchuzhina and Belyaev (1960) and Gerasimov and Belyaev (1958) have been found sufficiently discrepant, so that more work was desirable. Attempts to deduce the interfacial tension from the shape of frozen drops of metals were completely unsuccessful, due to distortion introduced by freezing. It was apparent that direct measurements on the liquid system were necessary. Scrap aluminum, earlier used for a new product, must be re-melted and refined. This technology consists of melting aluminum under a molten salt mixture in order to prevent oxidation and to enhance the coalescence and recovery of the molten metal. In this process, the interfacial tension between aluminum and the salt mixture plays a significant role in terms of both metal recovery and dross de-wetting. Since scrap metal always has an oxide layer, it is required, by either mechanical or chemical means, to break this layer to allow the metallic droplets to coalesce.

6.4.1. Experimental methods

There are several methods for interfacial tension measurement. However, at high temperatures, the choice of the measurement technique is limited. Since most high temperature liquids are corrosive and often non-transparent to visible light, the sessile drop technique can rarely be used. However, by the use of the X-ray beam, the shape of sessile drops immersed in another liquid may be determined. This technique was used by Utigard and Toguri (1985) in the measurement of interfacial tension of aluminum in cryolite melts. On the basis of the curvature of the drop and the density difference between the metal and the salt, X-rays lead to a fuzzy outline of the drop shape and together with the sensitivity of the drop outline on the interfacial tension, this technique is limited to an accuracy of about 5–10%. Another method for liquid–liquid interfacial tension measurements is the pin detachment method. The description of this method for the surface tension measurement is given in Section 6.2.3.1. The liquid metal must wet the pin itself very well in order to justify the assumption of a zero contact angle between the pin and the metal. Recently, Fan and Østvold (1991) used this method in the interfacial tension measurement between liquid aluminum and cryolite melts using a titanium diboride pin. A serious problem is that TiB2 tends to react with aluminum to form aluminum carbide and starts to dissolve, thereby changing the properties of the metal. Another problem is that impurities in the TiB2 may change the wetting properties and therefore the calculated interfacial tension.

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Physico-chemical Analysis of Molten Electrolytes

The drop weight method has been used to determine the interfacial tension in molten slag/metal systems by (El Gammal and Müllenberg (1980) and for aluminum/salt systems by Ho and Sahai (1990). This method is based on measuring the size of individual metallic drops, which are forced through a small orifice into the molten salt. The drop size is usually determined by recording the mass increase of the crucible holding the salt, as individual drops are forced through the orifice into the melt. However, as was found by Richardson (1982) and Utigard (1985), one problem with this technique is that the metal and the molten salt are not necessarily in thermodynamic equilibrium before the metal is forced into the melt. This may lead to rapid interfacial tension changes. It is known that while reactions take place at the interface, the interfacial tension may significantly decrease. The maximum bubble pressure method, which is a technique somewhat similar to the drop weight method, has also been used at high temperatures. With this technique, it is required to measure the pressure necessary to force small drops of a molten metal through an orifice into the molten salt. The interfacial tension between molten magnesium and the salts of the system MgCl2 –KCl–BaCl2 has been measured using this method by Reding (1971). On the basis of the density difference and the maximum “drop” pressure, the interfacial tension can be calculated using the Schroedinger (1914) equation. This method is very demanding and therefore rarely used. In addition, there is the problem of exchange reactions taking place as the two liquids come into contact, leading to interfacial tension gradients and possible wetting changes.

6.4.1.1. Capillary method

Among the simple, classical methods of determining surface or interfacial tensions is the capillary rise (or depression). It requires the use of a tube made of a material, which is not wetted either by the melt or by the metal, and, of course, to measure somehow the position of the interface, which has to be observed. The latter is difficult with opaque tubes and when the interface is below the surface of an opaque metal. Therefore, the relative movement of the liquid when the position of the tube is changed is transmitted to another liquid in a glass tube outside the furnace by means of a gas buffer. This method, which has been used to measure the aluminum–cryolite interfacial tension by Dewing and Desclaux (1977), the interfacial tension between aluminum and the chloride–fluoride melts by Silný and Utigard (1996), and in the systems NaF(KF)–AlF3 by Silný et al. (2004), is based on measuring the position of the metal–salt interface in the tube, moving through the interface. As the tube is moved down through the salt layer before reaching the metal surface, the molten salt entering the tube displaces a certain volume of gas (see Figure 6.17a). Since the metal does not wet the tube, no metal will enter the tube when it reaches and passes through the salt–metal interface and as long as the tube immersion is less than the capillary depression. Therefore, during this period

Surface Tension

309

Scale

b)

Meniscus position

a) a b c h

Capillary position

c)

Figure 6.17. Principle of the interfacial tension measurement using the capillary depression method.

there is no further displacement of gas within the tube (see Figure 6.17b). However, as the tube is pushed further down, the metal suddenly starts to enter into the tube, and the gas is again displaced from the tube (see Figure 6.17c). Connecting the capillary tube to a horizontal glass tube in which there is a small liquid drop, the movement of the drop meniscus can determine the position of the salt–metal interface. This method has the disadvantage that, as the capillary tube moves down into the crucible, the average temperature inside the tube increases, leading to gas expansion and additional movement of the measuring meniscus. Dewing and Desclaux (1977) tried to avoid this problem by moving the capillary rapidly to a certain immersion and then measuring the distance traveled by the meniscus during its initial rapid movement, assuming that the gas expansion will be reflected by a subsequent slower movement of the meniscus. By immersing the capillary tube to various depths, they were able to obtain a curve of the meniscus movement versus the depth of capillary immersion, which allowed them to determine the capillary depression. As the movement of the meniscus was measured visually, personal judgment had to be made as to when the rapid movement of the meniscus had stopped. One advantage of this method is that ceramic tubes made of sintered alumina are not wetted by most metals and are nearly inert in most low-temperature molten salts. Another advantage is that the metal and the salt can be kept in contact with

310

Physico-chemical Analysis of Molten Electrolytes

each other long enough before the start of the measurement, allowing for the chemical equilibrium to be established. Because of these advantages, Silný and Utigard (1996) decided to develop a revised and improved version of this technique for the measurement of the interfacial tension between aluminum and chloride-based molten salts. To eliminate the manual reading of the meniscus, the position of the capillary immersion tube as well as that of the measuring meniscus were continuously recorded by a video camera. The experimental setup is schematically shown in the original literature. The measuring capillary alumina tube had an inner diameter of 4.3 mm and an outer diameter of 6.35 mm, and was connected to the 2.5-mm inner diameter horizontal glass tube containing the meniscus of the liquid through a plastic tube. The capillary positioning device was designed to exhibit sufficient force to move the capillary smoothly through the gas-tight gasket, which is required in order to prevent air leakage into the reaction tube. The vertical movement of the alumina capillary was displayed by a pointer along the same scale as the meniscus position. By zooming in on this scale with a video camera set with a shutter speed of 1/1000 s, sharp images with a high resolution were obtained. These images were recorded on a super VHS VCR. To work in an inert atmosphere, the furnace compartment was sealed, allowing for evacuation followed by refilling with high-purity argon. A gas buffer made from polyethylene provided pressure equalization between the reaction tube and the surroundings, allowing operation with a slight overpressure inside during periods when no measurement was made. The relay unit performed one sweep, which consisted of lowering the capillary tube at a rate of 5 cm/s to the bottom position, followed by lifting it to the upper resting position. The whole sweep lasted approximately 2 s. Immersion rates above 10 cm/s caused increasing measurement scatter. This may be due to disturbances at the salt–metal interface or to the fact that the camera scanning speed was limited to only 30 frames/s, giving a poor resolution of the meniscus position. On the other hand, immersion speed below 1 cm/s caused a noticeable thermal expansion of the gas within the capillary tube, affecting the position of the recording meniscus. Figure 6.18 shows the results obtained for sweeps using 3, 6, and 12 cm/s immersion speeds. However, at immersion speed 5 cm/s the immersion takes about 1 s and approximately 25 measurement points were recorded for evaluation. During the computational procedure, several corrections have to be made. To decrease the magnitude of these corrections, it is preferable to use as large a diameter of the crucible as possible. The first correction must be made due to the displacement of some metal when the tube is immersed into the metal, which results in an increase of the metal level in the crucible. To compensate for this error, the depth of immersion measured # $ from the point where the tube first touched the metal must be divided by 1 − (r /rc )2 , where r is the

Surface Tension

311

9

Rel. meniscus position (cm)

8 12 cm/s

7 6 cm/s

6

3 cm/s

5 4 3 2

0

5 2 3 4 Capillary position (cm)

1

6

7

Figure 6.18. Shape of the immersion curve for various immersion speeds.

external radius of the capillary and rc is the crucible radius. Before proceeding with the other corrections, the uncorrected capillary depression should be divided by this factor. A second correction is caused by the curvature of the interface before the tube touches it. The capillary depression has to be calculated based on a hypothetical flat, undistorted surface. The real interface lies below this hypothetical interface and the excess pressure that exists immediately below the interface is 2σ /r1 , where r1 is the radius of curvature of the interface at the point before the tube is immersed, and σ is the interfacial tension. The third correction is due to changes in shape of the interface caused by the capillary. As the tube enters the aluminum, an interfacial tension force is formed along the outer edge of the tube. The resulting interfacial tension force is spread evenly over the cross section of the crucible leading to a pressure change of 2σ r /rc2 . Since the immersion alumina capillary is completely non-wetted by aluminum, it was assumed that it forms a contact angle of 180◦ with the aluminum. The interfacial tension can then be calculated from the following equation

σ = 2

gρh 1 1 r − − 2 r r1 rc

(6.100)

where σ is the interfacial tension, r is the internal radius of the measuring capillary, r1 is the radius of the aluminum surface curvature at the center of the crucible, r is the external

312

Physico-chemical Analysis of Molten Electrolytes

radius of the capillary, rc is the internal radius of the crucible, g is the acceleration due to gravity, ρ is the density of aluminum, and h is the capillary depression. To estimate r1 for the 65-mm diameter crucible, the following relation was used

ρ (kg/m3 ) r1 (mm) = 53 + 75 σ (N/m)

2 (6.101)

ρ is the density difference between the melt and aluminum, r1 varied between 100 and 180 mm. Equation (6.101) was derived by superimposing sessile drops of various shapes with an equatorial diameter of 70 mm (equal to crucible ID) and with a contact angle 90◦ . The sessile drop shapes were then digitized and the apex diameter was calculated using the program developed by Rotenberg et al. (1983). If these corrections had not been made, the maximum additional error in the calculated interfacial tension would have been 5%. To determine the effect of uncertainties of the various parameters on the calculated interfacial tension, a set of sensitivity calculations was carried out. It was found that a 1% uncertainty in the capillary diameter, the capillary depression, or the density of the aluminum, causes a 1.0–1.1% uncertainty in the calculated interfacial tension. The 1% uncertainty in the density of the melt or in the temperature, leads to an uncertainty in the interfacial tension of less than 0.1%. For evaluation of the experimental data a computer program written in Visual Basic was developed. This program allowed evaluating concurrently ten experimental curves, which were taken in one measurement cycle. The program allows fully automatic curve evaluation; it finds inflex points on the curve, and calculates the interfacial tension with appropriate experimental constants. 6.4.1.2. Pin detachment method

The pin detachment method as described in Section 6.2.3.3. can be used advantageously in the measurement of interfacial tension also. Such a measurement was carried out, for example, by Zhanguo Fan and Østvold (1991) in the study of the interfacial tension between liquid aluminum and the melts of the system NaF–AlF3 –Al2 O3 at temperatures of 1000–1100◦ C. They found that the interfacial tension between pure cryolite and aluminum at 1000◦ C is (508 ± 1) N·m−1 . In the system NaF–AlF3 –Al2 O3 the effect of alumina addition depends on the cryolite ratio. At CR < 3, the interfacial tension decreases linearly with increasing temperature. On the other hand, at CR > 3.5 with increasing temperature, an increasing interfacial tension was found.

Chapter 7

Vapor Pressure The study of vapor pressure is an important part of investigation on molten salts. In connection with the mass spectroscopy method, which enables us to determine the composition of the gas phase, it is a useful tool to determine the activities of components and thus the composition of both the liquid and vapor phases. Formation of vapor includes all the processes in which gas is created from a system of condensed phases. Measurement of vapor pressure is closely connected to the determination of equilibrium between the gaseous and liquid or solid phases.

7.1. THERMODYNAMIC PRINCIPLES

Pressure is one of the basic variables in thermodynamics. It is the force exerted by the system on the unit area of its wall. In the SI system, the basic unit of pressure is Pascal (Pa). One Pascal is equal to the force of one Newton exerted on one square meter, Pa = N · m−2 . An older unit is 1 atm, which equals 101 325 Pa.

7.1.1. Gas mixtures

For ideal gases or for gases at low pressures, the following laws are valid: Boyle’s law (PV = constant), Gay-Lussac’s law (V/T = constant), and the Equation of State (PV = nRT). For homogeneous gas mixtures again the Equation of State is valid PV =

n · RT ,

n = n1 + n2 + · · ·

(7.1)

Hence P =

n3 RT ni RT n1 RT n2 RT + + ··· + = Pi + V V V V

(7.2)

The expression Pi = ni RT /V is the partial pressure of the component, i.e. the pressure, which the gas component i considered would have, when it would, at the given temperature, occupy the whole volume of the mixture. The total pressure of the gas mixture

313

314

Physico-chemical Analysis of Molten Electrolytes

is equal to the sum of the partial pressures of all the components. This statement is the Dalton’s law introduced in 1801. Pi ni RT /V ni = = = xi ⇒ Pi = xi · P ni RT /V ni P

(7.3)

Analogously, Amagat introduced the concept of “partial volume”, which is the volume, which the gaseous component would occupy, if it were alone at the given temperature and the total pressure of the system V =

n1 RT n2 RT n3 RT + + + ··· = Vi P P P

where Vi =

ni RT P

(7.4)

The volume of the gas mixture equals the sum of the partial volumes of the individual components – Amagat‘s law (1880). For the gas mixtures, the same Equations of State as for pure gases is valid. However, further variables, expressing the composition arise. 7.1.2. Liquid–gas equilibrium

In 1886, Raoult empirically determined that in some systems, the partial pressure of the ith component in the gas phase, Pi , equals the product of the mole fraction of this component in the gas phase, yi , and the total pressure P, and also simultaneously to the product of the mole fraction of this component in the liquid phase, xi , and the saturated gas pressure of the pure component i at the temperature considered, Pi0 , Pi = yi P = xi Pi0

(7.5)

The relation between the temperature and the equilibrium pressure in the onecomponent system and two phases is defined by the Clapeyron’s equation dP H = dT T V

(7.6)

where H and V are the enthalpy and volume change at the phase transition, respectively. If we consider the equilibrium between the liquid and gas phases, the Clapeyron equation can be modified by introducing two simplifying assumptions • •

the molar volume of the gas phase is much larger than that of the liquid phase (V = Vgas – Vliq = Vgas ), at low pressure, the gas phase behaves according to the Equation of State of the ideal gas (Vgas = RT /P).

Vapor Pressure

315

Introducing the above simplifying assumptions we get vap H d ln P = dT RT 2

(7.7)

where P is the saturated vapor pressure and vap H is the molar heat of vaporization. Equation (7.7) is the differential form of the Clausius–Clapeyron equation. It is the basis of all relations between the saturated vapor pressure and the temperature. For practical use, Eq. (7.7) has to be integrated. In the simplest case, we assume that the heat of vaporization does not depend on temperature. After integration we get ln P =

vap H +A RT

(7.8)

In a narrow temperature range, this assumption is fulfilled and therefore Eq. (7.8) describes well the dependence of the saturated vapor pressure on temperature. The heat of vaporization is the energy, which must be added to a certain amount of liquid to pass into the gas state. It can be determined either by direct measurement or by calculating from the dependence of the saturated gas pressure on temperature using the differential form of the Clausius–Clapeyron equation 2 Hvap = RTvap

7.2.

d ln P dT

(7.9)

EXPERIMENTAL METHODS

A number of experimental methods have been used for the measurement of vapor pressure at high temperatures. They have been reviewed by Margrave (1967) and later by Kubaschewski and Alcock (1979). However, for the pressure range 1–101.325 kPa and the temperatures around 1000◦ C, only a few methods are available. Static methods have certain disadvantages connected with the difficulty of finding suitable refractory materials for the high-temperature part of the apparatus. The boiling point as well as transpiration methods have been extensively used in molten salt research. An excellent characterization of these methods was given by Kvande (1983). 7.2.1. The boiling point method

The boiling point method is also often called the effusion method in the literature. There are two variations of the boiling point method. In the first, the temperature is increased slowly at constant pressure, and in the second, the external pressure is decreased slowly at constant temperature. In the isothermal method, difficulties associated with superheating

316

Physico-chemical Analysis of Molten Electrolytes

and accurate determination of the boiling temperature are avoided. Furthermore, pressure equalization occurs very rapidly, whereas thermal transport may be rather slow. Boiling usually implies vapor bubble formation in a liquid. Since the method is also applicable to solid samples, it may seem inappropriate to call it the boiling point method. However, in this context, the term “boiling” may be taken to mean the evolution of vapor from a condensed sample when its saturated vapor pressure is equal to the external pressure. The term “boiling point” then refers to the temperature at which the vapor pressure equals an arbitrarily chosen external pressure, and not only 101.325 kPa. 7.2.1.1. Theory of the boiling point method

A consistent theory of the boiling point method was developed by Motzfeldt et al. (1977). They studied the transport processes taking place in a capillary based on transport equations for a binary gas mixture with gradients both in composition and in total pressure (coupled diffusion and viscous flow). The equilibrium vapor pressure is obtained by computer fitting of the resulting theoretical equation to the experimentally observed dependence of the mass loss rate on the inert gas pressure. A short description of this theory is given in the following paragraphs. A schematic drawing of the effusion cell is shown in Figure 7.1. The cell opening shape is a capillary with radius r and length l. The cell is suspended inside the furnace at temperature T. The inert gas is denoted by subscript “1” and the vapor by subscript “2”, while subscript “f” denotes the furnace space and the subscript “i” denotes the interior of the cell. The symbol c denotes concentration, x molar fraction, and P pressure. The following assumptions are made. (a)

(b) (c)

(d)

The system is maintained at steady state, with a net transport n˙ 2 mol/s of vapor through the capillary, while the net transport of inert gas, n˙ 1 = 0. The flux J2 = n˙ 2 /π r 2 . At the exit end of the capillary, the vapor dissipates quickly to the colder parts of the furnace due to diffusion and convection, so that x1f = 1, x2f = 0,P1f = Pf ,P2f = 0. The capillary is narrow in order to suppress the diffusive flux, and a significant pressure drop may exist along the capillary (Pi > Pf ). The rate of mass transport depends on the rate of viscous flow caused by this pressure drop. The steady-state evaporation lowers the temperature of the sample under the observed furnace temperature T, which means that the vapor pressure P2i inside the cell may be lower than the equilibrium vapor pressure P20 at the temperature T.

Regarding assumption (c), Wagner (1943), on the other hand, assumed that the pressure remains uniform along the capillary. However, this assumption limits the applicability of his results to certain experimental conditions; therefore a new treatment seemed to be necessary.

Vapor Pressure

317

P1f = Pf P2f = 0

r

l

Pi = P1,i + P2,i

Sample

Figure 7.1. Schematic diagram of the suspended effusion cell inside the furnace.

The total fluxes J1 and J2 for the case of gradients both of composition and total pressure can be expressed by general equations J1 = −D12 c

dx1 + cx1 v dz

(7.10)

J2 = −D21 c

dx2 + cx2 v dz

(7.11)

where v is the mean molecular velocity in the z direction. In order to derive the correct expression for the velocity v, we will first consider equations for the special case where the total pressure is uniform. The original experimental work of Graham showed that the diffusive fluxes are inversely proportional to the square root of the molecular masses. According to Graham’s law of diffusion, the diffusivities are expressed by the equation: √ −J1d /J2d = D1 /D2 = m2 /m1 = γ , in which m1 and m2 are the molecular masses and

318

Physico-chemical Analysis of Molten Electrolytes

the symbol γ was introduced for brevity. Since the interdiffusion coefficients are equal, it can be written that D12 = D21 = D. In this case, the net transport may be expressed in terms of a mean molecular velocity in the z direction and the velocity vd may be expressed by the equation vd =

D(1 − γ ) dx1 γ + x1 (1 − γ ) dz

(7.12)

A pressure gradient causes viscous flow, which can be described in terms of its mean linear velocity vvisc . For a straight, cylindrical tube with radius r, this velocity is given by the equation vvisc = −

r2 dP × 8η dz

(7.13)

where η is the viscosity of the gas mixture. The fluxes caused by diffusion and by viscous flow are additive, thus the velocity in Eqs. (7.10) and (7.11) is a sum of the two separate contributions v = vd + vvisc

(7.14)

In the experimental arrangement shown in Figure 7.1 at steady state, the net flux is equal to the mean molecular velocity multiplied by the total concentration. According to the kinetic theory, the viscosity of a gas is independent of pressure, while it is expected to vary with the gas composition. Substitution of Eq. (7.14) into Eqs. (7.10) and (7.11), with vd from Eq. (7.12) yields the expression for the fluxes caused by gradients in both composition and total pressure n˙ 2 = C

where A =

P2i 1 − exp(−n˙ 2 /A)

2 − Pf2

− A ln[γ + (1 − γ ) exp(−n˙ 2 /A)]

(7.15)

πr 2 D πr4 and C = · RT l 16RT lη

A further effect is caused by the fact that evaporation requires heat and the necessary heat flux causes the sample temperature to be slightly lower than the constant furnace temperature T. Assuming that a heat transfer coefficient K is specific to the given experimental arrangement, then we have K(T − Ts ) = n˙ 2 vap H

(7.16)

Vapor Pressure

319

where vap H is the molar heat of vaporization of the sample. The lowered sample temperature causes a lowered vapor pressure in accordance with the Clausius–Clapeyron’s equation vap H P2i ln 0 = R P2

1 1 − T Ts

vap H ∼ (T − Ts ) =− RT 2

(7.17)

Combining these two equations yields P2i = P20 exp(−n˙ 2 /B)

(7.18)

where B = KRT 2 /vap H 2 is a constant for a given experimental arrangement at constant temperature. Introducing Eq. (7.18) in Eq. (7.15), we get a complete equation for the rate of mass transport 

P20 exp(−n˙ 2 /B) n˙ 2 = C  1 − exp(−n˙ 2 /A)

2

 − Pf2  − A ln[γ + (1 − γ ) exp(−n˙ 2 /A)]

(7.19)

An experiment using thermobalance yields a record of the cell mass with sample as a function of time. From the slope of this curve, the rate of mass loss is determined. The inert gas pressure in the furnace is read on a manometer, and is lowered stepwise in the course of the experiment. Thus a set of corresponding values for n˙ 2 and Pf , which may be considered as “knowns” in Eq. (7.15) is obtained. On the other hand, the parameters A, B, C, P02 , and γ are generally unknown, since the molecular mass of the vapor, which is included in the parameter γ , is not known a priori. The problem may be handled by means of a suitable, non-linear least-squares analysis computer program, which fits Eq. (7.16) for the observed set of data. The course of the experiment may be roughly divided into two parts. At the onset Pf > P20 and the mass transport occurs mainly by the diffusion of vapor through a higher pressure of inert gas. The rate of mass loss is low and is determined primarily by the value of the parameter A. In the later stage of the experiment Pf < P20 and the rate of mass loss is high. It is determined mainly either by resistance to viscous flow through the capillary, or by the rate of heat transfer to the sample, or by a combination of both effects. These two effects are connected with the values of the parameters C and B, respectively. In the case when heat transfer is rate-determining, it is assumed that the mass transport occurs by diffusion with no (or negligible) pressure drop along the capillary. Formally it

320

Physico-chemical Analysis of Molten Electrolytes

corresponds to zero viscosity, which means that the parameter C is infinite. For this case, the following equation is valid Pf = P20

exp(−n˙ 2 /B) 1 − exp(−n˙ 2 /A)

(7.20)

It should be stressed that the above-described approach is strictly valid at temperatures above 900◦ C, when the viscous flow represents the prevailing restraints to mass loss. This conclusion is reasonable, as the efficiency of heat transfer increases rapidly with an increasing temperature. The boiling point method connected with the evaluation of experimental results according to the method of Motzfeldt et al. (1977) was applied to the determination of vapor pressure of various molten salt systems at a temperature range of 600–1200◦ C. The most accurate results were, however, obtained by computer fitting of the complete Eq. (7.19) using a sophisticated computer program developed by Hertzberg (1983). The boiling point method using the apparatus described was applied to a number of cryolite-based melts, in the following studies • • • • • • • •

Kvande (1983) – the vapor pressure above the system Na3AlF6 (l)–Al2 O3 (s)–Al(l), Guzman et al. (1986) – the influence of different fluoride additions on the vapor pressure of molten cryolite, Kvande (1986) – the structure of alumina dissolved in cryolite melts, Zhou et al. (1992) – the vapor pressure at the complex formation in NaF–AlF3 and Na3AlF6 –MgF2 melts, Gilbert et al. (1995) – the acid–base properties of cryolite-based melts with CaF2 , MgF2 , and Al2 O3 additives, Gilbert et al. (1996) – the structure and thermodynamics of NaF–AlF3 melts with additions of CaF2 and MgF2 , Robert et al. (1997a) – the structure and thermodynamics of alkali fluoride–aluminum fluoride–alumina melts; vapor pressure, solubility, and Raman spectroscopic studies, Robert (1997b) – the structure and thermodynamics of potassium fluoride–aluminum fluoride melts; Raman spectroscopic and vapor pressure studies.

7.2.1.2. Experimental setup

The boiling point method originated from Ruff and Bergdahl (1919) and was further developed by Ruff (1929). The principle of the isothermal method is as follows. The system to be investigated is taken in a cell (crucible) with a narrow capillary opening in the lid. The cell is suspended from a balance into a furnace with an inert gas atmosphere at constant temperature. The initial pressure of the inert gas is higher than the equilibrium

Vapor Pressure

321

vapor pressure of the system at the furnace temperature. Vapor transport through the capillary opening takes place by gaseous diffusion only, and the rate of mass loss is low. The inert gas pressure is then lowered stepwise and hence the vapor transport through the opening and the rate of mass loss of the cell increases. The rate of mass loss is particularly marked when the inert gas pressure eventually becomes lower than the equilibrium vapor pressure, because of the onset of direct flow through the capillary opening. This effect may be used to determine the vapor pressure. Ruff and Bergdahl (1919) and Fischer et al. (1932) recorded the change in the mass of the cell at increasing temperature and at constant inert gas pressure. They observed that the dependence of mass of the cell on time or temperature did not show a sharp break at the equilibrium vapor pressure. The results were not substantially improved by plotting the derivative of mass with respect to time (i.e. the rate of mass loss). Fischer et al. (1932) used the variant of recording the changes of mass at a stepwise decreasing pressure and constant temperature. However, even in this variant, a sharp break of the rate of mass loss versus pressure was not observed. A detailed description of the apparatus working in the temperature range 700–1100◦ C and the experimental procedure has been given by Herstad and Motzfeldt (1966) and Kvande (1979). The main component of the equipment is a cold wall vacuum furnace with a vertically mounted graphite heating tube element. A cylindrical cell made of graphite or sintered alumina, with a threaded lid is used as a sample container. Vapor outlet from the cell was a capillary opening drilled through the lid, 0.5 mm in diameter and 10 mm in length. The graphite cell contains an inner glassy carbon crucible to reduce instabilities in the recorded mass due to wetting properties at the graphite–melt interface. The thermobalance was of standard analytical, knife-edge type, but rebuilt for continuous electronic recording. The output from the balance could be monitored on a chart recorder or a computer. The loading capacity of the balance was 160 g with a sensitivity better than ± 0.2 mg. Temperature was measured using a Pt–Pt10Rh thermocouple, calibrated to the melting point of silver. The hot junction was placed about 3 mm below the bottom of the cell. Temperature was kept constant within ± 0.5 K by means of an electronic controller connected to a separate thermocouple. In order to measure mass losses at different inert gas pressures at constant temperature, the device was equipped with direct reading Penning gauge to control and measure the inert gas pressure quickly and accurately (better than 10 Pa). About 12 g of salt was used in each boiling point experiment. The apparatus was evacuated by a rotary fore-pump and an oil-diffusion pump, enabling to reduce pressure to about 10−3 Pa. The inert gas pressure was measured by a U-tube mercury manometer and could be adjusted by use of a valve connected to the mechanical pump. The sample was kept overnight at 200◦ C before measuring the mass losses at different inert gas pressures at a constant temperature.

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Physico-chemical Analysis of Molten Electrolytes

In some experiments, corrections of the measured vapor pressure have to be done due to changes in composition during measurement. The correction method proposed by Knapstad et al. (1981), with eventual modifications should be used.

7.2.2. The transpiration method

The transpiration method is a simple and versatile method for vapor pressure measurement at high temperatures. An inert carrier gas is passed over the condensed substance in a constant temperature furnace zone. The flow rate of the carrier gas is constant and sufficiently small so that the carrier gas is saturated with vapor, which condenses at some point downstream. The mass of vapor transported by a known volume of carrier gas is determined. If the total vapor pressure is known, from the boiling point method, the results from the transpiration method may be used to calculate the average molar mass of the vapor.

7.2.2.1. Theoretical background

A theoretical analysis of the roles of diffusion and viscous flow in a transpiration experiment was given by Kvande and Wahlbeck (1976). In contrast to the theories already existing at that time, they were able to avoid the somewhat unrealistic assumption that no pressure gradient exists in the apparatus. The theory developed by Kvande and Wahlbeck (1976) gives a more general and accurate description of the transpiration experiment. A short description of the theory is given in the following paragraphs. Figure 7.2 shows a schematic diagram of the isothermal part of the apparatus, which consists of the saturation chamber and the entrance and exit capillaries. Z is the direction of the carrier gas and vapor flow and the subscript “e” denotes the entrance capillary. The following assumptions were made: (i) (ii) (iii) (iv)

The flow rate of the carrier gas is sufficiently slow, so that it is saturated with the vapor when it leaves the saturation chamber through the exit capillary, At the end of the exit capillary, the vapor quickly dissipates to colder parts of the furnace because of convection and thermal diffusion, The forced flow of carrier gas causes a pressure drop through the capillaries, Diffusion may occur through the exit and the entrance capillaries.

Assumption (iii) is important, which avoids uniform total pressure in the apparatus, assumed in the former works. Similarly as in the theoretical treatment of the boiling point method discussed in Section 7.2.1.1., the correct transport equations for a binary gaseous mixture with gradients in both composition and total pressure are also applied to the case of the transpiration

Vapor Pressure

323

Sample region (saturation chamber)

Carrier gas

Sample vapor at pressure P2i and composition x2i

Entrance capillary n1, e < 0 n2, e

Exit capillary n1 > 0 n2

Condensed sample

Z

Z' l

0

0

l

Figure 7.2. Schematic diagram of the isothermal part of the apparatus used for transpiration experiments.

method. The exact theoretical treatment has been presented in detail by Kvande and Wahlbeck (1976), however, only the final equations will be given here. The relation between the flow rates n1 of carrier gas and n2 of vapor, the total pressure Pf at the exit end of the capillary, and the equilibrium vapor pressure P2i can be written in the form n˙ 1 + n˙ 2 A 2 Pf2 + (7.21) + ln[1 − x2i (1 − γ )] P2i2 = x2i C C The molar fraction x2i is given by the equation (n˙ 1 + n˙ 2 ) n˙ 2 1 − exp − x2i = n˙ 1 + n˙ 2 A

(7.22)

For the explanation of parameters A, C, and γ , look into the theory of the boiling point method. The above equations consider diffusion and flow of vapor only through the exit capillary of the apparatus. The equations become very complex when the diffusion “upstream” through the entrance capillary is taken into account. Their application is not practical, as they contain too many unknown parameters. At moderate to high flow rates, the effect of vapor diffusion can be neglected. Thus A = 0 in Eq. (7.21), or more correctly, A becomes a very small number. Thus, Eq. (7.21) gives n˙ 2 P2i = n˙ 1 + n˙ 2

Pf2

n˙ 1 + n˙ 2 + C

1/2 (7.23)

324

Physico-chemical Analysis of Molten Electrolytes

In order to calculate the average molar mass of the vapor, which will be denoted by M2 , the simple expression n˙ 2 = m ˙ 2 /M2 is introduced. Here, m ˙ 2 is the measured mass of vapor carried away from the sample pre-unit time. If we now assume that Pf2

n˙ 1 + C

n˙ 2 C

(7.24)

i.e. after deleting the term n˙ 2 /C inside the parenthesis of Eq. (7.23), substituting Eq. (7.24) in Eq. (7.23) gives after some rearrangement m ˙2 M2 = n˙ 1

(Pf2 + n˙ 1 /C)1/2 − P2i P2i

(7.25)

The average molar mass M2 then can be calculated from the measured values of m ˙ 2, Pf , and n˙ 1 , when the vapor pressure P2i and the parameter C are known at a fixed temperature T. From the experimental measurements performed by Kvande (1979), typical values for the individual parameters A = 0.015 mol · h−1 , C = 5.6 × 107 mol · h−1 · Pa−2 , Pf = 99.325 MPa, and n˙ 1 = 0.1 mol · h−1 were obtained. While these values showed little variation, n˙ 2 varied from 6.10−5 to 0.015 mol · h−1 . In using the transpiration method for determining the average molar mass of the vapor, one should carry out the following: (i)

(ii) (iii)

To experimentally determine the vapor flow rate n˙ 2 versus carrier gas flow rate n˙ 1 plot in a transpiration experiment. This requires measurement of m ˙ 2 and Pf at varying n˙ 1 at fixed temperature and hence, at fixed P2i . This plot determines the nearly linear range from diffusion range at small n˙ 1 up to the desaturation range at very large n˙ 1 . The average molar mass M2 and the parameters A and C can be determined by fitting Eq. (7.21) to the set of m ˙ 2 , Pf , and n˙ 1 data by non-linear least-square analysis. To determine the values of M2 and C by fitting Eq. (7.25) to the data acquired above by use of a non-linear least-square analysis computer program. To restrict the measurements for determination of the average molar mass to n˙ 1 values in the nearly linear range. Then use Eq. (7.25) to calculate the average molar mass of the vapor at any other temperature from the measured data for m ˙ 2 , Pf , and n˙ 1 , and the known values of P2i and C. The value of the parameter C, determined by the above procedure, may be corrected for the change in temperature according to Eq. (7.25). The viscosity depends on T1/2 , and thus C is proportional to T−3/2 .

Vapor Pressure

325

Compared to the former theories, the present theory gives a more complicated equation for the rate of vapor transport. Furthermore, it requires the use of computer fitting of this equation to the experimental data. In spite of this disadvantage, this approach gives more general and accurate results. 7.2.2.2. Experimental setup

The detailed description of the apparatus and the experimental procedure were given by Kvande (1979). Figure 7.3 shows the main features of the transpiration apparatus. A standard Kanthal-wound furnace is located horizontally and can be moved back-andforth along a tube made of Inconel 600 (an alloy consisting of 76% Ni, 15% Cr, and 9% Fe, and not reacting significantly with fluoride vapors at about 1000◦ C). The tube has an outer diameter 30 mm, an inner diameter 24.7 mm, and a length 1 m. Kvande (1979) has taken special care to reduce the effect of diffusion. A long tube was made, consisting of an end-to-end connection of a stainless-steel tube and three tubes were made of gas-tight graphite. The whole assembly could then be slid back-and-forth inside the Inconel tube. A part of the first graphite tube was fitted tightly into the steel tube and formed a permanent construction. The two other graphite tubes were threaded, and by careful machining of the tubes and the threads, they could be screwed on to each other to make gas-tight connections. Inside the graphite tube, two narrow capillary openings with the radius 0.5 mm and the length 10 mm were formed. In the space between the capillaries the sample was placed, contained in a boat made of graphite, with the inner dimensions: length 73 mm, width 10 mm, and height 9 mm. The carrier gas was passed through the steel tube and the three graphite tubes. Thus, the gas was forced through the two capillaries, into and out of the region, where the boat

1 2

3

4

5

6

7

8

5

9 10

11

Figure 7.3. Main features of the apparatus used for transpiration experiments. 1 – thermocouple, 2 – gas entrance, 3 – graphite tube, 4 – furnace, 5 – capillary openings, 6 – graphite shields, 7 – Kanthal heating, 8 – graphite boat, 9 – cold-finger, 10 – Inconel tube, 11 – gas exit.

326

Physico-chemical Analysis of Molten Electrolytes

was placed. The speed of the carrier gas, which was expected to increase significantly when passing through the narrow entrance capillary, was reduced by a graphite shield, positioned between the entrance capillary and the boat. A water-cooled cold-finger, made of copper, with a protective graphite cylinder on the outer side, was used to condense the vapor. The temperature was measured by a Pt–Pt10Rh thermocouple, calibrated against the melting point of silver. The thermo-voltage was indicated by a digital voltmeter. The temperature variation over the length of the boat was less than ±1◦ C. During the measurements, the temperature was kept constant within ± 0.5◦ C by the use of Eurotherm PID-controller, Model 070. The carrier gas used was argon or nitrogen. The flow rate was adjusted by a pressure reduction valve and kept constant, as indicated by a rotameter. The carrier gas was then passed through adsorption towers to remove H2 O and CO2 . The volume of the carrier gas was measured by a gas-flow meter. The reproducibility in the measured gas volume was of the order of 0.1%. The boat with a fresh sample was placed in the graphite tube between the two capillaries. The steel–graphite-tube assembly was positioned such that the boat came as close as possible to the entrance end of the Inconel tube, while the furnace was located at the opposite end of the Inconel tube. Carrier gas was then passed through the tube to remove all air. After ten to fifteen minutes the gas flow was stopped, and the steel–graphite-tube assembly and the furnace were pushed towards each other into their correct positions. When the sample reached the selected temperature, the experiment was started by passing carrier gas over the sample at constant flow rate. This temperature, being lower than the desired temperature, was found from accurately determined temperature–time curves. During the measurements, the temperature was read every five minutes. After exactly one hour, the gas was stopped. The steel–graphite-tube assembly was pulled quickly towards the entrance end of the Inconel tube, while the furnace was pushed towards the opposite end. Cooling of the sample was pursued by use of a fan. In less than one minute, the temperature was reduced below 800◦ C and then the mass loss was negligible. When the temperature of the boat had reached 25◦ C, it was removed and weighed, and the amount of vapor transport was determined from the mass loss of the sample during the measurement. The total gas volume, as read on the gas-flow meter, was reduced to standard conditions (0◦ C and 101.325 kPa) by use of the values for the ambient temperature and pressure. These values were read half-an-hour after the beginning of the experiment.

Chapter 8

Electrical Conductivity Electrical conductivity of molten salts is of considerable interest both from practical as well as theoretical points of view. By means of conductivity data, conclusions on the structure and transport theories of molten salts may be tested. Furthermore, the current and energy efficiencies of electrolytic processes are closely related to electrical conductivity of the electrolyte. Electrical conductivity of molten alkali and earth alkali metal halides increase by 2–4 orders at melting. The electrical conductivity of these melts is purely ionic and their electrolysis follows Faraday’s law. Deviations from this law are caused by secondary processes during the electrolysis, as for example dissolution or the back reaction of the electrolysis products. Electrical conductivity and thus also the mobility of ions is, in general, given by quantities like ionic charge, ionic mass, radius, polarizability, and the coordination number. The values of electrical conductivity of inorganic ionic melts range from 0.1 to 10 S · cm−1 , while the respective values for metals are around 105 S · cm−1 and the electrical conductivity of water at ambient temperature is 2.10−6 S · cm−1 . Conductivity of melts increases with increasing temperature as the process of charge transport in the direction of the electrical field gradient requires certain activation energy. Activation energy of electrical conductivity depends on temperature. As in the case of diffusion or viscous flows, the activation energy of electrical conductivity consists of two parts: of activation energy of the ionic motion and of energy connected with changes of the melt ordering with changes in temperature. In a number of articles, the Nernst–Einstein equation was used to correlate values of the electrical conductivity and the diffusion coefficients. The application of this equation to molten salts did not bring expected results as the electrical conductivity values calculated from the diffusion data are always higher than the experimental conductivity data. As a main reason for this difference, the presence of cavities in the melt is mentioned, which are sufficiently large so that both the cations and the anions could be placed in them. Such cavities are regarded as pair vacancies. If a pair jump of both kinds of ions using the pair vacancy takes place, both the atoms participate in the mass transfer and thus in the diffusion process, but not in the charge transfer.

327

328

Physico-chemical Analysis of Molten Electrolytes

8.1. THEORETICAL BACKGROUND

Conductivity of electrolytes is a scalar quantity in the relation between the current density j and the gradient of the electrical field ϕ j = κ · gradϕ

(8.1)

Conductivity is defined by the equation κ=

κi = F

i

zi u i c i

(8.2)

i

where F is the Faraday’s constant, ci is the molar concentration of ions with the charge zi and the mobility ui . The conductivity unit in the SI system is Sm−1 . The product of conductivity and molar volume, κ · Vm , or the ratio of conductivity and molar concentration, κ/c, is the molar conductivity of the electrolyte, λ. Its unit in the SI system is S · m2 mol−1 . When the molar conductivity is related to the unity of the amount of positive or negative charge we get equivalent conductivity. This enables direct comparison of the mobility of ions of different valence. The equivalent conductivity is the ratio κ/cz. The participation of cations and anions in charge transfer represents quantities which are called transference numbers. It is a part of conductivity of the given kind of ion in the total conductivity. The transference number of the ion i is thus defined by the relations κi i ui Di ti = = = = κi i ui Di

(8.3)

where Di is the diffusion coefficient of the ion i. Obviously it holds that ti = 1. The shape of the temperature dependence of the electrical conductivity depends on the system to be investigated. In general, it can be expressed in two ways. Either it is the polynomial equation κ = a + bt + ct 2 + · · ·

(8.4)

or it can also be described by the Arrhenius-type equation κ = A exp

Econd RT

(8.5)

Electrical Conductivity

329

where Econd is the activation energy of the electrical conductivity. A similar equation can also be written for the temperature dependence of the molar conductivity λ = A exp

Econd RT

(8.6)

In systems without any electronic conductivity, the conductivity versus temperature curve is concave to the x axis. This is due to the continuity of the liquid and gaseous state, where in the liquid state, the amount of non-dissociated molecules increases. When electrons also participate in the charge transfer, the dependence of conductivity on temperature is expressed by Eq. (8.4) as well, but the conductivity versus temperature curve is convex to the x axis. This difference in conductivity versus temperature curve is due to the equality of chemical potentials of the liquid and its vapor in which only non-dissociated molecules are present. In pure ionic liquids at increasing temperature, the amount of non-dissociated no charge-transferring molecules increases. When electronic conductivity is also present in the melt, caused by the presence of elements in different valence states (e.g. Na(0)/Na(I), Al(0)/Na(I), Fe(III)/Fe(II), Nb(V)/Nb(IV) etc.) in the melt enabling the jumping of electrons from the atoms in the lower oxidation state to the higher, with increasing temperature, the amount of delocalized electrons and thus the part of the electronic conductivity in the total, increases. 8.1.1. Electrical conductivity of “ideal” and real solutions

The study of electrical conductivity of molten salts is one of the indirect methods used for the determination of molten salts’ structure and of component interaction in molten mixtures. The change in composition of a molten mixture is often accompanied by structural changes, which affect the dependence character of the electrical conductivity on composition. Consequently, an analysis of this dependence should provide some information regarding the present ionic species and their arrangement in the melt. Supplementary information, i.e. concerning the formation and decomposition of complex ions, the character of the cation–anion bond, and the character of conductivity, cationic, anionic, electronic, etc., can be obtained from analysis of the dependence of the activation energy on composition. Two factors principally determine the shape of isotherms of the electrical conductivity of binary mixtures: (i) (ii)

the formation of new chemical compounds and consequently the formation of new complex ions, the values of electrical conductivity of the components and the newly formed compounds.

330

Physico-chemical Analysis of Molten Electrolytes

In order to get some information on the possible structure of the given molten system from the conductivity measurement, a suitable reference state should be defined. Since conductivity is a scalar quantity, no ideal behavior is given by definition. However, there were several attempts in the literature to present a model of electrical conductivity of molten salts, which would describe satisfactorily the course of the conductivity dependence on composition. 8.1.1.1. Model of Markov and Shumina

Theoretical interpretation of the concentration dependence of equivalent conductivity for simple binary mixtures was first presented by Markov and Shumina (1956). It should be emphasized that this theory, even when considering simple structural aspects, represents rather a method of interpretation of the experimental data than a genuine picture of the structure of the melt. In molten salts generally only ions and not molecules are present, hence the conception of Markov and Shumina (1956) is to be considered also from this aspect. Their theory is based on the assumption that the electrical conductivity of a mixture of molten salts varies with temperature like pure components. In this respect, general character of the electrical conductivity dependence on composition, indicating the interaction of components in an ideal solution, could be expected. In a mixture of univalent salts of the type AX–BX, the following interactions should be present: AA, BB, AXB, and BXA. The last two interactions are equal, thus they can be written as 2AXB. Considering that the probability of the interactions mentioned is proportional to their molar fractions, Markov and Shumina derived a relation for the composition dependence of the equivalent conductivity in a mixture of molten salts in the form λmix = x12 λ1 + x22 λ2 + 2x1 x2 λ2

(8.7)

where λi is the equivalent conductivity and xi is the molar fraction, with the presumption that λ1 < λ2 . Thus for the calculation of equivalent conductivity of a molten salt binary mixture, according to Eq. (8.7), only equivalent conductivities of components at the same temperature are to be known. 8.1.1.2. Model of Kvist

Kvist (1966) in the study of electrical conductivity of the system Li2 SO4 –K2 SO4 , proposed that the cations in molten lithium sulfate move in groups, each group containing k cations, and when x mole fractions of foreign univalent cations are added, it is possible to calculate k from the relation k=

1 λ u x λ u

(8.8)

Electrical Conductivity

331

where u/u is the relative mobility difference between the lithium cation and the foreign cation and λ/λ is the relative difference in equivalent conductivity between pure lithium sulfate and the mixture. For the Li2 SO4 –K2 SO4 system, Kvist (1966) has found that k = 2.7. Equation (8.8) is, however, only valid in dilute mixtures and it is a question whether a similar relation would be valid at all concentrations. To expand the model to the whole concentration region, Kvist (1967) chose the system Li2 SO4 –Ag2 SO4 , since there is a favorable situation as the anion is large compared with the cations and the mass of the silver ion is much larger than that of the lithium. For the model, the following assumptions were made: • • • •

cations are moving in groups, each group containing the same number k of cations, there are groups containing silver and lithium cations and groups containing solely lithium cations, groups containing silver cations have the same mobility uAg , regardless of the composition of the group, mobility u1 is a property of groups containing only lithium ions.

Due to the large difference in the mass of cations, the mobility of groups containing silver and lithium cations is given by the mobility of silver cations, regardless of the number of silver cations present. In the original description of the model, randomness of the distribution of cations was postulated. However, this assumption is in contrast with the formation of cationic groups and with the fact that there are two kinds of groups: one that contains both kinds of cations and the other that are formed solely of one kind of cations. This postulate was thus omitted. In one mole of mixture composed of x mole Li2 SO4 and (1 – x) mol Ag2 SO4 , the probability of a group containing k lithium cations will be x k , and therefore

λ/F = x k u1 + 1 − x k uAg

(8.9)

If in the model the following assumption is made that at all concentrations u1 = λLi2 SO4 /F

and

uAg = λAg2 SO4 /F

(8.10)

one gets from Eq. (8.9) that

λ = x k λLi2 SO4 + 1 − x k λAg2 SO4

(8.11)

The model can be tested by measuring λ as a function of x and looking for parameter k such that Eq. (8.11) reproduces the experimental curve. Kvist (1967) verified the model measuring electrical conductivity of the system Li2 SO4 –Ag2 SO4 . He found that the molar conductivity of this system can be described

332

Physico-chemical Analysis of Molten Electrolytes

well with k = 4.3. However, the model cannot explain the minimum of the molar conductivity in the system Li2 SO4 –K2 SO4 measured by Kvist and Lundén (1965). It should be noted that the model of Kvist is a generalized version of the model of Markov and Shumina (1956), since for k = 2, Eq. (8.11) transforms into Eq. (8.7). 8.1.1.3. Series and parallel models of electrical conductivity

A new model for electrical conductivity of molten salt mixtures was introduced by Fellner (1984). This model should also define the ideal conductance of molten mixtures. Consider a mixture of molten salts composed of nA moles of component A and nB moles of component B. For the volumes of these components, it holds VA =

nA MA = nA VA0 , ρA

VB =

nB MB = nB VB0 ρB

(8.12)

where MA and MB are the molar masses of compounds A and B, respectively, ρA , ρB are the densities of pure components, and VA0 , VB0 are the molar volumes of pure components. If these melts are placed in two cuboid conductivity cells with a base S and length lA and lB , the electrical resistances of corresponding cells are given by the relations RA,s =

lA , κA S

RB,s =

lB κB S

(8.13)

where κA and κB are the conductivities of pure components A and B. Let us first consider the series model (Figure 8.1). According to the proposed series model, it is assumed that the final resistance of the whole system after mixing both melts, Rmix, s , equals the sum of resistances RA, s and RB, s Rmix, s = RA, s + RB, s

(8.14)

Vmix VA VB = + 2 2 κmix, s S κA S κB S 2

(8.15)

where κmix, s is the conductivity of the molten mixture A + B (index “s” denotes the series model). Since we deal with an ideal model, we can assume that this approximation is also valid for the volume of molten mixture. Then we readily obtain the final relation xA VA0 + xB VB0 xA VA0 xB VB0 = + κmix, s κA κB

(8.16)

Now we shall consider the parallel connection of both conductivity cells. Let us assume that the base of the cell is a square. Then for the resistance of each compartment, it holds

Electrical Conductivity

(a)

333

(b)

A

A

B

B

Figure 8.1. Series (a) and parallel (b) model of the electrical conductivity of molten salt mixture.

(index “p” denotes the parallel model) RA, p =

S , κA V A

RB, p =

S κ B VB

(8.17)

The conductance of the system equals the sum of the reciprocal values of resistances of both the cells. Thus it follows 1 1 1 = + Rmix, p RA, p RB, p

(8.18)

κmix, p Vmix κA VA κ B VB = + S S S

(8.19)

Introducing the molar fractions into Eq. (8.19), we obtain (we again assume the ideal behavior of the mixture)

κmix, p xA VA0 + xB VB0 = xA VA0 κA + xB VB0 κB

(8.20)

where κmix, p is the calculated conductivity according to the parallel model. Since κA VA0 and κB VB0 are the molar conductivities of the pure components, it readily follows from Eq. (8.20) that the parallel model leads to the conception of additive behavior of molar conductivity of the molten salt systems. Fellner and Chrenková (1987) tested the proposed models in a number of binary and ternary systems. The authors found that the series model

334

Physico-chemical Analysis of Molten Electrolytes

describes well the conductivity of the systems, which are ideal from the thermodynamical point of view, while the parallel model fails. The model of Markov and Shumina (1956) gives rather similar results as the series model. 8.1.1.4. Dissociation model of electrical conductivity of molten salt mixtures

The model of electrical conductivity of molten salt mixtures based on incomplete electrolytic dissociation of components was proposed by Daneˇ k (1989). The dissociation degree of a component is affected by the presence of second component. Consequently, the dissociation degree of both components in the system is not constant, but changes with composition, affecting the concentration of the conducting particles in the electrolyte. This effect is caused by interactions of components, given by the nature of the repulsive forces between ions, determining their actual coordination sphere. Let us consider a binary system with a common anion of the type AX–BX. Let us further assume that each component in the molten mixture is not fully dissociated and that the equilibrium between the ionic pairs A+ · X− and B+ · X− and the “free” ions A+ , B+ , and X− constitutes in the melt A+ · X− ↔ A+ + X−

(8.21)

B+ · X− ↔ B+ + X−

(8.22)

If we denote the dissociation degrees of the components in the mixture by α1 and α2 , and their molar fractions by x1 and x2 , then in one mole of an arbitrary mixture the following amounts of particles are present nA+ = x1 α1 ;

nB+ = x2 α2

nX− = x1 α1 + x2 α2 nA+ ·X− = x1 (1 − α1 );

(8.23) nB+ ·X− = x2 (1 − α2 )

The total amount of all particles is then n = 1 + x1 α1 + x2 α2 . For the equilibrium molar fractions of individual particles we get xA + =

xA + · X − =

x 1 α1 ; 1 + x 1 α1 + x 2 α 2

x1 (1 − α1 ) ; 1 + x 1 α1 + x 2 α2

xB+ =

x 2 α2 ; 1 + x 1 α1 + x 2 α2

xX − =

x 1 α 1 + x 2 α2 1 + x 1 α 1 + x 2 α2

xB+ ·X− =

x2 (1 − α2 ) 1 + x 1 α 1 + x 2 α2

(8.24)

Electrical Conductivity

335

The equilibrium constants of the dissociation reactions (8.21) and (8.22) are given by the relations K1 =

K2 =

2 α01 2 1 − α01 2 α02 2 1 − α02

=

α1 (x1 α1 + x2 α2 ) (1 − α1 )(1 + x1 α1 + x2 α2 )

(8.25)

=

α2 (x1 α1 + x2 α2 ) (1 − α2 )(1 + x1 α1 + x2 α2 )

(8.26)

where α01 and α02 are the dissociation degrees of the pure components AX and BX at the given temperature. Rearranging Eqs. (8.25) and (8.26) we get for α1 and α2 in an arbitrary mixture the following relations

2 2 x1 α12 + x2 α1 α2 + α01 (1 + x2 α2 ) = 0 − α01

(8.27)

2 2 (1 + x1 α1 ) = 0 − α02 x2 α22 + x1 α2 α1 + α02

(8.28)

These implicit equations for the concentration dependence of α1 and α2 can be solved analytically. Separating α2 in Eq. (8.27) α2 =

2 − x α2 − x α α2 α01 1 2 1 01 1

2 x2 α1 − α01

(8.29)

and inserting it into Eq. (8.28) we get for α1 the cubic equation % 2

4

& 2 2 2 2 2 α13 x1 x2 α01 + α12 x2 x1 α01 + x2 α01 − α02 α02 − α01 − α02

+ α1 x2 (1 + x2 )

2 2 α01 α02

4 − α01

4 + α01

2 1 − α02

(8.30)

x2 = 0

Equation (8.30) can be solved either analytically or preferably using the Newton– Raphson’s method. As starting values for α1 and α2 it is advantageous to choose the values of the dissociation degrees of pure components α01 and α02 . So it is possible to calculate the values of α1 and α2 for an arbitrary composition of the mixture and for arbitrary values of the dissociation degrees of the pure components. For the conductivity of the electrolyte the general Eq. (8.2) is valid. For uni-univalent electrolytes zi = 1 and F · ui = λi , and Eq. (8.2) attains the form κ=

i

ci λ i =

ni i

V

λi

(8.31)

where ni are the amounts of conducting particles in the mixture and V is the volume of the mixture. In the case of molten electrolytes, it may be assumed that the ionic pairs

336

Physico-chemical Analysis of Molten Electrolytes

A+ · X− and B+ · X− , present in the mixture, are electro-neutral and do not contribute to the conductivity of the electrolyte. The whole charge is transported by “free” ions, i.e. by cations A+ and B+ , and anions X− . In such a case, Eq. (8.31) can be written in the form κ=

nA+ n + n − λ + + B λB+ + X λX− V A V V

(8.32)

If we consider one mole of mixture, then according to Eq. (8.25) nA+ = x1 α1 , nB+ = x2 α2 , and nX− = x1 α1 + x2 α2 . For the molar conductivity of the mixture we get λ = κV = x1 α1 λA+ + x2 α2 λB+ + (x1 α1 + x2 α2 )λX−

(8.33)

and after rearranging

λ = x1 α1 λA+ + λX− + x2 α2 λB+ + λX−

(8.34)

As the molar conductivities of the individual ions are not known a priori, their sum can be expressed on the basis of limiting conditions λA+ + λX− =

λ1 ; α01

λB+ + λX− =

λ2 α02

(8.35)

and for the molar conductivity of the molten mixture we get the final expression λ = x1

α1 α2 λ1 + x 2 λ2 α01 α02

(8.36)

From Eq. (8.36) it follows that when the dissociation degree does not change with composition, i.e. α1 = α01 and α2 = α02 , Eq. (8.36) is identical with the parallel model proposed by Fellner (1984). Thus the expression xi (αi /α0 i ) can be considered as the activity of component in the mixture. The calculation procedure is as follows: For the chosen values of the dissociation degrees of the pure components α01 and α02 , the values of α1 and α2 are calculated according to Eqs. (8.29) and (8.30) for each composition of the mixture with a known value of the molar conductivity, λi, exp . The theoretical value of the molar conductivity, λi, calc is calculated according to Eq. (8.36), in such a way for each couple of α01 and α02 that a set of theoretical values of molar conductivities for the given composition is obtained. The criterion for selection of the right values of α01 and α02 is given by the relation n

(λi, exp − λi, calc )2 = min

(8.37)

i=1

Chrenková and Daneˇ k (1990) applied the dissociation model of electrical conductivity in a number of binary univalent systems of alkali metal halides, alkali metal nitrates,

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337

Table 8.1. Calculated values of the dissociation degrees of pure components in binary systems with a common anion System

T (K)

α01

α02

LiF–NaF LiF–KF NaF–KF LiCl–NaCl LiCl–KCl LiCl–CsCl NaCl–KCl NaCl–RbCl NaCl–CsCl KCl–RbCl RbCl–CsCl LiI–NaI LiI–KI LiI–RbI LiI–CsI NaI–KI LiNO3 –NaNO3 LiNO3 –KNO3 NaNO3 –KNO3

1200 1200 1200 1100 1100 1100 1100 1100 1100 1100 1100 1000 1000 1000 1000 1000 640 640 640

0.21 0.17 0.47 0.28 0.19 0.24 0.42 0.19 0.19 0.37 0.41 0.23 0.21 0.19 0.13 0.41 0.42 0.43 0.53

0.42 0.91 0.80 0.52 0.83 0.95 0.81 0.50 0.63 0.57 0.58 0.45 0.56 0.84 0.74 0.74 0.52 0.85 0.81

0.052 0.116 0.065 0.050 0.104 0.158 0.054 0.075 0.109 0.022 0.034 0.044 0.090 0.109 0.139 0.047 0.044 0.092 0.048

and silver halides with a common anion as well as with a common cation. The required experimental data on the molar conductivity were excerpted from Janz et al. (1972, 1975, 1977, 1979), Janz and Tombins (1981) and Smirnov et al. (1971, 1973 a,b). The calculated values of the dissociation degrees of pure components of binary systems with a common anion are given in Table 8.1 and those for binary systems with a common cation are given in Table 8.2. In both the tables, the values of the geometrical parameter , introduced by Tobolsky (1942), are also given =

d 1 − d2 d1 + d 2

(8.38)

where d1 and d2 are the interatomic distances (the sum of the ionic radii of the cation and anion) of the individual components. The values of the ionic radii were taken from Waddington (1966) and Kleppa and Hersh (1961). As follows from Table 8.1, in systems with a common anion, always the component with a larger cation, i.e. with a lower field strength (the charge to radius ratio) has a higher dissociation degree. It means that more the electronegative cations bind the surrounding anions stronger, they possess a larger tendency to form ionic pairs or associates. In systems with a common cation (c.f. Table 8.2) always the component with the lower polarizability of anion, i.e. the component with the smaller anion, has a higher dissociation degree. It is

338

Physico-chemical Analysis of Molten Electrolytes Table 8.2. Calculated values of the dissociation degrees of pure components in binary systems with a common cation System

T (K)

α01

α02

LiF–LiCl LiF–LiBr LiF–LiI LiCl–LiBr LiCl–LiI LiBr–LiI KF–KBr KF–KI CsF–CsCl CsF–CsBr CsF–CsI AgCl–AgBr AgCl–AgI AgBr–AgI

1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 850 850 850

0.73 0.99 0.999 0.67 0.705 0.79 0.99 0.99 0.81 1.00 0.95 0.54 0.50 0.455

0.24 0.19 0.155 0.44 0.33 0.59 0.46 0.325 0.35 0.28 0.21 0.40 0.21 0.21

0.099 0.130 0.174 0.030 0.076 0.046 0.111 0.148 0.077 0.099 0.133 0.025 0.064 0.039

due to the fact that the entry of a strange ion into the pure molten component causes a nonrandom mixing and thus the lowering of the configurational entropy of the melt. The entry of a strange ion causes a change in the coulombic, polarization, and the dispersion energy of the system resulting from the substitution of ions of different size in the coordination sphere of ions and distortion of the symmetry of the electrical field of ions. As has been shown by Førland (1955), Lumsden (1961), and Blander (1962) on the basis of a simple geometrical model, and by Reiss et al. (1962) and Davis and Rice (1964) on the basis of the conformal solution theory, the measure of the energetical change is the dimensionless parameter (c.f. Eq. (8.38)), which represents the fractional difference in the ionic radii of equally charged ions. On the basis of the theories mentioned, negative values of the mixing enthalpy may be expected, the magnitude of which increases with increasing difference in ionic radii as follows mix H ≈ Ex1 x2 (U0 + U1 + U2 2 )

(8.39)

In Eq. (8.39) U0 is the change in the dispersion energy resulting from change in the induced dipoles of the second coordination sphere of cations, U1 is the change in the polarization energy caused by asymmetry of the electrical field of anions owing to different electro-negativity of cations with different radius, and U2 is the change in the coulombic energy resulting from a different cation–anion distance of both the components. It is evident that these effects affect also the concentration of the “free” ions in the mixture, which is determined by the values of α01 and α02 . Therefore a linear dependence of the difference of both the dissociation degrees on the geometrical parameter can be expected. In Figure 8.2, this dependence for all the systems studied is shown. Using the

Electrical Conductivity

339

1.0

0.8

α01− α02

0.6

0.4

0.2

0.0 0.00

0.04

0.08

d1− d2

0.12

0.16

0.20

d1+ d2 Figure 8.2. Dependence of the dissociation degree difference of components (α01 − α02 ) on parameter . ◦ – systems with a common anion; – systems with a common cation.

regression analysis, for this dependence the equation α01 − α02 = 5.095

(8.40)

with the standard deviation sd = 8.5×10−2 was obtained. A very good correlation of both the quantities, obvious from Figure 8.2, refers to the prevailing influence of change in the polarization energy on dissociation of both components at mixing. The change in the coulombic energy is not as important, as the regression analysis of the α01 −α02 = f (2 ) plot yields a lower value of the correlation coefficient. In spite of a good correlation of the dependence shown in Figure 8.2, the dispersion in the α01 − α02 values might be due to several factors: (i) (ii) (iii) (iv) (v)

precision and correctness of the molar conductivity values of the mixtures, uncertainty in the ionic radii values, negligence of the coulombic energy change between more distant ions, simplification of the three-dimensional to the linear interaction, (m−n) neglecting the dissociation of more complex clusters of the Am Xn type.

340

Physico-chemical Analysis of Molten Electrolytes

It is thus obvious that for deeper theoretical analysis of the mutual interaction of components, more precise and more intact experimental data on electrical conductivity of the molten salt systems are needed. The above-mentioned factors most probably also cause differences between dissociation degree values of pure components in dependence on the presence of the second component. From the results of the calculation, it further follows that there is a very close coherence between the course of the dependence of the molar conductivity and the enthalpy of mixing on composition. This follows also from the linear dependence of the difference of the dissociation degrees of pure components (α01 − α02 ) on the enthalpy of mixing at x1 = x2 = 0.5, which for some systems, is shown in Figure 8.3. The values of the enthalpy of mixing were taken from Kleppa and Hersh (1961), Lumsden (1964), and Melnichak and Kleppa (1970). The dissociation model of electrical conductivity of molten salt mixtures was further applied • •

in ternary univalent systems with a common ion by Daneˇ k et al. (1990), in binary reciprocal univalent systems by Daneˇ k et al. (1991), 1.0

0.8

α01 − α02

0.6

0.4

0.2

0.0

1

0

−1

−2

−3

−4

−5

−6

∆Hmix (kJ mol−1) Figure 8.3. Dependence of the difference of dissociation degree of components (α01 − α02 ) on enthalpy of mixing at x1 = x2 = 0.5 for some systems with a common ion.

Electrical Conductivity

• • •

341

in systems containing a bivalent cation by Chrenková et al. (1991a), in systems containing a trivalent cation by Daneˇ k and Chrenková (1991), in the KF–KCl–KBF4 system by Chrenková et al. (1991b).

It was found that in ternary univalent systems with a common anion, the dissociation degree increases with the increasing radius of the cation, while in systems with a common cation, the dissociation degree increases with decreasing radius of the anion. In binary reciprocal univalent systems, the most dissociated is always the stable pair of salts. However, the correctness of the calculation was conditioned by the consistency and correctness of the molar conductivity values and especially by the correctness of the value of the standard Gibbs energy of the metathetical reaction. In chloride systems containing a bivalent cation, the alkali metal chlorides MCl–MeCl2 (M = Li, Na, K, Rb, Cs; Me = Mg, Ca, Ba, Cd, Pb) are almost completely dissociated, while the dissociation degrees of chlorides of bivalent metals are substantially lower. The difference in dissociation degrees of pure components is directly proportional to the dimension of the alkali metal cation. This indicates that higher the electro-negativity of the alkali metal cation present, the higher is the tendency of the bivalent cation to form complex anions. Finally, the dissociation model of electrical conductivity was also successfully used in pseudobinary systems containing a trivalent cation MX–AlX3 (M = Li, Na, K; X = F, Cl) and Na3AlF6 –NaCl by Daneˇ k and Chrenková (1991). However, the real equilibrium composition should be considered. In such a way also, the relatively rare positive deviations from additivity of the molar conductivity may be rationally explained. Olteanu and Pavel (1995) presented theoretical premises of the dissociation model for electrical conductivity in molten salt mixtures. The authors gave a versatile numerical method together with its corresponding computing procedure and provided an easier and more precise way of calculation. Eliminating the sum (x1 α1 + x2 α2 ) from Eqs. (8.25) and (8.26), one obtains a relation between α1 and α2 independent of molar fractions x1 and x2 2 ) 2 ) α02 (1 − α01 α01 (1 − α02 1 1 + =1 2 2 2 2 (α01 − α02 ) α1 (α02 − α01 ) α2

(8.41)

from which α 2 can be expressed explicitly

α2 =

2 (1 − α 2 ) α1 α02 01 2 (1 − α 2 ) − (α 2 − α 2 ) α01 02 01 02

(8.42)

342

Physico-chemical Analysis of Molten Electrolytes

Inserting Eq. (8.42) into Eq. (8.25), a cubic equation for α 1 is obtained 2 2 2 2 2 4 2 − α02 ) + α12 x1 (α01 α02 − α01 ) + x2 (α01 − α02 ) α13 x1 (α01 (8.43) 2 2 α02 + α1 (1 + x2 )(α01

4 4 2 − α01 ) + α01 (1 − α02 )

=0

The key problem of the dissociation model is then the correct evaluation of α01 and α02 . In order to solve this problem, the Nelder–Mead (1964) numerical minimization algorithm was used. This algorithm represents an extension of the simplex method of Spendley et al. (1962). The analysis of the validity of the procedure was made using the data of Olteanu and Pavel (1995) for electrical conductivities and molar volumes. In two further articles, Olteanu and Pavel (1996, 1997) combined the dissociation model of Daneˇ k (1989) with the series and parallel models of Fellner (1984) and proposed sophisticated series-dissociation, parallel-dissociation, and series–parallel-dissociation hybrids. It should be pointed out that the small improvement in the fit with experimental data is attained by introducing further variables. The standard deviation of the fit, however, should always be compared with the experimental error of measurement. In the last article, Olteanu and Pavel (1999) partially eliminated the main drawback of the theoretical models of Daneˇ k (1989) and Olteanu and Pavel (1995, 1996, 1997), i.e. the varying values of the dissociation degree of the same pure salt depending on the nature of the second component. In the new approach Olteanu and Pavel (1999) proposed a model, in which the equilibrium constants of the dissociation processes were set equal to the probabilities of the two dependent processes in order to describe more successfully the incomplete dissociation and its effect on the electrical conductivity of the mixture. One of the versions of this model leads to almost the same value for the dissociation degree of a component regardless of the nature of the second value. 8.1.1.5. Association model in molten salts and mobility isotherms

A similar idea on the incomplete dissociation of molten salts was presented by Klemm and Schäfer (1996). Their model was stimulated by a qualitative explanation of the Chemla effect made by Klemm (1984), which is the crossing over of the mobility isotherms at a certain temperature. This means that at high concentrations of a larger sized cation, its mobility is greater than that of the smaller one. This effect was first observed in the LiBr–KBr system by Mehta et al. (1969), was also observed later in many other monovalent systems, and named after one of its discoverers. In molten lithium bromide, although it has a larger conductivity than potassium bromide, more ions are associated to neutral LiBr molecules than are in potassium bromide to neutral KBr molecules, because Li+ is smaller than K+ . In mixtures, Li+ and K+ compete in the formation of molecules LiBr and KBr, respectively, the smaller Li+ being more successful. Therefore in the mixtures, the internal mobility of lithium decreases

Electrical Conductivity

343

faster with an increasing content of potassium bromide than the mobility of potassium bromide, and thus the mobility isotherms may cross over. According to Klemm and Schäfer (1996) it is assumed that the binary mixture of molten salts M1 X–M2 X consists of five kinds of particles, three of which are charged + − and two are neutral: M+ 1 , M2 , X , M1 X, and M2 X. The sum of mole fractions of all the particles equals unity x1 + x2 + x3 + x4 + x5 = 1

(8.44)

The electro-neutrality of the melt is expressed by x1 + x2 = x3 ;

0 < x3 < 0.5

(8.45)

Also the mole fractions of the salts add up to unity x13 + x23 = 1

(8.46)

It is easy to show that x13 =

x1 + x4 , 1 − x3

x23 =

x2 + x5 1 − x3

(8.47a, b)

In order to define the composition of the melt, the mole fraction x23 will be used in the following text. The association constants K1 and K2 are defined by the relations K1 =

x4 , x1 x3

K2 =

x5 x2 x3

(8.48a, b)

From Eqs. (8.46), (8.47a, b), and (8.48a, b) it follows that x1 =

(1 − x23 )(1 − x3 ) , 1 + K 1 x3

x2 =

x23 (1 − x3 ) 1 + K 2 x3

(8.49a, b)

By adding Eqs. (8.49a) and (8.49b), and using Eq. (8.45) one obtains x3 =

1 − x23 x23 (1 − x3 ) + 1 + K 1 x3 1 + K 2 x3

(8.50)

and finally a cubic equation in x3 (x23 ) x33 (x23 ) + αx32 (x23 ) + βx3 (x23 ) + γ = 0

(8.51)

344

Physico-chemical Analysis of Molten Electrolytes

where 1 1 − x23 − α= K1 K2 2 1 1 1 β= + x23 − − K1 K2 K2 K1 K2

γ =−

1 2 + K1 K2

1 K1 K2

(8.52a) (8.52b) (8.52c)

The molar conductivity of the mixture expressed in terms of x23 is then given by the relation κ(x23 ) = [(1 − x23 )u1 (x23 ) + x23 u2 (x23 )].F /Vm (x23 )

(8.53)

where u1 (x23 ) and u2 (x23 ) are the mobilities of both the components expressed in terms of x23 , F is the Faraday’s constant, and Vm is the molar volume of the melt. 8.1.1.6. Model of electrical conductivity of silicate melts

An original approach to the characterization of electrical conductivity of silicate melts was proposed by Licˇ ko and Daneˇ k (1983). In silicate melts with a high SiO2 content (min. 40 mole %) the charge is transferred exclusively by cations. The authors used Eq. (8.2) for the calculation of the mobility of cations. For the system CaO–MgO–SiO2 , Eq. (8.2) attains the form κ = 2F (uCa2+ cCa2+ + uMg2+ cMg2+ )

(8.54)

Since the molar concentrations of cations in the investigated system could be calculated from density measurements performed by Licˇ ko and Daneˇ k (1982), the mobility of cations can be calculated. Using the multiple linear regression analysis, it was determined that for ci = 0 the electrical conductivity is not equal to zero, but attains negative values. The dependence of electrical conductivity of cations on concentration corresponds to the equation κ = −a + b1 cCa2+ + b2 cMg2+

(8.55)

where bi = 2F ui . However, the conductivity apparently cannot attain negative values. Thus, when the conductivity approaches zero, the concentration of the conducting particles attains a certain limiting value ci0 , below which the cations for a certain reason cannot participate in charge transfer. From this it follows that 0 0 a = b1 cCa 2+ + b2 cMg2+

(8.56)

Electrical Conductivity

345

Table 8.3. Mobility of cations in the CaO–MgO–SiO2 system at 1773 K and of cations and electrons in the system CaO–FeO–Fe2 O3 –SiO2 at 1723 K System CaO–MgO–SiO2 T = 1773 K CaO–FeO–Fe2 O3 –SiO2 T = 1723 K

µCa2+ × 104 (cm2 s−1 V−1 )

µMg2+ × 104 (cm2 s−1 V−1 )

µFe2+ × 104 (cm2 s−1 V−1 )

µFe3+ × 104 (cm2 s−1 V−1 )

µe− × 104 (cm2 s−1 V−1 )

1.85

1.57

–

–

–

2.0

–

1.8

1.1

225

Consequently, for such systems, Eq. (8.2) can be written in the form κ=F

zi ui (ci − ci0 )

(8.57)

i

Existence of a certain limiting concentration ci0 , at which the conductivity attains a zero value, was explained by the fact that a part of Ca2+ and Mg2+ cations is bonded into larger structural units, clusters, composed of silicate polyanions connected together by cations with polar covalent bonds. A similar behavior of cations was also found by Daneˇ k et al. (1986) in the CaO–FeO–Fe2 O3 –SiO2 system, where due to the presence of iron in two oxidation states, electronic conductivity is also present (c.f. Section 2.1.8.5.). The concentration of electrons was calculated from the molar fractions of FeO and Fe2 O3 . The mobility of electrons was found to be higher by two orders compared with that of the cations. The mobility of cations and electrons in both the investigated systems at 1723 K is given in Table 8.3. 8.1.2. Electrical conductivity in ternary systems

The ideal behavior in the case of electric conductivity is not defined physically, as we deal with scalar quantities, for which the total derivative does not exist and the simple additivity rule may thus not be used. However, the electrical conductivity is thermally activated and the additivity of activation energies of pure components is enabled. Based on this idea the additivity of logarithms of the electrical conductivity may be accepted as the “ideal” behavior. It should be, however, emphasized that there are two kinds of electrical conductivities, i.e. the conductivity, κ, and the molar conductivity, λ. The concept of the additivity of logarithms is recommended to apply to the molar conductivity, as the concentration course of the molar conductivity is smoothed by multiplying the conductivity with the molar volume. The “ideal” course of the electrical conductivity in the ternary system can be then expressed in the form ln λ = x1 ln λ1 + x2 ln λ2 + x3 ln λ3

(8.58)

346

Physico-chemical Analysis of Molten Electrolytes

or x

x

x

λ = λ11 · λ22 · λ33

(8.59)

For the excess electrical conductivity of the real ternary system, the validity of the general Redlich–Kister’s type equation can be proposed. For the description of the composition dependence of electrical conductivity in the ternary system, the following equation could then be used λ=

x λ11

x · λ22

x · λ33

+

3

xi · x j

i,j =1 i=j

k n=0

Anij · xjn

+

m

Bm · x1a · x2b · x3c

(8.60)

a, b, c

The first term represents the “ideal” behavior, the second one the binary interactions, and the third term the interactions of all the three components. Coefficients of the regression equation (8.60) are calculated using the multiple linear regression analysis. Omitting the statistically non-important terms on the chosen confidence level and minimizing the number of relevant terms, we try to get a solution, which describes the concentration dependence of the electrical conductivity with a standard deviation of the fit being comparable with the experimental error of measurement. For statistically important binary and ternary interactions, we look for appropriate chemical reactions and check their thermodynamic probability calculating their standard reaction Gibbs energies.

8.2.

EXPERIMENTAL METHODS

With respect to the ability to dissolve non-conducting materials, the measurement of electrical conductivity of molten salts, especially of fluorides, at high temperatures requires often the use of metallic conductance cells with low cell constant. When measuring with conductance cells of this type, resistances of the order of some tenths of ohm are measured, which requires the use of precise resistance bridges. For the metallic conductance cells with low resistance capacity, it is necessary to determine the dependence of the measured resistance on the current frequency used and the measurement must be realized at a frequency at which the resistance does not change anymore. In general, the measurement of electrical conductivity is not an absolute method and the conductance cell used has to be calibrated using the molten salt with a known conductivity. When the calibration of the cell is done at ambient temperatures, the accurate resistance measurement requires correction for the change in the resistance cell capacity with temperature. The total accuracy of measurement is then affected by knowing the coefficient of thermal dilatation of the material used from which the cell was made.

Electrical Conductivity

347

The measured resistance of most cells varies with the applied alternating current, which is due to the polarization of the cell electrodes. Consequently, the use of large electrodes in cells with a high cell constant (capillary cells) will minimize the relative variation. In highly corrosive fluorides, however, most investigators use all-metal cells with low cell constants and thus considerable polarizations, so far. All types of cells show the same general response to the alternating current flow. When the amplitude of the sinusoidal potential applied is less than the potential required for an electrochemical reaction, the conductivity cell can be represented approximately by the scheme shown in Figure 8.4. The frequency independent metal–salt interfacial capacitance (the double-layer capacitance) is charged and discharged according to the variation of the potential. At potentials sufficiently high for the occurrence of the reaction, the charge transfer across the interface may be represented by the impedance Zr in parallel with the doublelayer capacitance Cs . Figure 8.5 is thus an approximate representation of the conductivity cell during that part of each cycle in which the potential exceeds the potential of the reaction. The most complete representation of the cell is given in Figure 8.6. The impedance Zr is split into two parts: (a)

Rk , which is a resistive term arising from the finite rate of the electrode reaction. Randles (1947) has shown that 1 RT (8.61) Rk = 1−α α 2 2 k n F ACox Cred Re

Cs

Figure 8.4. The simplest representation of a conductivity cell. Re – ohmic resistance, Cs – double-layer capacitance.

Zr Re

Cs

Figure 8.5. Representation of a conductivity cell when the applied potential exceeds the potential for reaction. Re – ohmic resistance, Cs – double layer capacitance, Zr – impedance due to the electrode reaction.

348

Physico-chemical Analysis of Molten Electrolytes

Rk

Rw

Cw

Cs

Re Co

Figure 8.6. Complete representation of a conductivity cell. Re – ohmic resistance, Cs – double layer capacitance, Rw – Warburg’s resistance, Cw – Warburg’s capacitance, Rk – resistive component due to the finite rate of electrode reaction, Co – “stray” capacitance.

(b)

where k is the standard molar rate constant of the electrode reaction, Cox and Cred are the concentrations of the oxidized and reduced species of the electrode reaction ox +ne ↔ red, respectively, A is the effective electrode area, and α is the energy transfer coefficient. The Warburg’s impedance, the resistive and capacitive parts of which are defined according to Randles (1947) by the relations

CW

1/2 1 2Dω n2 F 2 ACox Cred 2D 1/2 = RT (Cox + Cred ) ω

RW =

RT (Cox + Cred ) n2 F 2 ACox Cred

(8.62)

(8.63)

where D is the diffusion coefficient for ox and red (assumed to be the same for both) and ω is the angular frequency of the alternating current applied. Finally, also the existence of a “stray” capacitance, C0 , for instance, between leads must be mentioned. As was shown by Hills and Djordjevic (1968), it is reasonable to assume that except at extremely high frequencies, C0 is negligible under the selected experimental conditions. The rate constant k in Eq. (8.61) will be very high at elevated temperatures and therefore Rk is assumed to be negligible. An analysis based on these approximations gives the series resistance component (the bridge arrangement allowed the series component of the unknown to be measured) of the cell impedance as Rs = Re +

ω1/2

a

a2

1/2

+ 2aCs

+ 2Cs2

(8.64)

Electrical Conductivity

349

where Re is the ohmic resistance of the electrolyte and a is the Warburg’s coefficient given by the relation a=A

n2 F 2 Cox Cred (2D)1/2 RT (Cox + Cred )

(8.65)

From Eq. (8.64) it is evident that unless the double-layer capacitance Cs is zero, Rs will not be a linear function of ω1/2 except at low frequencies. Many authors have assumed such linearity in the whole frequency range and obtained Re values from the linear extrapolation of Rs to ω1/2 = 0 (i.e. to infinite frequency). 8.2.1. Capillary cells

A capillary cell made of Pyrex glass and supplied with two electrode couples (A and B) made of Pt30Ir wire 0.5 mm in diameter was used by Matiašovský et al. (1971) for conductivity measurement of molten nitrates. Figure 8.7 shows the glass capillary cell and Figure 8.8 shows schematically the whole experimental assembly. The TESLA BM 344 generator supplied the alternating current and voltage, which were applied to the couple B of electrodes. The voltage drop across the large 200-k resistance was kept constant by means of the tube voltmeter TR–1202 EMG-1319, (TV1)

B

A

0

10

mm

20

30

Figure 8.7. Glass capillary cell. A – electrodes without current supply, B – electrodes with current supply.

350

Physico-chemical Analysis of Molten Electrolytes

S

TV1

M2

B

B A

A

TV2 Figure 8.8. Circuit diagram for capillary cell measurement. A – electrodes without current supply, B – electrodes with current supply, S – generator, TV1 – tube-voltmeter, TV2–nano-voltmeter, M2–resistance.

thus providing a constant current flow through the cell. The voltage drop along the cell was measured by means of the selective nano-voltmeter UNIPAM 208 (TV2) and voltage readings were made both by means of the electrode couples A and B in order to separate the polarization effect at the current supplying B electrodes. The very high inner resistance of the nano-voltmeter (>10 M) made the polarization of the A electrodes entirely negligible. Accordingly, the voltage values measured by means of the A electrodes did not vary with frequency. As the cell current was kept constant, these voltage values are directly proportional to the resistance of the electrolyte and could be used for the calculation of conductivity values after the cell constant has been determined. The cell constant C was determined using pure NaNO3 , for which the conductivity given by Janz et al. (1968) was accepted. The cell constant was calculated according to the equation C = κNaNO3 · UNaNO3

(8.66)

where U is the voltage drop between the A electrodes in molten NaNO3 . The electrical conductivity is then given by the relation κx =

C Ux

The observed C values ranged from 0.998 to 1.005·10−3 A/cm.

(8.67)

Electrical Conductivity

351

When the voltage drop across the capillary cell is measured by means of the currentsupplying electrode couple B, the resulting resistance values depend on frequency according to the equation Rs (ω) = Re + R(ω)

(8.68)

where Re is the true ohmic resistance of the electrolyte and R is the polarization resistance. By measuring the voltage drop across the cell by means of the electrode couple A, the resulting resistance values are frequency independent: Rs = Re . Some authors, Yim and Feinleib (1957) and Fellner et al. (1993), used the boron nitride (BN) capillary cell in the measurement of electrical conductivity of the fluoride melts. The cell used by Fellner et al. (1993) consisted of a pyrolytic boron nitride tube of inner diameter of approximately 4 mm and length of 100 mm (Figure 8.9). One electrode consisted of a tungsten rod of 2 mm diameter, which could be placed precisely in the same position in the tube, while a graphite crucible served as the other electrode. The crucible containing 10–15 g of the salt mixture was placed in a vertical laboratory furnace provided with inert argon atmosphere and heated up to the required temperature. The temperature was measured using the Pt1–Pt10Rh thermocouple.

1 3

2

4 5

6

7

8

Figure 8.9. Schematic representaion of the boron nitride (BN) capillary conductivity cell. 1 – steel tube, 2 – molybdenum contact rod, 3 – thermocouple, 4 – pressed BN cell holder, 5 – tungsten electrode, 6 – melt, 7 – pyrolytic BN tube, 8 – graphite crucible.

352

Physico-chemical Analysis of Molten Electrolytes

A Solartron 1250 Frequency Response Analyser was used for the measurement of cell impedance at variable frequency. The ac amplitude was 10 mV and a 10- standard resistor connected in series with the cell was used as the measuring resistance. An online PC was used for the control of the frequency analyzer and for the collection and processing of the data for the real and imaginary parts of the cell impedance. Extrapolating the measured data to infinite frequency and subtracting the resistivity of the leads and of the electrodes, which was determined by placing the tungsten electrode at the bottom of the cell, the resistance of the electrolyte was obtained. The cell constant was determined by calibration using NaCl. It was found that the cell constant did not vary with temperature and no gradual penetration of the melt into the BN tube was observed. 8.2.2. Conductivity cell with continuously varying cell constant

Wang et al. (1992) developed a new technique for the measurement of electrical conductivity of fluoride melts, which employs the principle of a Continuously Varying Cell Constant (CVCC) through a moving platinum disc electrode in a relatively large diameter capillary tube-type conductivity cell. At the same time, the real component of the circuit impedance, Rm , at a fixed high-frequency current is measured. Since the Rm versus the cell constant plot is linear, the electrical conductivity of the electrolyte is given by the relation dRm −1 κ= A dL

(8.69)

where κ is the conductivity of the melt, A is the inner cross-sectional area of the conductivity cell tube, and dRm /dL is the slope of the resistance of the measuring circuit versus the programmed variation of the conductivity cell length, L. The continuously varying cell constant is accomplished by linearly varying the length of the conductivity cell, L, by moving the Pt disc electrode downwards and upwards while keeping the cell cross-area, A, unchanged. The slope is therefore derived through a series of circuit resistance measurements versus the programmed lengths of the conductivity cell. The electrical conductivity derived from Eq. (8.69) is free of extraneous conductivity effects such as applied frequency and wire contact resistance. A schematic representation of the conductivity cell assembly is shown in Figure 8.10. The graphite crucible, which was used as the molten salt holder, was of a 3-cm inner diameter and was also used as the counter electrode connected to a LCR Impedance Meter through a thermocouple Inconel protection sheath. In the upper part of the graphite crucible, a BN holder of 3.8 cm inner diameter was placed, in which the pyrolytic BN tube-type conductivity cell was vertically fastened. The tube cell, made of vapor-deposited pyrolytic BN, was dense, non-conductive, with a consistent inner diameter, and able to resist corrosion attack of molten fluoride. The pyrolytic BN tube was immersed in

Electrical Conductivity

353

Inconel rod Thermocouple

Position control Resistance meter Pt rod BN insulator BN holder

Melt level Pyrolytic BN tube Graphite crucible L

Pt disc electrode

Figure 8.10. Schematic representation of the CVCC conductivity cell assembly.

the molten electrolyte for about 5.5 cm. A Pt disc electrode was connected to 0.16-cm Pt wire, insulated by a BN tube. The other end of the Pt wire was screwed into the 0.64-cm Inconel rod, which was vertically connected to the arm of a positioner measuring the actual conductivity cell length L. The upper end of the furnace was covered with a special split water-cooled lid. The furnace was purged with argon to prevent air from burning the graphite crucible and to ensure the accuracy of the measurement. The Model Unidex XI (AeroTech, Pittsburgh, PA) was vertically located on a vibration-free stand. It can move the Pt disc electrode in the center of the pyrolytic BN tube-type conductivity cell up and down to a known position with an accuracy of ± 0.001 cm. A programmable position controller controlled the position.

354

Physico-chemical Analysis of Molten Electrolytes

Electrical conductivity was measured using the Impedance Meter SP2596, Electro Scientific Industries, Portland, OR, with a fixed frequency of 1 kHz. All three components of the measuring circuit impedance, i.e. the real component (resistance), capacity, and induction, could be measured separately. In a study of the frequency effect, an HP 4274A Multifrequency LCR meter, ElectroRent Corp., Norcross, GA, with a frequency range from 100 Hz to 100 kHz and a measuring resistance with 5 significant digits, was utilized. The inner cross-sectional area, A, of the tube-type cell was determined by calibration, measuring the electrical conductivity of 1.00-M KCl aqueous solution. It was assumed that the thermal expansion of the tube’s inner diameter due to the increased temperature is insignificant and, therefore, was neglected. The frequency dependence of the measured resistance of the CVCC conductivity cell was tested using molten KCl and three different compositions of cryolite melts. The statistical analysis of the results indicated that the electrical conductivity of each electrolyte is independent of the applied frequency. Figure 8.11 shows the conductivity results as a function of the applied frequency. No variation of the conductivity values was observed within dispersion of ± 1%. This verifies the principle on which the technique is based, i.e. that the slope of resistance versus the distance L in the tube-type conductivity cell is independent of the applied frequency. Conventional methods, on the other hand, have to take into account the applied frequency and many conductivity values were derived or extrapolated to the infinite frequency of the measuring current. In another article, Wang et al. (1993) used the CVCC conductivity cell for measurement of a variety of molten cryolite melts with additives of aluminum fluoride, aluminum oxide, calcium fluoride, magnesium fluoride, and lithium fluoride. On the basis of the measured results, a multiple regression equation for the electrical conductivity of cryolite melts was derived. Influence of the bath composition on the electrical conductivity at different bath temperatures was discussed. A comparison of the measured results with the published electrical conductivity values for cryolite melts was made. The new regression equation can be used to calculate electrical conductivity of cryolite melts in modern industrial bath chemistry. The electrical conductivity of molten cryolite melts containing aluminum carbide, A14 C3 , was measured using the CVCC conductivity cell by Wang et al. (1994). Aluminum carbide can originate in an aluminum electrolytic reduction cell in any region where aluminum and carbon mutually contact. The electrical conductivity of the bath is decreased by A14 C3 but the quantitative effect has not been defined. The experimental data obtained were incorporated into a convenient mathematical model for determining the quantitative effect of A14 C3 on the aluminum electrolyte. 8.2.3. Two-electrode cell

The electrical conductivity measurement using a two-electrode cell is very sensitive to the calibration technique, since every minor change in the cell constant affects the results

Electrical Conductivity

355

3.0 2.8 A (2.754 ± 0.02) S.cm−1 ± 0.73%

κ(S.cm−1)

2.6

2.4

B (2.373 ± 0.003) S.cm−1 ± 0.17% (2.215 ± 0.005) S.cm−1 C

2.2

± 0.45%

D (2.042 ± 0.014) S.cm−1 2.0 1.8 0.1

± 0.64%

1.0 10.0 Frequency (kHz)

100.0

Figure 8.11. Dependence of the measured conductivity on applied frequency for four different melt compositions. A – Na3AlF6 at 1022◦ C; B – CR = 2.5 + 3% Al2 O3 + 3% LiF + 4% CaF2 + 2% MgF2 at 945◦ C; C – KCl at 818◦ C; D – CR = 2.2 + 3% Al2 O3 + 5% CaF2 at 959◦ C.

substantially. Aqueous KCl solutions are often used as conductivity standards. Problems with thermal expansion of the cell material and the change in the cell geometry during heating can substantially affect the values obtained. For calibration at elevated and high temperatures, Janz (1980) recommended pure KNO3 and NaCl as reference standards. A two-electrode cell made of two bright platinum discs 5 mm in diameter at a distance of 12 mm was used in the conductivity measurement of molten fluoride systems by Matiašovský et al. (1970). These authors showed that most of the Rs = f (ω1/2 ) plots for different molten salts are in full agreement with Eq. (8.64), distinctly non-linear and that an extrapolation to infinite frequency leads to incorrect resistance values. This is surely one of the reasons for the conflicting conductivity data in the literature, especially for molten fluorides, when full-metallic conductivity cells were applied. Thus, e.g. for molten NaF at 1020◦ C, the conductivity values ranging from 3.32 to 5.60 −1 · cm−1

356

Physico-chemical Analysis of Molten Electrolytes

were found. Matiašovský et al. (1970) found that for their cell, the “limiting” frequency, above which Rs does not change with frequency, is approximately 100 kHz. A cell similar to that used by Matiašovský et al. (1970) was used for conductivity measurements of chlorides and fluorides also by Winterhager and Werner (1956). These authors obtained constant Rs values between 20 and 50 kHz. The distinctly lower “limiting” frequencies are probably caused by the use of larger platinized electrodes, i.e. electrodes with a much larger effective surface area. This results in greater double-layer capacitance and higher a values, so that Rs = Re at lower frequencies, according to Eq. (8.64). 8.2.4. Four-electrode cell

The four-electrode method with double immersion for conductivity measurements was published by Ohta et al. (1981). It was derived from the four-electrode method used for the measurement of conductivity of the semiconductors originally developed at the Philips laboratories. The schematic drawing of the method’s principle is shown in Figure 8.12. The method demands a square orientation of the electrodes. Electrical shifting of the electrodes’ function by 90◦ eliminates the deviation of the electrodes’ position from this geometry. The principle of the method, which is based on the measurement of electrical field distribution in the investigated liquid, is as follows: In the first step, when the electrodes are in the upper position, the current I1 is applied through the electrodes a and b, while the voltage V1 across the electrodes c and d is measured. Then the electrode function is switched by 90◦ ; thus the current J2 flows through electrodes b and c and voltage V2 is read from electrodes d and a. From the values

V2 d

a I1 b

V1 c ∆W

I2 Figure 8.12. Basic geometry of the four-electrode method with double immersion. a, b, c, d – electrodes, W – distance in the immersion depth, V1 , V2 – voltage, I1 , I2 – current.

Electrical Conductivity

357

so obtained the first and second electrode orientation apparent resistances R1 = V1 /I1 and R2 = V2 /I2 , respectively, are calculated. From these two resistances, the average resistance in the upper position RU = (R1 + R2 )/2 is calculated. In the second step, the electrodes are immersed deeper by W and a similar process as in the first step is repeated, resulting in the value RL . By this arrangement actually only the resistance of the disk thickness W is measured. An extensive mathematical analysis of the method was given by Ohta et al. (1981), which results in the following conductivity equation κ=

ln 2(RU − RL ) 2π RU RL W

(8.70)

where RU and RL are the average resistances in the upper and lower position of the electrodes, respectively, and W is the distance between the upper and lower electrode position. As it follows from Eq. (8.70), only the distance in the electrodes’ depth of immersion must be known. When the temperature of the measured liquid is changed, the change in the surface level (new depths of immersions) need not be measured. Ohta et al. (1981) also discussed the role of different parameters, i.e. the ratio between the electrode diameter and their distances, the material of the crucible, electrode displacement from the center of the crucible, the depth of measured liquid, immersion depth, the frequency of the measuring current, etc. This method is rather cumbersome and time consuming when performed manually, as it is necessary to measure eight current/voltage data and change precisely the electrode immersion, to obtain a single conductivity value. In order to eliminate this disadvantage, a computerized device is needed.

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Chapter 9

Viscosity The viscosity of molten alkali metal halides ranges from 0.5 to 5 mPa · s. In a broader range of temperature (more than 200 K), the activation energy of viscous flow cannot be regarded as constant. Since the molten salts are composed of two kinds of particles that have a different radius, it may be expected that the viscosity value will be given by the motion of the bigger, thus the more sluggish ions. This effect may be observed especially in melts containing complex anions, which are much bigger than the present cation. Due to this fact, values of the viscosity and electrical conductivity of molten salts cannot be correlated. Similarly, the values of the activation energy of viscous flow are often higher than those of electrical conductivity. The viscosity of molten salts in simple systems, in which no compounds are formed, is in general lower than the values calculated according to the additive rule. The up to now only known case of additive change of viscosity with composition was observed in the system CdCl2 –CdBr2 . It is interesting that the minima of the viscosity dependence on composition are in general not in accordance with the minima of the conductivity dependences, or they do not respond to the composition of the complex compound, probably due to the change in the ionic structure.

9.1. THEORETICAL BACKGROUND

Viscosity is a scalar quantity, which characterizes the inner friction of liquids. The coefficient of the dynamic viscosity η is defined by the differential equation F = ηA

dv dr

(9.1)

where F is the force, which acts on the liquid layer, A is the layer area, v is the velocity of layer motion, and r is the thickness of the liquid layer. The unit of the dynamic viscosity in the SI system is Pa · s. The viscosity of water at 20◦ C is 1.009 mPa · s. The kinematic viscosity ν is the dynamic viscosity divided by density, ν = η/ρ.

359

360

Physico-chemical Analysis of Molten Electrolytes

The viscosity of liquids depends significantly on temperature; it can change even by some orders of magnitude. The dependence of viscosity on temperature is thus expressed most often in the form of exponential equations of the Arrhenius’ type Evis η = A · exp (9.2) RT Most of the liquids obey Eq. (9.2) if the temperature interval is not too large. However, for glass-forming melts, the viscosity of which ranges from 1 Pa to 1014 Pa, the Vogel– Fulcher–Tamman equation is frequently used B η = A · exp (9.3) T − T0 where B and T0 are constants. T is close to the glass transition temperature, which is the temperature at which the relaxation time of the melt equals infinity, i.e. when the structure of the melt does not change and becomes frozen. 9.1.1. Viscosity of “ideal” and real solutions

In order to get some information on the possible structure of the given molten system from viscosity measurement, a suitable reference state should be defined. Since the viscosity is a scalar quantity, no ideal behavior is given by definition. However, based on the validity of the Arrhenius equation for temperature dependence of viscosity and on the additivity of activation energies of viscous flow due to the general validity of Hess’s law, the additivity of logarithms of viscosity could be accepted as the “ideal” behavior, e.g. for a ternary system at constant temperature, it may be written as ln{ηid } = x1 ln{η1 } + x2 ln{η2 } + x3 ln{η3 }

(9.4)

x

(9.5)

or

x

x

ηid = η11 η22 η33

where ηi are viscosities of pure components and xi are their molar fractions. For real systems it can then be written x

x

x

η = ηid + ηex = η11 η22 η33 + ηex

(9.6)

where ηex is the viscosity excess, which can be expressed in the form of the Redlich– Kister’s type equation ηex =

3 i,j =1 i=j

xi xj

k n=0

Anij xjn

+

m a,b,c=1

Bm x1a x2b x3c

(9.7)

Viscosity

361

The first term represents interactions in the binary systems while the second one describes the interaction of all the three components. The calculation of coefficients Anij and Bm for the chosen temperature is then performed using the multiple linear regression analysis, omitting the statistically non-important terms on a chosen confidence level. The excess viscosity so defined and calculated already may yield relevant information on the structure of the investigated system.

9.1.2. Application to binary and ternary systems 9.1.2.1. Binary systems

In binary systems, the above-accepted “ideal” behavior of the concentration dependence of viscosity can be used e.g. in the estimation of the dissociation degree of the thermal dissociation of the additive compound AB, which are formed in the binary system A–B. As an example, the binary system KF–K2 NbF7 was chosen, in which the additive compound K3 NbF8 is formed. The concentration dependence of viscosity of the system KF–K2 NbF7 shows positive deviations from the “ideal” behavior, most probably due to formation of the congruently melting intermediate compound K3 NbF8 . This compound is, however, not stable and dissociates partially according to the reaction K3 NbF8 (l) ⇔ K2 NbF7 (l) + KF(l)

(9.8)

with the dissociation degree α 0 . Assuming that the ideal viscosity course could be described by Eq. (9.8), there is the possibility to calculate α 0 by simulating the real viscosity course in the binary system KF–K2 NbF7 at equilibrium using the equation x(KF) x(K2 NbF7 ) x(K3 NbF8 ) ηK NbF7 ηK NbF 2 3 8

ηcalc = ηKF

(9.9)

where η(KF) and η(K2 NbF7 ) is the viscosity of KF and K2 NbF7 , respectively, η(K3 NbF8 ) is the hypothetical viscosity of pure non-dissociated K3 NbF8 , and xi s are the equilibrium molar fractions of KF, K2 NbF7 , and K3 NbF8 in the mixture. η(K3 NbF8 ) can be roughly estimated from the values of viscosity of both the other components. For every composition, it is possible to calculate the equilibrium molar fractions of the constituents KF, K2 NbF7 , and K3 NbF8 for every chosen value of α 0 and η(K3 NbF8 ). The correct value of α 0 is given by the condition n i=1

(ηcalc, i − ηexp, i )2 = min

(9.10)

362

Physico-chemical Analysis of Molten Electrolytes

Table 9.1. Calculated values of dissociation degree, α 0 , the equilibrium constant of reaction (A), and the viscosity of non-dissociated K3 NbF8 T (K)

η (K3 NbF8 ) (mPas)

K

α0 *

α 0 **

α 0 ***

1050 1100 1150

4.01 3.55 3.20

0.468 0.510 0.556

0.42 0.45 0.49

0.61 0.55 0.38

0.44 – –

* from viscosity measurements. ** from density measurements. *** from phase diagram calculation.

The results of calculation are shown in Table 9.1 9.1.2.2. Viscosity in ternary systems

For a description of the viscosity dependence on composition in the ternary system, the following equation can be used

x

x

x

η = η11 · η22 · η33 +

3 i,j =1 i=j

xi · x j

k n=0

Anij · xjn +

m

Bm · x1a · x2b · x3c

(9.11)

a,b,c

The first term represents the “ideal” behavior, the second the binary interactions, and the third term the interactions of all the three components. Coefficients of the regression Eq. (9.11) are calculated using the multiple linear regression analysis. Omitting the statistical non-important terms on the chosen confidence level and minimizing the number of relevant terms, we try to get a solution, which describes the concentration dependence of the electrical conductivity with a standard deviation of the fit being comparable with the experimental error of measurement. For statistically important binary and ternary interactions, we look for appropriate chemical reactions and check their thermodynamic probability calculating their standard reaction Gibbs energies. 9.1.2.3. Viscosity of silicate melts

Viscosity is one of the most frequently measured properties of silicate melts. Viscosity data vary in the range 0.1–1016 Pa · s. To obtain precise and correct viscosity data is experimentally very difficult, especially at high temperatures. Contradictory data frequently found in the literature demonstrate this fact. This can be illustrated by the CaSiO3 melt, for which at 1873 K, values of 0.15 to 0.25 Pa · s are given.

Viscosity

363

The problem of anionic structure and the quantitative distribution of particles in silicate melts (see also in Chapter 2.1.9.) is discussed in several works. The simplest application of the Flory’s (1953) polymer theory to silicate melts is the theory of linear and branched chains proposed by Masson (1965, 1968, 1977) and Masson et al. (1970). In this theory, it is assumed that the silicate anions are composed exclusively of linear and branched chains 2(n+1)− of the general formula Sin O3n+1 . These chains arise by poly-condensation reactions of the type 2(n+1)−

SiO4− 4 + Sin O3n+1

2(n+2)−

⇔ Sin+1 O3n+4

+ O2−

(9.12)

with the equilibrium constant K1n =

xSi

2(n+2)− n+1 O3n+4

xSiO4− · xSi 4

· xO2− (9.13) 2(n+1)−

n O3n+1

where xi are the mole fractions of the corresponding anions. Equation (9.13) shows that K1n is closely related to the degree of polymerization of the melt at a given SiO2 content. Equation (9.13) may be written also in the form xSi

2(n+2)− n+1 O3n+4

xSi

2(n+1)− n O3n+1

=

K1n · xSiO4− 4

xO2−

=r

(9.14)

At constant temperature, pressure, and composition also r is constant, i.e. at thermodynamic equilibrium the ratio of the molar fractions of the successive members of the homologous series of silicate anions is constant. In the absence of data on the relationship between activity and composition, molar fractions of anions are used instead of activities in Eqs. (9.13) and (9.14). The support for this simplification comes from the observation of phosphate glasses, where the molar fractions of phosphate anions have been measured by Meadowcroft and Richardson (1965) using chromatographic methods. It is obvious that the constant K1n depends on the chain length. The dependence of K1n on n for linear and branched chains was derived by Masson et al. (1970). The equilibrium given by Eq. (9.12) can be described using a single equilibrium constant, e.g. the dimerization constant K11 . Using the same simplifying assumptions, Masson et al. (1970) derived equations for the calculation of the anionic distribution in binary silicate melts. The validity of the original Masson’s theory is limited to melts with xSiO2 < 0.5, as the presence of cyclic and spherical structural units was not considered. However, the validity of the polymer theory was extended to include cyclic and spherical particles, i.e also for more polymerized silicate melts, by Pretnar (1968), Baes (1970), and Esin (1973, 1974). Pretnar assumed that in binary silicate melts, short chains (n ≤ 5)

364

Physico-chemical Analysis of Molten Electrolytes 2(2n−f )−

and spherical particles of the general formula Sin O4n−fn n , where n ≥ 6, were present ( fn is the number of bonds between SiO4 tetrahedrons in the poly-silicate anion). Pretnar showed that for chains fn = n − 1

(9.15)

fn = 2n − 1.71n2/3 − 0.5

(9.16)

while for more condensed polyanions

is valid. He also assumed that the equilibrium given by Eq. (9.12) could be described by the general scheme 2O− ⇔ O0 + O2−

(9.17)

with the equilibrium constant K=

xO0 xO2− xO2 −

(9.18)

where O0 and O− represent the bridging and non-bridging oxygen atoms, respectively. The equilibrium, which determines the quantitative distribution of anions in binary silicate melts, can thus be described so using a single equilibrium constant. The average mole mass of silicate anions and the average number of SiO4 tetrahedrons in the particular anion can be calculated according to the equations nxn n= xn

(9.19)

Mn = 28.086n + 16(4n − fn )

(9.20)

M=

xn Mn ; xn

where Mn is defined as

and for xn Pretnar (1968) derived the equation xn = xSiO2 (1 − q)2 q n−1

(9.21)

where q = xn+1 /xn is the parameter of a geometrical series determining the concentration of polymerized anions in the form

n2/3 q n−1 = 1 +

' 3

4q +

( 3

9q 2 + · · ·

(9.22)

Viscosity

365

The degree of polymerization Pretnar (1968) further defined as the ratio between the number of oxygen bridges present in the real structure and the total number of bridges theoretically possible fn · x n n (9.23) P = 2xSiO2 where xn is the mole fraction of silicate anions with n SiO4 tetrahedrons. The relation between the degree of polymerization P and the equilibrium constant of Eq. (9.18) is given by the equation xMO 2 P +P −1 2xSiO2 K= (9.24) 4(1 − P )2 From Eq. (9.24) for the polymerization degree it follows

) * 1 xMO xMO xMO P = 1+ 1− − 8K + + 16K − 2 2 − 8K 2xSiO2 2xSiO2 2xSiO2 (9.25) Pretnar also correlated P with the ratio r defined by Eq. (9.14) as follows 5

∞

n=1

n=6

2P n−1 = (n − 1)r + (2n − 1.71n2/3 − 0.5)r n−1 1 − r2

(9.26)

The distribution of silicate anions can be then calculated from the equation xn = xSiO2 (1 − r)2 r n−1

(9.27)

On the basis of the calculated silicate anion distribution, the mean chain length (the mean number of SiO4 tetrahedrons in the species) n · xn (9.28) n= xn 2(2n−fn )−

and the mean mole mass of the silicate Sin O4n−fn M= can be calculated.

xn · Mn xn

anions in the given melt (9.29)

366

Physico-chemical Analysis of Molten Electrolytes

A special problem in the application of the polymer theory is the knowledge of the equilibrium constant of the poly-condensation reaction K11 . The values of the dimerization constants in some binary MeO–SiO2 systems (M = Ca, Mn, Pb, Fe, Co, Ni) were calculated by Masson (1977). Balta and Balta (1971) found a linear relationship between the logarithm of the equilibrium constant and the ionization potential of the metallic cation, which allows estimation of the equilibrium constant in systems, where the experimental data are missing. e.g. for cations with a ionization potential close to the second ionization potential of Mg, Masson (1977) published for Mn at 1773 K and Pb at 1273 K the values of the equilibrium constant K11 = 0.19 K and K11 = 0.196, respectively. The theory proposed by Pretnar (1968) was applied by Licˇ ko and Daneˇ k (1986) to the viscosity of the system CaO–MgO–SiO2 and by Daneˇ k et al. (1985b) to the four-component system CaO–FeO–Fe2 O3 –SiO2 . Based on the values of the equilibrium constants K11 published by Masson (1977) and Masson et al. (1970), the values 0.19 and 0.0016 were accepted for the systems MgO–SiO2 and CaO–SiO2 , respectively. In the system CaO–FeO–Fe2 O3 –SiO2 the following values were accepted: for the system CaO–SiO2 the value K11 = 0.0016, for the system FeO–SiO2 the value K11 = 0.7, and for the system Fe2 O3 –SiO2 , based on the third ionization potential of Fe, the value K11 = 20. The temperature dependence of the equilibrium constants was ignored. The equilibrium constant in the ternary system CaO–MgO–SiO2 was calculated from the additive contributions of the logarithms of equilibrium constants in the binary systems CaO–SiO2 and MgO–SiO2 xMgO xCaO log K11 = log K(C−S) + log K(M−S) xMeO xMeO

(9.30)

where xMeO = xCaO +xFeO , where xCaO and xMgO are the molar fractions of the oxides in the investigated melt. This approach is equivalent to the assumption that the standard Gibbs energy of the polycondensation reaction in the ternary melt is additively changed when CaO is replaced by MgO. The equilibrium constant K11 in the system CaO–FeO–Fe2 O3 –SiO2 was calculated similarly xFe O xCaO xFeO log K11 = log K(C−S) + log K(F(II)−S) + 2 3 log K(F(III)−S) xMeO xMeO xMeO (9.31) where xMeO = xCaO + xFeO + xFe2 O3 . From the calculation results, the linear dependence of dynamic viscosity on the calculated mean molar mass of silicate anions present in the melt was found. The slope of this dependence is an exponential function of the reciprocal thermodynamic temperature (see Figure 9.1). The lines η = f (M) for different temperatures intersect practically in

Viscosity

367

1.5 1673 K 1773 K

η (Pa.s)

1.2

0.9 1873 K

0.6 1973 K

0.3

0.0 0

200

400

600

800

1000

M (g.mol−1) Figure 9.1. Viscosity vs. mean molar mass of silicate anions in melts of the system CaO–MgO–SiO2 .

a single point corresponding to the molar mass of the basic building unit – the SiO4 tetrahedron. Thus the dependence of dynamic viscosity on the average molar mass of silicate anions and the temperature in the system, CaO–MgO–SiO2 could be described by a single equation −9

η = 2.285 × 10

2.349 × 104 K −1 M (g·mol ) − 92 × exp + 0.07 Pa·s (9.32) T

The standard deviations in the observed concentration and temperature ranges do not exceed 1.5 × 10−2 Pa · s. A similar equation was obtained also in the system CaO–FeO–Fe2 O3 –SiO2 (see Figure 9.2) η = 2.881 × 10

−9

2.293 × 104 K −1 + 0.05 Pa·s (9.33) M (g·mol ) − 92 × exp T

where the standard deviations of the viscosity values attained the value sd = 9 × 10−3 Pa·s. When comparing Eq. (9.33) with Eq. (9.32), a very satisfactory agreement could be seen, which implies a general character of this relationship and its general validity in silicate melts. From the dependence between the dynamic viscosity and the average molar mass of silicate anions the following conclusions could be made.

368

Physico-chemical Analysis of Molten Electrolytes

1.0 1723 K

η (Pa.s)

0.8

0.6 1823 K

0.4

0.2

0.0

0

200

400

M (g.mol−1)

600

Figure 9.2. Viscosity vs. mean molar mass of silicate anions in melts of the system CaO–FeO–Fe2 O3 –SiO2 .

(1)

(2)

(3)

The frictional force, which hinders the relative motion of the neighboring liquid layers, is brought about by the momentum transport between silicate anions. This means that the structural units, the discrete silicate anions, are also the flow units. From the steric point of view, silicate anions are similar and probably to a considerable extent isometric (if the opposite would be the case, η = f (M) would not be linear). It is obvious that both statements are valid at least in the range of the investigated concentrations (i.e. up to 60 mole % SiO2 ). In the system CaO–MgO–SiO2 , at least up to 60 mole % SiO2 , and in the system CaO–FeO–Fe2 O3 –SiO2 at least up to 61 mole % SiO2 , the melts are formed by discrete silicate anions. This means that the boundary for gel formation (i.e. “infinitely” large anions) is shifted to higher SiO2 concentrations than the given values.

Of considerable interest is the effect of the chemical nature of the present cations on the viscosity and thus also on the anionic structure of the melts. With increasing ionization potential, the viscosity, i.e. the tendency towards polycondensation, increases. At equal mean molar masses of silicate anions, the decisive role will obviously be played by the strength of the bonds between the cations and the negatively charged unshared oxygen atoms. The stronger the bonds, the more resistant will be the structure to the effect of shearing forces and the higher will be the dynamic viscosity of the melt. Since Mg2+

Viscosity

369

cations are more strongly bonded in the melt structure than the Ca2+ cations, it can be assumed that at equal mean molar mass of silicate anions, MgO–SiO2 melts will be more viscous than the CaO–SiO2 melts. These conclusions are in agreement with spectroscopic observations of CaO–MgO–SiO2 glasses performed by McMillan (1984). In view of the decreasing viscosity of the CaO–FeO–Fe2 O3 –SiO2 melts with an increasing content of iron oxides, the assumption on the entry of the Fe3+ cations into the polyanionic network and its participation in the formation of globular anions was not fully proved. However, in spite of this the tetrahedral coordination of at least some Fe3+ cations in the melt is on the whole logical and follows from the ratio of the ionic radii of ferric and oxygen ions. The ferric cations may then be present in the melt as isolated FeO5− 4 tetrahedrons, twinned Fe2 O4− 5 tetrahedrons, or participate in the structure of anions with 5− a low number of SiO4− 4 or FeO4 tetrahedrons. On the one hand, they thus acquire the character of complex anions, on the other they do not contribute to the increase of the melt viscosity. The Fe3+ content in the melt decreases with increasing temperature, producing a higher content of larger purely silicate polyanions. This temperature-activated process results in an increase of the mean molar anion mass with increasing temperature. Such an explanation of the Fe3+ cationic behavior in silicate melts is in agreement with the conception of Waff (1977) and with the results of Mössbauer spectra measured by Pargamin et al. (1972) and Levy et al. (1976), where complex compounds of the type Ca0.5 Fe3+ O2 with tetrahedral coordination of the ferric cation were considered.

9.2.

EXPERIMENTAL METHODS

For measurements of viscosity of molten salts and glasses at high-temperatures, several methods were proposed. The selection of a particular method depends in general on the viscosity of the liquids to be measured. A broad dispersion of experimental results reflects substantial experimental difficulties connected with viscosity measurement. In general, in the measurement of viscosity of molten salts the method of torsional pendulum is most frequently used, while in the measurement of viscosity of liquids, such as molten glasses, the falling body and the rotational methods are most suitable. Methods for viscosity measurement of liquids with a very high viscosity (above 108 Pa · s) will not be described here.

9.2.1. Method of torsional pendulum

One of the most used and also most suitable methods of viscosity measurement of molten salts is the torsional pendulum method, which could be developed by the introduction of automatic and computational methods. The pendulum is realized in general by an arbitrary rotational body suspended from a torsional wire. However, the exact

370

Physico-chemical Analysis of Molten Electrolytes

1

15

2

16

3

17

4

18

1 – deflecton unit, 2 – centric chucks, 3 – torsion wire, 4 – mirrors, 5 – 12 mm ∅ brass rod, 6 – 10/5 mm diameter steel rod, 7 – inconel rod, 8 – 4 mm diameter Pt rod, 9 – cooling water, 10 – heating Kanthal winding, 11 – electrical surface contact,

5

19

6

20

7

21

12 – electrical counter contact, 13 – Degussite furnace shaft, 14 – crucible carrier tube, 15 – oscillating link, 16 – inlet and outlet of the torison wire thermostat,

8

22

9

23

10

24

11

25

12

26

17 – transparent window, 18 – rings for the moment of inertia change, 19 – Base-board, 20 – air-tight copper folding connecting the furnace, base-board, and vacuum pump,

13

b

21 – outlets of thermocouples, 22 – cooled upper lid of the furnace, 23 – ceramic radiation rings, 24 – thermocouple for furnace temperature control, 25 m heating Kanthal

c

coil, 26a, b, c – measuring bodies, 27 – Inert 14

27

gas outlet

Figure 9.3. Schematic representation of the apparatus for viscosity measurement.

mathematical description of the viscosity calculation enabling an absolute viscosity measurement without the need of calibration was given only for a sphere and for a cylinder up to now. The high-temperature oscillation viscosimeter constructed and used by Silný and Daneˇ k (1993) is described in detail here. The measuring device, which is schematically shown in Figure 9.3, consists in general of five main parts: • • •

furnace with a programmable temperature control and the temperature control of the torsional wire, torsional pendulum, initiator of oscillations,

Viscosity

• •

371

system of oscillation detection, control PC with an interface.

9.2.1.1. Furnace and programmable temperature control

The platinum cylinder with a 15 mm diameter and 20 mm height was used as the measuring body. The measured melt of 25 cm3 in volume, placed in a platinum crucible, was inserted in a resistance furnace. After melting of the sample, the pendulum was immersed in the melt, the surface of the melt being kept always 2 mm over the top of the cylinder. The depth of immersion was continuously monitored and controlled using an electrical contact. The additional damping, caused by a cylinder-carrying rod, was eliminated by a computational procedure. The computer controlled the whole measuring system including the furnace temperature. After all the input data and the required temperature profile were inserted, the measurement of the viscosity at the desired temperatures was performed automatically. All the temperature-dependent variables (oscillation period in gas, cylinder dimension, damping in gas, density of the measured liquid, inertia of the oscillating system) were expressed in the form of polynomials and calculated for the actual experimental temperature. The experimental error in the viscosity measurement did not exceed 1%. The measurement of each sample was carried out in the temperature interval of approximately 100 K starting at 20–30 K above the temperature of primary crystallization. 9.2.1.2. Torsional pendulum

The torsional pendulum consists of the measuring body, which can be spherical, cylindrical, or cylindrical with conical ends. The surface of the body has to be as smooth as possible and the body must be centric. The body made of platinum is attached to the torsional pendulum using a 2-mm diameter rod made of Pt20Ir alloy, which is screwed into another 3-mm diameter rod, made of the same material. The upper end of the measuring body is immersed in to the measured liquid 2 mm below its surface. The Pt20Ir rod is outside the hottest furnace zone and is screwed into another 6-mm rod made of stainless steel. This 6-mm rod is in its upper part narrowed to 5-mm diameter. Here, calibration brass rings are placed, used to measure and change the moment of inertia of the torsion pendulum. The pendulum continues with a 12-mm diameter brass rod, where the mirrors, serving for reflection of the light beam to the detection system, are symmetrically attached on both sides of the rod. The length of the torsional wire 0.3 mm in diameter, made of the Pt8W alloy recommended by Kestin and Moszynski (1958), is 548 mm. This material shows a very low inner friction and a high stability of the torsional constant. The total mass of the torsional pendulum suspended from the wire was 241 g. The wire is, on both ends soldered into tubes with a 2-mm outer diameter, which are fastened with identical screws. After the wire was definitely built into the system, it was tempered at 1000◦ C for

372

Physico-chemical Analysis of Molten Electrolytes

several hours. The temperature of the wire was held at the viscosity measurement at 25◦ C using the double mantle in which water from a ultra thermostat circulated. One of the most important conditions for obtaining reliable results is the symmetry of the torsional pendulum, which should not surpass 0.05 mm. 9.2.1.3. Initiator of oscillations

The deviation mechanism is made of single-phased electromotor, which can rotate only in a definite angle. The system is kept always in one of the two exceeding positions. Deflection is realized by changing the current polarity. After initiation, the system returns to its original position. Due to the non-ideality of the torsional wire the period and the damping constant depend, even though only slightly, on the amplitude of oscillations. This is why the amplitude not exceeding 16–22 cm was used. 9.2.1.4. Oscillations detection system

The oscillation detection system is schematically shown in Figure 9.4. The beam source represents an optical system, which projects the chink illuminated by a halogen bulb with a cool 12V/100W reflector through an objective to the mirror fastened on the torsion

FD1

4 3

Amplitude

standstill position center of detectors FD2

2 1 t'2, n t'1, n

t'4, n t'3, n

t1, n

t'6, n t'5, n

t2, n

t'8, n t'7, n

t3, n

t'1, n+1

t4, n

t'6, n+1 t'8, n+1 t'2, n+1 t'4, n+1 t'1, n+2 t'5, n+1 t'7, n+1 t'3, n+1

t1, n+1

t2, n+1

t3, n+1

t4, n+1

Time Figure 9.4. Measured time intervals of the pendulum harmonic motion.

Viscosity

373

pendulum. The mirror, made of flat glass with an aluminum layer, reflects the light beam to the photo-detectors FD1 – FD3. Photo-detectors FD1 and FD2 are placed symmetrically to the standstill position of the light beam at a mutual distance of approximately 100 mm. The precise adjustment of the symmetry is not necessary, as the difference between the standstill position of the beam and the center of detectors is obtained with good precision by mathematical reconstruction of the dumping harmonic curve. The total course of the beam is approximately 3 m, i.e. the distance between the light source and the axis of the torsional pendulum is approximately 1.5 m. The photo-detector FD3 indicates the maximum amplitude of the beam deviation. After the start up of the deviation system when the beam crosses the photo-detector FD3, the deviation of the systems stops and the initiator of oscillations returns to the starting position. 9.2.1.5. Control PC with the interface

The SAPI-1 control computer was used. As it follows from Figure 9.4, the control device −t during every period. The space between must measure all the eight time sections t1,n 8,n horizontal lines 1, 2 and 3, 4 represents the thickness of the beam passing photo-detectors , t , t , andt . In order to FD1 and FD2, to which then belong the time intervals t1,n 7,n 3,n 5,n eliminate the thickness of the beam, each of these intervals is halved, whereby this half is added to the adjacent time interval, thus tm,n = 0.5tm−1,n + tm,n + 0.5tm+1,n

(9.34)

where m = 2, 4, 6, and 8. The whole process of reading the time intervals and saving them to the PC memory takes a maximum of 1µs, so there is no significant error in the time interval measurement. During the process of data storage, the process of data reading takes place. After reading and storage of the chosen number of periods, the interface is set into the starting state and the computer begins to process the stored data into the final viscosity value. 9.2.1.6. Calculation of viscosity from the measured time intervals

During the processing of time intervals, the mathematical reconstruction of parameters of the dumping harmonic oscillations of the torsional pendulum takes place. The most important variables are the logarithmic decrement of dumping and the period of oscillations. This process was also described by Ohta et al. (1975), but with regard to some errors in the original literature, substantial parts of the calculation are given also here. The period of oscillations can be calculated either starting from the maximum amplitude n up to n + 1 Tn, max = 0.5(t2, n + t2, n+1 ) + t3, n + t4, n + t1, n+1

(9.35)

374

Physico-chemical Analysis of Molten Electrolytes

or from one crossing of the zero to the next one Tn, zero = 0.5(t1, n + t1, n+1 ) + t2, n + t3, n + t4, n

(9.36)

The average period is then T =

Tn N

(9.37)

where N is the number of processed periods. The oscillation of the torsional pendulum represents damping harmonic oscillations, which can be described by the general equation 2π t δt A = A1 exp − sin T T

(9.38)

where A is the immediate amplitude in the time t, A1 is the maximum amplitude in time t = 0, δ is the damping constant (the logarithmic decrement), and T is the oscillation period. For apparent sinusoidal members A1,n and A3,n (the index 1 holds for the right-sided and 2 for left-sided amplitudes) it holds A1, n = a/ sin(π t1, n /T )

(9.39)

A3, n = a/ sin(π t3, n /T )

(9.40)

where a is the half distance between FD1 and FD2. The distances between the standstill position of the light beam and the center of the photo-detectors for individual oscillations are d2, n = A2, n sin π(1/2 − t2, n /T ) − a

(9.41)

d4, n = A4, n sin π(1/2 − t4, n /T ) − a

(9.42)

where A2, n = 0.5(A1, n + A3, n )

(9.43)

A4, n = 0.5(A3, n + A1, n+1 )

(9.44)

Viscosity

375

And the true distance between the standstill position of the light beam and the center of the photo-detectors is d=

1 (d2, n + d4, n ) 2N

(9.45)

where N is the number of processed periods. The corrected time intervals for nonzero d are ∗ 2 2 2 3/2 t1, n = t1, n − T ad /[2π(A1, n − a ) ]

(9.46)

∗ 2 2 2 3/2 t3, n = t3, n − T ad /[2π(A3, n − a ) ]

(9.47)

For corrected sinusoidal terms A∗1, n and A∗3, n then it holds A∗1, n = a/ sin(π t1, n /T )

(9.48)

A∗3, n = a/ sin(π t3, n /T )

(9.49)

and finally the logarithmic decrement can be expressed in the form δ=

ln A∗n − ln A∗n+N N

(9.50)

Practically, the logarithmic decrement is determined as a tangent to the linear dependence of the logarithms of the right-sided as well as the left-sided amplitudes on time, where the final value of the decrement is the average of both the values. Relations describing the dependence of viscosity on the logarithmic decrement valid for a spherical body were derived by Verschaffelt (1915). According to these relations η=

3δI 1 4π R 3 T0 (2 + b1 R + p)

(9.51)

where I is the moment of inertia of the oscillation system and for p it holds p=

b1 R + 1 (b1 R + 1)2 + b12 R 2

(9.52)

The parameter b1 can be calculated according to the equation + b1 =

πρ ηT

where ρ is the density of the measured liquids.

(9.53)

376

Physico-chemical Analysis of Molten Electrolytes

If the constant 3/4 in Eq. (9.51) is replaced by 2/5 this equation can also be used for a cylindrical measuring body, when the height of the cylinder is equal to its diameter. Daneˇ k et al. (1983) supposed that Eq. (9.51) can be applied for an arbitrary measuring body; in such case it will not be an absolute measurement. The value of this constant must be determined experimentally by calibration. Azpeititia and Newell (1958, 1959) derived later following equations for a cylindrical measuring body πρhR 4 0 [A(p − q)x −1 + Bx −2 + Cqx −3 ] = 2 − I ω

(9.54)

πρhR 4 1 0 [A(p − q)x + Bx + Cpx] = − 1 + − I ω ω

(9.55)

and

where A = 4 + R/h, B = [16(4π – 3(31/2 ))/(9(31/2 )π )]R/h + 6 = 2.407949 R/h + 6, C = (17/9) R/h + 3/2, ω = T0 /T, p = 1/{2[ + (1 + 2 )1/2 ]}1/2 , q = 0.5p, = δ/2π , 0 = δ 0 /2π, I is the moment of inertia of the oscillation system, R is the radius of the cylinder, h is the half height of the cylinder, and x=R

2πρ ηT

1/2 (9.56)

Viscosity can be calculated from the damping constant using Eq. (9.54) and from the increase of the period T compared with T0 using Eq. (9.55). All corrections of the measured damping are involved in the term 0 0 = 1 − 2 + 3 where,

(9.57)

1 is the damping constant of the whole oscillation system in gas, 2 is the damping constant of the only measuring body in gas, calculated from the known viscosity and density of the gas at the given temperature, and

Viscosity

377

3 is the damping constant of the part of the holding rod immersed into the measured liquid, calculated from the viscosity obtained without this correction. The accuracy of Eqs. (9.54) and (9.55) is better than 0.1%. Brockner et al. (1979) derived also an equation for a hollow cylinder. In this case, the melt is placed inside a hollow cylinder, which enables to measure the viscosity of liquids with a high vapor pressure at the given temperature. The moment of inertia of the whole system, I, is determined using an additional ring of the known moment of inertia calculated from the equation I=

M 2 (r − r02 ) 2

(9.58)

where M is the mass of the ring and r and r0 is the outer and inner radius of the ring, respectively. The period of oscillation is defined by the equation + T = 2π

I K

(9.59)

where K is the torsion constant of the wire. The moment of inertia of the torsional pendulum can be calculated from the equation I = I0

T2 T 2 − T02

(9.60)

where I is the moment of inertia of the additional brass ring and T and T0 are the periods of oscillation with and without the additional brass ring, respectively. 9.2.2. Falling body method

The method of falling body is used with advantage for the simultaneous measurement of density and viscosity of melts with the viscosity in the range from 1 to 103 Pa · s. The measurement is based on Stokes’ law. The principle of this method is to measure the speed of a partially counter-balanced body, which moves up or down in the melt, depending on whether the body is under-balanced or over-balanced. The method was used by Daneˇ k and Licˇ ko (1981). As the measuring body is alternatively a sphere with 15-mm diameter or a cylinder with conical ends with a 10-mm diameter and the total height of 15 mm, both made of the Pt10Ir alloy were used. The body was suspended from the scale of the analytical balance A3/200 Meopta using the Pt40Rh wire of 0.3 mm in diameter and immersed into the investigated melt. The speed of movement of the body in the melt at different off-balance positions was measured on the scale of the analytical balance provided by two fluctuant phototransistors that at the

378

Physico-chemical Analysis of Molten Electrolytes

movement of the balance optical beam, switched on and off the electronic stopwatch. The time was measured with an accuracy of 1 ms. Changing the off-balance of the analytical balance, the speed of movement of the body in the melt was changed too. As it follows from Stokes’ law, the viscosity of the investigated melt can be calculated from the slope of the body speed dependence on the value of the off-balance according to the equation η=k

m v

(9.61)

where m/v is the slope of this dependence and k is a constant of the apparatus, which depends on the dimension of the body and the crucible, and on temperature. The value of the constant k was determined for every body by calibration using the melt of known viscosity. At measurements in the temperature range 1000–1600◦ C as the calibration substance, the alkali-calcium glass NBS 710 was used. For viscosity around 1 Pa · s, glycerine was used as the calibration substance. The speed of the body movement was measured after the equilibrium speed was reached at its surfacing and immersion. The resulting viscosity was calculated as the average of both the values, which eliminated the effect of the surface tension on the melt/suspension wire interface. From the intercept of both dependencies of the speed of the body on the off-balance at emerge and immerse, the value of the equilibrium mass of the body in the investigated melt, mt , was obtained, from which the density of the melt could be calculated according to the equation ρ=

m0 − mt + δm V0 (1 + αt)3

(9.62)

where m0 is the mass of the body in air, V0 is the volume of the body at room temperature, and δ m is correction to the surface tension effect exerted on the wire, which is significant especially in melts with high surface tensions. The linear coefficient of expansion α was determined by dilatometric measurement. Daneˇ k and Licˇ ko (1981) verified their apparatus in a very broad temperature range, from room temperature up to 1600◦ C. They measured the density and viscosity of glycerine, boron oxide, sodium tetraborate, and sodium-calcium silicate glass with the composition 17 mass % Na2 O, 10% CaO, 73% SiO2 . The results of measurement are in agreement with the literature data regarding the precision of the apparatus used. For the measurement of density and viscosity using the falling body method, some effects, influencing the precision of the measured quantities, should be taken into account. First it is the dimension and the mass of the measuring body, eventually its shape. At the possible lowest off-balance of 10 mg, when the beam of the balance optical system still passes the distance between the two phototransistors, the effect of the moment of

Viscosity

379

inertia of the balance arms plays an important role in low viscosity liquids. This damping effect of the balance could be partially compensated using a bigger measuring body. As it was, however, found in the measurement of the viscosity of sodium tetraborate, the error increases rapidly at viscosities lower then 1.5 Pa·s. The effect of the shape of the measuring body exerts only in the realization of absolute measurements using Stokes’ law, which is valid for fall of a spherical particle in an infinite viscous environment. Thus, the relations for the calculation of viscosity could not be used when using a body with conical ends. In this connection also, the dimension of the crucible and the height of the level of the melt should be taken into account. These effects were studied especially by Francis (1933) and Hunter (1934). The next source of errors is the effect of the surface tension on the interface melt/suspension wire, which is manifested especially in density measurement. According to Riebling (1963), this effect can be eliminated introducing the correction δ m (c.f. Eq. (5.34)), which could be calculated from the relation δm =

0.46π dσ g g

(9.63)

where d is the diameter of the suspension wire in cm, σ is the surface tension of the investigated liquid in mN · m−1 , g is the constant due to gravity, and the constant 0.46 is the correction factor valid for thin wires introduced by Wilhelmy (1863). The value of this factor was verified in the measurement of density of the NBS 710 melt using a different measuring body (Pt10Ir sphere of 1.766 cm3 in volume and Pt40Rh cylinder with conical ends of 0.654 cm3 in volume). Practically the same value of the correction constant was obtained as by Wilhelmy (1863). It could be, however, expected that its value would depend on the contact angle of the suspension wire and the melt. The total experimental error at density measurement using the described experimental device did not exceed ± 0.005 g·cm−3 . In Figure 9.5, the results of the density and viscosity measurements of molten sodium tetraborate are shown. The dependence of the speed of the partially counter-balanced body on its mass, when it moves up and down in molten sodium tetraborate at 935◦ C, was registered. The mass of the body in air was 28.5958 g and its volume at the working temperature was 0.6755 cm3 . From the slope of the dependencies, when the body is moving up and down, the viscosities η1 = 0.1755 Pa·s and η2 = 0.1866 Pa·s, respectively, were calculated. The mean value is η = 0.1811 Pa·s, while the value given by Janz (1991) is η = 0.1789 Pa·s. The equilibrium mass in the liquid sodium tetraborate at 935◦ C has been found to be m0 = 27.221g. The correction due to the surface tension on the interface melt/suspension wire, δm = 0.017g, was calculated with regard to the surface tension value of sodium tetraborate given by Janz (1991). The density of the melt was then calculated as ρ = 2.064g·cm3 , while the value given by Janz (1991) was

380

Physico-chemical Analysis of Molten Electrolytes

27.30 27.28 m1 = (27.219 + 0.179ν) g

27.26

η1 = 0.1755 Pa.s

m(g)

27.24 27.22

m0 = 27.221 g

27.20 η2 = 0.1866 Pa.s

27.18 m2 = (27.222 − 0.1828ν) g

27.16 0.00

0.05

0.10

0.15 0.20 v (cm.s−1)

0.25

0.30

Figure 9.5. The dependence of the speed of the partially counter-balanced body on its mass when it moves down and up in molten sodium tetraborate at 935◦ C.

ρ = 2.066g·cm3 . As can be seen, the agreement between the measured and the literature data on viscosity and density of sodium tetraborate is very good. 9.2.3. Rotational method

For the measurement of the viscosity of liquids such as molten glasses, rotational method is frequently used. In the viscosity range 10 – 105 Pa·s the classical rotational devices can be applied. However, the same devices could also be applied for viscosities up to 107 Pa·s working in the so-called a-periodical mode of measurement. The principle of the rotational method consists of the shear stress measurement between two concentric cylinders, from which one is driven by a constant angle velocity and the viscous liquid situated in between them drags the second one. The shear stress, τ , depends on the geometry of the system according to the following equations τ=

τs (r/R)2

(9.64)

Viscosity

τs =

381

I 2π R 2 h

(9.65)

where r and R are the radius of the inner and outer cylinder, respectively, h is the height of the cylinders, and I is the moment of inertia of the system. In Eq. (9.65), the correction for the end effects is neglected. In Figure 9.6 the schematic arrangement of the experimental setup consisting of a cylindrical cup with a radius R rotating with a constant angle velocity ω, and inside a hollow cylinder of a radius r and the height h to be dragged by the liquid placed between them. Due to the shear stress of the liquid, the rotation of the inner cylinder is delayed by an angle α in comparison with the rotation of the outer one. The angle of delay α

α

A

A'

R h

r

ω

Figure 9.6. Schematic representation of the rotational method of liquids viscosity measurement. The meaning of symbols is given in the above text.

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Physico-chemical Analysis of Molten Electrolytes

depends on the viscosity of the melt according to the experimental relation η = α(a + bα)

(9.66)

where a and b are constants of the experimental setup obtained by calibration. The classical rotational method can be used for viscosities from 10−3 Pa·s up to approximately 105 Pa·s depending on the geometry of the measuring cylinders with a relative error ± 3 %. A wide-range rotational viscosimeter for measuring viscosity of glasses was described by Hamlík (1983). The author adapted a commercial viscosimeter RHEOTEST 2. This apparatus measures the viscosity, shear stress, and the shear velocity and is provided by different measuring cylinders enabling to measurement the of viscosity of liquids in the range 10−3 – 4.104 Pa·s and also a tempered vessel for measurement in the temperature range from –60 to 300◦ C. The angular momentum is read using a dynamometer and is indicated by an analog α-meter in the range 0–100 mV. The device is equipped by a 12-stage mechanical gearbox and a two-speed motor. This broad variability enables a sufficiently smooth adjustment of the device to the viscosity of glasses. The commercial viscosimeter was completed by the high-temperature vertical tube furnace with a control device and evaluation system of the firm NETZSCH, enabling to measure viscosity of glasses up to 1600◦ C. The viscosimeter and the furnace were joined by means of a pedestal, enabling the adjustment of the system to one common axis. The temperature was measured and controlled by two Pt30Rh–Pt6Rh thermocouples. The control thermocouple is placed at the outer side of the heating body and passes through the rim of the upper plate of the furnace. The measuring thermocouple passes through the lid of the furnace down to the surface of the melt. Both the thermocouples are protected by corundum capillary. Both the crucible and the inner hollow cylinder are made of Pt30Ir alloy. Two methods of measurement are used. The first one is the classical rotational method employing the original properties of the viscosimeter. Because the measurement of viscosity of molten glasses requires a configuration with a free spindle, the determination of the angular momentum I as the function of α and the shear stress was not taken into account. Direct calibration using the experimental relation η/α = f(α) showed to be very simple and relatively accurate. Based on experimental results, the linear function has been chosen in the form η = α(a + bα)

(9.67)

where constants a and b are determined by calibration. Using the rotational method described by Hamlík (1983), the viscosities of molten glasses in the range from 10 to 105.3 Pa·s can be measured. For lower viscosities, other

Viscosity

383

methods have to be used, as the dynamometer is not equally loaded and the values of α oscillate. Higher viscosities cannot be measured since α exceeds the limit of the dynamometer. An extension of the measuring range towards viscosities up to 107.5 Pa·s was possible using the aperiodic motion of the spindle. The measuring procedure is as follows. Using the driving unit, the dynamometer is twisted to α = 100 mV. When switching off the driving motor, the spindle turns back to zero α due to the force of the dynamometer spring. This motion is recorded by the α-meter. Choosing a constant short α interval, it is possible to regard the time of the motion in this interval as the function of viscosity. This function was determined experimentally in the form η = t (a + ln t)

(9.68)

where t is the time and a and b are constants for the time interval from 10 to 1500 s. The correct function of the viscosimeter and the chosen working procedures were checked by measuring the viscosity of standard glasses NBS 710 and NBS 717.

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Chapter 10

Direct Methods of Investigation Direct methods for studying the structure of molten salts are X-ray and neutron diffraction analyses, infrared and Raman spectroscopy, NMR (nuclear magnetic resonance) measurement, and also very recently, XAFS (X-ray Absorption Fine Structure) measurement in melts, were developed. However, the most frequently used direct methods are X-ray and XAFS measurements, Raman spectroscopy, and NMR measurements. Therefore these three methods of direct investigation will be briefly described here. Sometimes, molecular dynamics and Monte Carlo simulation also are regarded as direct methods. On the other hand, using these two methods, no direct measurement is made and thus they will not be discussed here.

10.1. X-RAY DIFFRACTION AND XAFS MEASUREMENTS

X-ray diffraction and XAFS (X-ray Absorption Fine Structure) measurements are powerful techniques to investigate the local atomic structure of solid and liquid phases and have been successfully applied to many types of materials such as solutions, catalysts, amorphous solids, etc. However, the X-ray diffraction and XAFS measurement of molten salts bring a number of common difficulties due to high temperature and their specific properties, as in the experimental work, (a) (b)

(c) (d)

very expensive and energetically demanding experimental devices are needed; the influence of oxygen and moisture must be avoided, since many salts gather oxygen or are hygroscopic; however, Okamoto et al. (1999), solved this problem using a quartz cell container; it is difficult to obtain a clear signal at high temperatures; a thin film of the melt must be prepared and kept during the measurement, which is the most difficult problem.

Only a few X-ray diffraction and XAFS studies at high temperatures can be found in the literature mainly due to technical difficulties in experimental work. E.g. Okamoto et al. (1998, 1999) measured the X-ray diffraction of some molten rare earth and uranium trihalides. The obtained X-ray diffraction data were analyzed using molecular dynamics technique. This procedure is almost standard in the structural analysis of molten salt systems. 385

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Physico-chemical Analysis of Molten Electrolytes

In particular, only very few papers were published on XAFS measurement of molten salts. The reason why the measurement of molten salts has not been widely used is again the peculiar experimental difficulty, mainly in the preparation of a liquid thin film. For example, the thickness of solid YCl3 to obtain a good XAFS data is estimated to be 140 µm. For high temperature XAFS measurements, up to 1273 K, the thickness of the liquid layer must be lower by one order. Mikkelsen et al. (1980) measured molten CuCl using a boron nitride cell and molten RbCl using a graphite cell. In the EXAFS measurements, Di Cicco et al. (1996) used samples of molten alkali bromides finely dispersed in a boron nitride pellet matrix. In the XAFS measurement of molten PbCl2 Ablanov et al. (1999) used a quartz cell with two glass pipes. They attained 10 µm thickness of the liquid film. A special quartz cell for XAFS measurements of molten salts was developed by Okamoto et al. (2002). The cell having a sand-glass form is shown in Figure 10.1. The solid sample is placed in the upper container and the cell is heated. When the sample melts, the fused sample runs down through the narrow measurement part of the cell, where the

Figure 10.1. Schematic arrangement of the inner compartment of the furnace for XAFS measurement using a quartz cell, according to Okamoto et al. (2002).

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XAFS spectrum is obtained, and the gathered melt is stored in the lower container. The measurement part of the cell is composed of two quartz sheets of 25 mm in perpendicular length, 5 mm in width, and with the 0.5 mm thickness, which are located parallel at an interval of 0.1 mm to form the melt path. The amount of the solid sample placed into the upper container is usually 2–5 g. The lower container has a relatively negative pressure with respect to the upper one because of the vapor pressure of the melt; thus, the melt passes along the measurement part into the lower container. Two kinds of cells differing in the width of the spacing were available to account for variations in viscosity, vapor pressure, concentration of absorber element, and temperature. The melt falling into the measurement part is detected by a change in the X-ray transmission intensity. The heating part with the quartz cell is placed inside the electric furnace provided with a water jacket. A kapton film 20 mm in diameter and 50 µm in thickness is used as the window for the X-ray beam. The furnace can be moved in horizontal and vertical directions with respect to the beam path to adjust the sample position. The atmosphere in the furnace could be controlled with respect to composition, pressure and gas flow. A thermocouple is set in the bottom of the heater part. The device could be used up to approximately 1000◦ C. The XAFS spectra were measured using transmission technique. A hard X-ray beam with energy of 5–20 keV was available by using the Si double-crystal monochromator in the beam-line. The absorption edges ranged from 5.965 to 17.038 keV. 1–3 s step-scan measurement for any energy was employed to obtain the XAFS spectra. The XAFS data analysis was performed using WinXAS ver.2.0, program developed by Ressler (1997). Structural parameters such as the inter-ionic distance, coordination number, and the Debye–Waller factor were obtained by the least-square fittings in k-space. Details of the data analysis were given by Okamoto et al. (2001). The influence of absorption by the quartz cell was examined by measurements of blank cells and some molten salt samples. In the blank test, the displacement of the absorption baseline was not observed. The lowest energy, which could be used in this system, was estimated to be 10 keV. Thereby, the XAFS spectra of some molten halides could be successfully obtained. The results showed that a small difference was observed among their results. It was concluded that the developed device is also suitable for the XAFS measurement of hygroscopic molten salts. Absorption spectra of molten YCl3 were obtained using a quartz cell with thickness of the slip 0.1 and 0.2 mm. The edge jump for the 0.1-mm-type cell was only 0.3, while it increased to 1.2 when the 0.2-mm-type cell was used. The first peak corresponding to the nearest Y–Cl correlation shows almost the same peak position in both types of cells and no significant difference was observed in the entire profile. The first peak in the result from the 0.1 mm type cell was slightly weaker than that from the 0.2 mm type. The curve-fitting results show that the difference in the structural parameters was very small. It was shown that this quartz cell could be used for an absorption edge around 17 keV.

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Physico-chemical Analysis of Molten Electrolytes

A detailed survey of the theoretical background, the experimental devices used for measurement, and the obtained results in the different types of molten salts was given recently by Ohno et al. (1994).

10.2. RAMAN SPECTROSCOPY

Raman spectroscopy has proved to be a very useful tool in the study of molten salts structure. The technical progress was marked by difficulties in the physical measurement as well as in the interpretation of the obtained spectra. Measurement of Raman spectra of molten salts using conventional spectrometers is not simple and often requires the application of a graphite windowless cell placed in a quartz tube in an argon atmosphere. Even when a good quality spectra was obtained, it is usually not easy to achieve a consistent analysis of the intensity data due to frequent band overlapping. However, advances made in the last decade, especially introduction of the deconvolution method, improved the evaluation of the experimental spectra. Laser Raman microprobes are commercially available, which provide improved tools for analytical purposes. The correct function of these devices was tested at low temperatures and consequently also high temperature furnaces were used in order to enable measurement at these conditions. At present, results of Raman spectroscopy studies were reported for a number of inorganic melts (Nakamoto, 1997). 10.2.1. Theoretical background

Spectroscopic methods provide specific information on the structure of molecules, on the chemical environment of atoms and their oxidation state. The basis of spectroscopy consists in monitoring changes that occur at the interaction of radiation with substances, eventually at radiation after the interaction of the excitation energy with the substance. Quite recently, an excellent review on the methods and recent results of Raman scattering from molten salts was given by Papatheodorou and Yannopoulos (2002). Important information about the structure and dynamics of molten salts is provided by the interaction between light and the system under investigation, which is called light scattering. In the inelastic or Raman scattering process, the energy of the scattered photons is different from the energy of the exciting radiation. Ferraro and Nakamoto (1994) summarized the fundamental physical principles of Raman scattering as follows. (a)

(b)

The electromagnetic radiation induces a dipole moment in the system, which depends on time. The oscillating dipoles emit secondary radiation or alternatively they “scatter” light. The major source of light scattering in the visible and the near ultraviolet parts of the spectrum is the electronic cloud of the molecules and not the quasi-static nuclei.

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(a)

389

(b)

Figure 10.2. Mechanism of the first-order Raman scattering; (a) anti-Stokes and (b) Stokes scattering.

(c)

Raman scattering is attributed to the coupling between the movement of the nuclei and the movement of the electrons. Alternatively, the deformability of the electron cloud (polarizability) depends on the nuclear configuration at any time.

In a quantum-mechanical treatment, Raman scattering is the result of an inelastic collision process between the photon and the elementary excitations of the medium. The photon either loses one or more quanta of vibration energy (Stokes lines) or gains one or more such quanta (anti-Stokes lines). In the first-order scattering, only one photon is involved, while in the second-order scattering, two photons are involved. The mechanism of the first-order Raman scattering is schematically shown in Figure 10.2. The scattering cross section may be divided into two parts, polarized and depolarized. Experimentally, the most common polarizations used in the study of molten salts and other isotropic liquids and glasses are those denoted as (i) (ii)

VV, indicating that the polarization direction of both the incident and scattered radiation are vertical to the scattering plane, HV, indicating that relative to the scattering plane the incident radiation is horizontal, while the scattered is vertical.

Both the VV and HV polarizations contribute to the isotropic scattering cross section while the anisotropic and HV scattering cross sections are proportional. The intensities of the isotropic and anisotropic scattered light are given by the equations Iiso (ω) = IVV (ω) − (4/3)IHV (ω) Ianiso (ω) = IHV (ω)

(10.1) (10.2)

where Ii (ω) (i = iso, aniso, VV, HV) is the experimentally determined intensity of scattered light, which in general can be expressed as Ii (ω) = Gexp ω(ωL ± ω)−4 Mi B −1 (ω, T )

(10.3)

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Physico-chemical Analysis of Molten Electrolytes

The first term, Gexp is a constant characteristic of the experimental set-up used, while the remaining terms are predicted theoretically. B(ω, T ) is a temperature function related to the Boltzman population factor n(ω) B(ω, T ) = n(ω)

anti-Stokes scattering

(10.4)

B(ω, T ) = n(ω) + 1

Stokes scattering

(10.5)

where n(ω) = exp

ηω −1 −1 kT

(10.6)

Finally, Mi is introduced for convenience and represents the integral function of polarizability. It is the only function, the value of which depends on the polarizability characterizing the scattering medium. Iiso (ω) Raman spectrum depends only on the vibration motion and Ianiso (ω) depends on both vibration and reorientation motions of the molecule. In addition to the familiar qualitative use of the depolarization ratio, ρ = IHV /IVV , to identify symmetric modes (i.e. ρ < 0.75), the quantitative studies of Iiso and Ianiso may provide detailed information about the vibration and reorientation relaxation. 10.2.2. Characteristic features of Raman scattering in melts

From the physico-chemical point of view, molten salts are a class of liquids having many microscopic and macroscopic properties similar to the corresponding properties of other (molecular, atomic) liquids. However, the experimental and theoretical evidence accumulated during the past 50 years shows that molten salts exhibit individual or group peculiarities that complicate the understanding of their morphology. The polarizability of ions forming the melt gives rise to light scattering either due to electrostatic field fluctuation around the ion and/or to polarizability changes due to bond formation within the melt (i.e. “complex” formation). An isolated n-atom molecule has 3n degrees of freedom and 3n−6 vibration degrees of freedom. The collective motions of atoms, moving with the same frequency and which in phase with all other atoms, give rise to normal modes of vibration. In principle, the determination of the form of normal modes for any molecule requires the solution of equation of motion appropriate to the n-symmetry. Methods of group theory are important in deriving the symmetry properties of the normal modes. With the aid of the character tables for point groups and the symmetry properties of the normal modes, the “selection rules” for Raman and IR activity can be derived. For a molecule with a center of symmetry, e.g. AX6 , octahedral molecule, a non-Raman active mode is also IR active, whereas for the BX4 tetrahedral molecule, some modes are simultaneously IR and Raman active.

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The vibration properties of a group of isolated molecules change drastically when these molecules are “condensed” to form a crystalline solid. The influence of the neighboring “molecules” and the surrounding crystalline lattice will alter the vibration modes of the molecules. The long-range order correlates the atoms in the crystal, and the vibrations are described in terms of lattice waves rather than by free molecular modes. Upon melting, the long-range order and space symmetry of the solids are destroyed. In principle, the vibration modes of the liquids can be considered as a long-wavelength limit of the solid vibrations and thus certain internal and/or external modes may be present in the vibration spectrum of the melt. The “internal” modes in melts have been investigated mainly by Raman spectroscopy in a variety of melt mixtures. The objective of these studies is the determination and characterization of possible discrete species (i.e. “complexes”) in the melt. It should be emphasized that in a condensed phase such as the melt, the vibration modes of the species cannot be treated as though they were isolated, i.e. without accounting for the perturbations due to the environment. Thus, in certain common anion mixtures of the type AX–NXn (N = polyvalent metal, A = alkali metal, X = halide) and at low concentrations of NXn , the forces within the species (e.g. FeCl4 , ScCl3− 6 ) are stronger than the forces between the species and the neighboring ions. It is then reasonable for interpretation of the spectra to consider isolated species as complex ions. On the other hand, when the interactions with the neighboring cations (e.g. A = Li or in mixtures rich in NXn ) are strong, the formation of discrete complex species will be drastically perturbed giving rise to other associated species and/or network-like structures. Finally, it is noteworthy that the vibration modes measured by Raman spectroscopy may arise from short-lived local structures in the melt. If the lifetime of the structure is long enough (10−12 s) so there is time for vibration and interaction with the exciting light and if the structure has “bonds” (with non-zero polarizability derivative), then Raman activity may arise. However, diffusion times in the melts are of the order of 10−11 s, and thus the local structure may not maintain its identity for long before exchanging ions with its immediate environment. It appears from the above discussion that caution is required in interpreting Raman (and IR) spectra of melts; otherwise misleading information on the structural properties of the melts and mixtures may be obtained. 10.2.3. Experimental techniques of measurement

A typical experimental set-up for measuring the Raman spectra of molten salts is shown schematically in Figure 10.3. The scattering plane is that of the page. A right angle (θ = 90◦ ) scattering geometry is generally used but back-scattering techniques (θ = 180◦ ) have been employed especially in cases of dark-colored melts. Lines from visible Ar+ and Kr+ ion laser, are usually sources for the incident light. CW laser power ranges from a few microwatts (5–10 mW) up to a few watts (1–3 W). The purpose of the focusing lens is to increase the laser power density at the scattering volume from where the scattered

392

Laser beam

Physico-chemical Analysis of Molten Electrolytes

Focusing lens

Melt

q

Collecting lens

Slit Dispersive System

Detector

Figure 10.3. Schematic diagram of the experimental set-up used for measuring Raman spectra of molten salts.

radiation is collected by a set of lenses. The aperture of the Collecting Lens (CL) determines the scattering collection angle . Voyiatzis et al. (1999) and Dai et al. (1992) have also used fiber optics as well as microscopy as CL systems for melts. The dispersive system is grating in a single, double, or triple monochromator. The detector system involves electronic amplification of signals obtained from a photomultiplier tube or a CCD detector. Quasi CW lasers, involving a chopper and lock-in amplifier, as well as pulsed lasers and gated techniques have also been used (Iida et al., 1997). Details for the currently used instrumentation can be found in Laserna (1966). It should be noted, however, that for a given melt with a specific scattering crosssection and at fixed laser power there are two main factors improving the intensity of the Raman signal. (i)

(ii)

The spectrometer and optics transmission. This implies careful alignment and matching of all optical components, a wide collection angle, and stability of the overall system. The quantum efficiency of the detection systems, e.g. the use of high efficiency PMT or intensified CCD detectors.

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High intensity Raman signals minimize the measurement time, which is a very important factor especially for the study of corrosive melts. Sample preparation for Raman studies occasionally present serious difficulties. Even high purity inorganic salts available commercially may contain traces of organic impurities, which upon melting the salt give a slight coloration and makes the melt fluorescent. Oxide formation during drying may also create serious problems. Filtration of fused salts through a sintered frit may be desirable, because the Tyndall scattering of solid particles usually increases the noise and background scattering will interfere with data at low Raman shifts. It is often used to treat the starting materials with activated charcoal in water to remove fluorescent and other organic impurities and then to re-crystallize the salt from the aqueous solution. Zone refining by melt crystallization is also necessary for high melting corrosive melts like metal fluorides. For most salts, however, the ideal means of purification are sublimation under vacuum. Fairly elaborate anhydrous preparation procedures are required for strongly acidic salts like halides of aluminum, zirconium, niobium, and zinc. Two types of Raman optical cells have been used so far. Fused silica in the form of cylindrical tubes of inner diameter 2–10 mm are the proper and simplest material for non-corrosive melts. Windowless cells made of graphite or noble metals have proved adequate for studying corrosive fluoride and/or oxide melts. Brooker (1997) measured the Raman spectra using a Laser Raman Microprobe Renishaw and a conventional spectrometer Coderg PHO. A super-notch filter served as a monochromator in front of the entrance slit of a single grating, which in turn disperses the Raman beam onto a 400 × 600 CCD detector. The Laser Raman Microprobe was equipped with a 632.8 nm helium–neon laser of 10 mW power and a 514.5 nm argon ion laser of 50 mW power with the appropriate super-notch filters. The laser beam was focused into the sample by a lens with an Olympus microscope and the back-scattered Raman light was collected by the same lens. Samples of molten salts were sealed in capillary tubes under dry nitrogen or vacuum. The conventional spectrometer consisted of a Coderg double monochromator equipped with a cooled PMT and was described in detail by Brooker et al. (1994). A 1 W laser was required to obtain spectra with adequate signal to noise ratio. A half-wave plate controlled the polarization of the incident beam. The 90◦ scattered light was analyzed with Polaroid films with accepted parallel or perpendicular polarized light. A quarter wave-plate in front of the entrance slit served to compensate for grating polarization preference. Gilbert et al. (1975) developed a special cell and furnace design for high temperature Raman spectroscopy, which was later substantially improved by Gilbert and Materne (1990). The cell is a modified version of the windowless cell used by Young (1964) for UV and visible spectroscopy. The graphite windowless cell is shown in Figure 10.4. The amount of pre-melted solid mixture added to the cell was adjusted so that all the formed liquid has to be retained in the space in between the windows and no melt is

394

Physico-chemical Analysis of Molten Electrolytes

∼4 cm

∼1 cm

Raman light Laser beam

Figure 10.4. Windowless graphite cell for Raman spectroscopy measurement of fluoride melts.

allowed to flow out of the cell. Such an arrangement enabled to insert the cell into a simple evacuated quartz tube, heated in a quartz furnace provided with the necessary openings and insulation. The cell was made of graphite and machined so that no contact can occur between the drop of melt and the quartz tube in which the cell was placed. Even after experiments lasting several hours, the tube remained clear. However, in some cases, depending on the volatility of the investigated melt, a white deposit on the quartz tube around the graphite cell could be observed, indicating some evaporation of the melt and attack of the quartz. The 488.0 nm Coherent Radiation Model 520-B argon-ion laser of 300 mW power was focused on the center of the drop and the Raman light is detected perpendicular to both the laser beam and the sample tube. Qualitative polarization measurements were made by rotating the plane of polarization of the laser beam with a half-wave plate. In most of the spectra, it was not necessary to use a spike filter to attenuate the plasma lines. The spectra were recorded using a modified Cary 81 spectrophotometer employing a 9558A EMI photo-multiplier counting detection system. The furnace was water-cooled, consisting of a stainless-steel block inside ensuring a good temperature control, and the heating elements were of Kanthal. In order to decrease temperature fluctuations, the furnace openings were closed with water-cooled optically flat quartz windows. The furnace and the laser were mounted on an optical bench allowing

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adjustment of the beam. Slit widths of 5 to 6.5 cm−1 and a time constant of 0.1 s were used in all spectra. Due to the strong evaporation of some melts, fast recording of the spectra was necessary. In this case, the Raman spectrometer was interfaced with an IBM PS/2 30 microcomputer. The software controlled the spectrometer, the data acquisition, and the extraction of useful information from spectra, such as subtraction, smoothing, de-convolution, etc. In order to present the data on the screen in real time, 2200 measurements per second were recorded. With one point per cm−1 , the scan rate as fast as 1000 cm−1 /min could be achieved with a reasonable signal to noise ratio.

10.2.4. Raman spectroscopy in various systems

Systems of metal halides have been extensively investigated by a variety of physicochemical methods and by numerous authors. The reason is the importance of these systems for many important metallurgical processes as well as the systematic investigation of different factors like charge, polarizability, and “complexing” in monovalent to pentavalent metal halides. A better understanding of molten salts can thus be obtained. In the following sections, the results of Raman spectroscopic studies of the systems containing monovalent to tetravalent cations will be presented. 10.2.4.1. Alkali metal halides

A series of experimental measurements in single alkali metal halides has been reported and their results have been discussed and interpreted in terms of dipole-induced-dipole and short-range order interactions between the ions in the melt. From these studies, it is evident that the intensity of the scattered light increases with increasing polarizability of both the anion and the cation. Raptis and McGreevy (1992), Papatheodorou et al. (1996), and Papatheodorou and Dracopoulos (2000) showed that for LiX melts containing high field and low polarizable Li+ cations, scattering arises mainly from the fluctuation anion polarizability, while for the CsX melts both the high polarizable Cs+ cation and X− anion contribute to the scattering intensity. The mechanism of light scattering from these melts is related to the fluctuations of the polarizability expressed in terms of the iso- and aniso-contributions. Raman spectra of alkali metal halide mixtures have also been reported and a detailed analysis was given by Papatheodorou and Dracopoulos (1995) and Papatheodorou et al. (1996). The studies involve systems LiCl–CsCl, LiF–KF, and LiF–CsF and the main findings can be summarized as follows. (i)

The scattering intensity from the LiX–MX (X = Cl, F; M = K, Cs) mixtures arises from contributions of the two cations, which appear to occupy individual sites (cages), each possessing a characteristic frequency ωLi and ωA . The observed

396

(ii)

Physico-chemical Analysis of Molten Electrolytes

intense scattering in the ωLi region and the frequency shifts, passing from the pure components into the mixture, are mainly due to the changes of short-range interactions and the loss of “symmetry” around the anion imposed by the local structure in the mixture. On the other hand, the observed intensity in the ωCs region is associated with the formation of highly polarizable configurations (clusters), which have a relatively long lifetime at low temperatures. The dramatic increase of the isotropic scattering intensity with increasing temperature is associated mainly with the Li–X interactions and the “symmetry” of the local structures. Thus, increasing the temperature strengthens the short-range overlap interactions and increases the local symmetry around the anion; both effects facilitate break-like fluctuations of the polarizability and lead to an increasing isotropic scattering intensity.

These findings provided a consistent picture of the investigated systems and helped in separating different interaction-induced polarizability mechanisms. Furthermore, the analysis pointed out the important role of the local structure symmetry around the anion to the variation of the isotropic scattering intensity with composition and temperature. 10.2.4.2. Halide systems containing divalent metals

Divalent metal halides and their mixtures with alkali metal halides have been among the first molten salt systems to be investigated by Raman spectroscopy. Studies of molten mixtures of the type MeX2 –MX (X = Cl, Br, I) have provided means of identifying and characterizing the species that may exist in the mixture and have information regarding the liquid structure of the MeX2 component. Binary mixtures containing MgX2 , CdX2 , and MeCl2 (Me = first row transition metals), which are known to form octahedral layer structures in solid state, are stabilized in alkali-halide-rich melts by the formation of MeX2− 4 tetrahedrons. For the same MeX2 –AX systems, four-fold coordination appears to prevail in the liquid structures of MeX2 -rich mixtures, including the structure of pure divalent halide. Even for NiCl2 , which is known for its high-octahedral ligand field-stabilization energy, it has been found by Badyal and Howe (1993) using thermodynamic and neutron diffraction measurements, that the fourfold coordination of nickel predominates in the structure of the NiCl2 –MCl melts in all compositions including pure NiCl2 . The situation is similar for the glass-forming ZnX2 (X = Cl, Br) melts, where the four-fold coordinated ZnX2− 4 species are stabilized in alkali-halide-rich mixtures, while a network-like structure of ZnX4 tetrahedrons bridged mainly by edges characterize the pure ZnX2 melts. Finally, the melting of HgX2 yields a molecular liquid involving X–Hg–X triatomic molecules, while in the mixtures with AX, − both tetrahedral HgX2− 4 and trigonal HgX3 species have been identified. One of the first studies of molten fluorides was that of the system BeF2 –MF (M = Li, Na). The four-fold coordination predominates in the melt structure with isolated BeF2− 4 species formed in

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mixtures rich in MF. With an increasing BeF2 content, bridging of tetrahedrons occurs and it was argued that, together, species like Be2 F3− 7 exist. ◦ The melting of α-BeCl2 at 415 C is followed by a ∼25% increase in molar volume giving a rather viscous liquid with a low conductivity of ∼0.01 S · cm−1 . These characteristics suggest that BeCl2 has properties similar to those of glass-forming inorganic liquids like ZnCl2 and As2 O3 . Cooling of the melt yields α-BeCl2 crystals and not glass. In contrast, the bulk glass is formed only through vapor transport, a situation, which has been also observed for As2 O3 . Spectra of liquid BeCl2 recorded up to 1100 cm−1 and temperature variations at the relative intensities of the bands at ∼328 and ∼275 cm−1 have been measured for both the glass and the melt. The intensities of the glass up to the devitrification temperature (∼200◦ C) do not change significantly in contrast to the liquid where rather fast and drastic changes take place. Pavlatou and Papatheodorou (2000) concluded that, for the glass/liquid the ∼328 cm−1 band is caused by stretching vibration of the BeCl4 tetrahedrons participating in edge-bridged chains, while the ∼275 cm−1 band is associated with the same species as involved in the construction of the cage-like structure through vertex bridging. Thus, in pure BeCl2 liquid/glass “chain” and “cluster” structures exist, participating in a temperature-dependent equilibrium. The high viscosity and low conductivity of the melt imply that the “chain” and “cluster” structures are neutral and have a high molecular mass. To ensure the electro-neutrality it is necessary to terminate these structures with BeCl3 -end units. The “chain” and “cluster” structures participate in different concentrations in glass and liquid. The glass and the low temperature liquid favor the “cluster” configurations, while at high temperatures, the edge-bridged “chains” dominate the liquid. 10.2.4.3. Halide systems containing trivalent metals

The structural properties of a large number of trivalent metal halide–alkali metal halide (LX3 –MX) melt mixtures have been investigated by Raman spectroscopy in different compositions including the pure LX3 component. Early studies of “model” systems like AlCl3 –MCl, AlF3 –MF, and YCl3 –MCl studied, (Papatheodorou, 1977; Brooker and Papatheodorou, 1983; Gilbert and Materne, 1990; Wilson and Ribeiro, 1999) have established certain common structural features as well as differences for these melts. For the AlCl3 –MCl melts, composition and temperature dependence studies have shown that pure AlCl3 is a molecular melt forming Al2 Cl6 dimers, which in the presence of alkali metal − halides, give in equilibrium AlCl− 4 and Al2 Cl7 species. For the AlF3 –MF system, changes of the relative Raman band intensities as a function of the melt composition suggested that at least two different coordination geometries were present for aluminum. Extensive 2− studies have shown that for binary melts rich in alkali metal fluorides, AlF− 4 , AlF5 , and 3− AlF6 are the predominant species at equilibrium. Due to the high corrosivity and volatility of AlF3 at high temperatures, binary mixtures with composition above 50 mole % AlF3

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Physico-chemical Analysis of Molten Electrolytes

cannot be practically investigated and thus the spectral aspects of pure molten AlF3 have not been revealed yet. The Raman spectroscopic studies of the YCl3 –MCl binary system have shown that the predominant species in melts rich in MCl (x(YCl3 ) < 0.25) are YCl3− 6 octahedrons. The thermodynamics of mixing has suggested high stability and thus long lifetimes of these species. With increasing YCl3 content, the structure of the mixture changes in a rather continuous way, where the YCl3− 6 octahedrons start sharing common chloride atoms (at x(YCl3 ) > 0.25) forming polynuclear structures. Finally, comparison of the solid-to-liquid spectral changes of pure YCl3 and the observation that there is practically zero change of molar volume upon melting YCl3 , suggested that the liquid structure is rather similar to that of the solid, i.e., it consists of distorted YCl3− 6 octahedrons sharing chloride atoms and forming a loose “network” structure. This structural model was further confirmed by Neilson and Adya (1977) by neutron diffraction measurements in liquid YCl3 , where a direct determination of the local structure suggested a coordination number for Cl around Y of 5.9. For the molten mixtures of the system FeCl3 –CsCl, no changes in the Raman spectra have been observed from the melts rich in CsCl up to 50 mole % FeCl3 . Characteristic spectra of FeCl− 4 have been obtained, indicating that “isolated” tetrahedral species are present in these compositions. The addition of iron (III) chloride into the equimolar FeCl3 –CsCl mixture drastically changes the FeCl− 4 spectra. Certain changes observed in the spectra resemble those measured for the AlCl3 –MCl systems. The bands with the maximum intensity at x(FeCl3 ) ≈ 0.66 fit well to the frequencies of Al2 Cl− 7 . These bands were thus assigned to Fe2 Cl− ion consisting of two tetrahedrons bound by an apex. 7 The Raman spectra of molten iron (III) chloride are best characterized as a structure, − where neutral Fe2 Cl6 and charged Fe2 Cl+ 5 and Fe2 Cl7 are the predominant species. + The Fe2 Cl5 species compatible with the measured frequencies would consist of a FeCl4 tetrahedron bound by edge with a trigonal FeCl3

Cl Cl

Fe

Cl Cl

+ Fe − Cl

Such a species could be formed by self-ionization of the molecular melt according to the reaction − 2Fe2 Cl6 ↔ Fe2 Cl+ 5 + Fe2 Cl7

(10.7)

The dissociation of Fe2 Cl6 and the presence of ionic species account for the almost ionic conductivity of the FeCl3 melt. The strong coulombic forces between the two ions involved and the spatial flexibility of the corner-connected tetrahedrons of Fe2 Cl− 7 will

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allow a more tight packing of the molecules and ions in FeCl3 , relative to pure AlCl3 , which gives an account of the molar volume differences between the two melts. In conclusion, out of the eight iron (III) centers participating in the reaction (10.7), seven are in four-fold coordination and one, in a three-fold coordination. In other words, the expected average coordination number for molten iron (III) chloride should be lower than four and this is in agreement with the finding of neutron diffraction data obtained by Price et al. (1998), which gives an average coordination number of 3.8. Information regarding the structure of molten rare earth metal chlorides has been derived either directly from the scattering experiments (neutron, X-ray, Raman) or indirectly from the thermodynamic and transport properties. It was first suggested by an extensive Raman study performed by Papatheodorou (1977) that the structure of molten YCl3 may be a weak network of distorted chlorine-sharing octahedrons. Raman studies of other rare earth metal fluorides, chlorides, and bromides, (performed by Metallinou et al. (1991), Dracopoulos et al. (1997, 1998), Photiadis et al. (1998), and Chrissanthopoulos and Papatheodorou (2000)), indicate that the octahedral network-like structure is a general feature of all LX3 (X = F, Cl, Br). Systematic Raman spectral changes with composition and temperature observed for the systems involving LaCl3 , NdCl3 , GdCl3 , DyCl3 , HoCl3 , YCl3 , LaBr3 , GdBr3 , NdBr3 , and YBr3 have been interpreted, as indicated that the six-fold coordination around the rare earth metal cation is preserved at all mole fractions. Detailed Raman spectroscopic measurements of a series of LX3 –MX (X = F, Cl, Br) indicate that the spectral behavior and structure of these melts are very similar, especially the dilute mixtures of rare earth metal halides. On the other hand, measurements of the total structure factor of molten LCl3 show certain small but systematic differences depending on the rare earth metal cation size. Raman spectroscopy of molten Na3AlF6 in CsCl was performed by Brooker et al. (1994, 1995), who have found two bands at 556 and 622 cm−1 . These bands − were assigned to the AlF3− 6 and AlF4 species, respectively. On the other hand, Tixhon et al. (1994) and Gilbert et al. (1996) found it necessary to include a third band at about 500 cm−1 and these authors invoked a model, which included an unusual pentavalent 2− − aluminum species with a revised assignment for the bands as AlF3− 6 , AlF5 , and AlF4 −1 −1 for the bands at 500, 556, and 622 cm . The envelope from 450 to 700 cm is very broad in pure sodium cryolite and the curve analysis may not produce a unique solution. The problem of existence of pentavalent aluminum species seems to be thus incompletely solved. Brooker (2000) has measured the Raman spectra of cryolite in molten FLINAK also with the goal to establish the identity of the aluminum fluoride species in this molten medium. He found that the Raman spectra from matrix-isolated aluminum ions in the LiF–NaF–KF eutectic exhibit the characteristic pattern of the discrete octahe3− dral AlF3− 6 ion both in the matrix-isolated solid and in the melt. This octahedral AlF6 ion seems to be the only species in the melt over the cryolite ratio range from 23 to 8 and up to 750◦ C.

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Physico-chemical Analysis of Molten Electrolytes

Gilbert et al. (1996) have shown that the aluminum ion prefers fluoride over chloride to such an extent that when Na3AlF6 is dissolved in NaCl, no chloride enters the aluminum coordination sphere and no evidence of mixed aluminum fluoro-chloro complexes is found. Raman spectra of (M, M )F–AlF3 (M, M = Li, Na, K) molten salts at 1293 K have been obtained by Robert and Gilbert (2000). The intensity ratios between the bands that are characteristic of the different complexes, are strongly affected by the M / M ratio, especially when one of the alkaline metal cations is Li+ . Its presence, together with another cation seems to produce an increase in the acidity of the melt. The result of the de-convolution of the spectra compares well with vapor pressure data, showing the same kind of deviation. Quantitative modeling has not been possible because there is a lack of thermodynamic data allowing a comparison with spectroscopic results. 10.2.4.4. Halide systems containing tetravalent metals

Raman spectroscopic studies regarding the structure of molten tetravalent metal halides and their mixtures with alkali metal halides are rather limited. Solutions of ZrF4 and ThF4 in molten mixtures rich in LiF–KF eutectics have been investigated in the early 1970s 4− by Toth et al. (1973). These studies have suggested the formation of ZrF2− 6 , ZrF8 , 3− and ThF7 complexes in the mixtures. Due to high melting points and volatility of these tetravalent metal fluorides, the investigation of these mixtures in a wide range of composition was impossible and the structure of the pure salts is unknown. Recently, systematic Raman spectroscopic investigations of the molten mixtures ZrCl4 –CsCl, ThCl4 –MCl, and ZrF4 –KF have been performed by Photiadis and Papatheodorou (1998, 1999) and Dracopoulos et al. (2001), providing structural information for both the melt-mixtures and the pure components. The main spectral characteristics of solid ZrCl4 do not change with increasing temperature. At 430◦ C, just below melting, seven bands are clearly seen. Upon melting, most of these modes appear to be transferred into the liquid. However, a new high intensity polarized band appears in the spectra at 375 cm−1 , which is presumably that of the stretching frequency of the ZrF4 tetrahedrons, indicating that monomers are also present in the liquid. From the liquid spectra, it follows that at least two different types of species are present in the liquid phase: monomeric ZrCl4 with a main polarized band at 375 cm−1 and a “polymeric-like” species with a main polarized band at 404 cm−1 . Measurement of the liquid-phase spectra at different temperatures confirmed the presence of the monomers and suggested that an equilibrium may exist in the melt of the type (ZrCl4 )n (l) ↔ nZrCl4 (l)

(10.8)

The extent of polymerization (value of n) could not be calculated, however, it is assumed that polymerization in the melt is rather small and more likely, that the molecular

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liquid is composed of either Zr2 Cl8 dimers or Zr6 Cl24 hexamers in equilibrium with monomers. Measurements of Raman spectra of solid and molten Cs2 ZrCl6 and CsZr2 Cl9 at − different temperatures showed that the “isolated” molecular ions ZrCl2− 6 and Zr2 Cl9 2− are present in both phases. The ZrCl6 octahedrons are the predominant species up to x(ZrCl4 ) < 0.33. In the composition range 0.33 < x(ZrCl4 ) < 0.66, the spectral changes with composition and temperature suggest an equilibrium involving three ionic species: − 2− − ZrCl2− 6 , Zr2 Cl9 , and Zr2 Cl10 (or ZrCl5 ). At mole fractions rich in ZrCl4 (x(ZrCl4 ) > 0.66), the spectra indicate an equilibrium between the ionic Zr2 Cl− 9 , the ZrCl4 monomers, and the (ZrCl4 )n polymer-like species. All data suggest that the value of n is small and most probably hexamers and/or dimers are the predominant “polynuclear” species in melts rich in ZrCl4 . Due to the fact that the ratio of ionic radii r(Zr4+ )/r(F− ) almost equals that of r(Th4+ )/r(Cl− ), the structural behavior of the corresponding binary melts is expected to be similar. Thus, it appears that an “isomorphism” exists in these melts. Raman spectroscopic measurements for molten ThCl4 –MCl (M = Li, Na, K, Cs) were possible at all compositions including pure ThCl4 melt. The data indicated that in molten mixtures rich in alkali metal chlorides, the predominant species are ThCl2− 6 octahedrons in equilibrium with ThCl3− pentagonal bipyramids. A similar picture was found for the alkali-fluoride7 2− 3− rich melts with ZrF4 . Both ZrF6 octahedrons and ZrF7 pentagonal bipyramids are formed. At mole fractions x(TX4 ) < 0.33 (TX4 = ZrF4 , ThCl4 ), an equilibrium between the two species exists 3− − TX2− 6 + X ↔ TX7

(10.9)

The equilibrium shifts to the right with decreasing TX4 mole fraction and increasing temperature. For mole fractions up to x(TX4 ) = 0.66, both binaries show similar behavior. Thus, at x(TX4 ) > 0.33, with increasing x(TX4 ), the frequency of the ν1 (A1g ) band shifts continuously to higher frequencies and new bands appear in the spectra. At about x(TX4 ) ≈ 0.66, the spectra is characterized by two polarized and two depolarized bands. The composition and temperature dependence of the Raman spectra for both the systems were interpreted by Dracopoulos et al. (2001) and formation of bridged octahedral species 2− 2− T2 X2− 10 and the T3 X14 species in equilibrium with “free” TX6 octahedrons in the melt was supported. In melts with x(TX4 ) > 0.7, measurements were possible only for the thorium system. The observed continuous band shifts support the view that the bridging of octahedrons by edge extends, yielding chains of the type (Tn X4n+2 )2− and (Tn X4n−2 )2+ , where the end T atoms of the chain are six- and four-fold coordinated for the chains of anions and cations, respectively. Finally, the chain octahedral ionic structures appear to be the predominant species in pure molten ThCl4 . A mechanism creating oppositely charged species having vibration

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Physico-chemical Analysis of Molten Electrolytes

frequencies verified by the Raman spectra could involve a self-ionization scheme of the type nThCl4 ↔

1 1 (Thn Cl4n−2 )2+ + (Thn Cl4n+2 )2− 2 2

(10.10)

Increasing the temperature induces breakage of the bridges between the octahedrons and shifts the reaction (10.10) to the left or/and lowers the n value. A similar self-ionization scheme probably occurs for pure molten ZrF4 .

10.3. NUCLEAR MAGNETIC RESONANCE

Nuclear magnetic resonance (NMR) is a powerful tool in the study of the structure of molten salts. It often offers significant advantages against diffraction methods and vibration spectroscopy, because it examines directly the properties and behavior of a specific element. With the arrival of high-field super-conducting magnets, high-temperature NMR, and the ability to examine nuclides with a quadruple moment, it was possible to investigate routinely a wide range of inorganic materials. A specific class of problems, which NMR has been applied to, includes the investigation of the static structure and dynamic behavior of inorganic melts. Specific problems concerning inorganic materials, where NMR has been applied, include the structure of amorphous materials, melts, and the detection of small amounts of phases in mixtures. 10.3.1. Theoretical background

The Nuclear Magnetic Resonance (NMR) spectroscopy is based on measurement of the high frequency radiation absorption by the sample placed in the magnetic field. Magnetically active samples using the NMR determinable atomic nuclei are only those, having a nonzero atomic spin. They are thus characterized by a nonzero spin quantum number, which can either be an integer (e.g. 1, 2, etc.) or a half-integer ( 12 , 32 , 52 , etc.). The spin of most nuclides is between 0 and 72 . The fundamental property of a nucleus is → the magnetic moment, − µ , given by − → − → µ =γ J

(10.11)

− → where γ is the gyromagnetic ratio, which is a constant for each nuclide, and J is the angular momentum of the nucleus. Each nuclide has 2I + 1 energy levels, characterized by a quantum number m, with values I , I − 1, I − 2, . . . , −I . In absence of the magnetic field, these energy levels

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have the same energy (they are degenerate), but the degeneracy lifts under a magnetic field interaction between them and the field. The energy difference between these states, E, is given by E = |γ ηB|

(10.12)

where η is the Planck’s constant and B is the magnetic field at the nucleus. A nucleus can increase or decrease its energy only by absorbing or emitting a photon with frequency ν, given by ν=

γ B 2π

(10.13)

The operative selection rules allow transitions only between adjacent energy levels. It is the frequency of this radiation, which is in the radio-frequency range that is measured in an NMR experiment. The NMR measurement is chemically and structurally useful, mainly because the electrons in the vicinity of the atom shield the nucleus from the applied magnetic field B0 . Nuclei in different structural environments show slightly different magnetic fields and consequently absorb and emit photons of slightly different frequencies. This shielding is characterized by the shielding tensor σ . In general, the shielding is anisotropic. Due to this shielding B = B0 − B0 σ

(10.14)

B = B0 (1 − σ )

(10.15)

and

because the absolute values of B0 are difficult to measure, sufficiently accurate absolute values of NMR frequencies are impossible to obtain. And the resonance frequencies are normally reported as chemical shifts, δ, relative to an experimentally useful standard δ=

ν − ν0 ν0

× 106

(10.16)

where δ is the chemical shift in ppm (percent per mile), ν is the frequency of the sample in Hz, and ν0 is the frequency of the standard in Hz. More negative or less positive chemical shifts correspond to larger shielding. For the structural determination of materials by NMR, the chemical shift is normally the most useful parameter.

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Physico-chemical Analysis of Molten Electrolytes

One of the main problems in obtaining the NMR spectra of solids is that there are a number of phenomena inducing a range of magnetic field strengths at individual nuclei of the same nuclide in a sample, which causes broad peaks. These phenomena include (a) (b) (c)

interaction of dipole moments of individual nuclei (dipole–dipole interaction), anisotropy of the electronic shielding at individual sites (chemical shift anisotropy), and interaction of the quadrupole moment of a nuclide with I ≥ 1 with the electric field gradient at the nucleus.

Other interactions, such as indirect nuclear interaction (angular momentum coupling) are usually insignificant peak-broadening mechanisms for inorganic solids. In low-viscosity fluids, atomic motion occurs at frequencies much higher than the resonance frequency, and all nuclei in a particular structural environment show the same average magnetic field. The resulting peaks are often extremely narrow. In solids, such motional narrowing does not occur normally, and the peaks for powdered samples are broad and often could not be interpreted. This peak broadening can be usually overcome by the use of an experimental technique called Magic-Angle Spinning (MAS). However, the peaks for amorphous samples are not as narrow, because of structural disorder in the sample. The MAS frequencies are typically between about 2 and 10 kHz. Since 81 out of the approximately 110 NMR active nuclides have a quadrupole moment, these second-order effects are extremely important for the investigation of inorganic materials. Most spectra of quadrupole nuclides in solids are obtained by observing only the central ( 12 , − 12 ) transition. Quadrupole effects are due to the interaction of the nuclear quadrupole moment (caused by a non-spherical distribution of charge on the nucleus) with the electric field gradient at the nucleus. Quadrupole effects cause peak broadening, displacement of the peak from the isotropic (true) chemical shift, and distortion of the peak shape. These effects decrease in magnitude with the square of the B0 field strength, and spectra of quadrupolar nuclides are usually recorded at the highest field available. The quadrupole interaction is described by the quadrupole coupling constant (QCC), which for a given nucleus, is a measure of the magnitude of the electric field gradient at the nucleus, and the asymmetry parameter, η, which is a measure of the deviation of the electric field gradient from axial symmetry. In general, the following properties of the MAS spectra could be defined: (a) (b) (c) (d)

increasing field strength decreases quadrupolar peak broadening, increasing QCC increases quadrupolar peak broadening, change in η changes the peak shape, and increasing inhomogeneous broadening reduces the sharpness of the singularity.

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Under static conditions, the isotropic chemical shift is about 13 of the way from the left edge of the peak. Under MAS conditions, the peak is narrowed about a factor of 3, the isotropic chemical shift is at the left edge of the peak, and the center of gravity is displaced to the right of the isotropic chemical shift. The values of QCC, η, and the isotropic chemical shift, δi , can be determined for a single, well-resolved peak or for up to about three overlapping peaks by trial and error using the methods of Ganapathy et al. (1982).

10.3.2. Experimental technique of measurement

Working with an NMR spectrometer is slightly more difficult than, for instance, with an electron microprobe. As the NMR frequencies are all in the radio frequency range, NMR spectrometers are computer operated radio transmitting and receiving systems controlled by a computer. A receiver and transmitter antenna surrounds the sample, which is located in the cavity of a super-conducting cryomagnet. The radio transmitter system consists of a tunable radio frequency generator, a pulse programmer, and pulse gating system, an amplifier, and band-pass filters. The generator produces a continuous wave radio frequency signal at desired frequency. This signal then passes through the pulse programmer and gating box, where it is converted to pulses of appropriate length, which appear in the system at the appropriate time. The signal then enters the amplifier, where it is boosted to a power large enough (often hundreds of watts) to excite the nuclear spin system of interest. The band-pass filters reduce noise. The output from the transmitter system then enters the sample probe, which contains tunable capacitors, the transmitting/receiving antenna, a mechanical assembly for spinning the sample (the stator), and the sample, placed in a rotor, in which it is spun. The antenna is usually a coil 1–2 cm long and about 1 cm in diameter. The stator is made up of an emitting high-pressure gas, forcing the rotor, which always has some grooves on it, to spin. The rotor is usually 0.5–1 cm in diameter and can be cylindrical or mushroomshaped. The attainable spinning speed depends on the gas pressure and the uniformity of the sample packing in the rotor, and increases with decreasing rotor diameter. The sample size is typically in the range of 100–500 mg and is usually powdered, but can be of large pieces of solids or even slurries or liquids. The magnetic field strength of the super-conducting cryomagnet is in the range of 1–14 Tesla. The receiving system is simply a very high quality radio receiver, again with appropriate band-pass filters, which sends the detected signal on to the computer, where it is stored and processed. In a simple 90◦ one-pulse experiment, the transmitter emits a radio frequency signal at the resonance frequency of the nuclide being observed a pulse long enough to equalize the populations of the higher and lower nuclear energy levels. Typical pulses are 1–15 µs long. After these pulses, the sample begins to emit a radio frequency signal at a resonance

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Physico-chemical Analysis of Molten Electrolytes

frequency as the spin system dephases (t2 relaxation time). After an instrumental deadtime period of 7–50 µs, the antenna picks up this signal and sends it on to the receiver. The spin system then relaxes without emitting additional signal (t1 relaxation time). After a time period, which is typically 0.1 to 5 times the t1 relaxation time, the process is repeated and the signal added in the computer to the signal from the previous pulses. Relaxation time t1 can be from milliseconds to several minutes or hours. This process is repeated as many times as needed to produce the desired signal to noise ratio, often hundreds or thousands of times. For analytical purposes, the most important characteristics in the nuclear magnetic resonance measurement are as follows: • • •

the value of the chemical shift in the peak’s maximum, which identifies the nature of the structural fragment, the half-width of the band, which characterizes the shape of the spectral band, the intensity of the band, which is characterized by its area. It is proportional to the number of nuclei. From the comparison of the relative intensities it is possible to determine the number of individual types of nuclei present.

The NMR technique is frequently used in measurement at room temperature or at a temperature below 0◦ C. Usually, (almost solely) a substance of the molecular character is used as the solvent and the dissolved investigated substance is molecular or ionic in character. In such cases, the measured spectrum has usually a narrow half-width of the spectral band (1–200 Hz). Peaks with such a half-width can be mathematically described by the Lorentzian function and we speak of spectral bands of the Lorentzian type. A difficulty in using NMR spectroscopy, especially for many natural samples, is that paramagnetic components or impurities (most commonly Fe or Mn) cause an extensive peak broadening. In the worst case, the peak can be so broadened that it is not detectable above noise. This broadening is due to a range of chemical shifts caused by magnetic field non-homogeneity due to the interaction of unpaired d- or f-electrons within the applied magnetic field. No systematic study of this well-known problem has been made. In general, phases containing Fe or Mn as the major components cannot be measured. However, the content of 1–2 mass% Fe in the sample does not destroy the spectrum totally. In principle, the presence of Fe2+ should cause a larger peak broadening effect than that of Fe3+ , because Fe2+ has more unpaired d-electrons. For quadrupolar nuclides with intrinsically broad peaks, paramagnetic effects may be less important.

10.3.3. High temperature NMR measurement

In the high-temperature spectra, where the solvent as well as the dissolved investigated substance are solely ionic in character, an extremely fast mutual exchange of nuclei,

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and acting as ligands in the present complex ions, takes place. The frequency of these exchange reactions is usually higher than the working frequency of the instrument by some orders. This fact causes on one hand the averaging of signals from all structural units and the broadening of the half-width of the band on the other hand. Broadening of the spectral band can be of the following origin: • •

•

the frequency of exchange reactions approaches the working frequency of the instrument, which may finally lead to separation of individual bands, the formation of solid substances in the liquid system either by crystallization of low soluble products or by the condensation of vapors on the crucible lid, where the temperature is substantially lower than the laser-heated bottom of the crucible; this effect can eventually lead to the separation of individual bands, too, other reasons, for e.g. the presence of paramagnetic substances or the kinetics of the exchange of non-coupled electrons.

Hence, high-temperature spectra bands of all structural units could not be observed, as it is the case of low-temperature spectra in molecular systems, and moreover, only one band averaging contributions from all structural units that can be expressively wide-spread. In the case of bands with higher half-width, it is thus necessary to use for their mathematical description, the weighted combination of the Lorentzian and Gauss functions. Molten salts are interesting as relatively simple liquids with ionic bonds. NMR studies have greatly contributed to understanding of their structure. In general, the NMR lines are usually averaged to a great extent. Most studies therefore involve isotopic chemical shifts and relaxation. Most molten salt systems are mobile enough to produce relatively narrow lines and they can provide a chemically rather simple view of the influence of structure and bonding on NMR chemical shifts. Several transitions in crystal structure have been described for individual molten salts and a variety of structural and order-disorder transitions in crystalline phase were investigated using high-temperature NMR measurements. Here only a few examples are shortly described. Massiot et al. (1990) studied the behavior of LiNaSO4 on heating. At 518◦ C, its low temperature trigonal phase transforms to a cubic structure. This phase transformation is accompanied with a 3-order increase of electrical conductivity. On the other hand, a much smaller increase in conductivity can be observed at the melting point of 620◦ C. Below the phase transition, the 23 Na line gradually narrows. At transition temperature, a new, narrower line appears. The sharp increase of electrical conductivity indicates nearly liquid-like motion of the cations. At the melting point, further line narrowing occurs. Similar results could be seen in 7 Li spectra. A phase transformation to a fast ionic conductor is exhibited also by KLiSO4 . Nuclear magnetic and quadrupole resonance studies of a wide variety of other structural phase transitions of inorganic salts and oxides at low and high temperatures have been fairly described by Rigamonti (1990).

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Physico-chemical Analysis of Molten Electrolytes

Recently, the most reliable high-temperature NMR experiments in fluoroaluminate melts have been performed using the laser-heated system developed by Lacassagne et al. (1997). The experimental device is schematically presented in Figure 10.5. Standard NMR measurement in fluoride melts was performed in the following way. The sample was contained in a high-purity boron nitride (BN AX05 Carborundum) crucible, tightly closed by a BN lid with a screw, and put inside the radio-frequency coil at the center of the super conducting cryomagnet. The NMR axial saddle coil was thermally isolated by a ceramic shield. The crucible was directly heated by a continuous CO2 laser (λ = 10.6 µm) passing axially through the probe head. With this design, temperature could not be measured by a thermocouple during NMR experiments. The temperature calibration was carried out in two steps; calibration of temperature versus laser power using a thermocouple located inside the BN crucible controlled by in situ observation of phase transition by NMR. The α → β phase transition and the melting of cryolite

Radiofrequency coil

BN crucible

Sample Heat shielding NMR probe

Spectrum

Cryomagnet Argon He/Ne

ZnSe window Laser beam

CO2 laser (120 W) Figure 10.5. Scheme of the instrument for the measurement of NMR spectra.

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Table 10.1. Typical acquisition conditions used for high-temperature NMR experiments Nucleus 19 F 27Al 23 Na 17 O

Frequency/ MHz (9.4 T)

Number of scans

Pulse length/µs

Recycle delay/s

Reference substance

376.3 104.2 105.8 54.2

8 64 64 64

π/2 π/8 π/8 π/2

1 0.5 0.5 0.5

1 M CFCl3 1 M Al(NO3 )3 1 M NaCl H2 O

were clearly marked by a net 27Al-signal modification. Repeating the same heating and cooling procedure, the temperature was accurate within ± 5◦ C. Since liquid fluorides are very corrosive and volatile, the experimental time spent in the liquid phase has been minimized (≈5 min) to avoid changes in the composition during the experiment. The chemical composition of the samples was checked by wet chemical analysis, highresolution NMR, and Rietveld analysis of their X-ray diffraction patterns before and after high-temperature NMR experiments. Weight loss measurements confirmed the absence of evaporation from the crucible. All the NMR experiments have been carried out using a Bruker DSX400 NMR spectrometer operating at 9.4 T. The NMR spectra have been obtained using single pulse excitation, and typical acquisition conditions for cryolite-based melts are given in Table 10.1. 27Al, 23 Na, 19 F, and 17 O chemical shifts at room temperature were related to 1 M aqueous solutions of Al(NO3 )3 , NaCl, CFCl3 , and H2 O, respectively, and were accurate to ± 0.5 ppm. The reported 19 F spectra were corrected for a broad probe-head signal due to the presence of Teflon in the probe assembly. 10.3.3.1. HT NMR spectra of the NaF–AlF3 melts

In alkali fluoroaluminate solid compounds, aluminum is present only in octahedral coordination with fluorine. According to Spearing et al. (1994) and Smith and Van Eck (1999), their 27Al chemical shifts range between −13 and −1.4 ppm and are typically more shielded than the AlO6 octahedrons in oxide compounds. Only a few studies report lower coordination numbers for Al in fluorides. Kohn et al. (1991) have described the 27Al MAS NMR spectra of glasses of jadeite mixed with cryolite in terms of the 5-fold and 6-fold coordination of aluminum at 22 and −5 ppm, respectively. Herron et al. (1993) reported a 27Al chemical shift at 49 ppm for the tetrahedral anion AlF− in a [1,8-bis-(dimethylamino) 4 naphtathalene H− ] [AlF− ] saturated solution. 4 In these liquids, for all the observed nuclei, the high-temperature NMR spectrum consists of a single, narrow line, characterized by its position (isotropic chemical shift) and its line width. This single sharp line reflects rapid exchange between the different available environments (rapid as compared to NMR time scales ranging from 102 to 108 Hz). Consequently, the observed peak position is the average of the chemical shifts of individual species, weighted by their respective populations.

410

Physico-chemical Analysis of Molten Electrolytes

Using the laser-heated NMR setup developed by Lacassagne et al. (1997), the 27Al chemical shift for a wide range of high-temperature liquid compositions of the NaF–AlF3 system was determined and the obtained results were compared with the existing Raman data published by Robert et al. (1999). It was shown that both the spectroscopic methods quantitatively agreed in describing the structure of the liquid phase in terms of AlF− 4, 3− AlF2− , and AlF anionic complexes. Bessada et al. (1999) also reported well-separated 5 6 27Al chemical shift ranges for the different types of coordination in individual fluoroaluminates, including AlF4 coordination, measured at 38 ppm on the high-temperature NMR spectrum of the NaAlF4 melt, where Raman spectroscopy gave evidence of only AlF− 4 species (see Table 10.2). The high-temperature NMR measurement of melts in the NaF–AlF3 system was also performed by Lacassagne et al. (2002). 27Al, 23 Na, and 19 F spectra of the different melts of the NaF–AlF3 system consist of a single Lorentzian peak, characteristic of rapid exchange between the different chemical species in the melt. The typical peak widths are constant, in the order of 100 to 200 Hz, and mainly due to the principal field non-homogeneity as could be checked by relaxation time measurement. Going from NaF to AlF3 , a continuous decrease in δ 23Na and increase in δ 27Al with the change in the slope between cryolite and chiolite was observed. On the other hand, δ 19F rapidly grows from liquid NaF (δ = −228 ppm), then between cryolite and chiolite, it keeps constant at δ = −192 ppm, and lowers with higher contents of AlF3 . From the evolution of the 27Al chemical shift, it is clear that the ionic structure of these melts consists of more than one type of aluminum-containing ionic species, their proportion depending on composition. However, from the measured chemical shift for aluminum in the 6-fold coordination with fluorine and the measured chemical shift for AlF− 4 complexes (38 ppm), it followed that every description of the structure involv3− ing only AlF3− 6 ionic species, even as distorted AlF6 complexes, is nonrealistic. The distribution of anions calculated directly from the experimental NMR chemical shifts 3− involving only AlF− 4 and AlF6 complexes did not coincide with any models described 3− previously. The calculated molar fractions of AlF− 4 and AlF6 species in the concentration range between cryolite and chiolite compositions are nearly identical following the Table 10.2. 27Al chemical shifts for different Al–O and Al–F coordination Aluminum coordination

δAl /ppm

AlO5− 4 AlO7− 5 AlO9− 6 AlF− 4 2− AlF5 AlF3− 6

90/55 30/40 20/−20 38 20 1.4/−13

Direct Methods of Investigation

411

same decreasing trend. However, the calculated equilibrium constant varies by one order of magnitude (0.02 to 0.21 at 1025◦ C) over the whole range of the AlF3 content. The observed chemical shifts were described in terms of the structural environment of the observed nuclei using the correspondence between the experimental chemical shifts and coordination described in Table 10.3 and established at room temperature for solid stable compounds. From 18 to 50% AlF3 , the average coordination number of aluminum with fluorine decreases from 5.5 to 4 and that of sodium increases from 8.5 to 11. The correlation between chemical shifts of 23 Na and 27Al indicates that these melts can be described from the Al–F as well as from the Na–F coordination point of view. It should be emphasized that the observed singularity in both the aluminum and sodium chemical shift closely corresponds to the behavior already observed for macroscopic properties such as conductivity, density, and viscosity. For instance, the conductivity decreases rapidly with increasing amount of AlF3 in the melt and coincides rather well with the evolution of coordination, increasing the average coordination number for Na and decreasing it for Al. A modification in this evolution is evidenced around 30 mol % AlF3 by a break in the slope. According to the Raman investigation of these melts, this particular composition corresponds to that with the lowest fraction of F− and the highest − one of AlF2− 5 , with emergence of AlF4 . 19 The chemical shift of F for individual fluoroaluminate anions can be directly deduced from experimental data. First, the chemical shift of δ = −228 ppm measured in pure NaF

Table 10.3. Coordination and experimental chemical shifts for Na and Al fluoride stable solid phases Nucleus

Compound

Site

AlF3

AlF3− 6 AlF3− 6 AlF3− 6 AlF3− 6 AlF− 4 NaF5− 6 NaF5− 6 NaF7− 8 NaF5− 6 NaF11− 12 FNa5− 6 FNa3Al5− FNa3Al5− FNa4Al6− FAl5− 2

δ/ppm (a)

27Al

Na3AlF6 Na5Al3 F14 Na5Al3 F14 NaAlF4 23 Na

NaF Na3AlF6 Na3AlF6 Na5Al3 F14 Na5Al3 F14

19 F

NaF Na3AlF6 Na5Al3 F14 Na5Al3 F14 Na5Al3 F14

−15 −1

(b)

(c)

(d)

−13.2

−

–

1.4

0

–

−1.5

−1

−1

–

−2.8

−3

−3

–

–

−

–

7

7.2

−

–

1

2.4

4

–

−12

−9.3

−8

–

−7

–

−6

–

−21

–

−21

–

−221

–

−

–

−189

–

−

−187

−

−

−189.5

−190

−

−

−191.4

−162

−

−

−165

38

–

412

Physico-chemical Analysis of Molten Electrolytes

was assigned to the free fluoride anion and that of δ = −200 ppm measured in pure NaAlF4 to AlF− 4 . Then it was possible to fit the shift evolution to individual anions over the whole composition range, providing the value of δ = −188 ppm was attributed to 3− the AlF2− 5 anion and δ = −176 ppm to the AlF6 ion. These results show the strong 19 dependence of the F chemical shift on the amounts of free fluoride ions in the melt. Hence, the interpretation of both 27Al and 19 F chemical shift variation in terms of the individual fluoroaluminate species confirms the coherence of such a description of cryolitic melts.

10.3.3.2. HT NMR spectra of the NaF–AlF3 –Al2 O3 melts

In a pioneering high-temperature NMR study, Stebbins et al. (1992) reported 27Al chemical shifts for four compositions of the NaF–AlF3 –Al2 O3 system that were systematically much higher in liquid than in the solid phases. They emphasized the dramatic effect of composition and temperature on the liquid structure and proposed an evolution of aluminum environment toward a higher fraction of a 4-fold coordinated environment of aluminum. The structure of molten mixtures of cryolite with 17 O-enriched Al2 O3 in the concentration range 0.6–8.2 mole % Al2 O3 using a multinuclear NMR measurement was investigated by Lacassagne et al. (2002). These authors found that, except for the 27Al high-temperature spectra, where broadening of the lines is observed with an increasing amount of dissolved alumina, for the 23 Na, 19 F, and 17 O nuclei, the evolution of the line width is not significant. A small, but significant 17 O signal at 26 ppm was observed at 0.6 mole % Al2 O3 . The intensity of the line increases with increasing amount of dissolved alumina, which indicates the increase of the number of 17 O nuclei in the melt up to a maximum value at saturation. The value of the chemical shift decreases from 26 ppm up to approximately 8 ppm. The 27Al chemical shift shows a strong evolution upon alumina addition. The chemical shift increases with increasing the amount of alumina dissolved. No significant evolution were observed for the 19 F and 23 Na chemical shifts over the whole range of compositions. The evolutions observed for 17 O and 27Al chemical shifts are nearly symmetrical except for the highest contents of alumina, where the 17 O chemical shift remains constant. These simultaneous evolutions indicate the existence of species containing Al–O–F oxygen bridges, in addition to the fluoroaluminate species already described for molten NaF– AlF3 binary system. The 27Al chemical shift increases from 20 ppm to 47 ppm over the range from 0 to 6.6 mole % Al2 O3 . This strong variation indicates that the average local environment of aluminum is highly modified by addition of alumina. The chemical shift becomes rapidly higher than 38 ppm, which is the maximum value measured for AlF− 4 in the NaF–AlF3 system. This means that in the melt, new chemical species with higher

Direct Methods of Investigation

413

chemical shift have to be considered. After 6.6 mole % Al2 O3 the chemical shift remains constant, which indicates the saturation of the melt with alumina. However, the exact anionic structure of the oxofluoroaluminate species is still a matter of discussion. A number of studies still converge clearly toward two major entities containing oxygen, namely Al2 OF2− 6 , which is dominating at low contents of alumina, and Al2 O2 F2− , being the prevailing species at high contents of Al2 O3 . 4 In the range of 0.6–3.8 mole % Al2 O3 , the 17 O chemical shift decreases from 25 to 8.5 ppm and above 3.8% Al2 O3 it remains constant up to saturation. High sensibility of the 17 O chemical shift to the content of added alumina showed that the average local environment of the oxygen atoms is strongly modified up to 3.8 mole % Al2 O3 . From the evolution of the 17 O chemical shift upon alumina addition, it was suggested that up to this concentration, more than one oxygen-containing species is present in the melt. Above that concentration, the chemical shift remains constant, which indicates that only one oxygen-containing species is present. The variation of the 27Al chemical shift upon alumina addition up to the solubility limit varies in the same manner, suggesting similar behavior. Lacassagne et al. (2002) assume that oxygen atoms in the oxofluoroaluminate com– plexes form most probably bridging bonds of the type Al–O–Al and Al O –Al, as it O was suggested by Robert et al. (1997a) from Raman spectroscopic measurements and by Daneˇ k et al. (2000b) on the basis of direct oxygen analysis using LECO measurements. The evolution of chemical shifts observed for 17 O and 27Al spectra (Lacassagne et al., 2002) really accounts for the existence of two different oxofluoroaluminate species, depending on the Al2 O3 content. Providing that at 0.6 mole % Al2 O3 , only 2− the Al2 OF2− 6 species is present, while at the saturation only the Al2 O2 F4 is in the 2− melt, the corresponding chemical shift for Al2 OF2− 6 is δ = 25 ppm and for Al2 O2 F4 is δ = 8.5 ppm. Assuming that only these two species are present, the variation in the 17 O chemical shift, δ(17 O), can be expressed by the equation O O O δ 17 O = XO (10.17) 2− δ 2− + X 2− δ 2−

– –

Al2 OF6

Al2 OF6

where XO

and X O are the atomic Al2 OF2− Al2 O2 F2− 6 4 Al2 O2 F2− 4 , respectively, for which it holds XO

Al2 OF2− 6

+ XO

Al2 O2 F4

Al2 O2 F4

fractions of oxygen in Al2 OF2− 6 and

Al2 O2 F2− 4

=1

(10.18)

From these relations and the experimentally measured chemical shifts, it is possible to calculate the fractions of each oxofluoroaluminate species over the whole range of alumina additions.

414

Physico-chemical Analysis of Molten Electrolytes

The interpretation of evolution of the 27Al chemical shift is more complex because of the presence of the different fluoroaluminate species in addition to the oxofluoroaluminate. In the simplest approach, it can be assumed that due to the relatively low alumina content, the relative proportions of the AlFX species in cryolite are not affected by alumina additions. The variation in the 27Al chemical shift, δ(Al), can then be expressed by the relation δ Al 2− Al2 OF2− Al2 OF6 6

+X

X

+X

δ(Al) = X

δ Al Al2 O2 F2− Al2 O2 F2− 4 4

Al + XAlFX δAlF X

(10.19)

with Al2 OF2− 6

Al2 O2 F2− 4

+ XAlFX = 1

(10.20)

From the variation of the experimental 27Al chemical shift in the system Na3AlF6 –Al2 O3 and using Eqs. (10.19) and (10.20), the 27Al chemical shift for the two oxofluoroaluminate species can then be derived δ Al

Al2 OF2− 6

= 50 ± 0.5 ppm;

δ Al

Al2 O2 F2− 4

= 58.5 ± 0.5 ppm

2− From the NMR point of view, the local structure of the anions Al2 OF2− 6 and Al2 O2 F4 can 27 be described as AlOF3 and AlO2 F2 tetrahedrons, respectively. The Al chemical shifts of the two oxofluoroanions are clearly in between the AlO4 chemical shift, δ = 80 ppm and for AlF4 , δ = 38 ppm. The substitution of one fluorine atom by one oxygen atom in the first coordination sphere of aluminum makes a difference of approximately 10 ppm in the 27Al chemical shift.

10.3.3.3. HT NMR spectra of the Na3AlF6 –Fex Oy melts

Iron is one of the most important impurities in the Hall–Héroult process of aluminum production. It causes the lowering of current efficiency and quality of the metal. Iron is introduced in the process mostly together with alumina in the form of oxides. According to Šimko (2004), a substantial part of iron entering the process comes also from iron tools used in the cell maintenance and iron spheres used in the blast cleaning of spent anodes. Investigation of the system Na3AlF6 –Fex Oy (Fex Oy = Fe2 O3 , FeO) using high temperature NMR spectroscopy method was performed quite recently by Šimko (2004). High-temperature NMR spectra of 27Al and 23 Na chemical shifts were obtained at 1020◦ C. 27Al and 23 Na high-temperature NMR spectra of molten Na

3AlF6 with Fe2 O3 content ranging from 0.5 up to 1 mole % are characterized by one detached Lorentzian peak. The values of the chemical shift increase linearly from 18.3 ppm for

System Na3 AlF6 –Fe2 O3 .

Direct Methods of Investigation Table 10.4.

27Al and 23 Na chemical shifts in the melts of the system Na

x (Na3AlF6 ) 1.000 0.950 0.925 0.900

415 3AlF6 –Fe2 O3

x(Fe2 O3 )

δ(27 Al)

δ(23 Na)

0.000 0.050 0.075 0.100

18.3 22.2 24.2 26.0

−6.5 −5.4 −5.0 −4.8

pure Na3AlF6 up to 26.0 ppm for the mixture with 1 mole % Fe2 O3 (see Table 10.4). This variation in the chemical shift can be due to the • •

change of the local environment of the aluminum nuclei caused by the presence of new particles, influence of the paramagnetic contribution of the uncoupled electrons of Fe(III) present as impurity.

The observed position of the peak’s maximum corresponds to the average chemical shift of particles containing the respective nucleus. In general, the resulting chemical shift is the additive sum of the chemical shifts of the present particles, δ = xi δi , where xi = 1 and xi is the molar fraction of the i-th particle with the chemical shift δ i . The increase of 27Al chemical shift is connected with the lowering of the coordination number of Al in the melt. Assuming that Fe2 O3 reacts with cryolite under the formation of fluoride and oxofluoride aluminum complexes according to the equation 4Na3 AlF6 + Fe2 O3 = 2Na3 FeF6 + Na2 Al2 OF6 + Na2 Al2 O2 F4 + 2NaF

(10.21)

in the mixture of 1 mol % Fe2 O3 and 99 mol % Na3AlF6 , these are the following resulting amounts of substances of the individual constituents n AlF3− = 0.95 mol 6 n FeF3− = 0.02 mol 6 n Al2 OF2− = 0.01 mol 6 n Al2 O2 F2− = 0.01 mol 4 n F− = 0.02 mol

416

Physico-chemical Analysis of Molten Electrolytes

Since the amount of all substances is approximately equal to one, n(i) ∼ = Xi . The following final chemical shift can be expected: Al Al Al δ 27 Al = X 2− δ 2− + X 2− δ 2− + XAlFX δAlFX Al2 OF6

Al2 OF6

Al2 O2 F4

Al2 O2 F4

= 0.01 × 50 + 0.01 × 58 + 0.95 × 18.3 = 18.5

(10.22)

The calculated chemical shift increased with respect to pure cryolite only by 0.2 ppm, while the experimental chemical shift increased by 7.7 ppm (c.f. Table 10.4). It may thus be supposed that there is an additional effect in the melt, which surpasses many times the composition change effect. This additional effect is caused most probably by uncoupled electrons on Fe(III) atoms present in the melt as impurities. The 23 Na high temperature NMR spectra of the system Na3AlF6 −Fe2 O3 show a similar, but not as pronounced an increase of the chemical shift from the value −6.5 ppm for pure cryolite up to −4.8 ppm for the mixture 1 mol % Fe2 O3 and 99 mol % Na3AlF6 . This low increase in the chemical shift can be explained by the fact that the aluminum atoms are not directly coordinated by the sodium atoms. High temperature NMR spectra of the molten system Na3AlF6 – FeO were obtained in melts containing 1–14 mole % FeO. The values of the chemical shift increase linearly from 18.3 ppm for pure Na3AlF6 up to 30.9 ppm for the mixture with 14 mole % FeO (see Table 10.5). This variation in the chemical shift has the same reason as in the system Na3AlF6 –Fe2 O3 . A similar analysis as in the previous system can be performed also in the Na3AlF6 –FeO system. However, here the estimation of the chemical shift evolution is more complicated, since due to the broader concentration range, also further chemical reaction could occur. 2− Since the chemical shifts of the Al2 OF2− 6 and Al2 O2 F4 anions are not too different, for simplicity, let us assume that at 14 mole % FeO, only the Al2 O2 F2− 4 anions are formed according to the reaction

System Na3 AlF6 –FeO.

1 Na3 AlF6 + FeO = Na2 FeF4 + Na2 Al2 O2 F4 2 Table 10.5. 27Al and 23 Na chemical shifts in the melts of the system Na3AlF6 –FeO x(FeO)

δ(27Al)

0.000 0.001 0.050 0.100 0.140

18.3 21.0 27.3 30.9 34.3

(10.23)

Direct Methods of Investigation

417

In the mixture of 14 mol % FeO and 86 mol % Na3AlF6 , the following resultant amounts of substances of individual constituents are present n AlF3− = 0.72 mol 6 n FeF2− = 0.14 mol 4 n Al2 O2 F2− = 0.07 mol 4 The amount of all substances is tuents are

ni = 0.93 mol, then the molar fractions of consti-

= 0.774 mol x AlF3− 6 x FeF2− = 0.151 mol 4 x Al2 O2 F2− = 0.075 mol 4 The following final chemical shift can be expected: Al Al δ 27 Al = X 2− δ 2− + XAlFX δAlFX Al2 O2 F4

Al2 O2 F4

= 0.075 × 58 + 0.774 × 18.3 = 18.5

(10.24)

The actual experimental value of the chemical shift is, however, δ(27Al) = 34.3 ppm. Thus, there is again a further effect influencing the position of the maximum of the band, which many times surpasses the composition change effect. As in the previous case, this difference will most probably be caused by the presence of uncoupled electrons on Fe(III) atoms that are present in the melt as impurities due to the oxidation of a part of Fe(II), which was confirmed by the EPR measurement. A similar poverty stricken increase of the 23 Na chemical shift has been observed also in the system Na3AlF6 −FeO. This can be ascribed to the fact that aluminum atoms are not directly coordinated by the sodium atoms. 10.3.3.4. HT NMR spectra of the KF–K2 NbF7 –Nb2 O5 melts

The 19 F, 93 Nb, and 17 O NMR spectra of the molten system K2 NbF7 −Nb2 O5 in the composition range from 7.5 upto 40 mole % Nb2 O5 have been studied by Cibulková (2005) using a high-temperature laser heating system developed by Lacassagne et al. (1997). Samples of about 60 mg were placed in the glove box under dried argon atmosphere into high purity boron nitride crucibles and tightly closed by a BN lid with a screw.

418

Physico-chemical Analysis of Molten Electrolytes

A continuous CO2 laser-heating beam was directly passed axially through the NMR probe, which allowed to record NMR spectra from room temperature up to 1500◦ C. Experiments were carried out under argon atmosphere, so that the oxidation of the crucible and of the sample was excluded. The time, spent by the sample in the liquid state, has been minimized to avoid evolution of the composition during the experiment. Measurement below 7.5 mole % Nb2 O5 was impossible, since at high concentration of K2 NbF7 interaction between boron nitride crucible and potassium fluoroniobate takes place according to the reaction 3K 2 NbF7 + 5BN → 3NbN + 5KBF4 + KF + N2

(10.25)

After heating, an intensive KBF4 signal at δ = −152.5 ppm was observed in the 19 F MAS NMR spectra. High temperature 19 F, 93 Nb, and 17 O NMR spectra of molten K2 NbF7 −Nb2 O5 mixtures consisted of a single peak due to the rapid exchange between the different species in the melt. The Lorentzian shape of the peaks indicated a complete dynamic averaging of the dipole–dipole and quadrupolar interactions. The position of peaks is hence the weighed average of isotropic chemical shifts of the individual species. The 19 F, 93 Nb, and 17 O chemical shift evolutions versus n(O)/n(NbV ) molar ratio in the K2 NbF7 −Nb2 O5 melts are shown in Figure 10.6. Two very distinctive regions with different slopes could be observed. When dissolving N2 O5 in K2 NbF7 , the following processes could be assumed. K2 NbF7 is electrolytically dissociated to two cations K+ and the anion [NbF7 ]2− . The dissolution of Nb2 O5 can be described by the following scheme − Nb2 O5 (s) → NbO+ 2 (l) + NbO3 (l)

(10.26)

In the next step, reactions between [NbO2 ]+ , [NbO3 ]− , and [NbF7 ]2− anions take place, for instance [NbO2 ]+ + [NbF7 ]2− = [NbOF2 ]+ + [NbOF5 ]2−

(10.27)

and other. Finally, three kinds of species will be present in the solution: the anions [NbF7 ]2− , all the oxygen- and fluorine-containing species that will be summarily marked as A, and the oxygen-containing species, which do not show any signal in 19 F spectrum. In the composition range 0.35 < n(O)/n(NbV ) > 0.9, the dependence of the 19 F chemical shift on the n(O)/n(NbV ) ratio can be expressed in the form δexp

19

n(O) F = a + b V n Nb

(10.28)

Direct Methods of Investigation

419

x(Nb2O5) 0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

60

0.45 −1200

50

790

−1250 770

30 −1350

750

d(17O)/ppm

−1300

d(93Nb)/ppm

d(19F)/ppm

40

20 730 −1400

10

0 0.2

0.4

0.6

0.8

1.0

1.2

1.4

−1450 1.6

710

n(O)/n(NbV) Figure 10.6. Evolution of the

19 F, 93 Nb,

and 17 O NMR chemical shifts versus n(O)/n(NbV ) molar ratio in K2 NbF7 –Nb2 O5 melts.

However, the linear dependence of the averaged chemical shift in the K2 NbF7 –Nb2 O5 melts can be also expressed as a contribution from only two different kinds of particles, those from the [NbF7 ]2− and those from the oxygen- and fluorine-containing particles δexp

19

F = x1 δ1 19 F + x2 δ2 19 F

(10.29)

where x1 and x2 are the molar fractions of the individual kinds of particles with the chemical shifts δ1 (19 F) and δ2 (19 F), respectively. Assuming that Eq. (10.28) equals to Eq. (10.29), and with respect to x1 + x2 = 1, we obtain δexp

19

n(O) F = a + b V = δ1 + (δ2 − δ1 )x2 n Nb

(10.30)

From Eq. (10.30) it follows that a = δ1 and b = δ2 − δ1 . It means that the value of the parameter a represents the chemical shift of particles with the n(O)/n(NbV ) ratio equal to zero, that is the pure K2 NbF7 melt. The parameter a represents thus the chemical shift

420

Physico-chemical Analysis of Molten Electrolytes

of the [NbF7 ]2− particles. The contribution of the other oxygen- and fluorine-containing particles present can be easily calculated from the parameter b. By fitting the data of the 19 F chemical shift to Eq. (10.28), the following equation was obtained

δexp

n(O) 19 F = 83.3 − 77.0 V n Nb

ppm

(10.31)

The value of parameter a = 83.3 ppm is in a rather good agreement with the chemical shift of the solid K2 NbF7 at room temperature, δ(19 F) = 74.5 ppm, obtained from 19 F MAS NMR spectrum. Du et al. (2002) found that in solid state, the chemical shift of the four equatorial fluoride atoms of the [NbOF5 ] octahedron attains the value 29.7 ppm, while that of one axial fluoride atom, the chemical shift is −145 ppm. The average 19 F chemical shift for [NbOF5 ]2− species should thus be δcalc

19

(4 × 29.7 − 145) F = ppm = −5.24 ppm 5

(10.32)

which is in agreement with the chemical shift obtained for the second species, δ 2 = 6.3 ppm. Unfortunately, from the NMR chemical shift measurement it is not clear, which anions are actually present in the K2 NbF7 –Nb2 O5 melts. Some supportive data from other techniques such as EXAFS or neutron scattering would be needed to solve this problem. Similar analysis could be done for the dependency of the 19 F chemical shift on the n(O)/n(NbV ) molar ratio in the range 0.9 < n(O)/n(NbV ) < 1.4. However, this situation is much more complicated, because there is no reference chemical shift as there is for the [NbF7 ]2− anions in the previous range. The above-described analysis was applied also to the 93 Nb NMR spectra. By fitting the data of the 93 Nb chemical shifts to Eq. (10.28), the following result was obtained δexp

93

n(O) Nb = −1534.9 + 310.7 V ppm n Nb

(10.33)

Of course, the interpretation of Eq. (10.33) has to be slightly modified. Similarly, the parameter a represents the chemical shift of the [NbF7 ]2− particles and parameter b represents contributions from all niobium-containing particles weighed by their concentration. However, now we must include also [NbO3 ]− anions, but exclude the F− anions. Again the parameter a = −1534.9 ppm is in good agreement with the chemical shift for pure K2 NbF7 , δ(93 Nb) = −1589 ppm, obtained from 93 Nb MAS NMR measurement at room temperature.

Direct Methods of Investigation

421

From the value of the coefficient b, the chemical shift of the further oxygen- and fluorine-containing species, δ 2 = −1224.2 ppm, was calculated, which is in a rather good agreement with the value δ(NbOF5 ) = −1310 ppm obtained by Du et al. (2002). The analysis of the second linear dependency meets the same complications as in the case of the 19 F chemical shift. According to Van et al. (2000) and Vik et al. (2001), the presence of the [NbF7 ]2− , [NbOF5 ]2− , and [NbO2 F4 ]3− anions were assumed. As can be seen from Figure 10.6, evolution of the 17 O chemical shift differs from that of the 19 F and 93 Nb. After fitting the experimental points, the 17 O chemical shift evolution follows the Boltzmann’s sigmoid course according to the equation  δ

17

  O = 730.32 + 

   48.15 * ) V  ppm n(O)/n Nb − 0.87799  1 + exp 0.07743

(10.34)

The inflex point lies at n(O)/n(NbV ) = 0.87799, which is in accordance with the break in the 19 F and 93 Nb chemical shift evolutions.

This Page Intentionally Left Blank

Chapter 11

Complex Physico-chemical Analysis The basic condition for the effective control of any electrochemical technology is the knowledge of the structure and properties of the given electrolyte and mechanisms of the electrochemical processes involved. The investigated electrolytes represent in general multi-component systems of inorganic salts and oxides or oxygen-containing compounds, in which the chemical reactions take place. The chemical equilibrium in the melt depends on the composition and temperature. Influence of composition plays the most important role, while the change in temperature does not affect the equilibrium dramatically. In the study of the structure, i.e. the ionic composition of the investigated molten electrolyte, the physico-chemical analysis, based on the results of measurements of phase equilibrium, density, surface tension, viscosity, and electric conductivity of melts, combined with X-ray phase analysis and IR, respectively, Raman spectroscopy of quenched melts, is used. In the last two measurements, it may be assumed that the high temperature composition is at least qualitatively conserved after quenching. In the investigation of the structure of the electrolytes, the so-called “chemical approach” is used. To draw conclusions on the structure of the electrolytes from the concentration dependencies of the particular properties, the following thermodynamic, statistical approaches, and material balance calculation were used. In many cases there are only three to four main components which define the physico-chemical properties of an industrial electrolyte. Minor components play only a marginal role. e.g. in the aluminum electrolysis, the main components of the electrolyte are cryolite, Na3AlF6 , aluminum fluoride, and aluminum oxide. The other components, like CaF2 and MgF2 , are present in low concentrations only and do not affect substantially the properties of the electrolyte. It is thus often sufficient to investigate only the system composed of these main components.

11.1. DILUTE SOLUTIONS

In the region of dilute solutions the following limiting law is valid lim

xi →1

dai = kSt dxi

423

(11.1)

424

Physico-chemical Analysis of Molten Electrolytes

where ai is the activity of the component expressed in terms of the mole fractions x i s according to any suitable model and kSt is the correction factor representing the number of foreign particles, which introduces the solute into the solvent at infinite dilution. The region of diluted solutions can be investigated preferentially by cryoscopic measurements. For lowering of the temperature of fusion of the solvent, fus T, the following equation holds fus T =

2 RTfus xB · kSt fus H

(11.2)

where Tfus and fus H is the temperature and the enthalpy of fusion of the solvent, respectively, xB is the mole fraction of the solvent and R is the gas constant. The resulting knowledge of kSt enables to deduce the possible ongoing chemical reaction between solvent and solute.

11.2. WHOLE SYSTEMS

In the study of the whole investigated system, two different approaches may be used. A. In the first approach, the structure (i.e. the ionic composition) is determined by the thermodynamic equilibrium composition, after all the chemical reactions taking place in the system are over. After reaching the chemical equilibrium, the ideal mixing of components is supposed. If the obtained standard deviation of the calculated property for the given chemical reactions is comparable with the experimental error of measurement, it is reasonable to assume that the structure of the electrolyte is given by the equilibrium composition determined by the calculated equilibrium constants. Besides, also information on e.g. the thermal stability and the Gibbs energy of the present compounds may be obtained. The task is solved by means of the material balance and use of the thermodynamic relations valid for ideal solutions. In general, this approach may be used in the evaluation of those properties for which the ideal behavior of the system is physically defined, e.g. for Gibbs energy of mixing and the molar volume. The procedure can be demonstrated by means of the calculation of equilibrium composition based on the measurement of density in the system A–B in which the intermediate compound AB is formed. The compound AB undergoes at melting a partial thermal dissociation. Let us consider 1 mol of mixture consisting of x1 moles of component A and x2 moles of component B. As the partial thermal dissociation of AB = A + B must be taken into account, the degree of thermal dissociation of the compound AB, α, has to

Complex Physico-chemical Analysis

425

be introduced. For x2 ≤ 0.5 the equilibrium amounts of individual constituents A, B, and AB can be expressed as follows (we consider that all B is first transformed into AB, which subsequently dissociates into A and B with dissociation degree α): n(A) = x1 − x2 + αx 2 n(B) = αx 2 n(AB) = x2 − αx 2 n(sum) = x1 + αx 2 and for the individual equilibrium mole fractions, we can write xA =

x1 − x2 + αx2 ; x1 + αx2

xB =

αx2 ; x1 + αx2

xAB =

x2 − αx2 . x1 + αx2

(11.3)

The degree of dissociation of AB is then given by the equilibrium constant K=

α02 α(1 − 2x2 + αx2 ) = (1 − α )(1 − x2 + αx2 ) 1 − α02

(11.4)

where α 0 is the degree of dissociation of AB of the pure AB. For each value of the equilibrium constant and for each composition, we can then calculate the equilibrium molar fractions of constituents. In the calculation of the phase diagram, the equilibrium molar fraction is inserted into LeChatelier–Shreder’s equation and the temperature of primary crystallization of every constituent is then calculated. For the optimized phase diagram, the Gibbs energy of mixing of the system is calculated using the condition n

[Ti (calc) − Ti (exp)]2 = min

(11.5)

i=1

Transforming the molar fractions xi into the mass fractions wi and introducing them into the equation ρ(calc) =

wB wAB wA + + ρA ρB ρAB

−1 (11.6)

where ρ A , ρ B are the densities of pure components A and B, respectively, and ρ AB is the density of the pure liquid non-dissociated compound AB, we get a set of density

426

Physico-chemical Analysis of Molten Electrolytes

values for every chosen equilibrium constant. Equation (11.6) is based on the additivity of specific volumes. The accepted value of K is determined by the condition n

[ρi (calc) − ρi (exp)]2 = min

(11.7)

i=1

B. In the second approach, which finds application in real systems, validity of the general Redlich–Kister equation for the excess property is supposed. For description of the composition dependence on the given property Y (molar volume, surface tension) in the system, the following equation is then used (e.g. for a ternary system)

Y =

3

Ai · x i +

i=1

3

xi · xj

i,j =1 i=j

k

Bnij · xjn +

n=0

m

Cm · x1a · x2b · x3c

(11.8)

a,b,c=1

The first term represents the additive behavior, the second the binary interactions, and the third term the interactions of all the three components. For the excess molar Gibbs energy of mixing in the real solution, the following equation may be supposed (e.g. for a ternary system)

Gex =

3 5 i,j =1 n=1 i=j

Aij n · xi · xjn +

3

Bij k · xia · xjb · xkc

(11.9)

i,j,k=1 i=j =k

where a, b, and c are integers in the range 1−3. In the case of transport properties like viscosity and electric conductivity, the ideal behavior is not physically defined, as we deal with scalar quantities, for which the total derivative does not exist and the simple additivity rule cannot be used. However, these properties are thermally activated and the additivity of activation energies is permissible. Based on this idea, the additivity of logarithms of these properties can be accepted as the “ideal” behavior. It should be, however, emphasized that there are two kinds of electrical conductivities, i.e. the conductivity, κ, and the molar conductivity, λ. The concept of the additivity of logarithms is recommended to apply to molar conductivity, as the concentration course of the molar conductivity is smoothed by the multiplication of conductivity with the molar volume. For description of the molar conductivity in a ternary system, the following

Complex Physico-chemical Analysis

427

equation will be then valid

λ=

x λ11

x · λ22

x · λ33

+

3 i,j =1 i=j

xi · xj

k n=0

Anij · xjn

+

m

Bm · x1a · x2b · x3c

(11.10)

a,b,c

Coefficients of the regression Eqs. ((11.8), (11.9), and (11.10)) are calculated using multiple linear regression analysis. Omitting the statistically non-important terms on the chosen confidence level and minimizing the number of relevant terms, we try to get a solution, which describes the concentration dependence of the investigated property with a standard deviation of the fit being comparable with the experimental error of measurement. For statistically important binary and ternary interactions, we look for appropriate chemical reactions and check their thermodynamic probability calculating their standard reaction Gibbs energies. The reaction products are identified using the X-ray phase analysis and IR spectroscopy of quenched melts. Interactions are mostly considered as chemical reactions between components, therefore the description of the method “chemical approach.” However, Van der Waals bonds and the formation of associates may not be excluded as interactions, in spite of the fact that they cannot be detected by spectroscopic measurements. The ongoing chemical reaction in most cases affects the course of the dependence of individual physico-chemical properties on composition. Consider a binary mixture AX–BX2 in which a complex compound A2 BX4 is formed. Let us now determine what happens with the individual physico-chemical properties compared with the situation 2− when no complex anion BX2− 4 would be formed. First of all, the originating BX4 anion will decrease the activity of the components AX and BX2 , while the activity of the originating compound A2 BX4 will increase, depending on the magnitude of the negative value of the Gibbs energy of reaction. 2AX + BX2 = A2 BX4

(11.11)

In the phase diagram of the system AX–BX2 , this behavior will be expressed either by the lowering of the eutectic temperature, or by the formation of incongruently or congruently melting compound. + 2+ The complex anion BX2− 4 , being bigger in volume than the remaining A , B , and – X ions, will increase the molar volume of the mixture. The electrical conductivity will decrease since fewer smaller ions with higher mobility are present. Surface tension will decrease too, since the BX2− 4 ions are preferentially adsorbed on the surface layer due to a higher ratio of covalent bonds compared with BX2 . The heavier BX2− 4 anion will increase the viscosity of the mixture.

428

Physico-chemical Analysis of Molten Electrolytes

From the analysis of every property, we obtain in such a manner a certain picture on the structure of the investigated system. When the majority of the pictures coincide, then we can accept such a picture as the very probable structure of the molten system. The result of such an investigative method should be, however, confirmed by some of the direct methods of investigation. As an example for the first approach, the calculation of the dissociation degree of different additive compounds performed by Daneˇ k and Proks (1999) can be mentioned. As an example for the second approach, the complex physico-chemical analysis of the system LiF–KF–K2 NbF7 –K2 O performed by Daneˇ k et al. (2000a) can be cited.

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Index Adiabatic calorimeter, 232 Alkali metal borates, 152 Amagat’s law, 314 Archimedean method, 258, 266 “asymmetrical” ternary systems, 211 Auto-complexes, 18 BiCl3 –Bi, 83 Boiling point method, 315 Boiling point method, theory of, 316 Boric acid anomaly, 103 Boron nitride, 351 Boyle’s law, 313 Calculation of phase diagram, 208 Calorimeter Setaram, 248 Calorimetric method, 231 CaO–Al2 O3 –SiO2 , 147, 186 CaO–FeO–Fe2 O3 –SiO2 , 96, 144 CaO–MgO–SiO2 , 142 CaO–TiO2 –SiO2 , 149 Capillary cell, 347, 349, 351 Capillary method, 290 Capillary depression method, 309 Cation–cation repulsion, 15 Cation–cation repulsion energy, 22 CeCl3 –Ce, 83 Cell constant, 346 Chemical shift evolutions, 418 Chemla effect, 342 Clapeyron’s equation, 314 Complex anions, 10 Complex compound, 8 Conductance cells, 346 Conductivity, 35 Conformal solution theory, 24

Congruently, melting compound, 162 Constant heat flow, Calorimeter with, 232 Contact angle, 305 Continuous solid solutions with a minimum, 161 Continuously Varying Cell Constant, 352 Coordination number, 11, 13 Cryolite-based melts, 28 Cryoscopic measurements, 204 Cryoscopic method, 29 Cryoscopy, 191 Crystallization process at cooling, 173 Cs2 ZrCl6 , 401 CsF–MgF2 , 8 CsZr2 Cl9 , 401 Current efficiency, 86 Current frequency, 346 Dalton’s law, 314 Degree of conversion, 32 Degree of dissociation, 26 Density, 34, 35, 49, 52, 255 Detachment force, 302 Differential thermal analysis, 205 Discrete silicate anions, 368 Dissolution, enthalpy of, 225 Dissolution, integral enthalpy of, 225 Double hard core model, 12 Double immersion, 356 Double-layer capacitance, 347 Dynamic viscosity, coefficient of, 359 Dystectic melting, 195 Dystectic point, 73 Eberhart’s model, 289 Effusion method, 315 Electrical conductivity of ionic silicate, 97

445

446

Electrical conductivity, 13, 20, 34, 327 Electrical conductivity, density, viscosity, and, 92 Electronic conduction, 79 Electronic conductivity, 93 Enthalpy interaction parameter, 21 Enthalpy of mixing, 15, 17 Enthalpy, formation of, 224 Enthalpy, measurement of, 221 Equilibrium constant, 26 Equivalent conductivity, 85, 328 Eutectic system, 156 Excess viscosity, 361 Falling body method, 377 Faraday’s law, 327 Field strength, 7 Figurative point, 109 First approach, 424 First coordination sphere, 7 FLINAK, 65, 66, 399 Four-electrode cell, 356 Fugacity, 116 Fusion, enthalpy of, 225 Gay-Lussac’s, 313 General equation, Redlich-Kister’s, 258 Gibbs energy of mixing, 17 Gibbs equation, 273, 284 Gibbs phase law, 107 Glass-forming oxides, 101 Glass-forming systems, Double calorimetry of, 251 Guggenheim’s equation, 286 Heat capacity determination, 239 Heats of fusion of eutectic mixtures, 230 Henrian activities, 122 Henry’s constant, 119 Hess’s law, 222 Heteropolyanions, 55 High temperature Raman spectroscopy, 393 High-temperature density measurement, 266

Index

High-temperature NMR, 407 High-temperature oscillation viscosimeter, 370 Hollow cylinder, 377 Hopping mechanism, 79 Ideal solutions, 118 Incomplete electrolytic dissociation, 334 Incongruently, melting compound, 164 Infrared and Raman spectroscopy, 385 Infrared spectra, 68, 70 Interfacial phenomena, 306 Interfacial tension measurement, 307 Interfacial tension, 304 Inter-ionic distances, 11 Internal mobility, 342 Ion, Lifetime of the, 7 Ionic charge, 327 Ionic mass, 327 Ionic pairs, 337 IR spectroscopy, 49, 60, 67 Isolated FeO5− 4 tetrahedrons, 369 Isoperibolic calorimeter, 232 Isothermal calorimeter, 232 K2 NbF7 , Surface adsorption of, 283 K2 NbF7 –Nb2 O5 , 417 K2 SO4 –PbSO4 –K2 WO4 –PbWO4 , 181 KCl–KF–K2 TiF6 , 217 KCl–CdCl2 , KCl–PbCl2 , 10 KCl–K2 SO4 –MgCl2 –MgSO4 , 184 KCl–KBF4 , 33 Kelvin’s equation, 273 KF–K2 MoO4 –SiO2 , density, 262 KF–KCl–K2 TiF6 –KBF4 , 218 KF–KCl–KBF4 –K2 TiF6 , 213 KF–B2 O3 , 73 KF–K2 MoO4 , 3 KF–K2 MoO4 –B2 O3 , 51 KF–K2 NbF7 , 361 KF–KBF4 , 33 KF–KBF4 –B2 O3 , 78 KF–KCl–KBF4 , surface tension in, 285 KF–MgF2 , 8

Index

Kinetic theory of liquids, 13, 14 Kirchoff’s law, 223 KX–K (X = F, Cl, Br, I), 82 Laplace equation, 273 LeChatelier–Schreder’s equation, 112, 193 LECO, 62 Lever rule, 108 Li2 SO4 –Na2 SO4 –K2 SO4 , 95 Li2 SO4 –Na2 SO4 –K2 SO4 –CoSO4 , 96 LiF–KF–K2 NbF7 , 47 LiF–NaF–K2 NbF7 system, Excess molar volume of, 261 LiF–NaF–K2 NbF7 system, Molar volume of, 261 LiF–B2 O3 , 71, 72 LiF–KF–B2 O3 , 76 LiF–KF–B2 O3 –TiO2 , 77 LiF–LiBO2 , 73 LiF–MgF2 , 8 LiF–NaF–B2 O3 , 73 LiF–NaF–K2 NbF7 , 48 Ligand field stabilization energy, 8 Liquid phase, limited miscibility in, 159 Local structure symmetry, 67 M3 LnCl6 , 39 Magnesium, 184 Magnetically active samples, 402 Mass spectroscopy, 313 Maximum bubble pressure method, 292 Mean activity coefficient, 121 MeCl2 –Me (Me = Ca, Sr, Ba), 83 MeF2 –Me (Me = Ca, Ba), 83 Melting point depression, 202 Melts in NaF–AlF3 , NMR measurement of, 410 Metallurgical slags, 104 Metathetical reaction, 178 MF–ZrF4 , 41 Miscibility gap, 76, 115 Mixing enthalpy, determination of, 244 Mixing, enthalpy of, 224

447

Mixture, Molar Gibbs energy of, 116 Mobility of ions, 327 Molar conductivity, 328 Molar entropy of mixing, 118 Molar volume, 255 Molecular dynamics computer simulations, 9 Molecular model of molten salt mixtures, 135 Molten salts, Raman spectra of, 391 Molten salts, Viscosity of, 359 Molten salts’ structure, 329 Monte Carlo method, 60 Multiple linear regression analysis, 362 NaCl–PbCl2 , 6 NaF–AlF3 , 9 NaF–B2 O3 , 73 NaF–MgF2 , 8 NaF–NaBF4 –B2 O3 , 78 Nernst–Einstein equation, 327 Network structures, 19 Neumann–Kopp’s rule, 226 Neutron diffraction analyses, X-ray and, 385 Neutron diffraction, 38 NMR (nuclear magnetic resonance), 385 NMR spectrometer, 405 NMR studies, 103 NMR, 65 NMR-Na3AlF6 –Fex Oy , 414 Non-random mixing, 134, 338 Non-reacting systems, Calorimetry of, 238 (“non reconstructive”) phase transition, 40 Nuclear magnetic resonance (NMR), 402 O/Nb molar ratios, 66 Oxide solubility, 58 Oxofluoroaluminate anions, 59 Oxofluoroaluminate, 413 Oxofluoro-complexes, 56 Pair vacancies, 327 Parallel model, 333 Partial molar volume, 256 Pascal (Pa), 313

448

Peritectic point, 164 Peritectic reaction, 165 Peritectic temperature, 164 phase diagrams of two-component systems, 155 Phase diagrams, 213 Phase diagrams, experimental determination of, 189 Phase Equilibria Diagrams Database, 41 Pin detachment method, 296, 297, 312 Planck function, 113 Polarizability, 327, 337 Polarization ability, 8 Polarization, 347 Polycondensation, 55 Polymeric structure, 19 Polymorphic transformation, 166 Polymorphic transformation, enthalpy of, 225 Quadrangle rule, 109 Quadrupole effects, 404 Quasi-binary system, 73 Quasi-crystalline structure, 10 Quasi-lattice theory, 133 Quaternary reciprocal systems, 184 Quaternary systems, 184 Radial distribution function, 12 Raman and IR activity, 390 Raman optical cells, 393 Raman scattering, 388, 389 Raman spectra of (M, M)F–AlF3 (M, M = Li, Na, K), 400 Raman spectra of alkali metal halide mixtures, 395 Raman spectra, 70 Raman spectroscopy, 20, 38, 60 Raman spectroscopy, NMR, MAS NMR, 2 Raoult’s and Henry’s laws, 119 Raoultian activity, 122 RbCl–PbCl2 , 10 RbF–MgF2 , 8 Reacting systems, Calorimetry of, 236

Index

Real solutions, 119 Reciprocal salt mixtures, 131 “reconstructive” phase transition, 40 Redlich–Kister’s type equation, 346 Regular solutions, 118, 125 Relative enthalpy, 252 RF–R (R = Li, Na, K, Rb, Cs), 82 Ring method, 296 Rotational method, 380 Scalar quantity, 328 Second approach, 426 Second coordination sphere, 7 Series model, 332 Sessile drop, 304 Shear stress measurement, 380 Silicate melts, 135 Silicate melts, thermodynamic model of, 136 Silicate melts, Viscosity of, 362 Silicate, Electrical conductivity of, 344 Simple ternary eutectic system, 168 Single ion activity coefficient, 120 Solid and liquid phases, unlimited solubility in, 161 Solid solutions, 157 Solubility of Al2 O3 , 61 Solubility of alumina, 66 solution calorimeter, 253 Stable diagonal, 181 Stortenbeker, 44 Storkenbeker’s correction factor, 29, 130, 195 Stray capacitance, 348 Structural classification, 5 Surface adsorption, 274 Surface entropy, 295 Surface Gibbs energy, 272 Surface tension, 20, 34, 47, 55, 271 “symmetrical” ternary systems, 211 Systems MF–AlF3, Surface tension of, 295 Temkin’s ideal solution, 118 Temkin’s model, 9, 127 Ternary compound, 177

Index

Ternary peritectic points, 177 Ternary reciprocal system, 178 The density, 74 The surface tension, 49 Thermal dissociation, 26, 48, 195 Three-component systems, 167 TiB2 , 42 Titanium diboride, 42, 70 Torsion pendulum method, 53 Total conductivity, 85 Transpiration apparatus, 325 Transpiration experiment, 322 Triangle rule, 109 Twinned Fe2 O4− 5 tetrahedrons, 369 Two-electrode cell, 354 Types of phase diagrams, 16 Utigard’s model, 290 V2 O5 –Na2 O, 89 V2 O5 –NaVO3 , 91 Vanadium bronzes, 90

449

Vapor pressure of, 20 Vapor pressure, 313 Vapor pressure, Measurement of, 315 Viscosity, 13, 14, 20, 34, 35, 48, 49, 53, 55 Viscosity, Calculation of, 373 Viscosity, measurements of, 369 Viscous flow, activation energies of, 360 Voltammetry, 67 Wilhelmy slide method, 296 Wilhelmy, 300 XAFS (X-ray Absorption Fine Structure), 385 X-ray diffraction analysis, 2 X-ray diffraction, 38 X-ray phase analysis, 49 X-ray powder diffraction analysis, 60 X-ray structural analyses, 103 ZnCl2 –NaCl–KCl, 20 ZrCl4 , 400

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