Molecular Mixed Crystals (Physical Chemistry in Action) 3030687260, 9783030687267

Table of contents :
Contents
Contributors
Part I Introductory Part
1 Introduction
1.1 Scientific Setting
1.2 About the Chapters
1.3 About the REALM
1.4 Audience
2 Molecular Homeomorphism and Crystalline Isomorphism
2.1 Background and Controversy
2.2 Isomorphism After Mitscherlich
2.3 Thermodynamic Theory of Mixed Crystals: Solid Solutions
2.4 Isomorphism Following the Discovery of X-Ray Diffraction and the Atomic Structure of Crystals
2.5 Isomorphism in Organic Compounds
2.6 Geometric Model of Molecular Compounds
2.7 Coefficients of Molecular Homeomorphism and Crystalline Isomorphism
2.8 Updating the Concept of Isomorphism
2.8.1 Case (a)
2.8.2 Case (b)
2.8.3 Case (c)
References
3 Thermodynamics
3.1 Introduction
3.2 Presentation of the ABΘ Model
3.3 From Model Parameters to Phase Diagram
3.4 A Transparent Treatment of Phase Diagram Characteristics
3.5 The Route from Experimental Data to the Numerical Values of the Parameters
3.6 The Output: Discovered Regularities and Empirical Relationships
Appendix: Computer Software Developed at Utrecht University
References
4 Polymorphism
4.1 A Short History of Crystalline Polymorphism, the Emergence of a Controversial Concept
4.2 The Gibbs Energy: Pressure and Temperature as Natural Variables
4.2.1 First- and Second-Order Phase Transitions
4.2.2 The Gibbs Energy in Relation to Pressure–Temperature Phase Diagrams
4.2.3 The Clapeyron Equation
4.2.4 From Two-Phase Equilibria to Triple Points
4.3 Topological Pressure–Temperature Phase Diagrams
4.3.1 Pressure–Temperature Phase Diagrams
4.3.2 Phase Equilibria
4.3.3 The Pressure Coordinate
4.3.4 The Clapeyron Equation
4.3.5 The Le Chatelier Principle
4.3.6 The Clausius–Clapeyron Equation
4.3.7 Triple Points and the Alternation Rule
4.3.8 Extrapolation of Equilibrium Lines
4.3.9 Redundancy in the Topological Method
4.3.10 The Implications of the Word “Topological”
4.3.11 Crystalline Dimorphism in Four Possible Phase Diagrams
4.3.12 Higher Order Polymorphism
4.4 Some Examples of Real Cases of Enantiotropy and Monotropy as a Function of Pressure and Temperature
4.4.1 Rimonabant: Overall Monotropy
4.4.2 Cysteamine Hydrochloride: Overall Enantiotropy
4.4.3 Benfluorex Hydrochloride: Enantiotropy Turning into Monotropy with Pressure
4.4.4 Ritonavir: Monotropy Turning into Enantiotropy with Pressure
4.4.5 Isomorphism in Halomethane Compounds: Similar Systems with Changing Hierarchies
4.5 The Consequences of Polymorphism in Molecular Alloys
4.5.1 Polymorphism and Molecular Alloys
4.5.2 Eutectic Systems
4.5.3 The Effect on Cocrystals: Polymorphism and Hydrate Formation
4.5.4 Solid Solutions and Polymorphism
References
Part II Facts and Features
5 Aromatics
5.1 Introduction
5.2 The para-Dihalobenzene Family
5.3 Polymorphism of p-Dichlorobenzene
5.4 The System p-Dichlorobenzene + p-Dibromobenzene
5.5 Tri- and Tetrasubstituted Halobenzenes
5.6 The Family of the 2-Substituted Naphthalenes; the Existence of Two Subfamilies
5.7 The System Naphthalene + 2-Naphthol
5.8 The Halo + Halo Systems
5.9 The Methyl + Halo Systems
5.10 Crossed Isodimorphism
5.11 The System trans-Azobenzene + trans-Stilbene
5.12 The System Thiophene + Benzene
5.13 The System Thianaphthene + Naphthalene
5.14 Ternary Systems, as an Extra
References
6 Chains
6.1 Introduction
6.2 n-Alkanes
6.2.1 The Family
6.2.2 Polymorphism
6.3 Solid-State Miscibility
6.4 A Perfect Family
6.5 Thermal History of Alkane Alloys
6.6 n-Alkanols
6.6.1 The Family
6.6.2 Polymorphism
6.6.3 Solid-State Miscibility
6.7 Fatty Acids
6.7.1 The Family
6.8 Polymorphism
6.9 Solid-State Miscibility
6.10 Melting Point Alternation
References
7 Plastic Crystals
7.1 Introduction
7.2 Two-Component Systems Involving Hydrogen Bonds
7.2.1 The ABθ Model
7.3 Two-Component Systems Involving Components with Weak Intermolecular Forces
7.3.1 The ABθ Model
7.4 Pressure–temperature Phase Diagrams and Their Relation with Two-Component Systems at Normal Pressure
7.5 Concluding Remarks
References
8 Liquid Crystals
8.1 Introduction
8.2 Liquid Crystalline States
8.3 Nematic-To-Isotropic and Smectic A-To-Nematic Phase Transitions
8.4 Re-entrant Nematic Behaviour
8.5 The EGC Method Applied to Liquid Crystal Binary Systems
8.5.1 Tricritical Behaviour in Binary Systems
8.5.2 Re-entrant Nematic Behaviour in Binary Systems
8.6 The Heptyloxycyanobiphenil (7OCB) + Nonyloxycyanobiphenil (9OCB) Binary System: An Experimental Study
8.7 The Heptyloxycyanobiphenil (7OCB) + Nonyloxycyanobiphenil (9OCB) Binary System: A Thermodynamic Analysis
8.8 Other Binary Systems Exhibiting Re-entrant Nematic Behaviour
8.9 Other Binary Systems Exhibiting Tricritical Behaviour
References
9 Enantiomers
9.1 Systems of Enantiomers
9.2 Racemates and Conglomerates
9.3 Formation of Mixed Crystals, Pseudoracemates
9.4 Adriani’s Investigation
9.5 The Carvoxime System Following Adriani
9.6 Intermezzo
9.7 The Carvoxime System, Tammann’s View
9.8 The Carvoxime System, More Recent Advances
9.9 X-ray Studies
9.10 Calorimetry
9.11 Thermodynamic Analysis
9.12 Carvoxime: Concluding Observations
9.13 Carvone and Limonene
9.14 The Carvone System
9.15 The Limonene System
9.16 Racemic Mixtures and Pressure
9.17 The Temperature-Composition Phase Behaviour of Camphor
9.18 The Camphor System Under Pressure
9.19 Mandelic Acid
9.20 Ibuprofen Under Pressure
9.21 Wallach’s Rule
9.22 Concluding Remarks
References
10 Complexes
10.1 From Random to Ordered Alloys
10.2 Mean-Field Approach in Molecular Alloys
10.3 Liquid Complexes
10.4 The Benzene-Hexafluorobenzene System
10.5 The Halogen-Substituted Benzene-Hexafluorobenzene Systems
10.6 Influence of the Methyl Group on the Complexation in the Benzenic Family
10.7 The Naphthalene Parent System: C10H8-C10F8
10.8 Solid Complex and Associated Melt
10.9 Concluding Remarks
References
11 Triacylglycerols
11.1 Introduction
11.2 Basic Terms on TAGs Polymorphism
11.3 Polymorphic Behaviour of Mixed-Acid TAGs
11.4 Influence of Kinetic Factors on the Polymorphic Behaviour of TAGs
11.5 Phase Behaviour of TAGs Binary Mixtures
References
Part III Applications
12 Phase Change Materials
12.1 Introduction
12.2 Some Basic Matters
12.3 An Example
12.4 Other Examples
References
13 Crystallization
13.1 Introduction
13.2 Multicomponent Equilibrium Model of Crystal Growth
13.3 Analytical Kinetic Models
13.3.1 Linear Kinetic Segregation Model
13.3.2 Linear Effective Kinetic Segregation Model
13.3.3 Mean Field Kink Site Kinetic Segregation (MFKKS) Model
13.3.4 Combined Analytical LKS-MFKKS Model
13.4 Determination of an Equilibrium Phase Diagram from a Kinetic Analysis
13.5 Some Additional Information and Observations on Melting and Crystallization
References
Subject Index
Substances Index

Citation preview

Physical Chemistry in Action

Miquel Àngel Cuevas-Diarte Harry A. J. Oonk   Editors

Molecular Mixed Crystals

Physical Chemistry in Action

Physical Chemistry in Action presents topical volumes which outline essential physicochemical principles and techniques needed for areas of interdisciplinary research. The scope and coverage includes all areas of research permeated by physical chemistry: organic and inorganic chemistry; biophysics, biochemistry and the life sciences; the pharmaceutical sciences; crystallography; materials sciences; and many more. This series is aimed at students, researchers and academics who require a fundamental knowledge of physical chemistry for working in their particular research field. Each volume begins with an introductory chapter aimed at the novice, which provides background and a valuable perspective on the respective field. The following chapters discuss the physicochemical concepts and methods used, as well as applications of the stated methods in the field. Volumes are edited and include contributed chapters from researchers working in the field. Contributions by authors from all of the various disciplines are encouraged.

More information about this series at http://www.springer.com/series/10915

Miquel Àngel Cuevas-Diarte · Harry A. J. Oonk Editors

Molecular Mixed Crystals

Editors Miquel Àngel Cuevas-Diarte Grup de Cristal·lografia Aplicada Universitat de Barcelona Barcelona, Spain

Harry A. J. Oonk Department of Earth Sciences Utrecht University Utrecht, The Netherlands

ISSN 2197-4349 ISSN 2197-4357 (electronic) Physical Chemistry in Action ISBN 978-3-030-68726-7 ISBN 978-3-030-68727-4 (eBook) https://doi.org/10.1007/978-3-030-68727-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

Part I

Introductory Part

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. À. Cuevas-Diarte and H. A. J. Oonk

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2

Molecular Homeomorphism and Crystalline Isomorphism . . . . . . . . Y. Haget, N. B. Chanh, H. A. J. Oonk, and M. À. Cuevas-Diarte

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3

Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. A. J. Oonk, T. Calvet, and M. H. G. Jacobs

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Polymorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. B. Rietveld, R. Céolin, and J. Ll. Tamarit

41

Part II

Facts and Features

5

Aromatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. R. van der Linde and H. A. J. Oonk

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Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 D. Mondieig, E. Moreno-Calvo, and M. À. Cuevas-Diarte

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Plastic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 J. Ll. Tamarit, M. Barrio, L. C. Pardo, and Ph. Negrier

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Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 J. Salud and D. O. López

9

Enantiomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 H. A. J. Oonk and I. B. Rietveld

10 Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 A. Marbeuf and D. Mikaïlitchenko 11 Triacylglycerols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 L. Bayés-García, M. À. Cuevas-Diarte, and T. Calvet

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Contents

Part III Applications 12 Phase Change Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 M. A. Cuevas-Diarte and D. Mondieig 13 Crystallization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 H. P. C. Schaftenaar, M. Matovi´c, and J. H. Los Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Substances Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

Contributors

M. Barrio Grup de Caracterització de Materials, Universitat Politècnica de Catalunya, Barcelona, Spain L. Bayés-García Grup de Cristal·lografia Aplicada, Universitat de Barcelona, Barcelona, Spain T. Calvet Grup de Cristal·lografia Aplicada, Universitat de Barcelona, Barcelona, Spain N. B. Chanh Université de Bordeaux, LOMA, Talence, France M. À. Cuevas-Diarte Grup de Cristal·lografia Aplicada, Universitat de Barcelona, Barcelona, Spain R. Céolin Grup de Caracterització de Materials, Universitat Politècnica de Catalunya, Barcelona, Spain Y. Haget Université de Bordeaux, LOMA, Talence, France M. H. G. Jacobs Technische Universität-Clausthal, Clausthal, Germany J. H. Los Ecole Normale Supérieure, Paris-Saclay, France D. O. López Grup de Recerca de les Propietats Físiques dels Materials (GRPFM), Universitat Politècnica de Catalunya, Barcelona, Spain A. Marbeuf CNRS-Université de Bordeaux, Talence, France M. Matovi´c Openbaar Lyceum Zeist, Zeist, The Netherlands D. Mikaïlitchenko CNRS-Université de Bordeaux, Talence, France D. Mondieig LOMA, UMR 5798, Université de Bordeaux, Talence, France E. Moreno-Calvo Grup de Cristal·lografia Aplicada, Universitat de Barcelona, Barcelona, Spain Ph. Negrier LOMA, UMR 5798, Université de Bordeaux, Talence, France H. A. J. Oonk Universiteit Utrecht, Utrecht, The Netherlands vii

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Contributors

L. C. Pardo Grup de Caracterització de Materials, Universitat Politècnica de Catalunya, Barcelona, Spain I. B. Rietveld Laboratoire Sciences et Méthodes Séparatives, Université de Rouen Normandie, Mont Saint Aignan, France J. Salud Grup de Recerca de les Propietats Físiques dels Materials (GRPFM), Universitat Politècnica de Catalunya, Barcelona, Spain H. P. C. Schaftenaar Universiteit Utrecht, Utrecht, The Netherlands J. Ll. Tamarit Grup de Caracterització de Materials, Universitat Politècnica de Catalunya, Barcelona, Spain P. R. van der Linde Universiteit Utrecht, Utrecht, The Netherlands

Part I

Introductory Part

Chapter 1

Introduction M. À. Cuevas-Diarte and H. A. J. Oonk

Abstract This chapter is an introduction to the book Molecular Mixed Crystals – a monograph on mixed crystals, and on a limited extent on stoichiometric compounds, formed between molecular substances. The emphasis is on the structural and thermophysical properties of binary systems. Summaries of the chapters are presented, and a view is given of the REALM – the network in which the authors of the chapters have joined their forces.

1.1 Scientific Setting This book, in the first place, is a monograph on molecular alloys—mixed crystals of the substitutional type between two or more molecular substances. In addition, and to a limited extent, attention is given to molecular compounds—stoichiometric complexes (Chaps. 10 and 11). The emphasis is on binary systems and their structural and thermophysical properties as a function of composition, temperature, and pressure. Within this context, the book is a treasury of information: crystal structures; polymorphic changes; heat effects of melting and transition; thermodynamic mixing properties; phase diagrams; empirical relationships; and the like. Apart from its scientific significance, the book is a reflection of almost half a century of concerted research, carried out by the REALM: Réseau Européen sur les Alliages Moléculaires (European Network on Molecular Alloys); see below. Right from its start, the philosophy of the REALM has been to study families of systems rather than a number of isolated ones. Such a family consists of systems of which the composing pure components are from a chemically coherent group. For instance, one can think of the family of the n-alkanes. M. À. Cuevas-Diarte (B) Grup de Cristal·lografia Aplicada, Universitat de Barcelona, Barcelona, Spain e-mail: [email protected] H. A. J. Oonk Universiteit Utrecht, Utrecht, The Netherlands e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. À. Cuevas-Diarte and H. A. J. Oonk (eds.), Molecular Mixed Crystals, Physical Chemistry in Action, https://doi.org/10.1007/978-3-030-68727-4_1

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M. À. Cuevas-Diarte and H. A. J. Oonk

The study of families of systems has been enormously fruitful. Several empirical relationships have been established: between different thermodynamic mixing properties; and also between thermodynamic mixing properties and exo-thermodynamic parameters. In particular parameters that are related to the structural mismatch between the components of the binary systems. The book is intended to become and to be a reference point for everyone interested in mixed crystals in general, and molecular mixed crystals in particular.

1.2 About the Chapters Including this Introduction chapter, the book has 13 chapters. Chapters 2–13 fall into three sections: (A) Introductory Part (Chaps. 2–4); (B) Facts and Features (Chaps. 5– 11); and (C) Applications (Chaps. 12 and 13). The following is a short characteristic of each of the chapters, to begin with Chap. 2. (A) 2.

Introductory Part Molecular homeomorphism and crystalline isomorphism Haget Y., Chanh N. B., Oonk H. A. J., Cuevas-Diarte M. À.

In order to form mixed crystals of the substitutional type between two substances A and B, the component molecules A and B must be similar in size and shape. The measure of similarity is expressed by the coefficient of molecular homeomorphism. In order to form a continuous series of mixed crystals, the component substances must be isomorphous. The chapter starts with an historical overview and ends with a precise statement of the conditions for isomorphism. 3.

Thermodynamics Oonk H. A. J., Calvet T. and Jacobs M. H. G.

Much of the research, detailed in the chapters ahead, has been carried out on binary systems under isobaric conditions. The investigations have revealed that the thermodynamic mixing properties of the mixed crystalline state comply with a relatively simple thermodynamic model—with three system-dependent parameters. In the text, the model is detailed, and a demonstration is given of its power, which, at the same time, is surprising and outstanding. 4.

Polymorphism Rietveld I. B., Céolin R. and Tamarit J. Ll.

Molecular substances, as a rule, manifest themselves in more than one crystalline form. And it may happen that two members of a given family, under given conditions of temperature and pressure, do not adopt the same form. Polymorphism, for that matter, is a fascinating and at the same time complicated phenomenon. For the treatment of the polymorphism of a given substance, it is a sine qua non to take into account the influences of temperature and pressure. The chapter starts with an historical overview.

1 Introduction

(B) 5.

5

Facts and Features Aromatics van der Linde P. R. and Oonk H. A. J.

This is the first of seven chapters in which the outcome is summarized of a large number of studies on a large number of (families of) systems. A start is made with the family of the para-dihalobenzenes, including the key system para-dichlorobenzene + para-dibromobenzene. The family of the dihalobenzenes is followed by the group of the 2-R-substituted naphthalenes, which falls apart into two subfamilies. The naphthalene group includes naphthalene itself (R = H) and the substances with R = F, Cl, Br, SH, CH3 , OH. In addition, a number of isolated systems are treated, one of them being trans-azobenzene + trans-stilbene. Especially, worth mentioning is the melting behavior of mixed crystalline samples prepared by zone leveling. And evidence is given of an extra attractive effect between substituted methyl and substituted halogen. 6.

Chains Mondieig D., Moreno-Calvo E. and Cuevas-Diarte M. À.

Starting from a simple aliphatic hydrocarbon chain, a study has been made of the effect on structural properties caused by the incorporation of an increasing number of hydrogen bonds. In reality, this comes down to investigations into the structural characteristics—including polymorphism—and the phase behavior of binary mixtures in the following groups of substances: the n-alkanes; the 1-alkanols; the α,ω-alkanediols; the alkanoic acids; and the alkanedioic acids. The results that have been obtained clearly show that two ‘parameters’ have a crucial influence on the structural and thermodynamic properties. These are (i) the parity (odd vs. even) of the carbon chain; and (ii) the aim at realizing as many as possible hydrogen bonds. 7.

Plastic crystals Tamarit J. Ll., Barrio M., Pardo L. C. and Négrier P.

This chapter describes binary systems that involve components with at least one orientationally disordered (plastic-crystalline) phase. Ordered and disordered molecular alloys are described for two families that differ from one another as regards their molecular interactions (hydrogen bonds vs. van der Waals). In addition, special attention is given to the relationship between stable and/or metastable polymorphs and the pressure–temperature phase diagram for pure components. 8.

Liquid crystals Salud J. and López D. O.

The tricritical and the reentrant nematic behaviors are two of the most relevant features of the Smectic A (SmA)-to-Nematic (N) phase transition in binary mixtures of liquid crystals. Both of these concepts are studied from a theoretical and an experimental point of view for some two-component systems whose members are calamitic liquid crystals belonging to the alkylcyanobiphenyl (nCB) or alkoxycyanobiphenyl

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M. À. Cuevas-Diarte and H. A. J. Oonk

(nOCB) series, n being the number of the carbon atoms in the alkyl or alkoxy chain, respectively. 9.

Enantiomers Oonk H. A. J. and Rietveld I. B.

Only in exceptional cases, a pair of enantiomers do form a series of mixed crystals. The most well-known of the exceptions is the system laevorotatory carvoxime + dextrorotatory carvoxime; its properties are discussed in some detail. Also, attention is given to recent work on the polymorphism of optically active drugs. 10.

Complexes Marbeuf A. and Mikaïlitchenko D.

In certain cases, two molecular substances A and B, having a high degree of molecular homeomorphism, give rise to the formation of a complex AB, rather than producing a series of mixed crystals. Complexes are formed when short-range Van der Waals forces are overruled by long-range coulomb forces or by hydrogen bonds. Two groups of binary systems have been studied: (i) the group of benzene and benzene derivatives, and (ii) the group of naphthalene and naphthalene derivatives. 11.

Triacylglycerols Bayés-Garcia L., Cuevas-Diarte M. À. and Calvet T.

Triacylglycerols (TAGs) are the main components of edible fats and oils. TAGs are widely employed in cosmetics and pharmaceutical formulations. TAGs exhibit a complex pattern of polymorphism. In this chapter, an account is given of their polymorphic crystallization and transformation behavior—and so from pure TAG components to more complex lipid systems. Special attention is given to the effects caused by the application of dynamic thermal treatment. These effects are the key to the design of end products that have the physical properties required for them. (C)

Applications

12.

Phase change materials Cuevas-Diarte M. À. and Mondieig M.

Apart from a purely scientific interest in molecular mixed crystals, the REALM continuously has been interested in finding applications—especially in the field of phase change materials for thermal protection and storage of thermal energy. The central actors are the heat of melting of the material and the thermal window, which is the temperature range in which the change from solid to liquid takes place. Applications are possible in the range of temperature from −50 to +200 °C. The composition of the material is one of the parameters that can be used to tune the thermal window to the desired temperature. 13.

Crystallization of molecular mixed crystals Schaftenaar H. P. C., Matovi´c M. and Los J. H.

1 Introduction

7

The crystallization of mixed crystals from a liquid mixture of the components is a complex event. Mass-transport and heat-transport limitations prevent the crystallizing system from adopting through and through thermodynamic equilibrium: equilibrium phase diagrams are making place for kinetic phase diagrams. The theoretical background of non-equilibrium crystallization is the main subject of the chapter.

1.3 About the REALM The member groups of the REALM are from the Université de Bordeaux (UBx), the Universitat de Barcelona (UB), the Universitat Autonoma de Barcelona (UAB), the Universitat Polytecnica de Catalunya (UPC), and Utrecht University (UU). The origin of the REALM goes back to about 1973 as a cooperation of crystallographers from UBx, UB, and in 1981 from UAB. Starting from 1984, thermodynamic expertise was brought in by physical chemists from UU. From 1988, the network was extended by physicists from UPC interested in plastic crystals, and their colleagues studying liquid crystals. Starting with the Université de Bordeaux, the following is an enumeration of names of the people, who have contributed to the research of mixed crystals, during their whole (scientific) career, or for a short period of time as a Ph.D. student. Université de Bordeaux. Yvette Haget, Nguyen Ba Chanh, Louis Bonpunt, Jany Housty, Alain Marbeuf, Denise Mondieig, Philippe Négrier, Valérie Métivaud, François Michaud, Abdou Belaaraj, Didier Mikaïlitchenko, Fazil Rajabalee, Philippe Espeau, Laurence Robles, Gabin Gbabode. Universitat de Barcelona. Miquel Àngel Cuevas-Diarte, Teresa Calvet, Mercedes Aguilar, Esperança Tauler, Manuel Labrador, Lourdes Ventolà, Màrius Ramírez, Evelyn Moreno, Xabier Novegil, Raquel Cordobilla, Raúl Benages, Laura Bayés, Mercé Font-Bardia. Universitat Autonoma de Barcelona. Eugènia Estop, Xavier Alcobé, Angel Alvarez. Universitat Politècnica de Catalunya. J. Muntasell; Josep Lluís Tamarit, David López, Maria del Barrio, Josep Salud, Luis Carlos Pardo. Guests from University Paris Descartes: René Céolin, Ivo Rietveld. Utrecht University. Harry Oonk; Cees van Miltenburg, Kees de Kruif, Hans Kolkert, Harrie Govers, Aad van Genderen, Paul van Ekeren, Jan Huinink, Tjibbe Kuipers, Gerrit van den Berg, Koos Blok, Michel Jacobs, Ineke van Ginkel, Joke Bouwstra, Mark van Bommel, Wybe van der Kemp, Peter van der Linde, Margot Vlot, Hannah Gallis, Marija Matovi˙c. Guests: Jan Los, Harald Schaftenaar, Günter Figurski.

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1.4 Audience The book addresses itself to an audience interested in mixed crystals in general, and molecular mixed crystals in particular—from a point of view of crystallography, thermodynamics and phase theory, and physical chemistry in general. Acknowledgements The editors are grateful to all of those who are mentioned above. Our thanks are due to the army of master students who contributed so much—to the research efforts of the REALM; the scientific output of our research groups; and, not to forget, the pleasant atmosphere on the shop floor. In addition, we should like to take the opportunity of expressing our gratitude to the following people in particular. Mercedes Aguilar (Universitat de Barcelona) for fine artwork. Petra van Steenbergen (Springer, Dordrecht, The Netherlands) for years and years of assistance in matters of publishing. Dr. Angeliki Athanasopoulous (Springer Nature, London) and Mr. Boopalan Renu and Mr. Ritu Chandway (Springer Nature, Chennai, India) for patiently and professionally guiding us on the route from manuscript to printed book. We also are very anxious to express our thanks to the following external Institutions and Industrial Companies for their interest and financial support. In France: Ministère de l’Education Supérieur, ANVAR, Isos, France Télécom, Thomson, Euracli, Sofrigam. In Spain: Ministerio de Ciencia e Innovación, Generalitat de Catalunya, FECSA, DUCASA, Simón Coll. In The Netherlands: the Stichting Technische Wetenschappen, the Dutch Technology Foundation STW. Dedication We gratefully dedicate this book to Yvette Haget and Nguyen Ba Chanh, who are the founding scientists of our line of research–the research on molecular alloys that started in Bordeaux and then fanned out to Barcelona, Utrecht, and Paris. Their scientific skills and their amiability have inspired us and all of our colleagues mentioned above. Spring 2021 Barcelona: Miquel Àngel Cuevas-Diarte Utrecht: Harry A. J. Oonk

Chapter 2

Molecular Homeomorphism and Crystalline Isomorphism Y. Haget, N. B. Chanh, H. A. J. Oonk, and M. À. Cuevas-Diarte

Abstract In order to form mixed crystals of the substitutional type between two substances A and B, the component molecules A and B must be similar in size and shape. The measure of similarity is expressed by the coefficient of molecular homeomorphism. In order to form a continuous series of mixed crystals, the component substances must be isomorphous. The chapter starts with an historical overview and ends with a precise statement of the conditions for isomorphism.

2.1 Background and Controversy Since ancient times, humanity has recognised the importance of composition in the physical and chemical properties of materials. As early as the third century BC, Archimedes used scientific methods to determine the composition of a sword and established a linear relationship between variations in specific weight and composition in the chrome-silver system. He was the first scientist to establish a correlation between a physical property and composition in a continuous series of solid solutions. For a long time, discoveries were limited to other fields of science, and it was not until the end of the eighteenth and beginning of the nineteenth centuries that significant advances were made in the field of the composition of solid materials with the emergence, towards the end of the century, of the study of phase diagrams. It was during this period that the most serious controversies arose. In 1800, Proust asserted that compounds could be characterised by the constant proportions of their

Y. Haget · N. B. Chanh Université de Bordeaux, LOMA, UMR 5798, Talence, France H. A. J. Oonk Universiteit Utrecht, Utrecht, The Netherlands e-mail: [email protected] M. À. Cuevas-Diarte (B) Grup de Cristal·lografia Aplicada, Universitat de Barcelona, Barcelona, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. À. Cuevas-Diarte and H. A. J. Oonk (eds.), Molecular Mixed Crystals, Physical Chemistry in Action, https://doi.org/10.1007/978-3-030-68727-4_2

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elements, whilst in 1803, Berthollet claimed that these elements could be combined in any proportion [1]. The controversy ended in 1870 with the publication of the research carried out by an American physicist, Gibbs, on phase equilibrium, where it became apparent that the debate was simply a question of terminology. Proust was correct as regards stoichiometric compounds, whereas Berthollet was right as regards mixed crystals. Great strides were also made in the field of the natural sciences. The seventeenth century witnessed the seminal syntheses of botanical information, the appearance of binomial nomenclature by genus and species, and the development of a new systematics. It was during the eighteenth century that the first attempts to classify minerals were made, provoking one of the most serious debates in the study of solid materials, between mineralogists and chemists. For the former, the observation of crystalline forms was fundamental, whilst, for the latter, it was chemical analysis which determined the mineral species. Regardless of these differences, a mineral species was defined as “an assemblage of substances with the same composition”, although many cases were known to exist where the proportions of the constituent parts were variable within determined limits. Thus, in 1782, Bergman [2] produced a mineral classification scheme based on chemical characteristics. One year later, Romé de Lisle [3] reported the first findings on isomorphism, heralding the discovery of isomorphic mixtures with his demonstration that copper sulphate and ferrous sulphate can be mixed to produce crystallisation. His work laid the foundations of modern crystallography. Later, in 1810, the renowned chemist Berzelius [4] produced a new, more methodical classification scheme; his stoichiometric laws were based on the number of saline constituents identified by chemical analysis. He observed that each salt possessed two groups of constituent oxides, one acid and the other basic, whilst the water of crystallisation was neutral. Based on these observations, he defined two important laws: the “rule of oxides” and the “law of unitary oxygen”. With the first, he established that the oxygen of the acid component was equal to or a multiple of the basic component, whilst with the second, he stated that the oxygen of the constituent oxide was a unit, or that the oxygen of each of the other constituents was a multiple. He then used the defined proportions to determine the different species. Many mineralogists, such as Pusch in 1815 [5] and Hausmann in 1911 [6], were opposed to this classification scheme, considering it anti-mineralogical since certain constituents might be essential chemically but not mineralogically, in other words, they did not produce sufficient differences in external characteristics to justify separation into different species. A purely stoichiometric approach frequently distinguished separate species which a mineralogist considered simply varieties of the same substance. In contrast, for mineralogists such as Wérner in 1774 [7], the fundamental criterion was crystalline form. In 1801, Haüy [8], the leading figure in crystallography at the beginning of the nineteenth century, combined his knowledge of crystallography, physics and geometric calculations to arrive at the conclusion that crystallisation was the most reliable criterion for distinguishing between different species. A mineral

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species consisted of a collection of substances whose constituent molecules were very similar and composed of the same elements in the same proportions. Haüy established the principle according to which two minerals of a different composition could not have the same form, unless one of them was what was called a borderline form (cube, octahedron, regular tetrahedron and rhombic dodecahedron). Crystallographers maintained that substances whose forms derived from the same crystal system belonged to the same species, whilst chemists argued that a species was formed by substances with the same composition. It was Haüy who, due to his great prestige, ended the debate in 1809 when he asserted that all the substances identified by mineralogists as belonging to the same species frequently presented considerable variations in composition. These differences in composition were explained by the assumption of the existence of mixtures which had combined with the truly constituent parts. Haüy established the fundamental basis of crystallography, and his opinions received widespread acceptance in the period. To add an interesting anecdote, following the French Revolution, his condition of priest was respected, and he was allowed to continue wearing his clerical clothing [9]. This concept of species entered into contradiction with the tradition represented by Berthollet, who asserted that in order to establish that a constituent molecule belonged to a species, and that the composition of the latter was constant, Haüy would be obliged to consider all the differences which analysis had shown to be present in minerals with the same form as a heterogeneous substance. In earlier studies, Vauquelin [10], in 1797, and Leblanc in 1801 [11] had observed considerable variations in the proportions of the constituents of salts, without finding a corresponding difference in their crystalline form. In fact, concerned by the relationship between chemical composition and crystal structure, Haüy implicitly acknowledged the principle of the continuity of solid materials, but confused the constituent molecule with the chemical molecule. It would be necessary to wait for Delafosse and the concept of the unit cell before Haüy’s tenets progressed further. In 1818, Beudant [12] had observed that by mixing a small quantity of ferrous sulphate hydrate with zinc sulphate hydrate, he obtained crystals with the rhombohedron shape of ferrous sulphate hydrate. He conducted many experiments, mixing this sulphate with zinc and copper sulphate hydrate and studying the form the crystallisation took. He reached the conclusion that even in small quantities one component could have a great impact on the properties of the compound, and that ferrous sulphate could take many different forms depending on the type and proportions of salts with which it was mixed. In some cases, when two substances were mixed, one imprinted its crystalline form on the other, whereas in others, the results could only be explained in terms of a combination of the two compounds, giving rise to a particular kind of composition with crystals that presented the forms of both substances. In a later study of 1858, Delafosse [13] referred to the law established by Beudant, based on his observations of the crystals of different carbonates and some other salts. According to this law, the angles of a mixed crystal would have an intermediate value

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between the initial products and they would be proportional to the quantity of each of them. According to Beudant, the form of the carbonates was explained by the presence of small quantities of calcium carbonate, undetected in chemical analyses. Using his reflective goniometer, in 1812, Wollaston [14] found small differences in the angles of these forms, which had not been acknowledged by Haüy for crystallographic reasons. These authors considered the substances identical. It was in this context, in 1819, that the first study by Mitscherlich [15] appeared on diverse sulphates—previously studied by Beudant—phosphates, arsenates and ferrous and calcium carbonates. This great German scientist had first studied Oriental languages, turning later to medicine and combining his medical studies with the study of ancient Persian texts. Shortly afterwards, he became interested in chemistry, and in 1818, he began to work in the field of crystallography. He observed that the crystals of potassium phosphate and potassium arsenate were almost identical in form. At the same time, he demonstrated that the sulphates of different metals (Fe, Co, Mn, Cu, Zn, Ni and Mg) could crystallise with the same form when they had the same quantity of water of crystallisation. One year later, he met Berzelius and moved to his laboratory in Stockholm where he conducted the research which would lead him to his formulation of isomorphism, and his confrontation with Haüy. On his return to Germany, he continued his brilliant research career, eventually becoming a professor at the University of Berlin. In 1832, he extended the research he had carried out in Sweden, broadening it to include other compounds. He became member of the Berlin Academy of Science and directed his own laboratory. He conducted innumerable studies in inorganic chemistry, ranging from synthesis of new acids to determining the vapour density of numerous substances. His achievements were no less notable in the field of organic chemistry: synthesis of benzene and derivatives of the same, determination of its chemical formula, etc. His formulations comprised the forerunners of Berzelius’ theory of catalysis. He was the first to demonstrate the existence of the phenomenon that we know as dimorphism. He established that NaH2 PO4 ·H2 O and NaH2 AsO4 ·H2 O normally exist in two different forms, and furthermore, that phosphate could present in a form identical to that of arsenate. He also developed methods for analytical organic and inorganic chemistry. Lastly, he was interested in geology and mineralogy, particularly, in the synthesis of minerals based on the fusion of silicon with various metal oxides. In his study of 1821 [16], by this time immersed in the chemistry of Berzelius, Mitscherlich examined the composition-form relationship of various phosphorous and arsenic acid salts. It was in this article that the term isomorphs appeared for the first time: “…An equal number of atoms combined in the same way produce the same crystalline forms, and these do not depend on the chemical nature of the atoms but on their number and form of combination”. The chemical elements could be classified into groups, and he named those elements pertaining to the same group, isomorphs. The small differences in the angles of the crystals of isomorphic substances were due to their chemical affinity. He clearly and explicitly stated that the fundamental tenet of isomorphic substances was that they could crystallise in any proportion.

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It should be noted that Mitscherlich used Berzelius’ theory of stoichiometry to conduct his studies, that is, he based his work on the case of type A1−x Bx C, ionic crystals, where an anion (C) accepted a metallic cation (A) or another cation (B) or a mixture of both. Isomorphism was defined by three properties; analogous form or identical crystalline form, analogous chemistry and, lastly, the possibility of syncrystallisation occurring in any proportion. These results contradicted the axiom established by Haüy, according to which a given substance was characterised by the angles of its crystals. Nevertheless, the law established by Mitscherlich was enormously helpful to Berzelius in determining the atomic weight of elements, and later, in clarifying the composition of many minerals where crystallography appeared to contradict chemical analysis (pyroxenes and amphiboles, for example). The discovery of isomorphism showed that Haüy’s proposal was too absolute and imprecise, but it did not demolish it totally, since in the examples which best fitted the definition of isomorphism, their chemical differences translated into differences in the value of the crystal angle. However, this was not appreciated at the time. Similarity of crystalline form refers to the external form of crystals, namely, their morphology. Nevertheless, the parametric relationship was determined during this period, and this enabled scientists to suggest hypotheses about the content of crystal unit cells. In 1980, Melhado [17] provides an extensive description of the events and controversies of this period. Haüy always maintained his opposing stance, despite the fact that Mitscherlich always accorded him respect: on observing that Haüy’s law did not hold true, for example, he remarked of the law that this was not general. During the debate which ensued, Mitscherlich had no direct contact with Haüy, as reflected by the fact that the debate was conducted entirely through letters between Haüy and Berzelius, with the latter always affirming his support for the work of Mitscherlich.

2.2 Isomorphism After Mitscherlich After Mitscherlich, the idea of isomorphism rapidly began to lose its currency, being relegated to the position of a control criterion for determining atomic weights, and research in the fields of chemistry and mineralogy began to diverge. Nevertheless, some scientists continued to show great enthusiasm for the subject of isomorphism: following the line of research initiated by Beudant’s law, Kopp, in 1843 [18] demonstrated that isomorphic substances had an approximately equal molecular volume (relationship between molecular weight and specific weight), and that the true characteristic of the isomorphism of two compounds was the ability to form mixed crystals. Towards the end of the century, Retgers stated a generalisation concerning the proportionality of the physical properties of mixed crystals with respect to composition, and later, Vegard applied the generalisation to crystal parameters.

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During the same period, another author, Laurent, in 1845 [19], stated that two substances from different systems could be isomorphs. In contrast, in 1848, Pasteur [20, 21] did not agree since this would not fulfil an essential condition of crystallisation, namely that maximum occupation of the space should occur, and that total syncrystallisation in this case could only arise if dimorphism was present. He would appear to have been the first to speak of isodimorphism: the phenomenon that each of two substances A and B can crystallise in two different forms, such that each of the forms of A is isomorphous with one of the forms of B. A new concept was introduced by Delafosse in 1851 [22], plesiomorphism. For this author, isomorphism implied that the composition of substances could be reduced to a single formula. Similarity between crystalline forms was the consequence of a pre-existing similarity in molecular type. He distinguished two kinds of isomorphism, that of Mitscherlich, which preserved the crystallographic system, and that of Laurent, in which compounds belonged to different systems. Faced with the wide variety of examples in which the three conditions defined by Mitscherlich were not fulfilled, each author gave priority to a different condition. For some, the essential characteristic was similarity in crystalline form, whilst for others this was merely a reflection of analogous chemistry, and the existence of mixed crystals was simply a consequence of the first two. Still others affirmed that it was this latter characteristic which truly demonstrated that two compounds were isomorphs. Another name which stands out in the study of isomorphism is that of Groth, who in 1870 [23] studied the relationship between crystalline form and chemical composition in derivatives of benzene and naphthalene. His research led him to conclude that for some atoms and groups of atoms, substitution by hydrogen did not modify the shape of the structure except in one direction, and he named this phenomenon morphotropism. The conditions on which the appearance of morphotropism was dependent included: (1) the morphological properties of the substituting atom, (2) the chemical nature of the compound in which the substitution took place, (3) the crystal system and (4) the relative position of the atom or group of atoms. Numerous studies were carried out during this period on the physical and chemical properties of mixed crystals. Of interest was the research conducted by Baumhauer in 1870 [24] concerning corrosion figures, in which he stated that in isomorphic substances, these figures would have the same position and form on the corresponding faces. In 1811, Arago [25] discovered the polarisation of light, enabling him to use optics as a means of research. Special mention should be made of the optical study by Sénarmont in 1851 [26] of isomorphic compounds and their alloys. Bodländer, in 1860, [27] determined the proportionality between composition and rotatory power, forming a straight line when represented graphically; in 1880, Schuster [28] observed that the angle of extinction presented continuous variation according to composition. In 1886, Wyrouboff [29] carried out numerous optical studies of isomorphism, basing his research on the idea that, according to Mitscherlich, isomorphism was the property of certain substances with a similar chemical composition and geometric shape, according to which they would crystallise together in any proportion. He examined

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the meaning of geometrical symmetry, and proposed a difference of angles of 2–3°. He rejected those cases where, although possessing a strong similarity in terms of chemical composition and crystal parameters, the substances belonged to different crystal systems. He established the law of variation in the optical axis angle of mixed biaxial crystals according to their chemical composition. Of particular note was his work on dimorphism in neutral thallium and ammonium racemates and tartrates, in which he found that although there were no common faces in their crystalline forms, they crystallised in all proportions. Today, we know that a single group can present various external forms. The same year, in 1886, Mallard [30] was to reject this definition of isomorphism. He proposed the term syncrystallisation to refer to the property of crystallising together, and relegated isomorphism to an etymological definition of the word, that is, to the more or less close similarity between crystalline forms. Towards the end of the nineteenth century, in 1889, an extraordinary and wellstructured study by Retgers [31] appeared, in which he reported extensive research on isomorphism. After examining many cases, above all those which were unclear, he concluded that the true expression of the isomorphic character of two compounds was similarity of form and chemical composition, together with the property of producing an intimate mixture, demonstrated by the continuity of all properties, both chemical and physical (specific weight, elasticity coefficients, thermal and electrical constants, etc.). It was necessary to diversify research methods and focus on those properties which presented marked differences in pure salts. Chemical and crystalline similarities alone were extremely elastic characteristics, given that they were subject to gradation (for example, the substitution of one atom for others of lower valency, of equivalent elements, similarity in crystal face angles, position and development of faces, etc.). He employed variation in specific volume (the inverse of specific weight) according to composition expressed as a weighted fraction to determine the presence of isomorphism and isodimorphism. In the former case, this variation was a straight line, whilst in the latter, two straight, non-parallel lines were obtained. Discontinuity could be present in both cases, but whereas in the first, the two parts of the straight line would be a prolongation of each other, in the second this regularity did not appear. This representation enabled isomorphism to be distinguished from isodimorphism with relative ease. Isomorphism implied chemical and crystalline similarity, where two compounds could yield homogeneous mixed crystals. Substances which presented the first two properties but not the third he described as morphotropic, and those which only presented similarity in form, isogonic. Morphotropy described changes in form due to successive chemical substitutions. He suggested that the study of isomorphism should be based on mixed crystals rather than on comparison of pure products.

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2.3 Thermodynamic Theory of Mixed Crystals: Solid Solutions It was a period in which both an important qualitative leap and great progress was made: for the sake of brevity, only three important advances will be mentioned. Gibbs’ theory of phase equilibrium [32] was widely accepted in the scientific community. In his research, he defined the concepts of phase, components and the conditions for thermodynamic stability of mixtures, and also established the basis for a classification of phase diagrams. These bases enabled Van der Waals, and later, in 1899, Roozeboom [33] to derive all possible phase diagrams in more depth. Le Chatelier explained the possibility of determining phase diagrams using thermal techniques, measuring the fusion point of mixtures with a thermocouple. In 1898, Van’t Hoff [34] established that properties such as miscibility between two substances would depend on pressure and temperature conditions and could change according to these conditions. As regards isomorphic mixtures, he introduced the term solid solutions, widely used nowadays, and stated that isomorphism existed even in those cases in which two compounds did not mix in all proportions. Similar to the solubility of liquids, in solid state, the ability to mix depended on temperature, and it was possible that at a given temperature partial miscibility would become total in compounds with the same crystalline form. The term solid solution was explained through analogy with the theory of solutions developed by this same author; the solid constituent assimilated with the solvent in higher proportions, whilst in the solute it was present in a lower proportion. A period commenced in which studies of mixed systems, and above all of metal alloys, multiplied, and the concept of eutectic systems emerged (from the Greek, eutektos, “easily melting”). At the beginning of the twentieth century, in 1905, Tammann [35] published a series of articles on metallic diagrams, in which he presented an exhaustive analysis of cooling curves, using these as the basis for constructing the diagrams. In his book on crystallography published in 1909, Wallerant [36] dedicated various chapters to isomorphism. Using examples, he conducted a separate analysis of each of the three conditions necessary for compounds to be defined as isomorphs, concluding that the existence of two of these conditions did not inevitably imply that of the third, and stated that before converting a definition into law, more detailed study was required. This text constituted an exhaustive compilation of all that was known at the time, and also presented an analysis of the characteristics of compounds known as isomorphs: crystalline form, exfoliation, optical properties, crystallisation from saturated solutions, etc. In none of these cases evidence was found of a common behaviour. As regards the property of syncrystallisation, he summarised the ideas on solutions proposed by Van’t Hoff and described the methods for obtaining solid solutions and possible phase diagrams. All the physical properties of mixed crystals analysed varied continuously with composition, and it was precisely this continuity which distinguished them from a

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chemical combination. As for their structure, Van’t Hoff did not agree with those authors who believed they were composed of alternate layers of pure compounds, nor those who, based on Bravais’ theory, claimed that some of the molecules from one of the substance’s unit cells were replaced by the molecules of the second, since according to this author, they would lose their symmetry. He saw the edifices of isomorphic mixtures as individual edifices, rather than more or less intimate mixtures of two edifices. The elementary particles of pure compounds would have the same number of similarly arranged molecules, and by substituting the elementary particles, it would be possible to pass from one crystalline edifice to another. Wallerant thus defined the existence of a continuous series of mixed crystals between two isomorphic compounds in the strictest sense of the word. Where the elementary particles of two substances did not contain the same number of molecules, or these were arranged differently, two series of crystals would necessarily be produced; isomorphism was no more than a theoretical concept which provided an easy explanation for the relationship between two substances susceptible to being mixed in order to crystallise.

2.4 Isomorphism Following the Discovery of X-Ray Diffraction and the Atomic Structure of Crystals In 1912, the discovery by Max Von Laue and his collaborators Friedich and Knipping of X-ray diffraction by crystals demonstrated that it was possible to determine the atomic order of solids, heralding a new phase in the development of a theory of mixed crystals. The application of X-ray diffraction to a microcrystalline powder discovered by Bragg and Debye was rapidly adopted to determine phase diagrams. With this new technique, it became possible to discover the number of system phases and to determine both the crystal parameters of the same and their dependence on composition. However, despite the information it provides, this technique is not routinely used these days to determine phase diagrams, often yielding partial information of the same. Throughout this book, it will be observed how diffraction can constitute an indispensable technique for demonstrating the presence of a given phenomenon and complementing the information obtained with other analytical techniques. With a simple structure, it was possible to determine the position of the atoms and calculate interatomic distances. Resolving structures held great interest for mineralogists since, as we have seen, they looked to the isomorphism of structures to provide an explanation for the numerous regularities observed in geochemistry. It was found that many minerals could be explained as solid solutions. One of the first to apply X-ray diffraction to the study of mixed crystals was Vegard in 1917 [37]. As we saw earlier, some authors, such as Retgers, believed that these were simple mechanical mixtures formed by fine, homogeneous and alternating layers of the constituent compounds. However, the studies of Wulf in 1906 [38],

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Grossner in 1907 [39] and Tutton in 1910 [40] demonstrated that the difference in molecular volume of the constituents played a fundamental role in the formation of mixed crystals, and that these were the consequence of a more intimate union of the components. Vegard studied the diffraction spectra of solid solutions of KBr-KCl and KClNH4 Cl (all three of which are cubic) and compared them with those of their components. He demonstrated that these mixed crystals were homogeneous substances rather than being composed of different layers, since they each produced single reflections situated between the respective values of the constituents. He formed an empirical law from his experimental results: the linear variation of the crystal parameters of alloys according to composition. As we will see later, the most frequent case was continuous variation of these parameters, but presenting deviation in linearity. Today, we know that Vegard’s general rule is, in fact, the exception. In a chapter of his book “Leçons de Cristallographie” of 1926, Friedel [41] undertook an extensive analysis of the three properties given as an indispensable condition for the existence of isomorphism: homeomorphism, or the close similarity between crystalline forms, total or partial syncrystallisation and identical chemical composition, except for the replacement of certain elements. For a long time, it was believed that the first two characteristics implied the third, and that syncrystallisation was a sign of identical chemical composition. However, nowadays, we know that this is not the case; even when isomorphic series, in the strict sense of the word, exist (for example, silicates, carbonates and oxides), there are numerous examples where only two of the three conditions are fulfilled. Homeomorphism and syncrystallisation can occur without identical chemical composition (for example, the substitution of the Si4+ Na1+ group by the Al3+ Ca2+ group in the plagioclases); homeomorphism and identical formulas can occur without syncrystallisation (between RbCl and NaCl); syncrystallisation occurs without homeomorphism, interpreted as isodimorphism, or that the compounds share a common multiple unit cell; and homeomorphism can occur with or without syncrystallisation whilst showing chemical analogies between the compounds (for example, the notable homeomorphism between NO3 Na and CO3 Ca, in the form of calcite, which do not syncrystallise). Homeomorphism is best defined by determining unit cells and crystal parameters. In the majority of cases, species which syncrystallise have a similar type of structure and are strictly homeomorphic; when they do not syncrystallise, it is due to a difference in molecular volume and, consequently, in parameters. In a compound series, when the differences in volume and parameters increase, syncrystallisation ceases to be possible, although not abruptly so; rather, the two compounds syncrystallise only in determined proportions, giving rise to a gap similar to that which occurs in isodimorphism. It would appear that any difference in crystal parameters of below 10% is not prohibitive. As regards the structure of solid solutions, Friedel was perhaps the first to speak of random substitution. He found that mixed crystals are statistically homogeneous, and X-ray diffraction gives an average of all the periods. Subsequent development of these techniques facilitated the discovery of the superstructure phenomenon, which emerges when a disordered solid solution is cooled,

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and the first theories describing the phenomenon began to be formulated. The appearance of high-temperature X-ray cameras led to the systematic study of metal alloy structures. The introduction of computing, neutron diffraction, electronic microscopy, etc., and the diversity of materials and fields of study (metals, ceramics, polymers, biological complexes, etc.) have both increased and become more specialised. It is neither possible here to enumerate them all here, but rather to examine certain specific aspects in the history of science which are most directly related to the subject matter in question. The following paragraphs will focus on the field of organic compounds, and more precisely, on molecules with a low molecular weight.

2.5 Isomorphism in Organic Compounds The first organic system, between oleic acid and margaric acid, was studied in 1823 by Chevreul [42]. Other systems (fatty acids, urea, etc.) were studied with the aim of establishing a relationship between their properties and their composition. In the previous section, mention was made of the work conducted by Provostaye in 1841 [43], Laurent in 1841 [44] and Groth in 1870 [23], in the study of the modifications to benzene and naphthalene crystalline forms, among others, brought about by the substitution of an atom. Particular mention should be made of those scientists who continued the line of research initiated by Retgers. In the nineteenth century, these directed their efforts to the study of isomorphism, determining binary diagrams among the para-dihalogen derivatives of benzene. The work of Bruni and Gorni of 1900 [45, 46], Küster in 1905 [47] and Nagornow in 1911 [48], among others who will be mentioned in later chapters, demonstrated that these derivatives, when disubstituted with chlorine or bromine, possessed the same symmetry and a similar parametric relationship. Numerous phase diagrams were also obtained using thermal analysis methods in the field of organic chemistry at the beginning of the twentieth century. Examples would include the research of Efrenov of 1913 [49], Smits in 1923 [50] and Timmermans in 1936 [51]. At this time, organic molecules were considered atomic systems with a determined configuration. Attempts to establish a relationship between phase diagrams and molecular structures led to nothing until information about organic crystal structure was obtained by X-ray diffraction. In general, organic compounds present low symmetry; the crystal unit cells contain few molecules but many atoms in general positions. As a consequence, the structure can only be determined from single crystals, which in many cases are difficult to obtain, and by employing sophisticated methods to analyse the diffractograms.

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2.6 Geometric Model of Molecular Compounds Following the discovery of the structure of penicillin, great strides were made from 1940 onwards in the study of the structure of organic molecules. A few years later, in 1957, Kitaigorodskii [52] laid the foundations of modern organic crystal chemistry. In brief, he established two empirical rules: molecules are arranged in a crystal in such a way as to occupy the minimum space whilst presenting maximum symmetry. In this dense packing model, the form and size of molecules are described by geometric criteria (bond length, bond angles and intermolecular distances, determined using known structures). This packing is characterised by the molecular coordination number; in many crystalline substances, this is twelve. It should also be stressed that all organic crystals are composed of successive layers, each of which is characterised by the partial coordination number of six. Where the intermolecular distances and the bond angles are known, molecular volume (V o ) can be determined; with this, the crystal unit cell volume (V ) and its content (Z), it is possible to determine the packing coefficient (K) of organic structures. K = Z · Vo /V

(2.1)

Kitaigorodskii observed that this coefficient K varies between 0.6 and 0.8.

2.7 Coefficients of Molecular Homeomorphism and Crystalline Isomorphism According to Kitaigorodskii [53], one condition necessary to form a true solid solution, that is, for one molecule to substitute another is the possibility of forming a large number of contacts between these molecules. In other words, a high packing index is required: the replacement of one molecule for another should not alter the intermolecular distances by more than 0.4 or 0.5 Å. In contrast to what occurs in inorganic chemistry—where a spherical atom/ion is substituted by another—the isomorphism between organic molecules is only approximate. For that reason, we prefer to use the term molecular homeomorphism rather than molecular isomorphism. In Kitaigorodskii’s view, there are three types of molecular homeomorphism: • Homeomorphism by atomic substitution: the molecules only differ from one another in the substitution of an atom. In many cases, isomorphic crystals are formed. An example would be the series of paradisubstituted benzene derivatives. • Homeomorphism by radical substitution: the molecules differ from one another in a radical of much smaller dimensions than the rest of the molecule. Here, it is difficult to know to what extent such homeomorphism will imply crystalline

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isomorphism. Included within this group is the ß-substituted naphthalene series, where ß is equal to OH, NO2 , NH2 . • Homologous homeomorphism: is found in those series whose members only differ in the length of one of their axes, for example, the aliphatic and paraffin series. Kitaigorodskii [54] proposed a coefficient, εk , to quantify the degree of molecular homeomorphism. This coefficient—the coefficient of molecular homeomorphism—is defined by superimposing two molecules in such a way that the included, or common, volume  is maximised, and where the excluded volume is . The coefficient is given by the following expression, in which the subscript “k” is from Kitaigorodskii: εk = 1 − / 

(2.2)

The closer the value of εk is to 1, the more similar the molecules are in form and size. Kitaigorodskii attempted to relate geometric similarity with degree of miscibility, “we can assume that the volume ratio plays a decisive role in solid-state solubility” [52]. The numerous phase diagrams determined by Timmermans [51] demonstrate that there is no miscibility when εk < 0.8, whilst extensive miscibility indicates that εk > 0.9. Kraftchenko compared those molecular sections which are more or less flat and similar in form for various systems in which diphenyl sulphide intervenes with anthracene, phenanthrene, etc. He found that solubility was continuous when εk is >0.95. In the case of the degree, the coefficient of molecular homeomorphism, a geometric comparison is made between two molecules. In a similar way, one can make a geometric comparison between the crystalline cells of two isomorphous crystals and introduce the coefficient of crystalline isomorphism [55, 56]. The coefficient of crystalline isomorphism, εm , is defined as εm = 1−m / m

(2.3)

2.8 Updating the Concept of Isomorphism To conclude this chapter, we consider the three cases that are represented by the sections (a), (b) and (c) in Fig. 2.1 [57, 58]. The three sections pertain to binary systems, and each section consists of three parts: on top the isobaric solid–liquid phase diagram; in the middle, the representation of a certain property P (such as molar volume, or one of the unit cell parameters) as a function of composition and for a given temperature (T 1 ); and at the bottom, the isothermal section, for T = T 1 , of the molar Gibbs energies of solid solutions as a function of composition.

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Fig. 2.1 Three cases of binary systems in which the molecules of the substances A and B can replace one another in the crystal lattice. Case (a): the solids A and B are isomorphous and form solid solutions in all proportions; case (b): the solids A and B are isomorphous but do not form solid solutions in all proportions; case (c); the solids A and B are not isomorphous, and, as a consequence, do not form solid solutions in all proportions

In all three cases, the molecules of the substances A and B have a degree of molecular homeomorphism elevated enough to replace one another in the crystal lattice.

2.8.1 Case (a) Solid A and solid B are isomorphous, and together they form solid solutions in all proportions. In the middle part of Fig. 2.1a, it is shown that the value of a property P is represented by a continuous, smooth curve between the values of P of the pure solids A and B. The other way round, the notion that the values of P be represented by a continuous curve is a condition sine qua non for isomorphism [58].

2 Molecular Homeomorphism and Crystalline Isomorphism

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A special kind of property P is the molar Gibbs energy—it is represented by the curve at the bottom of the figure. For a continuous series of solid solutions, the molar Gibbs energy is a function that is convex over the whole composition range. Moreover, the cigar-like phase diagram is an indication that the solid solutions do not have a large positive deviation from ideal-mixing behaviour (see Chap. 3).

2.8.2 Case (b) Solid A and solid B are isomorphous, but do not form a continuous series of solid solutions: there is a miscibility gap. The Gibbs energy function is not convex over the whole composition range, due to the fact that the mixing of A and B goes together with a considerable positive deviation from ideal mixing. The compositions of the solid phases, that are in equilibrium at T = T 1 , follow from the points of contact of the double tangent line. In spite of the fact that solid A and B do not mix in all proportions, the values of property P displayed by solutions that are rich in A are fully in line with the values displayed by the solutions that are rich in B. Again, a condition sine qua non for being isomorphous!

2.8.3 Case (c) Solid A and solid B do not mix in all proportions. There is a series of solid solutions rich in A and another series of solid solutions that are rich in B. The most important observation is the fact the values of property P displayed by the solutions rich in A do not line up with de values displayed by the solutions rich in B. And because of this fact, solid B is not isomorphous with solid A. There is a Gibbs energy function for the solid solutions rich in A (in Fig. 2.1c denoted by α) and another Gibbs energy for the solid solutions rich in B (denoted by β). The compositions of the solid phases that are in equilibrium at T = T 1 follow from the points of contact of the common tangent line. In conclusion for this case: in spite of the fact that the molecules of A and B display an elevated degree of molecular homeomorphism, the crystalline solids of A and B are not isomorphous, and for that reason do not form a continuous series of solid solutions.

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References 1. Fujii K (1986) The Berthollet-Proust controversy and Dalton’s chemical atomic theory 1800– 1820. Br J Hist Sci 19:177–200 2. Bergman TO (1782) Sciagraphia Regni Mineralis. In: Ferber JJ (ed). Leipzig & Dessau 3. Romé de Lisle (1783) Cristallographie ou Description des formes propres à tous les corps du règne minéral dans l’état de combinaison saline, pierreuse ou métallique, 4 vols. In: Imprimerie de Monsieur, vol 1. Paris, p 379 4. Berzelius (1810) Versuch, die bestimmten und einfachen Verhältnisse aufzufinden, nach welchen die Bestandtheile der unorganishen Natur mit einander vernden sind. In: Afhandlingar I fysik, kemi och mineralogi, vol 3, pp 162–276 5. Pusch GG (1815) Kritische Betrachtung über das, auf die electrochemische Theorie und die chemische Proporzionslehre gegründete, Mineral-System des Herr Professor Berzelius. Leonhard’s Taschenbuch Gesammte Mineral 11:2 6. Hausmann JFL (1811) Specime de relatione inter corporum naturalium anorganicorum idoles chemicas atque externas. Soc Wissenschaften Göttingen Comment 2:1–47 7. Wérner AG (1774) Von den äusserlichen kennzeichen der Fossilien. In: Crusius AL (ed). Leipzig 8. Haüy RJ (1801) Traité de minéralogie, 1st edn. 2nd edn in 1822. Four volumes & atlas. Courcier, Paris 9. Amorós JL (1959) Notas sobre la historia de la Cristalografía. I. La controversia Haüy— Mitscherlich. Bolet Real Soc Hist Nat 57:5–30 10. Vauquelin NL (1797) Une Mémoire sur la nature de l’Alun du commerce, l’existence du Potassium dans cette salt et de diverses combinassions simples ou triple. Combinaison de l’alumine avec l’acide sulfurique. Ann Chim 518 11. Leblanc C (1801) De la Cristallotechnie. Bull Soc Philomath Paris 11–12 12. Beudant FS (1818) Sur les causes qui peuvent faire varier les formes cristallines d’une même substance minérale. Ann Chem Phys 8:5–52 13. Delafosse G (1858) Nouveau cours de minéralogie, comprenant la description de toutes les espèces minérales avec leur applications directes aux arts. 3 volumes and atlas, Livre I, Chapitre II. Roret Editors, Paris, pp 64–74 14. Wollaston WH (1812) On the primitive crystals of carbonate of lime, bitter-spar, and iron-spar. Philos Trans 102:159–162 15. Mitscherlich E (1820) Sur la relation qui existe entre la forme cristalline et les proportions chimiques. Premier Mémoire sur l’identité de la forme cristalline chez plusieurs substances différentes, et sur le rapport de cette forme avec le nombre des atomes élémentaires dans les cristaux. Ann Chim Phys 14:172–190 16. Mitscherlich E (1821) Sur la relation qui existe entre la forme cristalline et les proportions chimiques. Deuxième Mémoire sur les Arséniates et les Phosphates. Ann Chim Phys 19:350– 419 17. Melhado EM (1980) Mitscherlitch’s discovery of isomorphism. University of California, pp 87–123 18. Kopp HFM (1843) Geschichte der Chemie, 4 vols. Vol 1, 1843. Vol 2, 1844. Vols 3–4, 1845. Friedrich VieWeg und Sohn, Braunschweig 19. Laurent MA (1845) Sur l’isomorphisme et sur les types cristallins. C R Acad Sci Paris 357–366 20. Pasteur ML (1848) Note sur un travail de M. Laurent intitulé: Sur l’isomorphisme et sur les types cristallins. Ann Chim Phys 294–295 21. Pasteur ML (1848) Sur les relations qui peuvent exister entre la forme cristalline, la composition chimique et le sens de la polarisation rotatoire. C R Acad Sci Paris 442–459 22. Delafosse G (1851) Sur le plésiomorphisme des espèces minérales, c’est-à-dire sur les espèces dont les formes offrent entre elles le degré de ressemblance qu’on observe dans les cas d’isomorphisme ordinaire, sans que leurs compositions atomiques puissent se ramener à une même formule. C R Acad Sci Paris 535–539

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23. Groth P (1870) Sur les relations entre forme cristalline et constitution chimique dans le cas de quelques composés organiques. Poggendorf Ann 141:31–43 24. Baumhauer HA (1870) Die Beziehungen zwischen dem Atomgewichte und der Natur der chemischen Elemente. Brunswick 25. Arago FJD (1811) Sur une modification remarquable qu’éprouvent les rayons lumineux dans leur passage à travers certains corps diaphanes, et sur quelques autres nouveaux phénomènes d’optique. Mém Inst Part I 12:93–134 26. Sénarmont MH (1851) Sur les propriétés optiques biréfringentes des corps isomorphes. Ann Chim Phys 33:391–437 27. Bodländer M (1860) Ueber das optische Drehungsvermögen isomorpher Mischungen aus den Dithionaten des Bleis und des Strontiums. Z F Kryst 9:309 28. Schuster M (1880) Ueber die optische Orientierung der Plagioklase. Tscher Mineral Petrogr Mitt 3:117–284 29. Wyrouboff G (1886) Sur deux cas embarrassants d’isomorphisme. Bull Soc Franc Minéral 9:102–128 30. Mallard FE (1886) Discussion à la suite de la communication de M. Wyrouboff. Bull Soc Min 9:115–121 31. Retgers JW (1889) Contribution à l’étude de l’isomorphisme. Ann Ecole Polytech Delft 5:143– 242 32. Gibbs JW (1873) On the equilibrium of heterogeneous substances. Trans Conn Acad Arts Sci 3(108–248):343–524 33. Roozeboom HWB (1899) Stoechiometrie und Verwandtschaftslehre. Z Phys Chem 28:494 34. Van’t Hoff JH (1898) Leçons de Chimie Physique. Librairie Scientifique A. Hermann, Paris, pp 26–79 35. Tammann C (1905) Über die Anwendung der thermischen Analyse in abnormen 36. Wallerant F (1909) Cristallographie. Déformation des corps cristallisés groupements. Polymorphisme-isomorphisme. Librairie Polytechnique CH. Béranger, Editeur, Paris, pp 329–490 37. Vegard L, Schjederup H (1917) The constitution of mixed crystals. Phys Z 18:93–96 38. Wulf F (1906) Untersuchungen im Gebiete der optischen Eigenschaften isomorpher Krystalle. Z Krystallogr 42:558–586 39. Grossner J (1907) Über Isomorphie. Z Krystallogr 43:130–147 40. Tutton A (1910) Crystalline structure and chemical constitution. London 41. Friedel G (1926) Leçons de Cristallographie. Berger-Levrault éditeurs, Paris, pp 539–559 42. Chevreul ME (1823) Recherches Chimiques sur les corps gras d’origine animale. Levrault. FG, Paris 43. de la Provostaye MF (1841) Isomorphisme de l’oxaméthane et du chloroxaméthane. C R Acad Sci Paris 322–325 44. Laurent MA (1841) Sur l’isomorphisme de certains corps liés entre eux par la loi des substitutions. C R Acad Sci Paris 876 45. Bruni G (1900) Soluzioni solide e miscele isomorfe. Gazz Chim Ital 30 II:140–151 46. Bruni G, Gorni F (1900) Il fenomeni di equilibrio fisico nelle miscele di sostenanze isomorfe. Gazz Chim Ital 30:127–140 47. Küster FW (1905) Beiträge zur Molekulargewichtsbestimmung and “festen Lösungen.” Z Phys Chem 50:65–73 48. Nagornow NN (1911) Isomorphe Gemische der p-Dihaloidbenzolderivate. Z Phys Chem 75:578–584 49. Efremov NN (1913) Camphor and phenols. Zh Russ Fiz Khim Obshch 45:348–362 50. Smits A (1923) Théorie de l’allotropie. Gauthier-Villars éditeur, Paris 51. Timmermans I (1936) Les solutions concentrées. Massons, Paris 52. Kitaigorodskii AI (1957) Concepts of organic crystallochemistry. Sov Phys Cryst II 4:454–462 53. Kitaigorodskii AI (1973) Molecular crystals and molecules. Academic Press, London 54. Kitaigorodskii AI (1984) Mixed crystals. Springer Verlag, Berlin

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55. Haget Y, Chanh NB, Meresse A, Cuevas-Diarte MA (1985) Le degré d’isomorphisme cristallin en tant que critère prédictif de la syncristallisation organique. In: The second CODATA symposium. Critical evaluation and prediction of phase equilibria in multicomponent systems. Paris, pp 169–173 56. Meresse A, Chanh NB, Housty JR, Haget Y (1986) J Phys Chem Solids 47(11):1019 57. Haget Y, Bonpunt L, Cuevas-Diarte MA, Oonk HAJ (1989) Syncrystallization and the concept of isomorphism. Disorder in molecular solids. Garchy, P-10 58. Haget Y, Oonk HAJ, Cuevas-Diarte MA (1990) Cristallographie, Thermodynamique et réactualisation du concept d’isomorphisme. In: 16èmes Journées d’Etude des Equilibres entre Phases. Marseille, pp 35–36

Chapter 3

Thermodynamics H. A. J. Oonk, T. Calvet, and M. H. G. Jacobs

Abstract Much of the research, detailed in the chapters ahead, has been carried out on binary systems under isobaric conditions. The investigations have revealed that the thermodynamic mixing properties of the mixed crystalline state comply with a relatively simple thermodynamic model—with three system-dependent parameters. In the text, the model is detailed, and a demonstration is given of its power, which, at the same time, is surprising and outstanding.

3.1 Introduction This chapter is on the thermodynamics of mixed crystals of two substances: their thermodynamic description; their stability; and their equilibrium with a liquid phase; see [1–3] for details. The equilibrium with liquid is of special importance: the temperature versus composition diagram of the equilibrium between a solid and a liquid phase is a pre-eminent tool for detecting the formation of mixed crystals. In what follows, the thermodynamic description of mixed crystals is presented in terms of the ABΘ model for the excess Gibbs energy. The A, B, and Θ are systemdependent parameters, and the simple reason for opting for this model is the fact that it has proven its outstanding suitability and success. Most of this chapter is reproduced with permission from Ref. [1, Chaps. 212 and 213] and Ref. [2, Chaps. 304 and 305]; ©2008 Springer and ©2012 Springer, respectively. H. A. J. Oonk Universiteit Utrecht, Utrecht, The Netherlands e-mail: [email protected] T. Calvet Grup de Cristal·lografia Aplicada, Universitat de Barcelona, Barcelona, Spain e-mail: [email protected] M. H. G. Jacobs (B) Technische Universität-Clausthal, Clausthal, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. À. Cuevas-Diarte and H. A. J. Oonk (eds.), Molecular Mixed Crystals, Physical Chemistry in Action, https://doi.org/10.1007/978-3-030-68727-4_3

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Apart from an appendix on computer software, the chapter is subdivided into five sections: • • • • •

Presentation of the ABΘ model The route from model parameters to phase diagram A transparent treatment of phase diagram characteristics The route from experimental data to the numerical values of the parameters The output: discovered regularities and empirical relationships.

Accordingly, this chapter has a twofold function: (i) a priori it presents the thermodynamics needed to assess experimental data on mixed crystals; and (ii) a posteriori it highlights the overall significance of the results obtained by the research described in the chapters ahead.

3.2 Presentation of the AB Model The key quantity in matters of stability and phase behavior is the Gibbs energy G. The quantity is composed of the (system’s) fundamental properties, which are the internal energy U, the entropy S, and the volume V, and the variables temperature T, and pressure P: G = U + P V −T S = H −T S,

(3.1)

where H is the enthalpy (H = U + PV ). The other way round, the properties S, V, H, and U follow from the Gibbs energy as S = −(∂G/∂ T ) P ;

(3.2)

V = (∂G/∂ P)T ;

(3.3)

H = G−T (∂G/∂ T ) P ;

(3.4)

U = G−T (∂G/∂ T ) P −P(∂G/∂ P)T .

(3.5)

For the purpose of this monograph, it is useful and convenient to define the systems on a molar base. For a system, which is one mole of substance A the molar Gibbs energy—represented by G ∗A with the asterisk superscript for molar property of pure substance—is a function of temperature and pressure only. For a system which consists of (1 − X) mole of substance A + X mole of substance B, such that A and B are separated from one another (not mixed), the (molar) Gibbs energy—obviously—is given by

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G = (1 − X ) G ∗A + X G ∗B .

(3.6)

When this system is changed to a homogeneous mixture of the substances, it will have a different Gibbs energy. The Gibbs energy, after mixing, can be expressed as G = (1 − X ) G ∗A + X G ∗B + m G.

(3.7)

The mixture of the two substances is referred to as an ideal mixture if the Gibbs energy change on mixing, m G, is independent of the pressure exerted on the system, and is given by m G = RT {(1 − X ) ln(1 − X ) + X ln X }.

(3.8)

The R in this expression is the gas constant; R = 8.314472 kg m2 s−2 K−1 mol−1 . For non-ideal systems, the Gibbs energy change on mixing, as a rule and as a function of temperature, pressure, and composition, is formulated as m G(T, P, X ) = RT {(1 − X ) ln(1 − X ) + X ln X } + G E (T, P, X ).

(3.9)

The last term in the expression, GE , is the excess Gibbs energy. In the following treatment, and because mixed crystals for the most part are studied at ambient pressure, the dependence on pressure is not taken into account. Next, realizing that the excess Gibbs energy vanishes for the pure components of the system, i.e., for X = 0, and for X = 1, one can represent the excess Gibbs energy by the expression G E (T, X ) = X (1 − X ) f (T, X ).

(3.10)

Two molecular substances, that are capable of giving a continuous series of mixed crystals, generally are that much alike that their liquid mixtures behave virtually ideal. For that reason, in this text, any excess Gibbs energies of the liquid state will be ignored. In terms of Eq. (3.10), the most simple formula for the excess Gibbs energy is obtained when f (T, X) is taken linear in both T and X: G E (T, X ) = AX (1 − X )(1 − T /)[1 + B(1 − 2X )].

(3.11)

In this expression, referred to as the ABΘ model, the A; B; and Θ are systemdependent parameters. The parameter A (expressed in J mol−1 ) represents the magnitude of the excess function; Θ (in K) its dependence on temperature; and B (dimensionless) its asymmetry. The use, between the square brackets, of the form (1 − 2X) rather than just X has the advantage that for the equimolar mixture the excess function does not contain the parameter B.

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For T = Θ, the excess function goes through zero; this is shown in Fig. 3.1, which is taken from [1], and which is representative of the mixed crystalline state in NaCl + KCl [3]. In terms of Eq. (3.11), and through Eqs. (3.2) and (3.4), respectively, the expressions for the excess enthalpy (the heat of mixing) and the excess entropy are H E (X ) = AX (1 − X )[1 + B(1 − 2X )], and

(3.12)

S E (X ) = (A/)X (1 − X )[1 + B(1 − 2X )].

(3.13)

The ABΘ model is such that the excess enthalpy and excess entropy are independent of temperature; or, in other terms, the model ignores the possible existence of excess heat capacities. The quotient of excess enthalpy and excess entropy is Θ. At T = Θ, the two excess functions compensate one another, and in the sense that the excess Gibbs energy is zero. For that reason, the temperature Θ is often referred to as compensation temperature. Another characteristic of the model is the fact that the excess Gibbs energy at zero K and the excess enthalpy are identical (this is of importance when it comes to the fitting of experimental data).

Fig. 3.1 Cross-sections of the function GE (T, X), Eq. (3.11), for A = 18.2 kJ mol−1 ; B = 0.2; and Θ = 2565 K. Right: for T = 875 K, as a function of mole fraction. Left: for X = 0.5, as a function of temperature. Experimental data fall in the temperature range bounded by the two vertical dashed lines. The small rectangle is representative of the range of melting. Reproduced from Ref. [1] with permission; © 2008 Springer

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3.3 From Model Parameters to Phase Diagram Prior to investigating how the stability of mixed crystals, and their equilibrium with liquid, is related to the numerical values of the constants of the ABΘ model, it is expedient to have a look at a calculated phase diagram. The phase diagram as shown in Fig. 3.2 has been calculated for a hypothetical system, whose liquid mixtures are ideal, and whose mixed crystalline state is defined by A = 20 kJ mol−1 ; B = 0.2; and Θ = 2000 K. The solid + liquid two-phase region at the upper side of the diagram has a minimum at which solid and liquid in equilibrium have the same composition. The two-phase region is bounded by the liquidus at its upper side and the solidus at the lower side. Together, liquidus and solidus are the loci of the compositions of the liquid and solid phases that, for given temperature, are in equilibrium.

TAo = 900 K ΔS *A = 3R = 24.94 J·K −1 ·mol−1

TBo = 1100 K ΔS *B = 3R = 24.94 J·K −1 ·mol−1

H E sol = 20000 X (1 − X ) [1 + 0.2(1 − 2 X ) ] J·mol−1 S E sol = 10 X (1 − X ) [1 + 0.2(1 − 2 X ) ] J·K −1 ·mol −1 Fig. 3.2 Calculated TX phase diagram with solid-liquid loop, including the equal-G curve (EGC); and region of demixing, including the spinodal (SPIN) Reproduced from [1] with permission; © 2008 Springer

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In geometric terms, the Gibbs energies, for a given temperature, of the solid and liquid mixtures are represented by two curves. The compositions of the solid and liquid phases in equilibrium are given by the points of contact of the common tangent line. The points of contact are on either side of the point of intersection of the two G-curves. At the intersection, solid and liquid have equal Gibbs energies. The locus in the TX diagram of the points of intersection is the equal-G curve, as shown in Fig. 3.2, the curve marked EGC. The EGC is the solution of the equation G(T, X ) = 0,

(3.14)

in which the operator  is for value of quantity in liquid minus value of the same quantity in solid such that solid and liquid have the same X value. For temperatures at the lower side of the phase diagram, the mutual miscibility of the two components of the system is limited: the phase diagram displays a miscibility gap, also referred to as region of demixing. In this case, there is just one Gibbs energy curve; the curve has a concave part for intermediate compositions; the two phases in equilibrium follow from the points of contact of the double tangent line. The locus of the points of contact is the binodal (curve) (BIN) and the boundary of the region of demixing. The spinodal (curve) (SPIN) as shown in Fig. 3.2 is the locus of the points of inflexion: the spinodal is the solution of the equation ∂ 2 G(T, X )/∂ X 2 = 0

(3.15)

Invariably, the spinodal is inside the region (of demixing) bounded by the binodal. Binodal and spinodal have a common extremum, which is the critical point with coordinates T c and X c . At the critical point, the third derivative of the molar Gibbs energy is zero.

3.4 A Transparent Treatment of Phase Diagram Characteristics Clearly, the EGC and SPIN on their own reflect much of the real phase diagram: the two are inside the two-phase regions and any extremum in them is an extremum in the real phase diagram (as long as the two do not interfere). Mathematically speaking, the EGC and SPIN, for given temperature, are the solutions of one equation with one unknown—in contrast to the real equilibrium compositions, which, for given temperature, are the solution of two equations with two unknowns. For reasons of transparency, therefore, we start the treatment of stability, and the treatment of the change from solid to liquid, by deriving the formulae for the SPIN and EGC, respectively.

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  Omitting the linear part, (1 − X )G ∗A + X G ∗B , the molar Gibbs energy is represented by the following expression, which is linear in temperature. G(T, X ) = RT {(1 − X )ln(1 − X ) + X lnX } + AX (1 − X )(1 − T /)[1 + B(1 − 2X )].

(3.16)

The first derivative with respect to mole fraction is ∂G/∂ X = RT ln[X/(1 − X )] + A(1 − T /)[(1 − 2X ) + B(1 − 6X + 6X 2 )]. (3.17) And the second derivative ∂G 2 /∂ X 2 = RT /[X (1 − X )] + A(1 − T /)[−2 − 6B(1 − 2X )].

(3.18)

By substitution of Eq. (3.18) into Eq. (3.15), the equation for the spinodal is obtained as TSPIN (X ) =

2 A · X (1 − X )[1 + 3B(1 − 2X )] . R + 2A X (1 − X )[1 + 3B(1 − 2X )]

(3.19)

By means of Eq. (3.19), and for a given set of values of A, B, and Θ, the spinodal temperature can be calculated as a function of X—and so the complete spinodal can be constructed. From the constructed spinodal, the coordinates (X c and T c ) of the critical point can be read off. In a different and exact manner, the coordinates X c and T c can be found starting from the condition dT SPIN /dX = 0. It is easily shown that from this condition it follows that X c is the solution of the equation (18B)X 2 −(2 + 18B)X + (1 + 3B) = 0.

(3.20)

Next, after the substitution of X c in Eq. (3.19), the value of T c is found. Note that from Eq. (3.20) it is obvious that X c is fully determined by the value of only one (B) of the three parameters. For the hypothetical system implied in Fig. 3.2, the value of B = 0.2 corresponds to X c = 0.377. Next, with A = 20 kJ mol−1 and Θ = 2000 K, the critical temperature is calculated as T c = 787 K. The full spinodal, calculated by means of Eq. (3.19), is shown in Fig. 3.2. Next, for isobaric equilibria between a non-ideal, low-temperature form α (=solid) and an ideal, high-temperature form β (=liquid) the molar Gibbs energies are ∗β

∗β

G β (T, X ) = (1 − X )G A (T ) + X G B (T ) + RT {(1 − X ) ln(1 − X ) + X ln X }; (3.21)

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∗α G α (T, X ) = (1 − X )G ∗α A (T ) + X G B (T ) + RT {(1 − X ) ln(1 − X ) + X ln X }

+ G Eα (T, X )

(3.22)

The equal-G curve (EGC) is the solution of the Eq. (3.23): G(T, X ) = G β (T, X ) − G α (T, X ) = 0.

(3.23)

After substitution of Eqs. (3.21) and (3.22), Eq. (3.23) becomes G(T, X ) = (1 − X )G ∗A (T ) + X G ∗B (T ) − G Eα (T, X ) = 0.

(3.24)

For mixed crystals, and molecular ones in particular, the change from solid to liquid is an event in a relatively small range of temperature—such as is reflected by the small rectangular in Fig. 3.1, left-hand side. The import of this fact is that the thermodynamic treatment is not weakened, if, within the small range, the Gibbs energy properties of the pure components are taken to be linear. And, in view of Eq. (3.1), this comes down to saying that the pure component enthalpy and entropy properties can be treated as constants. That being the case, Eq. (3.24) can be formulated as     G(T, X ) = (1 − X ) H A∗ − T S ∗A + X H B∗ − T S B∗   − H Eα (X ) − T S Eα (X ) = 0.

(3.25)

Equation (3.25) can be abbreviated to G(T, X ) = H (X ) − T S(X ) = 0,

(3.26)

in which H(X) is the heat effect of the change as a function of mole fraction and S(X) the entropy effect: H (X ) = (1 − X )H A∗ + X H B∗ − H Eα (X )

(3.27)

S(X ) = (1 − X )S ∗A + X S B∗ − S Eα (X )

(3.28)

And from Eqs. (3.26)–(3.28), the temperature of the EGC is the quotient of the enthalpy effect of the change divided by the entropy effect of the change: TEGC (X ) =

(1 − X )H A∗ + X H B∗ − H Eα (X ) H (X ) = , S(X ) (1 − X )S ∗A + X S B∗ − S Eα (X )

(3.29)

in which the pure component properties H*, the heat of melting, and S*, the entropy change on melting, are related through the melting temperature; e.g., for component A, H A∗ = T Ao · S ∗A .

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The EGC in Fig. 3.2 has been calculated with Eq. (3.29) introducing the pure component and excess properties shown at the foot of the figure. The fact that Eq. (3.29) has an excess function in the numerator and another excess function in the denominator makes that the equation lacks some transparency. To obtain a transparent relationship between the EGC and the excess Gibbs energy directly, one can realize that the expression in Eq. (3.25), between the two signs of equality, for given X, will be zero at T = T EGC (X). For the excess part of the expression one can write: Eα (X ), H Eα (X ) − T S Eα (X ) ⇒ H Eα (X ) − TEGC (X ) · S Eα (X ) ≡ G EGC

(3.30)

in which the right-hand side of the identity represents the excess Gibbs energy along the EGC as a function of mole fraction. When this modification is introduced in Eq. (3.25), the solution of the equation takes the form TEGC (X ) = TZERO (X ) −

Eα (X ) G EGC , (1 − X )S ∗A + X S B∗

(3.31)

where TZERO (X ) =

(1 − X )H A∗ + X H B∗ . (1 − X )S ∗A + X S B∗

(3.32)

In the TX diagram, the distance from zero line to EGC is given by (minus) GEα divided by a system property, namely the weighted entropy of transition of the pure components. The zero line coincides with the straight line between the melting points of the two pure components A and B, when they have the same entropies of melting (S ∗A = S B∗ ), and also when they have the same melting point (T Ao = TBo ).

3.5 The Route from Experimental Data to the Numerical Values of the Parameters In the preceding section, it has been shown how the thermodynamic mixing properties of substitutional mixed crystals, and the phase equilibria they are involved in, are related to the ABΘ model. In the chapters ahead, the route is followed in the inverse direction: for a given system, or set of systems, the numerical values of the parameters A, B, and Θ are derived from experimental data. At this place, a few remarks are made about the inverse route. From Eq. (3.11), it follows that (four times) the equimolar excess Gibbs energy is related to the model parameters A and Θ as 4G E (X = 0.5) = A(1 − T /).

(3.33)

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From Eq. (3.33), it follows that for zero K 4G E (X = 0.5; T = 0 K) = A.

(3.34)

And from Eq. (3.12), for four times the equimolar excess enthalpy 4H E (X = 0.5) = A.

(3.35)

In other words, in terms of the ABΘ model, the excess enthalpy (the heat of mixing) is identical with the excess Gibbs energy at zero K. Before (say) the year 1950, the most of the knowledge about the change from solid to liquid (or rather the change from liquid to solid) was derived from cooling experiments, in which the temperature of a sample of given composition is recorded as a function of temperature (cooling curves). Cooling curves yield accurate liquidus temperatures, but most of the times, it yields unreliable solidus temperatures. In practice, for that matter, the excess Gibbs energy of the mixed crystalline state is derived from the liquidus, in a series of computations—by the routine LIQFIT — in which the calculated liquidus is made to pass through the experimental liquidus points [4]. By the advent of the methods of microcalorimetry (DTA, DSC), the cooling curve method has become obsolete cooling has made way for heating; and gram for milligram to specify the mass of a sample. Microcalorimetric experiments, as a rule, start from the solid state, and it means that the outcome of an experiment, read the characteristics of the experimental recording, depend on how the solid sample is prepared. Generally, the liquidus temperatures are more reliable than the solidus temperatures. And this observation implies that LIQFIT is the appropriate method for the thermodynamic analysis of the data. Data on regions of demixing (miscibility gaps), as a rule, come from equilibration experiments, in which a sample of given overall composition is kept at a specified temperature for a long enough time to separate into the two coexisting (see, e.g., [5]). For molecular materials, in contrast to the ionic ones, data on the mutual solid solubility of the components are scarce. For a certain part, this is due to the fact that for molecular materials the regions of demixing are below room temperature, and for that reason rather inaccessible to experimental study. In the realm of molecular mixed crystals, the introduction of the modern methods of microcalorimetry has been extremely fruitful. Especially so, because the experimental recordings yield at the same time (i) the temperature characteristics of the transition from solid to liquid and (ii) the heat effect of the transition as a function of composition. The temperature characteristics yield GE , the heat effect yields H E , and the two together yield S E . And all of the information is furnished by samples that often have a mass of no more than a few milligrams.

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3.6 The Output: Discovered Regularities and Empirical Relationships Above, the notion of compensation temperature came up, and it was stated that the compensation temperature of the mixed crystalline state is given by the parameter Θ of the ABΘ model. Enthalpy-entropy compensation is one of the guiding principles in experimental thermodynamics. The reason is in the fact that it is often found that within a class of similar systems the quotient of an enthalpy change (H) and the corresponding change in entropy (S) is system-independent. In other terms, all systems within the class have the same value for the quotient H/S (see, e.g., [6]). For the systems at hand, the H and S are the changes in enthalpy and entropy when the system is transferred from the hypothetical ideal mixture of the components to the real mixture of the same composition. These changes are the excess enthalpy and the excess entropy, respectively. A beautiful example of a class of similar systems is provided by the mixed crystals of n-alkanes having the rotator I form, see Chap. 6 of this work. In a graph of equimolar excess enthalpy against equimolar excess entropy, the data points of the individual systems are on a straight line, directed to the point 0,0. The compensation temperature of the class is 320 K. The research on mixed crystals—molecular, metallic, and ionic—has revealed that their compensation temperatures are above the melting temperatures. And such that the quotient of compensation temperature and melting temperature increases with increasing melting temperature. The relationship between the two temperatures, coming from some 50 systems, of which the melting temperatures range from about 100 K (Ar + Kr) [7] to about 2800 K (MgO + CaO) [8], can be represented by the log-log expression [2, 3 for details and references] log[/K] = (1.10 ± 0.05) log[TEGC (X = 0.5)].

(3.36)

Note that by this equation the composition temperature can be calculated, once the equimolar EGC temperature is known say, after the experimental determination of the phase diagram. The inconvenience of knowing the EGC temperature is circumvented by the relationship formulated by van der Kemp et al. [9], and which is based on the melting temperatures, T Ao and TBo , of the pure components of a given system:  = (4.00 ± 0.16)T Ao TBo /(T Ao + TBo ).

(3.37)

The compensation temperature connects the excess entropy of a given system with its excess enthalpy. The excess enthalpy is given by Eq. (3.12) as a function of mole fraction, and in terms of the model parameters A and B. Concentrating first on parameter A, one can observe that within a family/class of similar systems its value, which is four times the equimolar excess enthalpy, changes from systems to system. Most of the times

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the excess enthalpy is positive and such that its value, read the parameter A, can be related to the relative difference in size of the molecules of the two components of the system. For the aforementioned rotator I form of n-alkane systems, as an example, the value of parameter A can be represented by the formula [10]   A = 51.09 m + 376.5 m2 kJ mol−1 .

(3.38)

In this expression, the property m is a so-called mismatch parameter. It represents the relative difference in chain length of the two pure component molecules of a given system. For the system hexadecane + heptadecane, as an example, the mismatch parameter m = (17 − 16)/16.5.

Appendix: Computer Software Developed at Utrecht University The development of computer software at Utrecht University goes back to the end 1960s. The source codes of some of the programs were initially written for use on main frame computers and for these programs source code listings are available in the literature. Later, these programs were further developed on personal computers, which had the advantage that user-friendly applications were made. For small enough projects, even software was written for use on pocket calculators. That was not only beneficial for supporting our research but also for teaching purposes. A large variety of software was constructed with application in the field of calorimetry, vapor pressure measurements and phase diagram calculation. For details see [2]. All of the developments are reflected in the software package XiPT. It is written in FreePascal, runs in Linux and Windows operating system environments, and is at present limited to ternary systems. To date, program XiPT is used for the construction of a thermodynamic database for materials relevant to geophysics, aiming at predicting thermophysical and thermochemical properties at the extreme conditions prevailing in planetary mantles. A free download of a small source code, including a database for geophysically relevant materials, based on the method by Jacobs et al. [11], to calculate thermodynamic properties in pressure-temperature space is found on website https://www. geo.uu.nl/~jacobs/Downloads/. The code is written in FreePascal and ForTran and runs in Linux and Windows operating system environments. A further interesting and challenging focus, which has been initiated recently, is implementing the developed methods in ‘Open Calphad,’ which is an effort to make thermodynamic computation widely available and accessible to scientists, see https://www.opencalphad.org.

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References 1. Oonk HAJ, Calvet MT (2008) Equilibrium between phases of matter. Phenomenology and thermodynamics. Springer, Dordrecht 2. Jacobs MHG, Oonk HAJ (2012) Equilibrium between phases of matter. Supplemental text for materials science and high-pressure geophysics. Springer, Dordrecht 3. Oonk HAJ (2001) Solid-state solubility and its limits. The alkali halide case. Pure Appl Chem 73:807–823 4. Bouwstra JA, van Genderen ACG, Brouwer N, Oonk HAJ (1980) A thermodynamic method for the derivation of solidus and liquidus curves from a set of experimental liquidus points. Thermochim Acta 38:97–107 5. Bunk AJH, Tichelaar GW (1953) Investigations in the system NaCl-KCl. Proc K Nedl Akad Wet Ser B 56:375–384 6. Boots HMJ, De Bokx PK (1989) Theory of enthalpy-entropy compensation. J Phys Chem 93:8240–8243 7. Walling JF, Halsey GD (1958) Solid solution argon-krypton from 70 to 96 °K. J Phys Chem 62:752–753 8. Van der Kemp WJM, Blok JG, van der Linde PR, Oonk HAJ, Schuijff A, Verdonk ML (1994) Binary alkaline earth oxide mixtures: estimation of the excess thermodynamic properties and calculation of the phase diagrams. Calphad 18:255–267 9. Van der Kemp WJM, Blok JG, van der Linde PR, Oonk HAJ, Schuijff A, Verdonk ML (1993) On the estimation of thermodynamic excess properties of binary solid solutions. Thermochim Acta 225:17–30 10. Rajabalee F, Metivaud V, Oonk HAJ, Mondieig D, Waldner P (2000) Perfect families of mixed crystals: the “ordered” crystalline forms of n-alkanes. Phys Chem Chem Phys 2:1345–1350 11. Jacobs MHG, Schmid-Fetzer R, van den Berg AP (2013) An alternative use of Kieffer’s lattice dynamics model using vibrational density of states for constructing thermodynamic databases. Phys Chem Miner 40:207–227

Chapter 4

Polymorphism I. B. Rietveld, R. Céolin, and J. Ll. Tamarit

Abstract Molecular substances, as a rule, manifest themselves in more than one crystalline form. And it may happen that two members of a given family, under given conditions of temperature and pressure, do not adopt the same form. Polymorphism, for that matter, is a fascinating and at the same time complicated phenomenon. For the treatment of polymorphism of a given substance, it is a sine qua non to take into account the influences of temperature and pressure. The chapter starts with an historical overview.

4.1 A Short History of Crystalline Polymorphism, the Emergence of a Controversial Concept Crystalline polymorphism is a natural phenomenon and its journey toward scientific recognition commenced when Klaproth discovered that aragonite has the same chemical composition as calcite [1]. By doing so, he corrected the analysis of Werner, who had classed aragonite as apatite [2]. Unfortunately, this correction was also the beginning of a not always courteous controversy opposing René-Just Haüy and his scientific followers against Berzelius and Mitscherlich, as Cahn and Lima-de-Faria recount [3, 4]. Abbé Haüy had postulated that a crystal consists of a three-dimensional stacking of tiny polyhedrons called “molécules intégrantes.” Through this postulate, he founded the new science of geometrical crystallography [5–8]. Unfortunately, he insisted on I. B. Rietveld (B) Laboratoire Sciences et Méthodes Séparatives, Université de Rouen Normandie, Mont Saint Aignan, France e-mail: [email protected] R. Céolin · J. Ll. Tamarit Grup de Caracterització de Materials, Universitat Politècnica de Catalunya, Barcelona, Spain e-mail: [email protected] J. Ll. Tamarit e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. À. Cuevas-Diarte and H. A. J. Oonk (eds.), Molecular Mixed Crystals, Physical Chemistry in Action, https://doi.org/10.1007/978-3-030-68727-4_4

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a second postulate that each chemical species could only exist as one molécule intégrante, and, following this second postulate, it had become inconceivable that calcite and aragonite are different forms of the single chemical entity calcium carbonate. Nonetheless, the chemical composition of calcite was systematically found to be the same as that of aragonite [9, 10]. Wallerant writes that “this result, in apparent disagreement with the ideas developed by Haüy, was confirmed by many chemists, who all came to the conclusion that aragonite and calcite had the same composition and that the same chemical compound could have different primitive forms” [11]. Despite the undeniable results, Haüy remained skeptical until the end of his life and, while finally recognizing the chemical identity of both minerals in 1817, he queried: “Hence, it remains to be discovered how two such different substances as lime carbonate [Calcite] and aragonite, considering their crystalline forms, nevertheless can be perfectly identical in composition.” In December 1819 at the Royal Academy of Sciences in Berlin, Eilard Mitscherlich established the existence of isomorphism by presenting the “first report on the identity of the crystalline form in several different substances and the relationship of this form with the number of elementary atoms in the crystals,” which was soon after published in the French “Annales de Chimie et de Physique” [12]. The memoire drew the attention of Berzelius who recognized that isomorphism—at that time limited to inorganic compounds having the same “primitive form” (structure) as other inorganic compounds—could be a means to determine molecular weights and he invited Mitscherlich to Stockholm, who worked with him for 2 years. The idea of isomorphism and polymorphism faced strong opposition from many crystallographers of Haüy’s school. Fortunately, owing to the support of Berzelius, Mitscherlich demonstrated in the end that polymorphism is indeed a physical reality. In his third report, also read at the Royal Academy of Sciences at Berlin and published in 1823, he first evoked the difficulty in establishing the proof of the existence of dimorphism for compounds which purity is questionable (“it has always been recognized that traces of a foreign substance may produce the difference in its [primitive] form”). He solved the problem by taking the element sulfur and producing its polymorphs (Fig. 4.1): “I choose,” Mitscherlich says, “an element, i.e., a simple chemical substance, that lends itself best to demonstrate that chemical substances, whether elements or compounds, can exist in two different crystalline forms” [13]. Thus, a year after the death of Abbé Haüy in 1822, the long controversy about the chemical identity of calcite and aragonite is finally over owing to the general acceptance of the concept of crystalline polymorphism. In the realm of organic chemistry, the first case of crystalline polymorphism was revealed in 1832 by Wöhler and von Liebig [14], who described the crystalline modifications of benzamide. However, it took no less than a century, after the advent of classical thermodynamics, crystallography, and other techniques, before the description of polymorphism on the atomic scale was achieved. At that stage, it had also become possible to consider the stability conditions of polymorphs from a thermodynamic point of view, as envisioned by Tobern Bergman (1735–1784) about whom

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Fig. 4.1 Mitscherlich’s drawings in his publication on the dimorphism of sulfur [13]. Top: four drawings of crystals of natural sulfur crystallized from solutions (orthorhombic polymorph, stable at room temperature). Bottom: crystals grown from molten sulfur (monoclinic polymorph stable at “high” temperature)

Metzger [15] wrote: “he considered the crystalline form as the result of the conditions in which they formed and he believed it could inform us about the history of their formation.” And how right he was.

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4.2 The Gibbs Energy: Pressure and Temperature as Natural Variables 4.2.1 First- and Second-Order Phase Transitions Once it was accepted that chemical substances could exist in more than one crystal structure, the question arose, which of those crystal structures would be the most stable one and whether or not this depended on external conditions. The theoretical groundwork for thermodynamics has been laid by Carnot, Thomson, Clausius, Clapeyron, and many others, but it has really been integrated into a coherent theory by Josiah Willard Gibbs. To him, we owe our understanding that two phases in equilibrium must have the same Gibbs free energy and that the most stable phase (i.e., crystalline solid, liquid, or vapor) possesses the lowest Gibbs free energy. Phase changes can be monitored by observing the crystal structure as a function of pressure and temperature. When equilibrium between two phases is reached, the two phases coexist. However, whereas the phases possess the same Gibbs free energy, their specific volumes and heat contents will be different. It implies that a phase change gives rise to a discontinuity in the volume and in the heat content of the system. Such phase changes are called first order as the first derivative of the Gibbs free energy with respect to T and P is discontinuous. It is possible, however, that a solid phase gradually transforms into another solid phase with temperature or pressure; in that case, the two phases, strictly speaking, never coexist. According to the Landau theory, such transitions involve an orderdisorder transformation, and the progress of the transition can be described with an order parameter. With a continuous phase change from an ordered state to a disordered state (or vice versa), no discontinuous volume change or change in heat content will occur; however, the second-order derivatives of the Gibbs free energy, the thermal expansion and the heat capacity, will exhibit a discontinuity. Such phase transitions are called second-order phase transitions. Theoretically, higher order phase transitions exist too. In the rest of the chapter, however, most of the text applies to first-order phase transitions, which involve clear discontinuities in easily measurable quantities such as volume and heat.

4.2.2 The Gibbs Energy in Relation to Pressure–Temperature Phase Diagrams To discuss the current developments in isomorphism and polymorphism, the relevant Gibbs free energy functions will be discussed. The Gibbs energy function, Gα , for a simple, compressible thermodynamic hydrostatic PVT system (characterized by the variables pressure, volume, and temperature) in a given phase α at equilibrium at pressure P and temperature T is a monotonic function, i.e., a smooth surface in the G-P-T tridimensional space, defined as

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G α (T, P) = U α (T, P) + P V α (T, P)−T S α (T, P) = H α (T, P)−T S α (T, P) (4.1) where H α (T, P) is the enthalpy, H α (T, P) = U α (T, P) + PV α (T, P), U α (T, P) is the internal energy, S α (T, P) the specific entropy, and V α (T, P) the specific volume of phase α. At a fixed pressure and temperature, the criterion for equilibrium is given by the minimum of the Gibbs energy: dG α (T, P) = 0

(4.2)

The first-order differential of the Gibbs energy demonstrates the dependence on the natural variables, pressure, and temperature of the Gibbs energy function, dG α (T, P) = −S α (T, P)dT + V α (T, P)dP

(4.3)

leading to the thermodynamic properties S(T, P) and V (T, P):  ∂G α ∂T P  α ∂G V α (T, P) = ∂P T

S α (T, P) = −



(4.4) (4.5)

The latter equations demonstrate that the slope of the Gibbs function versus temperature in an isobaric section is always negative, whereas in an isothermal section as a function of pressure, it is always positive. Applying the equilibrium condition Eq. (4.2) to two phases α and β in thermal and hydrostatic equilibrium, it is clear that coexistence is maintained, even if the Gibbs free energy of the system as a whole changes, as long as both phases have the same Gibbs free energy, i.e., when  dG(T, P) =

∂G ∂n α



 dn α + T,P

∂G ∂n β

 dn β

(4.6)

T,P

With the constraint that the exchanges of matter between the two phases, dnα and dnβ , are inversely related: dn α + dn β = 0

(4.7)

the solution of the preceding equations in combination with Eq. (4.2) becomes 

∂G ∂n α



 = T,P

∂G ∂n β

 (4.8) T,P

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Fig. 4.2 Monotonic Gibbs free energy surfaces for phases α and β of a simple compressible thermodynamic PVT system as a function of pressure P and temperature T. The projection of the intersection of the two surfaces on the pressure–temperature plane provides the two-phase equilibrium line [α + β] and the stability domains of the phases α and β

or μα (T, P) = μβ (T, P)

(4.9)

i.e., phases α and β would coexist when their chemical potentials, μα (T, P) and μβ (T, P), at given P and T are equal. The preceding mathematical equations can be visualized in the tridimensional space G-T-P depicted in Fig. 4.2. The intersection between the two surfaces representing the Gibbs free energies of the phases α and β corresponds to Eqs. (4.8) and (4.9) and represents the conditions at which the two phases coexist (i.e., are in equilibrium). Its projection on the pressure–temperature plane results in the two-phase [α + β] equilibrium line, and because the equilibrium line is the projection of two intersecting monotonic Gibbs free energy surfaces, the line itself is also monotonic. As can be seen in Fig. 4.2, Gα (T, P) < Gβ (T, P) is valid in the stability region of phase α, and similarly, Gβ (T, P) < Gα (T, P) is valid in the stability region of phase β.

4.2.3 The Clapeyron Equation At the α + β equilibrium line, the Gibbs energy of phases α and β are equal, therefore: dG α (T, P) = dG β (T, P) Then, according to Eq. (4.3), for each phase, we have:

(4.10)

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−S α (T, P)dT + V α (T, P)dP = −S β (T, P)dT + V β (T, P)dP

(4.11)

From this equation, the slope dP/dT of the α + β equilibrium can be obtained:   dP/dT = S β (T, P) − S α (T, P)]/[V β (T, P) − V α (T, P) = α→β S/α→β V (4.12) where α→β S and α→β v are the entropy and volume changes at the α → β phase transition. The entropy change α→β S can be expressed as α→β S = α→β H/T α→β for a first-order phase transition, and thus, Eq. (4.12) is commonly written: dP/dT = α→β S/α→β V = α→β H/[T α→β · α→β V ]

(4.13)

This is the well-known Clapeyron equation, which will be used below for the construction of topological phase diagrams.

4.2.4 From Two-Phase Equilibria to Triple Points When three phases are brought into play, the intersections of the Gibbs free energy surfaces or the two-phase equilibrium lines would intersect in a single point in which the three phases would have the same Gibbs free energy and thus be in equilibrium. Such a point is referred to as a triple point, and Fig. 4.3 demonstrates how it is related to the Gibbs free energy surfaces.

(a)

(b)

Fig. 4.3 a Monotonic Gibbs free energy surfaces of phases I, II, and III of a simple compressible PVT system as a function of pressure P and temperature T. b The projection of the intersection of the surfaces on the pressure–temperature plane provides the two-phase equilibria [I + II], [I + III], and [II + III], as well as the three-phase coexistence triple point

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Fig. 4.4 Isobaric sections of Gibbs-free-energy versus temperature diagrams (c) for a pure substance displaying two polymorphs (α and β) with (a) enantiotropic and (b) monotropic behavior. Solid lines represent stable equilibria (minimum Gibbs free energy), whereas dashed lines represent metastable equilibria. Full and open circles represent stable and metastable three-phase equilibrium points, respectively

Let us consider a pure substance consisting of a pure liquid (L) and two possible solid phases α and β or polymorphs. At a given pressure, two possible stability hierarchies exist as a function of temperature as shown in Fig. 4.4. The scheme in Fig. 4.4a represents section ‘a’ in the phase diagram of Fig. 4.4c, in which phases α, β, and L have each a stability domain. The Gibbs surfaces of the α and β phases intersect at the temperature T α−β , which is the α to β transition temperature and it is located below the melting temperatures, T α−L and T β−L . The metastable extensions of the α and L Gibbs functions give rise to the metastable melting point of the α phase. In this case, T α−β < T α−L < T β−L , the α and β phases are related by an enantiotropic relationship. In Fig. 4.4b, which represents the Gibbs free energy as a function of temperature of section ‘b’ in the phase diagram of Fig. 4.4c, phase β is metastable over the entire temperature range, and phase α is the only stable solid below its melting point T α−L . In such a case, the metastable α to β transition T α−β exists at temperatures higher than the melting points of the phases α and β: T β−L < T α−L < T α−β , which is referred to as monotropic behavior of the β phase with respect to α. More about enantiotropy and monotropy will follow in the next part of this chapter.

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4.3 Topological Pressure–Temperature Phase Diagrams 4.3.1 Pressure–Temperature Phase Diagrams Whether an aggregation state of a chemical compound (crystalline solid (CS), liquid (L), or gas (V (vapor))) is stable or not depends on the temperature and pressure of the system as explained in the previous paragraphs. In pressure–temperature phase diagrams (Fig. 4.5), the gas phase is located at the lowest pressures and will be stable over the entire temperature domain (within the stability limits for the chemical compound itself). The maximum pressure, at which the gas phase is stable, increases with temperature. The solid can be found at low temperatures, but at higher pressures than the gas phase. The stable domain for the liquid is sandwiched between the solid and the gas phase. It is generally stable under similar pressure ranges as the solid, but at higher temperatures. At a given pressure, but with increasing temperature, the gas phase will become stable again (see Fig. 4.5a). Under even higher pressures and temperatures, above the critical point (not depicted in Fig. 4.5), the distinction between liquid and gas ceases to exist. The interfaces between two domains are the equilibrium lines. With three domains representing the stable conditions for three phases (CS, L, and V), three equilibrium lines exist, one between CS and L related to the crystallization and fusion transitions, one between L and V with condensation and vaporization transitions and one between CS and V with deposition and sublimation transitions. In the case of a chemical compound possessing three stable states, the three equilibrium lines will cross in one single point, the triple point, where all three phases are in equilibrium with each other. This has been schematically illustrated in Fig. 4.5a. As indicated in the introduction to this chapter, solids generally have more than one possibility to stack and form crystals; they exhibit polymorphism. Polymorphs will have different structures with different unit-cell parameters, and the structures may

Fig. 4.5 (a) Schematic pressure–temperature phase diagram of a solid, a liquid, and a gas. (b) An example of a possible pressure–temperature phase diagram of a chemical compound exhibiting crystalline dimorphism

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have different symmetry elements. For example, the two known crystal structures of ritonavir are monoclinic P21 and orthorhombic P21 21 21 [16]. The latter has a different packing, and its unit cell contains more symmetry elements. When more than a single crystalline phase is known to exist for a given chemical compound, the question arises which of them is more stable and under which conditions. Once the conditions for the phase boundaries between specific phases are known, they can be represented in the pressure–temperature phase diagram as illustrated in Fig. 4.5b. How to determine the stability domains of the different polymorphs experimentally, while making use of the theoretical constructs introduced above, will be the main subject in the following part.

4.3.2 Phase Equilibria The best approach to determine the stability hierarchy, i.e., the stability ranking of the phases known to exist, is by investigating the stability conditions for the two-phase equilibria, because they are related to a phase transition, which can be observed. As mentioned previously, phase transitions involve an exchange of heat and a change of structure and specific volume that can be quantified. Provided the transition occurs at the equilibrium conditions, it will be possible to use the information to establish the phase diagram, as will be shown below. A phase equilibrium and a phase transition are not equivalent, and this is a real experimental issue. A phase equilibrium is defined as the conditions for which the Gibbs energies of two different phases are equal. In a pressure–temperature phase diagram, a phase equilibrium between two phases will appear as a single line. However, the fact that the Gibbs energy of one phase is equal to that of another phase will not necessarily trigger the phase transformation. Phase changes may only occur after an activation energy has been overcome necessary to weaken interactions and turn molecules. This is often the case for solid-to-solid transitions. For other phase transitions, a stochastic process may determine the position of the molecules, and these positions may not necessarily coincide with those of the new phase. This occurs during freezing or recrystallization from a solution. It implies that phase transitions can be delayed in relation to their phase equilibrium conditions. While heating a solid, even though it has exceeded its stability conditions in relation to another solid phase, it may still melt normally, if the system has no time to rearrange itself. This is illustrated in Fig. 4.6, where the solid–solid transition of benfluorex hydrochloride is plotted as a function of the heating rate of the differential scanning calorimeter (DSC). It can be seen that if the heating rate is too fast, the solid–solid transition does not occur and the metastable phase melts. While liquids may be undercooled below their melting points, if they do not get an opportunity to stack in the stable crystalline phase, melting is most often not delayed, because the formation of a liquid is not dependent on a particular structure and the necessary heat to release the individual molecules is often not very high and can be provided by the surroundings of the system. Undercooled liquids and delayed solid–solid transitions represent two

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Fig. 4.6 An example of a solid–solid transition (benfluorex hydrochloride forms I and II) that depends on the heating rate used for its observation. Once the heating rate reaches 1 K min−1 , the solid–solid transition is not observed at all and the metastable solid melts instead. However, it can be observed that the melting transition temperature of form II still depends on the heating rate, which is rarely the case (cf. the melting transition temperature of form I at 435 K). Reproduced with permission from Wiley, Barrio et al. [17]

causes for the delay in phase change. The first is too much freedom, and therefore, the system cannot find the stable state rapidly (undercooled liquids), and the latter is too little freedom, which causes the system to be stuck in a particular structure and in need of a large amount of energy to overcome a large energy barrier. In reality, both causes may play their part in each delayed phase transition.

4.3.3 The Pressure Coordinate It is clear from the foregoing that it has to be verified that the observed transition conditions coincide with the phase equilibrium conditions. This can be done by studying the system under different kinetic conditions and measuring its response, as illustrated in Fig. 4.6. Once the position of an equilibrium is known, its pressure and temperature can be indicated in the pressure–temperature phase diagram. However, how exactly can these coordinates be obtained and what do they stand for? The equilibrium temperature is generally measured by calorimetry and reflects the temperature at which two solid phases (or other phases) are in equilibrium. However, the pressure is generally not determined. It is often misunderstood to be 1 bar, because of the atmospheric pressure of the air around the sample. The thermodynamic pressure of a system, however, is its vapor pressure, and when in the presence of other gases, it will be its partial vapor pressure and not the sum of the

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pressures of all gases present. It is true that the pressure of the other gases in the system do have an effect on the vapor pressure of the compound under consideration, but at a total pressure of 1 atm or 1 bar, the increase of the vapor pressure of molecular substances is extremely small, and can be neglected in most if not all cases. In addition, atmospheric gases can be considered impurities of the system that may, to a very small and mostly negligible extent, modify the transition temperature. Calorimetric measurements are generally carried out in a capsule with enough dead volume for the sample to evaporate, even if the vapor pressure of the sample is very low. It implies that while heating the sample in a DSC capsule, its thermodynamic (i.e., vapor) pressure follows the sublimation curve of the solid. Thus, the pressure under normal calorimetric measurement conditions will be the sublimation pressure of the solid. The response to pressure by the system can be studied by imposing a hydrostatic pressure and measuring the shift in the transition temperature. As mentioned above, in the presence of the vapor phase, the thermodynamic pressure of the system will be that of the vapor. Thus, if one would like to subject the system to higher pressures, air bubbles, which may allow the system to establish a vapor phase, should be eliminated from the system. It also becomes obvious when one considers unary pressure–temperature phase diagrams (Fig. 4.5); once the pressure increases above the sublimation line of the solid, only one phase is stable and that is the solid. Thus, to study the response of a solid under pressure, a hydrostatic pressure should be applied, while excluding the presence of any gas phase.

4.3.4 The Clapeyron Equation The equilibrium temperature of a solid–solid transition can be determined by calorimetry, and it is now clear that the thermodynamic pressure is the (partial) vapor pressure of the chemical compound under study (“the system”). These two coordinates in the pressure–temperature phase diagram represent one point on the equilibrium line between two condensed phases (CS1 –CS2 or in the case of fusion CS–L). As it has been stated above on the pressure coordinate, in calorimetric measurements the condensed phase is in equilibrium with its vapor phase. It implies that the point on the equilibrium line is actually a triple point at which two condensed phases and the vapor phase are in equilibrium. To determine the position of the equilibrium at higher pressures, one can make use of the Clapeyron equation, which has been derived above (Eqs. 4.10–4.13): S H dP = = dT V T V

(4.14)

It provides the slope in the pressure–temperature phase diagram dP/dT of an equilibrium with S the entropy change, and V the volume change both associated with the phase transition, and because the Clapeyron equation is valid at the equilibrium,

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S = H/T is valid, and the entropy can be replaced by the enthalpy change, H, at the temperature T on the equilibrium line where the enthalpy change has been determined. Calorimetric methods generally provide the enthalpy change of a transition, and the volume change can be determined by X-ray diffraction or density measurements, for example. With the slope obtained from the Clapeyron equation together with the coordinates of a point on the equilibrium line, an equation of a straight line can be obtained obtained describing the equilibrium line as a function of P and T.

4.3.5 The Le Chatelier Principle One of the simplest but also most powerful principles for the construction of phase diagrams is the Le Chatelier principle. It states that a system tends to compensate effects caused by external influences. As two simple and fitting examples, we will discuss the increase of pressure and the increase of temperature. In the presence of a phase equilibrium, it is merely logical that the densest phase will be the one stable at the higher-pressure side; by becoming denser, the system absorbs part of the external work by decreasing its specific volume. Providing heat to the surroundings of the system will increase the temperature, and as a straightforward effect of equilibration, the heat will be absorbed by the system. Here again, in the presence of a phase equilibrium, one of the two phases will have a higher capacity to absorb heat than the other; thus, the phase with the higher heat content will be the stable one at higher temperature and vice versa. Liquids, in which molecules freely move, have a higher heat content than solids, and therefore, solids will never melt on decreasing their temperature. Thus, once the position of an equilibrium line is known, the Le Chatelier principle will determine on which side the respective phases are stable. Nonetheless, although the Le Chatelier principle is very useful, it does not resolve, whether a phase transition will occur. Thus, a denser phase may never become stable because another even denser phase may exist, or a solid with a high heat content may never be stable as the solid with a lower heat content melts before the phase transition into the other solid could occur.

4.3.6 The Clausius–Clapeyron Equation The Clapeyron equation can be used to obtain an expression for equilibria between condensed phases. Two-phase equilibria that involve the gas phase are rather curved and cannot easily be approximated by a straight line; however, in this case, the difference in volume between the gas and the condensed phase is so large that the Clapeyron equation can be integrated neglecting the volume of the solid phase. This leads to the Clausius–Clapeyron equation, which provides the vapor pressure, P, of a condensed phase as a function of the temperature, T:

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 P = A · ex p

−vap H RT

 (4.15a)

A is a constant, vap H is the heat of vaporization (sublimation) of the condensed phase and R is the gas constant (8.3145 J K−1 mol−1 ). This equation is often written in its logarithmic form, which is easier from an experimental point of view: ln(P) = −

vap H +B RT

(4.15b)

in which vap H and B, which is ln(A), are fitting constants for a line representing the natural logarithm of the (measured) vapor pressure of a condensed phase as a function of the inverse temperature. In the absence of vapor pressure data, it is often possible for molecular compounds to calculate (or estimate) the boiling point temperature and the heat of vaporization at the boiling point. Boiling occurs when the vapor pressure of the liquid equals the external pressure, which at sea level is in the order of 1 atm, because at that point, the vapor pressure is strong enough to counteract the external pressure. Taking 1 atm for the boiling pressure, the value of B can be calculated for the liquid–vapor phase equilibrium. The equation obtained in this way is obviously approximate, in particular because the enthalpy of vaporization is considered constant over the entire temperature and pressure range. Nevertheless, these vapor pressure lines can play their part in the elucidation of the phase stability hierarchy, as we will discuss below.

4.3.7 Triple Points and the Alternation Rule The triple point has already been introduced as a point in the phase diagram, where three phases are in equilibrium and where as a consequence three two-phase equilibrium lines meet. It is important to understand that pressure–temperature phase diagrams are in fact projections on the pressure–temperature plane of the intersections of the Gibbs free energy surfaces of the respective phases, as explained above. Thermodynamics dictates that the phase with the lowest Gibbs energy is the most stable one. When two intersecting surfaces meet a third surface, on one side, the intersecting surfaces will be lower, and on the other side, the third surface must be lower (see Fig. 4.4). Projected on the pressure–temperature plane, a phase equilibrium, which is stable on one side of the triple point, must become metastable on the other side of the triple point, because at that side the phase represented by another plane will have the lowest Gibbs energy (Fig. 4.4). This thermodynamic given that an equilibrium line must change one level in its stability hierarchy when passing a triple point is also called the alternation rule, as it leads necessarily to the property of a triple point that the equilibrium lines coming

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into the triple point must be alternatingly stable and metastable. This is an extremely useful property for the construction of phase diagrams, as the diagram is locked in once the stability hierarchy of a few key phase equilibria has been determined.

4.3.8 Extrapolation of Equilibrium Lines Once the transition temperature, enthalpy, and volume change have been determined for a two-phase equilibrium, it can be placed in the pressure–temperature phase diagram using the Clapeyron equation. Slopes of phase equilibria involving two condensed phases are in the order of MPa/K, thus rather steep in relation to most vapor pressures. Those pressures are generally much less than 1 atm at room temperature, which is much less than 0.1 MPa. In other words, the pressure coordinate can be safely set to zero for a transition obtained under atmospheric conditions (such as X-ray diffraction and DSC measurements). Having established the transition temperature at the approximate pressure of 0 MPa, the slope obtained by the Clapeyron equation can now be used to extrapolate the equilibrium line toward negative and positive pressure. With a slope and a single point, one can only define a straight line and that means that extensive extrapolation will always lead to inaccuracy, but how much will be too much? Experimental evidence appears to indicate that solid–solid transitions are straight over a rather long pressure–temperature range. In particular, for the pressure, these lines can safely be considered straight up to values of at least 300 MPa. This has been experimentally verified for a large number of organic compounds such as piracetam, benzocaine, and cysteamine hydrochloride [18–20]. Solid–liquid equilibria tend to be more curved because the volume change in liquids is affected to a larger extent by pressure than solids are. The problem of extrapolation of curved lines is that a limited number of experimental points, which necessarily contain some scatter, in a limited pressure–temperature range, will never allow an accurate estimate of the curvature outside the measurement range. Simple mathematical expressions for curves tend to exaggerate curvature on extrapolation. In such cases, it will be more convenient to use a straight line, the simplest case of a monotonic curve, as the uncertainty over the extrapolated part will not be much larger (provided the extrapolation is not into infinity) and straight lines are very easy to define. It facilitates the extrapolation of equilibrium lines, and it will not affect the global mapping of the topological phase diagram.

4.3.9 Redundancy in the Topological Method Topological pressure–temperature phase diagrams can be constructed because of the redundancy in the information for these diagrams. This redundancy can be used to verify the consistency of the diagram. For example, in the case of a triple point,

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it is clear that three two-phase equilibria must intersect in this point, as that is a thermodynamic given. However, only two two-phase equilibria need to be determined to find the triple point. This implies that determining the triple point by pairing up each time two two-phase equilibria, an estimate of the uncertainty in the position of the triple point can be obtained. On the other hand, if one of the phase equilibria is not known, with the position of the triple point in combination with thermodynamic cycling discussed below, the missing equilibrium can be placed in the phase diagram. An important thermodynamic given is that around a triple point the sum of thermodynamic quantities such as the entropy, but also the volume equals zero, the initial state and the final state being the same. Taking the triple point I-II-L as an example, around its triple point I→II H + II→L H + L→I H = 0, and also I→II V + II→L V + L→I V = 0. This means that not all the inequalities need to be known as one of the volume changes or enthalpy changes can be obtained from the other values. Considering that most of this data can be measured, missing data can be calculated or at least estimated. It can be used to determine the specific volume of the liquid phase for example. Obviously, measuring pressure–temperature curves directly increases the redundancy in the phase diagram, and therefore, its reliability, but in principle topological phase diagrams, can be reliably constructed from DSC and ambient pressure X-ray data only, provided the data are reliable.

4.3.10 The Implications of the Word “Topological” In the forgoing paragraphs, it has been alluded upon that the topological phase diagram is not necessarily a precise depiction of the phase behavior of a chemical compound. When using the above-mentioned redundancy properly, topological phase diagrams will correctly map the phase behavior of a system, but the diagram may be off in the predicted position of a triple point or transition point (when found by extrapolation). Nonetheless, it will depict the phase behavior and in broad lines represent the domains of the stable and metastable phases even if the positions of the borders of the different domains may possess some uncertainty. Topological implies in this context that the equilibrium lines connecting the triple points may be stretched or compressed, but that the relative position of the triple points in relation each other does not change. Two non-parallel monotonic curves can only cross once, which entails that only four triple points can exist in the case of crystalline dimorphism, because the four phases give rise to four two-phase equilibria. In the case of trimorphism, ten triple points exists and so on. This has already been demonstrated in 1890 by Riecke [21], who combined three by three, the existing phases of a one-component system, and thus showed that the number of two-phase equilibria ne and the number of triple points nt in a one-component system involving n phases are related in the following way:

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Two-phase equilibria: ne =

n! 2!(n − 2)!

(4.16)

nt =

n! 3!(n − 3)!

(4.17)

Triple points:

Taking only the relative positions of the four possible triple points into consideration for crystalline dimorphism reduces the number of possible pressure–temperature phase diagrams to four as will be shown below.

4.3.11 Crystalline Dimorphism in Four Possible Phase Diagrams Crystalline dimorphism for a chemical compound is defined by the existence of two different crystalline forms. In terms of a phase diagram, four possible domains may be present that of solids I and II, and those of the liquid and the vapor phases. The solid phases will occupy the high-pressure, low-temperature domain; however, the positions of the individual solid phases depend on their basic properties: heat content and specific volume. Bakhuis-Roozeboom has demonstrated in 1901 that there are only four possible ways in which two solid phases could be thermodynamically related to each other (see Fig. 4.7). In first instance, Lehmann had defined enantiotropy in relation to the temperature by a form I changing into form II at increasing temperature and reverting back into form I with decreasing temperature [22]. Enantiotropy signifies that both solid forms have a stable domain under ordinary conditions. Lehmann defines monotropy

Fig. 4.7 The four cases of Bakhuis-Roozeboom: 1 overall monotropy, 2 overall enantiotropy, 3 a monotropic system turning enantiotropic at high pressure, and 4 an enantiotropic system turning monotropic at high pressure. Reproduced with permission from Elsevier Masson, Céolin and Rietveld [23]

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by a form I that transforms into form II with increasing temperature without reverting back into form I, when the temperature is lowered again. Disregarding very slow or highly hindered transformations and considering this only from a thermodynamic standpoint, form II should in that case be the stable form over the entire temperature range and form I would be monotropically related to form II. Adding pressure as an additional variable, the system may remain “overall” monotropic (Fig. 4.7-1), “overall” enantiotropic (Fig. 4.7-2), or change from monotropic to enantiotropic with increasing pressure (Fig. 4.7-3), or from enantiotropic to monotropic with increasing pressure (Fig. 4.7-4). These changes in behavior are merely due to the position and the slope of the different two-phase equilibria involved.

4.3.12 Higher Order Polymorphism The four diagrams by Bakhuis-Roozeboom define the four topological possibilities for dimorphic systems, that is, for systems with two crystal structures. The actual differences in heat and volume between these phases and also between them and the liquid and the vapor will define the diagram in more detail. For trimorphism or higher, phase diagrams will quickly become more complicated. However, because stability is defined by the Gibbs free energy surfaces as a function of pressure and temperature, all polymorphs can be studied in relation to one other polymorph in the same system, and the final polymorphic phase diagram will be a superposition of the individual dimorphic diagrams as defined by Bakhuis-Roozeboom. Thus, even if a polymorphic phase diagram may be rather complicated, it can be represented by a series of dimorphic diagrams that can be much simpler to study. In the next chapter, an example for each dimorphic diagram proposed by Bakhuis-Roozeboom will be given featuring organic compounds.

4.4 Some Examples of Real Cases of Enantiotropy and Monotropy as a Function of Pressure and Temperature 4.4.1 Rimonabant: Overall Monotropy Rimonabant, an active pharmaceutical ingredient developed by Sanofi exhibits dimorphism; both forms have almost the same melting temperatures, melting enthalpies, and specific volumes [24]. The system is overall monotropic, and form II is the more stable solid form. The phase diagram in Fig. 4.8 has been obtained by direct measurements and by DSC and X-ray diffraction data in combination with the application of the Clapeyron equation. It can be seen that the melting curves, I-L and II-L, and the solid–solid transition I-II diverge with increasing pressure and

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Fig. 4.8 Topological pressure–temperature phase diagram of rimonabant. The system is overall monotropic with form II the only stable form. Solid lines are stable equilibria, dashed lines are metastable equilibria, and dotted lines are supermetastable. Solid circles are stable triple points, and gray circles are metastable triple points. For readability, the axes are not to scale. The coordinates of the triple points can be found in the original publication. Reproduced with permission from Wiley, Perrin et al. [24]

that their intersection is located at negative pressure.1 It implies that the triple point I-II-L is metastable. Overall monotropy only contains one single stable triple point, that of the highest melting solid, here II-L-vap. Interestingly, the more stable form of rimonabant does not possess any hydrogen bonds, whereas the less stable one does [24].

4.4.2 Cysteamine Hydrochloride: Overall Enantiotropy Cysteamine hydrochloride is used in the treatment of nephropathic cystinosis and has been granted orphan designation by the European Commission. It was investigated with calorimetry, high-pressure thermal analysis, and X-ray diffraction as a function of temperature [19]. The commercial form I and form III possess an overall enantiotropic phase relationship, with form I stable at room temperature and form III stable above 37 °C (see Fig. 4.9). The melting temperature of form III was found at 67.3 ± 0.5 °C. Form II exists at temperatures below 200 K; however, the exact phase relationships with form I and form III are not yet known. Comparing Figs. 4.8 and 4.9, it can be seen that in the cases of overall monotropy and overall enantiotropy, the 1

Negative pressures are real and exist as expanded condensed phases, which occur in for example turbulent flow. From a physical point of view, the direction of the force has changed: Positive pressure is the system pushing against the exterior, whereas negative pressure is the system pulling inward.

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Fig. 4.9 Topological pressure–temperature phase diagram of cysteamine hydrochloride. The system is overall enantiotropic in relation to forms I and III, which have each a stable domain over the entire pressure range. See Fig. 4.8 for the legend. The axes are not to scale. Triple point coordinates can be found in the original article. Reproduced with permission from ACS, Gana et al. [19]

triple point of the three condensed states (solid 1, solid 2, and the liquid) is metastable, and it can be found in the vapor phase (and mostly at negative pressure). The differences are the stability hierarchy of the solid–solid equilibrium and, particularly, its position in the phase diagram.

4.4.3 Benfluorex Hydrochloride: Enantiotropy Turning into Monotropy with Pressure Benfluorex hydrochloride is enantiotropic under ambient conditions. Both the melting temperature of form I and that of form II have been measured under pressure [17]. This system was particularly complicated due to a solid–solid transition that would not occur when the measurements were carried out too rapidly leading to the melting of form II and leaving the experimenter in the dark about the solid–solid transition (see Fig. 4.6). Calculation of the slope of the II-I transition in combination with

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Fig. 4.10 Topological pressure–temperature phase diagram of benfluorex hydrochloride. The system is enantiotropic at low pressures and becomes monotropic with form II the only stable form above about 150 MPa. Because of the scale, the equilibria involving the vapor phase coincide with the x-axis. Reproduced with permission from Wiley, Barrio et al. [17]

the measurement of the melting transitions under pressure have led to the conclusion that benfluorex hydrochloride turns monotropic under pressure (Fig. 4.10). The triple point I-II-L can be found at positive pressures and always possesses the highest stability level in the diagram together with the highest melting solid. This is in contrast with the previous two examples, where the triple point, at which the three condensed phases are in equilibrium, is metastable and can be found in the domain where the vapor phase is stable or below at negative pressures.

4.4.4 Ritonavir: Monotropy Turning into Enantiotropy with Pressure The last dimorphic example is that of ritonavir. The drug that has become notorious, because its more stable form II unexpectedly crystallized out in the initial formulation based on the less stable form I [16, 25]. Making use of the calorimetric and Xray diffraction data available in the literature, it can be deduced that ritonavir is monotropic under ambient conditions with form II the only stable form [26]. The melting point of form II, triple point II-L-vap (O1 in Fig. 4.11), is stable. Increasing the pressure leads to form I becoming stable too. An interesting detail in this system is that form I is the denser form, which according to the articles of Burger and Ramberger should be the most stable form. It can be seen here that those rules are not always correct. The construction of a topological pressure–temperature phase diagram uses the same data as the Burger and Ramberger rules, but it delivers a more complete picture of the phase behavior.

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Fig. 4.11 Topological pressure–temperature phase diagram of ritonavir. The system is monotropic at low pressures and becomes enantiotropic above about 17.5 MPa. This value is extraordinarily low in comparison with other systems. Solid lines are stable equilibria, dashed lines are metastable equilibria, and open lines are supermetastable. Solid circles are stable triple points, and open circles are metastable triple points. For readability, the axes are not to scale. The coordinates of the triple points can be found in the original publication. Reproduced with permission from Elsevier Masson SAS, Céolin and Rietveld [26]

4.4.5 Isomorphism in Halomethane Compounds: Similar Systems with Changing Hierarchies Enantiotropic and monotropic behavior for the rhombohedral (R) and face-centered cubic (FCC) polymorphs of the halomethane compounds CBr(4−n) Cln , n = 0, 2, 3, 4 can be found in Fig. 4.12. Whereas, CCl4 (case a) is overall monotropic with the FCC remaining metastable, CBrCl3 exhibits a small enantiotropic domain where FCC is stable and the system turns monotropic at higher pressures. In CBr2 Cl2, the enantiotropic domain has become much larger, and in CBr4 , FCC is the stable

Fig. 4.12 Experimental pressure–temperature phase diagrams for the compounds a CCl4 [27], b CBrCl3 [28], c CBr2 Cl2 [29], and d CBr4 , schematic according to Bridgman [30]. The inset of b corresponds to a magnification of the low-pressure domain of the p–T phase diagram involving the R, FCC, and L phases for CBrCl3 . Red dashed lines correspond to the extrapolation of the two-phase equilibria to normal pressure. Pressure is found at the x-axis in this series of diagrams. Reproduced with permission from IOP Publishing, Tamarit et al. [31]

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phase in a monotropic relationship with R at low pressure, the latter system turning enantiotropic only at higher pressures. Hence, even if the phase diagrams of these isomorphic systems are topologically similar; there are many different ways in which the stability hierarchy influences the final outcome. Furthermore, in all four diagrams, the monoclinic phase (M) possesses an enantiotropic phase relationship with the phases it borders with.

4.5 The Consequences of Polymorphism in Molecular Alloys 4.5.1 Polymorphism and Molecular Alloys The consequences of polymorphism for molecular alloys can be considered in two ways. On the one hand, polymorphism can occur in new phases that only exist within molecular mixtures, the so-called cocrystals or binary compounds, which include hydrates and racemates. Cocrystal polymorphism can be considered similar to that of pure compounds, in particular from a thermodynamic point of view. Racemates represent a group of model cocrystals for which the thermodynamic assessment is relatively easy, as their phase behavior is symmetric with respect to concentration with a mirror plane at 0.5 mol fraction, and a few examples of topological phase diagrams of racemates will be discussed in a later chapter on enantiomers. In the case of cocrystals made up of different chemical substances, a thermodynamic assessment will most likely become more complicated; however, in essence, it remains similar to that of a pure compound. Except for the racemates in a later chapter, this will not be discussed in detail in this book. On the other hand, polymorphism in the pure components that make up a binary mixture can have consequences for the behavior of the mixture. In that case, one should consider the three mixing behaviors of chemical substances: eutectic mixtures, cocrystals, and solid solutions. This will be discussed in the last part of this chapter.

4.5.2 Eutectic Systems For dose form development, the interactions of pharmaceuticals with water are often very important, first of all, because of water’s role as a medium for drug dissolution and secondly, because water may affect the integrity of a drug formulation. Eutectic equilibria depend on the solid phases that are involved. As an example, the dimorphism of cysteamine hydrochloride will be used. Above, it has been shown that form I is stable at room temperature and turns into form III at 310 K, which possesses a stable melting transition. Cysteamine hydrochloride exhibits a eutectic transition in the presence of water at 240 K. A eutectic equilibrium results from the

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mixing in the liquid of two different molecules that are unmixed in the solid state. Entropic effects, and often enthalpic interactions too, cause the eutectic equilibrium to be lower than the melting points of the pure compounds. The eutectic point can be estimated by calculating the intersection of the ideal liquidus lines, i.e., neglecting enthalpic contributions, using the Schröder equation: −RT ln x +  S→L S(T − TS→L ) = 0

(4.18)

This equation is valid for the right-hand liquidus (cysteamine hydrochloride in Fig. 4.13); for the left-hand liquidus, the x needs to be replaced by (1 − x). In this equation, R is the gas constant, 8.3145 J K−1 mol−1 , T is the temperature in kelvin of the liquidus at the mole fraction of cysteamine hydrochloride x, S→L S is the melting entropy of pure cysteamine hydrochloride (form I: 50.7 J K−1 mol−1 , form III: 47.3 J K−1 mol−1 ), or water (22.0 J K−1 mol−1 ) depending on the liquidus, and T S→L is the melting point of the respective pure compound (water: 273.15 K, form I of cysteamine hydrochloride: 338.3 K, or form III: 340.4 K) [19]. The description of the equation already hints at the issue at hand. To calculate the eutectic point, one needs to know the melting temperature and melting entropy of the two phases involved in the eutectic equilibrium. In the case of cysteamine hydrochloride, which exhibits a phase transition at 310 K, the melting point and the melting enthalpy determined by DSC will not be those of the phase involved in the eutectic equilibrium as a phase change has occurred before melting. Thus, to calculate the estimated eutectic point with the ideal liquidus, the metastable melting

Fig. 4.13 Binary phase diagram of water and cysteamine hydrochloride. Reproduced with permission from ACS, Gana et al. [19]

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point of form I and its melting enthalpy should be used. These data can be obtained through the topological approach for a unary phase diagram described above. When points on the liquidus line are available, obtained by DSC or DVS, the liquidus lines can be fitted, and the excess Gibbs energy can be determined using for example a Redlich–Kister expression for the excess function H E − TS E :   −RT ln x +  S→L S(T − TS→L ) = H E − T S E (1 − x)2

(4.19)

In the case of dimorphism, two liquidus lines will need to be fitted, one which starts at the stable melting point down to the polymorphic transition point and the other one which starts at the metastable melting point and which becomes stable below the polymorphic transition. This has been illustrated in Fig. 4.13, where the broken line is the liquidus of form III, stable above the solid–solid transition, and the solid line is the liquidus of form I. Both liquidus lines intersect the liquidus line of water, however, at a different temperature and concentration. The difference may not be very large in this case, but can differ more extensively depending on the properties of the two polymorphs involved. For the fit of the two liquidus lines, different sets of points are allotted to different equilibria. The open squares belong to the liquidus line of the stable melting point, whereas the data points obtained below the polymorphic transition point, solid squares, are ascribed to form I, with a different melting temperature and melting enthalpy. It results in two separate liquidus lines. Nonetheless, the excess function, which for a eutectic transition (immiscible solid) fully depends on the liquid phase, is the same for both liquidus lines, the stable and the metastable one. This limits the parameters involved and makes the fit of the liquidus lines more stable. In this approach, it was assumed that no solid solution exists with water in cysteamine hydrochloride. The presence of a solid solution could alter the phase diagram dramatically, and the polymorphic transition from form I to form III may be displaced and occur at a different temperature. In that case, a straightforward calculation of the liquidus line using Eqs. (4.18) and (4.19), as has been carried out for this system, will not be possible.

4.5.3 The Effect on Cocrystals: Polymorphism and Hydrate Formation The effect of polymorphism on the formation of cocrystals is from a thermodynamic point rather limited. A stable phase will be stable, no matter the starting point of its components. However, the difference may be apparent in the conversion rate into the binary compound or cocrystal. Two examples will be shown on hydrate formation, as hydrates are of importance to the pharmaceutical industry. From a thermodynamic point of view, the observations will be entirely valid for any other binary compound.

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Triethylenetetramine dihydrochloride possesses two polymorphs. The unary system is monotropic with form I the only stable form. In the binary mixture with water, a non-congruent melting dihydrate exists [32]. When the dihydrate ceases to exist in a peritectic transition, it turns into liquid and the solid phase I of the pure component triethylenetetramine dihydrochloride (see Fig. 4.14). No solid–solid equilibrium between form II and the dihydrate exists, also not a metastable one. In particular, from differential vapor sorption (DVS) experiments, it can be seen that the transformation of form II into the hydrate is very haphazard (not shown). Other experimental evidence indicates that form II either first dissolves or turns into form I upon which the dihydrate appears. In the case of triethylenetetramine dihydrochloride, it is clear that form II is the monotropic metastable form, which has in any case a tendency to turn into form I. Form I will subsequently change into the dihydrate depending on the conditions. For l-citrulline, several polymorphic forms exist of which the stability hierarchy is not entirely resolved; however, form δ appears to be the most stable one, and form α is thought to be metastable [33]. Solubility measurements exhibit the typical spring and parachute behavior in the case of form α (Fig. 4.15), which is a behavior sought after for dosage forms as it ensures a rapid dissolution, which facilitates the uptake of a pharmaceutical in the body. The decrease in the concentration of l-citrulline after the rapid increase is related to a recrystallization, which in this case is the formation of a hydrate. The δ form, which is understood to be more stable than the α form, in part as a result of the behavior depicted in Fig. 4.15, causes the concentration of l-citrulline to rise more gradually. Moreover, once the concentration has reached a plateau, the δ form

Fig. 4.14 Binary phase diagram of water and triethylenetetramine dihydrochloride. Reproduced with permission from Elsevier, Henriet et al. [32]

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Fig. 4.15 Solubilization of forms α and δ l-citrulline in water. The decrease in the l-citrulline concentration after the dissolution of the α phase is due to hydrate formation. Reproduced with permission from ACS, Allouchi et al. [33]

does not appear to change into a hydrate; its conversion into the hydrate has never been observed [33]. Thus, it appears to be that the stable phase, δ, resists conversion even if hydrates are very often the most stable states. Hence, the rate of formation of binary compounds depends in part on the polymorphism of the constituents, but most often, the most stable state will appear eventually, even if it may take a long time as in the case of δ l-citrulline.

4.5.4 Solid Solutions and Polymorphism Isomorphism was considered a useful property to determine atomic masses by Berzelius, while it is also an important requirement for the presence of extensive solid solutions in binary phase diagrams. When isomorphic components of a binary system have similarly shaped molecules too, such as for example the spherical (CH3 )3 CCl and CBrCl3 [34], those molecules can “dissolve” into each other’s crystal structures. For truly isomorphic phases, such as the FCC phases of (CH3 )3 CCl and CBrCl3 , solid solutions exist that expand over the entire composition range of the binary phase diagram (Fig. 4.16). It is clear that mixtures of the systems depicted in Fig. 4.12 will all form solid solutions in particular between their FCC and R phases, which are all more or less isomorphic and contain at least some disorder, which favors the introduction of foreign species in the crystal network. Depending on the similarity between the phases, these solid solutions may stretch the entire concentration range like the one of the (CH3 )3 CCl and CBrCl3 system or exist within a limited concentration range.

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Fig. 4.16 Binary phase diagram of (CH3 )3 CCl and CBrCl3 . Reproduced with permission from ACS, Barrio et al. [34]

Isomorphism between two similar phases of different compounds implies continuity in the Gibbs energy as a function of composition and in any other physical property. Among these properties, lattice parameters (or lattice volumes) are the most typical and accessible experimentally. Figure 4.17 shows the continuity for the whole composition range of the binary mixture (CH3 )3 CCl and CBrCl3 . By changing one Cl atom in the previous binary mixture by a Br atom ((CH3 )3 CBr), an even more complex example is obtained with the binary system Fig. 4.17 Lattice parameters for the FCC and R mixed crystals between (CH3 )3 CCl and CBrCl3 as a function of molar fraction. Reproduced with permission from ACS, Barrio et al. [34]

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Fig. 4.18 Binary phase diagram of (CH3 )3 CBr and CCl3 Br. Reproduced with permission from ACS, Barrio et al. [34]

(CH3 )3 CBr and CBrCl3 (Fig. 4.18). The [R + FCC] equilibrium is truncated due to other two-phase equilibria of low-temperature, ordered phases that interfere. Nevertheless, lattice parameters (Fig. 4.19) for both R and FCC mixed crystals change in a continuous fashion as a function of the mole fraction. This experimental fact confirms the isodimorphism between the mutual R and the mutual FCC phases of both compounds, and it allows therefore a thermodynamic assessment through the use of a single Gibbs function for each of the molecular alloys. Another example exhibiting solid solutions over the entire binary concentration range is the enantiomer system of camphor, which phase diagram is discussed in more detail in another chapter. Obviously, the two camphor enantiomers are isomorphic, i.e., they have the same crystal structures; however, isomorphism alone is not enough to guarantee the formation of solid solutions, demonstrated by the fact that not all enantiomer systems form solid solutions. In the case of camphor, the molecule is spherical as is the case for (CH3 )3 CCl and CBrCl3 . The sphericity causes orientational disorder in the crystal structures, and therefore, the difference between the two enantiomers becomes less apparent; hence, they mix easily forming solid solutions, which due to their isomorphism, stretch over the entire concentration range (Fig. 9.13). Camphor is even an example of isodimorphism,2 and the binary system of the two optical antipodes contains two distinct solid solutions stretching over the entire concentration range both with orientational disorder (Fig. 9.13 in Chap. 9). However, because the two camphor enantiomers are very similar due to their spherical shape, no solidus or liquidus has been observed in this binary phase diagram. Only when 2

Or even isotrimorphism, however, the low temperature isomorphic phase strictly speaking does not (or only for a very limited concentration range) contain a solid solution. The low temperature part forms a cocrystal with two symmetric equilibria similar to eutectic equilibria.

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Fig. 4.19 Lattice parameters for the FCC and R mixed crystals between (CH3 )3 CBr and CCl3 Br as a function of molar fraction. Reproduced with permission from ACS, Barrio et al. [34]

the orientational disorder diminishes, at about 200 K, does camphor form a racemate and even this phase still contains considerable disorder [35]. Isodimorphism can complicate matters even more in binary systems if the isomorphic phase is stable for one component and metastable for the other and vice versa; this is called crossed isodimorphism. Well-known and well-studied compounds in this respect are the 2-R-naphthalenes (R = F, Cl, Br, SH, CH3 ; called for convenience ‘group 1’) that form crossed isodimorphic binary systems with naphthalene and 2napthol (R = OH), which are called here ‘group 2’, see for example Fig. 4.20. Both groups crystallize in the P21 /a space group with small differences in lattice parameters between the two groups [36]. Thus, binary mixtures between naphthalene and the 2-R-naphthalenes (except for R = OH) will form systems with limited solid solutions and necessarily a two-phase region in the binary phase diagram. Only between members of the same group, such as naphthalene and 2-naphthol or 2-Cl-naphthalene and 2-methyl-naphthalene, solid solutions exist over the entire concentration range

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Fig. 4.20 Naphthalene—2-chloronaphthalene system. Reproduced with permission from IUCr, Haget et al. [36]

(see Fig. 4.21). Thus, the two groups are not considered isomorphous, despite the identical space group of their solid phases [36]. The rationale behind the non-isomorphism can be found in the packing differences in both groups, referred to as type 1 and type 2 for the respective groups. Figure 4.22 presents the lattice parameters as a function of composition for mixed crystals of the type 1 packing, and the continuity of the packing in the mixtures can be clearly observed. In the case that two members of the two different groups are mixed, it emerges (see Fig. 4.23) that the lattice parameters are not continuous as a function of composition. In Fig. 4.23, the left-hand side represents pure naphthalene, and the right-hand side contains many different members of group 1 with type 1 packing. On the right-hand side, limited type 1 solid solutions exist, and their lattice parameters can be extrapolated to pure naphthalene on the left-hand side, leading to potential lattice parameters of a type 1 packing crystal of naphthalene, i.e., a presumably metastable phase with a packing strictly isomorphous to the 2-R-naphthalene derivatives of group 1. The potential lattice parameters of the presumably metastable form give rise to a smaller volume cell, so one could expect the polymorph to be stable at high pressure. This study clearly highlights that investigating isomorphic relationships can provide new polymorphs, which have previously not been obtained under “ordinary conditions.” Another peculiarity of solid solutions is that they may retain higher order transitions. 2-chloronaphthalene exhibits a disordered phase at high temperature with

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Fig. 4.21 2-chloronaphthalene—2-methylnaphthalene system. The transition from α to β at the 2-chloronaphthalene (left-hand) side is second order, and it continues in the binary solid solution up to 0.12 mol% [37]

space group P21 /a, Z = 2, and 426 Å3 . With decreasing temperature, at 309 K, the structure starts to change gradually. The new space group is the same as the previous one, P21 /n; however, Z = 4, and the cell volume has obviously doubled. Nevertheless, even if Z has been interpreted to double, no discontinuity in the specific volume is observed. The transition continues down to about 253 K [38]. Mixed with 2-methylnaphthalene, 2-chloronaphthalene forms a continuous solid solution in the same way as discussed above, because their high-temperature phases are very similar. However, the second-order transition also exists in a part of the concentration range in the solid solution as can be seen in Fig. 4.21, where the ordered phase is marked by β and the disordered phase is marked by α. A similar behavior can even be observed for the binary mixture of naphthalene and 2-chloronaphthalene, where

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Fig. 4.22 Lattice parameters (in Å and in ° for β) as a function of mole fraction for the mixed crystals (type 1 packing) formed between 2-F-naphthalene and 2-chloronaphthalene (circles), 2-methylnaphthalene (horizontal ellipses), 2-thionaphthol (squares), and 2-bromonaphthalene (diamonds). Reproduced with permission from IUCr, Haget et al. [36]

β can be found again in the limited solid solution on the 2-chloronaphthalene rich side, see Fig. 4.20. Thus, second-order transitions emanating from a pure phase can be conserved in part of the solid solution in the binary system.

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Fig. 4.23 Lattice parameters (in Å and in ° for β) as a function of mole fraction for the mixed crystals formed between 2-naphthalene (type 2 packing) and 2-chloronaphthalene (circles), 2-methylnaphthalene (horizontal ellipses), 2-thionaphthol (squares), and 2-bromonaphthalene (diamonds), all of them with type 1 packing. Pure 2-naphthalene crystal (solid inverted triangles, type 2 packing) and the extrapolation proposed for the supposedly metastable form of 2-naphthalene (open inverted triangles, type 1 packing). Reproduced with permission from IUCr, Haget et al. [36]

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References 1. Klaproth MH (1788) Phosphorsäure, ein Bestandteil des Apatits (digitally available from the Münchener Digitalisierungszentrum, https://www.mdz-nbn). Berg J 1:294–300 2. Werner AG (1788) Geschichte, Karakteristik, und kurze chemische Untersuchung des Apatits. Berg J 1:76–96 3. Cahn RW (1999) Slaying the crystal homonculus. Nature 400:625 4. Lima-de-Faria J (1990) Historical atlas of crystallography. Kluwer, Dordrecht 5. Abbé Haüy R-J (1784) Essai d’une théorie sur la structure des crystaux, appliquée à plusieurs genres de substances crystallisées. Gogué & Née de la Rochelle, Paris 6. Haüy R-J (1817) Comparaison des formes cristallines de la stontiane carbonatée avec celle de l’arragonite. Ann Chim Phys 5:439–441 7. Haüy RJ (1792) Exposition abrégée de la théorie sur la structure des crystaux. l’Imprimerie du Cercle Social, Paris 8. Haüy RJ (1801) Traité de minéralogie. Louis, Paris 9. Thenard L, Biot J (1809) Mémoire sur l’analyse comparée de l’arragonite, et du carbonate de chaux rhomboidal. Mém Phys Chim Soc D’Arcueil 2:176–206 10. Stromeyer J (1814) De la différence chimique entre l’arragonite et le spath calcaire rhomboïdal. Ann Chim 92:254–299 11. Wallerant FFA (1909) Cristallographie: Déformation des corps cristallisés–groupements–polymorphisme–isomorphisme. Librairie polytechnique, Paris 12. Mitscherlich E (1820) Sur la relation qui existe entre la forme cristalline et les proportions chimiques. Premier mémoire sur l’identité de la forme cristalline chez plusieurs substances différentes et sur le rapport de cette forme avec le nombre des atomes élémentaires dans les cristaux. Ann Chim Phys 14:172–190 13. Mitscherlich E (1823) Sur le rapport qui existe entre les proportions chimiques et la forme cristallin. IIIme mémoire: sur les corps qui affectent deux formes cristallines différentes. Ann Chim Phys 24:264–271 14. Wöhler F, von Liebig J (1832) Untersuchungen über das Radikal der Benzosäure. Ann Pharm 3:249–282 15. Metzger H (1969) La genèse de la science des cristaux. A. Blanchard, Paris 16. Bauer J, Spanton S, Henry R, Quick J, Dziki W, Porter W, Morris J (2001) Ritonavir: an extraordinary example of conformational polymorphism. Pharm Res 18:859–866 17. Barrio M, Maccaroni E, Rietveld IB, Malpezzi L, Masciocchi N, Ceolin R, Tamarit JL (2012) Pressure-temperature state diagram for the phase relationships between benfluorex hydrochloride forms I and II: a case of enantiotropic behavior. J Pharm Sci 101(3):1073–1078 18. Toscani S, Céolin R, Ter Minassian L, Barrio M, Veglio N, Tamarit J-L, Louër D, Rietveld IB (2016) Stability hierarchy between piracetam forms I, II, and III from experimental pressuretemperature diagrams and topological inferences. Int J Pharm 497:96–105 19. Gana I, Barrio M, Ghaddar C, Nicolai B, Do B, Tamarit J-L, Safta F, Rietveld IB (2015) An integrated view of the influence of temperature, pressure, and humidity on the stability of trimorphic cysteamine hydrochloride. Mol Pharma 12(7):2276–2288 20. Gana I, Barrio M, Do B, Tamarit J-L, Céolin R, Rietveld IB (2013) Benzocaine polymorphism: pressure-temperature phase diagram involving forms II and III. Int J Pharm 456:480–488 21. Riecke E (1890) Spezielle Fälle von Gleichgewichterscheinungen eines aus mehreren Phasen zusammengesetzten Systemes. Z Phys Chem (Munich) 6:411–429 22. Lehmann O (1891) Die Krystallanalyse oder die chemische Analyse durch Beobachtung der Krystallbildung mit Hülfe des Mikroskops. Engelmann, Leipzig 23. Ceolin R, Rietveld I-B (2016) X-ray crystallography, an essential tool for the determination of thermodynamic relationships between crystalline polymorphs. Ann Pharm Fr 74:12–20 24. Perrin MA, Bauer M, Barrio M, Tamarit JL, Ceolin R, Rietveld IB (2013) Rimonabant dimorphism and its pressure-temperature phase diagram: a delicate case of overall monotropic behavior. J Pharm Sci 102:2311–2321

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25. Chemburkar SR, Bauer J, Deming K, Spiwek H, Patel K, Morris J, Henry R, Spanton S, Dziki W, Porter W, Quick J, Bauer P, Donaubauer J, Narayanan BA, Soldani M, Riley D, McFarland K (2000) Dealing with the impact of ritonavir polymorphs on the late stages of bulk drug process development. Org Process Res Dev 4:413–417 26. Céolin R, Rietveld IB (2015) The topological pressure-temperature phase diagram of ritonavir, an extraordinary case of crystalline dimorphism. Ann Pharm Fr 73:22–30 27. Maruyama M, Kawabata K, Kuribayashi N (2000) Crystal morphologies and melting curves of CCl4 at pressures up to 330 MPa. J Cryst Growth 220:161–165 28. Parat B, Pardo LC, Barrio M, Tamarit JL, Negrier P, Salud J, López DO, Mondieig D (2005) Polymorphism of CBrCl3 . Chem Mater 17:3359–3365 29. Barrio M, Tamarit JL, Negrier P, Pardo LC, Veglio N, Mondieig D (2008) Polymorphism of CBr2 Cl2 . New J Chem 32:232–239 30. Bridgman PW (1970) The physics of high pressure. Dover Publications, New York 31. Tamarit JL, Barrio M, Pardo LC, Negrier P, Mondieig D (2008) High-pressure properties inferred from normal-pressure properties. J Phys Condens Matter 20:244110 32. Henriet T, Gana I, Ghaddar C, Barrio M, Cartigny Y, Yagoubi N, Do B, Tamarit J-L, Rietveld IB (2016) Solid state stability and solubility of triethylenetetramine dihydrochloride. Int J Pharm 511:312–321 33. Allouchi H, Nicolai B, Barrio M, Ceolin R, Mahe N, Tamarit J-L, Do B, Rietveld IB (2014) On the polymorphism of L-citrulline: crystal structure and characterization of the orthorhombic delta form. Cryst Growth Des 14:1279–1286 34. Barrio M, Negrier P, Tamarit JL, Mondieig D (2011) From high-temperature orientationally disordered mixed crystals to low-temperature complex formation in the two-component system (CH3 )3 CBr + Cl3 CBr. J Phys Chem B 115:1679–1688 35. Nagumo T, Matsuo T, Suga H (1989) Thermodynamic study on camphor crystals. Thermochim Acta 139:121–132 36. Haget Y, Chanh NB, Meresse A, Bonpunt L, Michaud F, Negrier P, Cuevas-Diarte MA, Oonk HAJ (1999) Isomorphism and mixed crystals in 2-R-naphthalenes: evidence of structural subfamilies and prediction of metastable forms. J Appl Crystallogr 32:481–488 37. Meresse A (1981) Doctoral Thesis: Implications de polymorphisme dans la formation des alliages moléculaires: syncristallisation en série naphtalénique beta substituée. Université de Bordeaux I, Bordeaux 38. Meresse A, Chanh NB, Housty J-R, Haget Y (1986) Polymorphism of β R-substituted naphthalene derivatives—review and comparative study. J Phys Chem Solids 47:1019–1036

Part II

Facts and Features

Chapter 5

Aromatics P. R. van der Linde and H. A. J. Oonk

Abstract A start is made with the family of the para-dihalobenzenes, including the key system p-dichlorobenzene + p-dibromobenzene. Special attention is given to the melting behaviour of mixed materials prepared by zone levelling. The family of the dihalobenzenes is followed by the group of the 2-R-substituted naphthalenes which includes naphthalene itself (R = H), and the substances with R = F, Cl, Br, SH, CH3 , and OH. It appears that the naphthalene group falls apart into two structural subfamilies. Furthermore, evidence is given of an extra attractive effect between substituted methyl and substituted halogen. In addition, a number of stand-alone systems are discussed, on the basis of which afore-mentioned phenomena are further illustrated. One of these systems is trans-azobenzene + trans-stilbene for which the outcome of lattice energy calculations is also presented.

5.1 Introduction The chapter starts by treating a family of binary systems whose components belong to a chemically coherent group of substances: the family of the para-dihalobenzenes, with halo = Cl, Br, I. With the exception of the diiodo compound, the members of the family are monoclinic and have the same space group, which is P21 /a. The five members with the same space group comprise ten binary systems; nine of these systems form, at room temperature, a continuous series of mixed crystals. The thermodynamic mixing properties of the ten binary systems are correlated with the coefficient of crystalline isomorphism, εm . This coefficient expresses the geometric similarity between the unit cells of the constituting components; its value is given by εm = 1 −

m m

(5.1)

P. R. van der Linde (B) · H. A. J. Oonk Universiteit Utrecht, Utrecht, The Netherlands e-mail: [email protected] H. A. J. Oonk e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. À. Cuevas-Diarte and H. A. J. Oonk (eds.), Molecular Mixed Crystals, Physical Chemistry in Action, https://doi.org/10.1007/978-3-030-68727-4_5

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in which  m is the common (included) volume and m the excluded volume when the two unit cells are brought to maximal overlap.1 Special attention is given to the system p-dichlorobenzene + p-dibromobenzene, which, through the years, has been a key system for the study of mixed crystals. Next, the group of the 2-R-substituted naphthalenes is considered. The members of the group are naphthalene itself (R = H); the halo-substituted members (R = F, Cl, Br, and including R = SH); 2-methylnaphthalene (R = CH3 ); and 2-naphthol (R = OH). In this case, there are two structural subfamilies, and it means that there is a subtle role of polymorphism. In addition, and in contrast to the para-dihalobenzenes, a number of the naphthalene systems have components that do not belong to a chemically coherent group; this means that their thermodynamic properties are not determined by geometric similarity (i.e. εm ) alone. This aspect is elaborated for the combinations of naphthalene with 2-fluoronaphthalene and 2-naphthol and the combinations of 2-methylnaphthalene with the 2-halonaphthalenes. Apart from the two families of systems, a number of stand-alone systems make their appearance. Some of these systems are just mentioned; some others are treated in some detail. The former include 1,3,5-trichlorobenzene + 1,3,5-tribromobenzene and 1,2,4,5-tetrachlorobenzene + 1,2,4,5-tetrabromobenzene. The latter are transazobenzene + trans-stilbene, thiophene + benzene, and thianaphthene + naphthalene. Throughout the chapter, emphasis is on phenomenology and experimental methodology. Moreover, for the thermodynamic mixing properties of the mixed crystalline state (superscript sol), the ABΘ model (see Chap. 3) is adopted. In this model, the molar excess Gibbs energy, as a function of temperature T and composition X, is given by   T · [1 + B · (1 − 2 · X )]; G E,sol (T, X ) = A · X · (1 − X ) · 1 − θ

(5.2)

in which A, B, and Θ are system-dependent parameters; Θ is the so-called compensation temperature (see Chap. 3). The composition variable X stands for the mole fraction of the second component. Invariably the second of the two components is the one with the larger molar volume. The fact that the model expression for GE is taken linear in temperature implies that the excess enthalpy, H E , is independent of temperature. Hence, the corresponding expression for H E , the heat of mixing, is H E,sol (T, X ) ⇒ H E,sol (X ) = A · X · (1 − X ) · [1 + B · (1 − 2 · X )].

(5.3)

Note: as for instance shown by van der Linde [1] and Yamamuro et al. [2], molecular systems that form mixed crystals have virtually ideal liquid mixtures. It means that H E,sol and GE,sol virtually are equal to minus H E and minus GE , respectively, A counterpart of εm is εk , the coefficient of molecular homeomorphism. The definition of εk is similar to the above definition of εm : ‘unit cells’ has to be replaced by ‘molecules’ (see also Chap. 2).

1

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where the operator  stands for the difference between liquid and solid: G E = G E,liq − G E,sol ⇒ −G E,sol

(5.4)

H E = H E,liq − H E,sol ⇒ −H E,sol

(5.5)

In the text, the difference properties GE and H E quite often make their appearance, because, strictly speaking, these are the ones that follow from the experimental information.

5.2 The para-Dihalobenzene Family The five substances p-dichlorobenzene (ClCl), p-bromochlorobenzene (BrCl), p-chloroiodobenzene (ClI), p-dibromobenzene (BrBr), and p-bromoiodobenzene (BrI) are crystalline at room temperature, all having space group P21 /a, with two molecules per unit cell. With the exception for ClCl, the crystal structure remains the same on heating to the melting point. In discussing the binary phase diagrams, the existence of ClCl’s polymorphism is ignored; it is discussed in Sect. 5.3. The five substances include ten binary systems, and from thermal analytical work, it is known, for a long time, that the substances are miscible in the solid state. In the case of nine systems, miscibility is complete; limited miscibility is shown by ClCl + BrI. In 1991, Calvet et al. [3] presented a study in which for the ten binary systems the change from solid to liquid was investigated by differential scanning calorimetry (dsc) and subsequently subjected to a thermodynamic analysis in terms of liqfit [4]. Liqfit allows for the calculation of thermodynamic excess functions on the basis of (experimental) liquidus data. The outcome of the study is summarized in Table 5.1. At the time of writing the 1991 paper, there was some confusion about the nature of the solid state, evoked by X-ray observations that could be taken as evidence of a demixing phenomenon; see e.g. Haget et al. [5]. In Table 5.1, the systems that were suspected of demixing have numbers marked with an asterisk. Today, it is known that the X-ray observations are in conflict with the state of true equilibrium, read minimal Gibbs energy. In the case of the system ClCl + BrI, with its eutectic phase diagram, there is a thermodynamically real miscibility gap [8]. Notwithstanding the experimental uncertainties in measured enthalpy effects and computed excess Gibbs energies, the information contained in Table 5.1 gives evidence of two important facts. In the first place, as follows from Fig. 5.1, the equimolar excess Gibbs energy effect of the mixed crystalline state (from the available data calculated for T = 333 K) is proportional to the equimolar excess enthalpy effect. This observation is evidence of the fact that the systems correspond to a class of similar systems in terms

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Table 5.1 Results of a thermodynamic analysis of ten binary para-dihalobenzene systems: equimolar excess enthalpy differences (H E ) and excess Gibbs energy differences (GE ) at the mean temperature of the solid-liquid region (T mean ) [3], along with the geometric mismatch between the components in terms of the coefficient of crystalline isomorphism εm [6] and the type of phase diagram. System

Number

(1 − εm )

ClCl + BrCl

1

0.033

[0]/[–]

329

220

120

ClCl + BrBr

2*

0.064

[0]/[–]

338

1090

360

ClCl + ClI

3

0.121

[–]

312

2230

720

ClCl + BrI

4

0.143

[e]

312

1990

850

BrCl + BrBr

5

0.029

[0]/[–]

346

130

170

BrCl + ClI

6

0.085

[–]

326

980

330

BrCl + BrI

7*

0.107

[–]

340

1960

540

BrBr + ClI

8*

0.054

[0]

341

760

110

BrBr + BrI

9

0.075

[–]

356

1100

290

ClI + BrI

10*

0.020

[0]

343

520

50

Type

Tmean (K)

−H E (X = 0.5) (J mol−1 )

−G E (X = 0.5) (J mol−1 )

The asterisks in the second column refer to systems that were suspected of showing demixing phenomena. For types of phase diagram and the symbols used in the fourth column, see Oonk and Calvet [7]. The indication [0]/[–] is for systems that (tend to) have a minimum close to the melting point of the lower melting component. ClCl’s polymorphism is ignored. (adapted from Ref. [3], © 1991, with permission from Elsevier)

of enthalpy/entropy compensation (see Chap. 3). The compensation temperature that corresponds to the diagonal in Fig. 5.1 is Θ = 500 K. The second fact, more obvious than the first, is that the equimolar excess enthalpy effect increases with decreasing geometric similarity (εm ) between the components of the member system; see Fig. 5.2.

5.3 Polymorphism of p-Dichlorobenzene The substance p-dichlorobenzene has three solid forms, which are in order of their appearance on the temperature scale: the monoclinic form III(γ ), the monoclinic form II(α), and the triclinic form I(β). Form II(α) is the one which is isomorphous with the solid form taken by BrCl, ClI, BrBr, and BrI. The transition temperatures and the heat effects of the transitions are shown in Table 5.2. In the binary phase diagrams, measured for temperatures above room temperature, and with ClCl as the first component, ClCl’s polymorphism gives rise to an almost negligible single-phase field for the form I(β). It means that, for the thermodynamic analysis of the phase diagrams, the existence of form I(β) can be ignored. However, for the analyses, the metastable melting properties of form II(α) are needed. The

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Fig. 5.1 Para-dihalobenzene family of systems. Excess Gibbs energy difference, GE (X = 0.5), as a function of the excess enthalpy difference, H E (X = 0.5), for systems with complete miscibility. The numbering is in line with Table 5.1 (reproduced from Ref. [3], © 1991, with permission from Elsevier)

Fig. 5.2 Para-dihalobenzene family of systems. Excess enthalpy difference, H E (X = 0.5), as a function of the coefficient of crystalline isomorphism, εm . ◯: data from Table 5.1; numbering in line with Table 5.1. ♦: improved value for system ClCl + BrI, taking into account the extension of the miscibility gap; see Sect. 5.3 (adapted from Ref. [3], © 1991, with permission from Elsevier)

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Table 5.2 Temperatures and enthalpies of transition of p-dichlorobenzene as obtained by adiabatic calorimetry [1, 9] Transition

T (K)

H (J mol−1 )

III(γ ) → II(α)

275.0 ± 0.2

1238 ± 7

II(α) → I(β)

306 ± 1

181 ± 6

I(β) → liquid

326.24 ± 0.03

17,907 ± 15

metastable melting point (α–liquid) calculated from the data presented in Table 5.2 is 326.05 K; the heat of melting is 18,027 J mol−1 [8]. Oonk et al. [8] in their analysis of the system ClCl + BrI, used the calculated values of 326.03 K and 18,088 J mol−1 , respectively (based on the data by Dworkin et al. [10]). In that analysis, the extension of the miscibility gap was also taken into account, as a result of which the improved value of −2900 J mol−1 (see Fig. 5.2) was obtained for H E (X = 0.5). In addition, the three-phase equilibrium solid I + solid II + liquid was estimated at T = 324 K, along with X(I) = 0.005; X(II) = 0.010; X(liq) = 0.049.

5.4 The System p-Dichlorobenzene + p-Dibromobenzene In 1938, Deffet [11] published a paper on the influence of high pressure on the melting curves of binary mixtures. The study included the system p-dichlorobenzene + pdibromobenzene, which was investigated at atmospheric pressure and at pressures of about 500 and 1000 atm. The solidus and liquidus curves in Deffet’s phase diagram, at atmospheric pressure, are in line with the results of the earlier investigations by Küster [12], Bruni and Gorni [13], Beck and Ebbinghaus [14], Beck [15], and Kruyt [16]. In Deffet’s phase diagram, the distance between solidus and liquidus at equimolar composition is 9 K. For the determination of the solidus points (which, unlike liquidus points, are not clearly discernible on cooling/heating curves), Deffet employed the visual heating method discussed by Beck and Ebbinghaus [14]. In 1948, Campbell and Prodan [17] published the description of ‘an apparatus for refined thermal analysis’, along with the results obtained for the ternary system p-dichlorobenzene + p-dibromobenzene + p-bromochlorobenzene. In Fig. 5.3a, it is shown how Campbell and Prodan derived liquidus and solidus temperatures from their carefully obtained cooling curves. As regards the binary subsystem p-dichlorobenzene + p-dibromobenzene, Campbell and Prodan’s liquidus data are in agreement with the results of the earlier investigations; their solidus data, however, are not (at the equimolar composition the distance between solidus and liquidus is less than 5 K). In spite of all this, Campbell and Prodan’s liquidus data are of unparalleled accuracy. Next, Campbell and Prodan’s liquidus data, along with the melting properties of the pure components, were used by van Genderen et al. to calculate the solidus of the system using liqfit. The calculated phase diagram is shown in Fig. 5.3b. In the phase

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Fig. 5.3 System {(1 − X) p-dichlorobenzene + X p-dibromobenzene}. a Typical cooling curve obtained by Campbell and Prodan [17], showing how liquidus (●) and solidus (◯) points were derived from it (X = 0.3003, T liq = 332.64 K). b Phase diagram calculated by liqfit using Campbell and Prodan’s liquidus data (●) [18] (a reproduced from Ref. [17], © 1948, with permission from American Chemical Society)

diagram obtained by liqfit, the distance between solidus and liquidus at equimolar composition is 8 K. Apart from the thermodynamic phase diagram analysis, van Genderen et al. [18] studied, by dsc, the melting behaviour of samples prepared by zone levelling, employing the instrument designed by Kolkert [19, 20]. They found that the zonelevelled material is changing from solid to liquid in a quasi-isothermal manner, just like the pure components of the system; see Fig. 5.4. For comparison, the melting curve labelled ‘c’ (Fig. 5.4) is representative of an everyday dsc recording of mixed crystalline material prepared by melting and quenching in the instrument. Most interestingly, the melting points of the levelled samples are situated on the equal-G curve of the calculated phase diagram; see Fig. 5.5. The equal-G curve (EGC) is the locus in the TX plane of the temperatures at which solid and liquid of the same composition have equal molar Gibbs energies (see Chap. 3). Surprisingly,

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Fig. 5.4 System {(1 − X) p-dichlorobenzene + X p-dibromobenzene}. Typical melting curves measured with dsc with a heating rate of 0.62 K min−1 and a sample mass of 1 mg. a pdichlorobenzene pure; b zone-levelled sample; c quenched sample; d p-dibromobenzene pure (reproduced from Ref. [18], © 1977, with permission from De Gruyter)

Fig. 5.5 System {(1 − X) p-dichlorobenzene + X p-dibromobenzene}. The ‘melting points’ of zone-levelled mixed crystals () registered by dsc, are situated on the equal-G curve (—). Dashed lines represent the solidus and liquidus calculated using liqfit (adapted from Ref. [18], © 1977, with permission from De Gruyter)

the EGC is more than an auxiliary curve in theoretical work — in that its course, like the course of the liquidus e.g., can be revealed by experimental methods. It may be clear that the experimental determination of the equal-G curve is possible only if a number of severe conditions are simultaneously satisfied. These are, for the sample, compactness and a high degree of homogeneity, and for the experiment, the use of a high heating rate. When studied by adiabatic calorimetry, mixed crystalline material prepared by zone levelling neatly starts to produce liquid at the temperature of the solidus, and

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the material is fully liquid at the temperature of the liquidus. A beautiful example is shown in Fig. 5.6. The experimental data in Fig. 5.6 (open circles) also show that quenching a liquid mixture does not yield a mixed crystalline material that is suitable for determination of reliable solidus temperatures. As can be seen in Fig. 5.7, the same holds for mixed crystalline material that is prepared by slow cooling (3 K per day) after annealing for five days at a temperature just below the liquidus temperature. To circumvent the laborious task of zone levelling, van der Linde prepared mixed crystalline material by rapid evaporation of a volatile solvent, and, subsequently, studied the melting behaviour of the material by adiabatic calorimetry [1, 23]. A typical example from van der Linde’s work is the heat-capacity melting curve (C p –T graph) as shown in Fig. 5.8. Van der Linde developed a method, referred to as ultracal, with the help of which one can assess the heat-capacity melting curve of the kind shown in Fig. 5.8, and at the same time, establish the solidus and liquidus temperatures of the phase diagram. The method was tested on the system p-dichlorobenzene + p-dibromobenzene, and subsequently applied to the system 1,3,5-trichlorobenzene + 1,3,5-tribromobenzene; see Sect. 5.5. For the system p-dichlorobenzene + p-dibromobenzene, mixed crystalline material prepared by rapid evaporation of a solvent was also used by Haget et al. in a dsc study [5]. The derived solidus and liquidus data were found to be, within experimental uncertainty, in full agreement with the calculated diagram as shown in Figs. 5.3b and 5.5 (see also van der Linde et al. [23]). Overviewing the history of the system p-dichlorobenzene + p-dibromobenzene, it is surprising, and at the same time reassuring to know, that the dsc methodology, which is fast and uncomplicated, is capable of revealing not only the system’s liquidus curve but also its solidus curve.

Fig. 5.6 System {(1 − X) trans-azobenzene + X trans-stilbene}. Heat-capacity melting curves (C p – T graph) of a zone-levelled mixture (●) and a quenched mixture (◯) having the same composition of X = 0.503 [21, 22] (reproduced from Ref. [22], © 1985, with permission from Elsevier)

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Fig. 5.7 System {(1 − X) p-dichlorobenzene + X p-dibromobenzene}. Heat-capacity melting curves of a mixture prepared by slow cooling (3 K per day) of the liquid; X = 0.519. ●: Experimental data; – – –: calculated ideal melting curve [1]

Fig. 5.8 System {(1 − X) p-dichlorobenzene + X p-dibromobenzene}. Heat-capacity melting curves of a mixture of prepared by rapid evaporation; X = 0.515 [1, 23]. ●: Experimental data; —: outcome of the simulation by ultracal; – – –: calculated ideal melting curve (reproduced from Ref. [23], © 2002, with permission from Elsevier)

5.5 Tri- and Tetrasubstituted Halobenzenes The binary systems 1,3,5-trichlorobenzene + 1,3,5-tribromobenzene and 1,2,4,5tetrachlorobenzene + 1,2,4,5-tetrabromobenzene can be regarded as model systems in the study of the mixing properties and phase diagrams of 1,3,5-tri- and 1,2,4,5tetrasubstituted halobenzenes.

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The substances 1,3,5-trichlorobenzene and 1,3,5-tribromobenzene are isomorphous: orthorhombic, space group P21 P21 P21 , with four molecules per unit cell. As a part of his thesis work, van der Linde [1, 23] showed on the basis of (i) direct measurement of the excess Gibbs energy of the mixed liquid phase (GE,liq (X = 0.5) = 94 J mol−1 ) and (ii) an ultracal analysis of four heat capacity melting curves that the system simply has a phase diagram of type [0], just like the systems 1, 2, 5, 8, and 10 of the para-dihalobenzenes (see Table 5.1), and the systems trans-azobenzene + trans-stilbene and thiophene + benzene (see Fig. 5.19). The solid-liquid phase behaviour of the tetrasubstituted system, however, is much more complex than the behaviour of the trisubstituted system. In the case of the system 1,2,4,5-tetrachlorobenzene + 1,2,4,5-tetrabromobenzene, three solid forms make their appearance: (i) (ii)

the form α, which is the low-temperature form of the chloro compound; the form β, which is the high-temperature form of the chloro compound, and at the same time, the low-temperature form of the bromo compound; and

Fig. 5.9 Phase diagram of {(1 − X) 1,2,4,5-tetrachlorobenzene + X 1,2,4,5-tetrabromobenzene}. Experimental data and calculated phase diagram assuming crossed isodimorphism between the two components. ●: dta; : xrd; ♦: Guinier-Lenné, Guinier-Simon [24]; —: calculated phase diagram [25] (reproduced from Ref. [25], © 1991, with permission from Elsevier)

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the form γ , which is the high-temperature form of the bromo compound.

At room temperature, β is the stable form of the chloro as well as the bromo compound. Experimental data, X-ray diffraction and thermal analysis, are from Mondieig et al. [24]. An unorthodox thermodynamic assessment has been presented by van Genderen et al. [25]. It was assumed that the γ form of pure 1,2,4,5-tetrachlorobenzene is identical with the β form, thus circumventing the problem of determining the metastable γ –β transition temperature and transition entropy. Furthermore, the low-temperature α form was neglected. The outcome of the assessment is a model that yields a phase diagram that is in adequate agreement with the experimental data and accounts for the various phase fields. The result, along with the experimental data, is shown in Fig. 5.9.

5.6 The Family of the 2-Substituted Naphthalenes; the Existence of Two Subfamilies Naphthalene and its 2-R-substituents, with R = F, Cl, Br, SH, CH3 , and OH, all crystallize, at temperatures near their melting point, in the space group P21 /a. All have two molecules per unit cell, and for the derivatives this involves molecular disorder, in that, on a given lattice site, there are four possible orientations of the molecules [26]; see Fig. 5.10. In spite of the fact that the seven substances have similar cell dimensions, there are two structural subfamilies, denoted by f1 and f2 , with slightly different types of molecular arrangement. This circumstance was described by Kitaigorodsky [27] as ‘all the naphthalene β-derivatives are supposed to possess similar energy surfaces which have two minima approximately of the same depth. One of them leads to the fluoronaphthalene packing, the second to the naphthalene type packing’. Kitaigorodsky’s view is reflected in Fig. 5.11. A member of subfamily f1 , say 2-fluoronaphthalene, is not isomorphous with a member of subfamily f2 , say naphthalene. This implies that the two do not give rise to a continuous series of mixed crystals; see Fig. 5.12. The system naphthalene + 2-fluoronaphthalene is an example of crossed isodimorphism (see also Sect. 5.10).

Fig. 5.10 Four orientations of the molecules of 2-substituted naphthalenes

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Fig. 5.11 2-Substituted naphthalenes. Imaginary (Gibbs) energy function G versus packing parameter P for the P21 /a unit cell with two molecules. —: f1 subfamily with R = F, Cl, Br, SH and CH3 ; – – –: f2 subfamily with R = H (and OH). The stable form of naphthalene (R = H) is different from the stable form taken by the members of the f1 subfamily [28, 29] (reproduced from Ref. [29], © 1991, with permission from EDP Sciences)

5.7 The System Naphthalene + 2-Naphthol At the time of writing the paper on 2-substituted naphthalenes and the existence of two subfamilies (1991; Ref. [29]), there was some confusion about the status of the substance 2-naphthol (R = OH): f1 or f2 ? In a sense, the confusion is related to the fact that 2-naphthol’s P21 /a form with two molecules per unit cell (α) has a stable existence of just 0.6 K; from T = 392.6 K to the melting point T = 393.2 K. A key system, with 2-naphthol as one of the components, is naphthalene + 2-naphthol. The literature about this system has been reviewed by Oonk and Tamarit [31]; at this place, a short summary is given. The experimental determination of the phase diagram goes back to 1895 and 1909 when Küster [32] determined liquidus temperatures and Rudolfi [33] determined both liquidus and solidus temperatures. The diagram published by Rudolfi has the cigartype of solid–liquid loop, type [0], characteristic of complete subsolidus miscibility. The same type of diagram was published by Vetter et al. in 1963 [34]. Vetter’s work is unique in that, by means of a technique related to zone melting, the true equilibrium compositions of the solid and liquid phases in equilibrium had been determined. The system was used as an example in a paper on the derivation of distribution coefficients from phase diagrams by Oonk and Pleijsier [35] who demonstrated the agreement between the calculated phase diagram and Vetter’s experimental data. Kolkert [19, 20] used the system to demonstrate the performance of his zone-levelling instrument for the growth of homogeneous mixed crystals (see also Sect. 5.4). After Baumgarth et al. [36], in 1969, had published their phase diagram, including the solid–solid two-phase region, a number of conflicting diagrams were published [37–40] until in 1998 Michaud et al. [41], by careful experimentation, confirmed the basic structure of the diagram by Baumgarth et al. [36]; see Fig. 5.13. The substance 2-naphthol, as a result, unmistakably is a member of the subgroup f2 .

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Fig. 5.12 Phase diagram of {(1 − X) naphthalene + X 2-fluoronaphthalene}. Experimental points and calculated phase diagram assuming crossed isodimorphism. ●: dta/dsc; : xrd; ♦: GuinierLenné [40]; —: calculated phase diagram; – – –: calculated metastable parts of the solid–liquid equilibrium curves [30] (reproduced from Ref. [30], © 1989, with permission from Elsevier)

Another system, found by Rudolfi [33] to show complete subsolidus miscibility — phase diagram with a minimum, type [–] — is the combination of naphthalene and 2-aminonaphthalene. Accordingly, 2-aminonaphthalene would be the third member of the subgroup f2 .

5.8 The Halo + Halo Systems The 2-halonaphthalenes (halo = F, Cl, Br) belong to the subfamily f1 . The same holds true for 2-thionaphthalene, whose properties are such that it can be taken here as a member of the group of the 2-halonaphthalenes. The four substances comprise six binary combinations, all of them showing complete subsolidus miscibility [40].

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Fig. 5.13 Phase diagram of the system {(1 − X) naphthalene + X 2-naphthol}. ●: dsc on heating; : rt-xrd; ˛: Guinier-Simon heating analysis; ◯: stable melting point of naphthalene (α–liquid); ♦: γ –α transition temperature of 2-naphthol; —: calculated phase diagram (reproduced from Ref. [41], © 1998, with permission from EDP Sciences)

Quite like the para-dihalobenzene systems (see Calvet et al. [3] and Fig. 5.2), the binary 2-halonaphthalene systems have mixing/excess properties whose magnitude increases with an increasing geometric mismatch (decreasing value of εm ); see Fig. 5.14.

5.9 The Methyl + Halo Systems The three methyl + halo 2-substituted naphthalene systems all have complete subsolidus miscibility; the phase diagrams for halo = Cl, Br have a maximum, type [+], which is quite exceptional, whereas methyl + F has a phase diagram of type [0]. The system methyl + SH also has a type [0] phase diagram [40].

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Fig. 5.14 Family of the 2-substituted naphthalenes. Equimolar excess Gibbs energy difference, GE (X = 0.5), as a function of the coefficient of crystalline isomorphism, εm (after Oonk et al. [29]). For numerical values of εm , see Haget et al. [26] (reproduced from Ref. [29], © 1991, with permission from EDP Sciences)

This time the components of the system not only differ as regards the size of their molecules but also as regards their chemical nature. As a result, the thermodynamic excess properties of the system have or may have, apart from a ‘geometric contribution’, also a ‘chemical contribution’. In the case of the methyl + halo systems there is, indeed, a chemical contribution, as can be inferred from Fig. 5.14. In terms of GE , the chemical contribution at equimolar composition has a value of about +0.5 kJ mol−1 (in Fig. 5.14 given by the intersection with the vertical axis, where the contribution of geometric mismatch is zero; εm = 1). Ignoring any excess behaviour of the liquid mixtures, the positive GE represents a negative excess Gibbs energy of the mixed crystalline state, read, an extra attraction between halo and methyl (geometric mismatch, on the other hand, corresponds to an extra effect of a repulsive nature). Note that in the case of the methyl + fluoro system the geometric and the chemical contribution virtually compensate one another. A detailed analysis of the methyl + halo systems has been presented by Calvet et al. [42], whose study includes a computer search on the statistics of intermolecular contacts in pure substances containing methyl and halogen — to find out in which way the extra attraction is reflected by radial frequency distributions f (R): f (R) =

N (R) (4 · π · R 2 · R) · NT

(5.6)

in which R represents the distance between the atoms and N the number of contacts. The property f (R) ‘corrects’, for a given contact such as F–CH3 , N(R) — in fact the number of contacts between (R − 0.5·R) and (R + 0.5·R) — for the geometrical factor 4π ·R2 ·R and for the total number of contacts, N T , for which R < 10 Å. The class width of the distribution, R, had been set at 0.2 or 0.4 Å. It

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Fig. 5.15 Radial frequency distributions f (R) in 2-substituted naphthalenes. Left graph: between F and CH3 (●), F and F (), CH3 and CH3 (◯); right graph: between F and Br (●), F and F (), Br and Br (◯) (reproduced from Ref. [42], © 1999, with permission from AIP Publishing)

was found, among other things, that the peak of the distribution of F–Br contacts is more or less the average of F–F and Br–Br; and that, for F–CH3 , on the other hand, the peak for F–CH3 is considerably higher than each of the two peaks F–F and CH3 –CH3 ; see Fig 5.15. The attractive nature of the combination substituted methyl and halogen, expressed by a phase diagram with a maximum, is also found in the system 1-methyl-2,3,5tribromobenzene + 1-methyl-2,4,6-tribromobenzene; see Fig. 5.16, system 5. The experimental data, on which Fig. 5.16 is based, were collected by Jaeger in 1904 [43]; a thermodynamic analysis was made by Oonk [44]. 1-Methyl-2,3,5-tribromobenzene and 1-methyl-2,4,6-tribromobenzene (components II and V in F. M. Jaeger’s classification of tribromotoluene binary systems) show the same crystal system with the same characteristics. NB. A theoretical underpinning of any extra attractive effect between methyl and halogen is lacking. Computations using modern methods of theoretical chemistry — on the combination –CH3 and –Cl — failed to reveal the ‘desired’ effect [45].

5.10 Crossed Isodimorphism In terms of chemically coherent groups, naphthalene is not a substance like the 2-halonaphthalenes. Neither, the 2-halonaphthalenes, which all belong to the f1 group, are isomorphous with naphthalene, which is a member of the f2 group. As a consequence, complete subsolidus miscibility for a naphthalene + 2-halonaphthalene system is out of the question. The stable solid–liquid phase diagram will show a threephase equilibrium situation where three two-phase branches come together, such as is seen in Fig. 5.12, for the combination of naphthalene and 2-fluoronaphthalene, and with the phase symbols α for f2 and β for f1 . The system is a case of crossed isodimorphism (see also Chap. 4): the stable phase diagram is the result of two,

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Fig. 5.16 F. M. Jaeger’s classification of the 15 tribromotoluene binary systems [43]. ●: CH3 ; ◯: Br (reproduced from Ref. [44], © 1992, with permission from Elsevier)

each other crossing solid–liquid loops. Each of these two loops emanates from a stable melting point and ends in a metastable melting point. The assessment of the metastable melting points is an essential part of the thermodynamic analysis of such a system. The thermodynamic analysis, presented by van Duijneveldt et al. [30], comprises the five systems naphthalene + 2-R-naphthalene with R = F, Cl, Br, SH, CH3 . The outcome of van Duijneveldt’s analysis, as regards the equimolar GE involving the f1 mixed solid, is included in Fig. 5.14, from which it follows that the naphthalene + 2-halonaphthalene systems line up with the 2-methylnaphthalene + 2-halonaphthalene systems. The outcome for the naphthalene + 2methylnaphthalene system is more or less in line with the halo + halo systems, of which the mixing effects come from geometric mismatch. Besides, instructive examples of thermodynamic phase diagram analysis in the case of crossed isodimorphism are the ones applied to the system naphthalene + 2-fluoronaphthalene (see Fig. 5.12) and the system thianaphthene + naphthalene (see Fig. 5.20).

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5.11 The System trans-Azobenzene + trans-Stilbene The text presented here for the system trans-azobenzene + trans-stilbene is inspired by the thesis work of Bouwstra [21, 22, 46–50]. Special attention is given to the outcome of lattice energy calculations, because this part of Bouwstra’s work can only be found in her thesis.2 The substances trans-azobenzene and trans-stilbene, see Fig. 5.17, are isomorphous and give mixed crystals in all proportions. Their space group is P21 /c with four molecules per unit cell on two independent sites, referred to as site A and site B. On site B, there is orientational disorder, such that there are two distinct orientations of the molecules; on site A, this kind of disorder is absent; see Fig. 5.18, for the case of trans-stilbene. In the mixed crystals of the two substances, there is a complex interplay of substitutional disorder, due to the interchange of trans-azobenzene and trans-stilbene, and the orientational disorder on site B. To give an idea of how the sites are occupied, as derived from X-ray structure refinement, the case is taken of a part of a mixed crystal, which has 740 molecules of trans-azobenzene and 260 molecules of trans-stilbene (X = 0.26). On site B, there are 318 azobenzene molecules and 182 stilbene molecules. Out of the 182 stilbene molecules, 160 have orientation B1 and 22 orientation B2 (and out of the 318 azobenzene molecules, 264 have the orientation B3 and 54 the orientation B4). On site A, there are 422 azobenzene and 78 stilbene molecules. These numbers show that the fraction α of trans-stilbene molecules that are on site A is equal to 78/(78 + 182) = 0.30. For mixed crystals of composition X = 0.56, the fraction α is found as α = 0.39. The important thing to note is that the values of the parameter α, found by X-ray structure refinement, are beautifully reproduced by the computations of the lattice free energy (≈ Gibbs energy). For X = 0.26, the lowest free energy is obtained for value α = 0.33; and for X = 0.56, the lowest free energy was found for α = 0.40. In Table 5.3, the enthalpies of sublimation of trans-azobenzene and trans-stilbene and two mixed crystals are given. The values under the heading ‘experimental’ follow from vapour pressures as a function of temperature [48]. The enthalpy of sublimation values in the third column were obtained by calculation of the lattice energies, using the formalism and the data set constructed by Govers [53, 54]. The enthalpies of sublimation allow one to calculate the excess enthalpies of the mixed crystals; their values are given in the fourth column. Fig. 5.17 Chemical structures of trans-azobenzene (left) and trans-stilbene (right)

2

Although not discussed in this chapter, lattice energy calculations for para-dihalobenzenes have been presented by Oonk et al. [51] and van Eijck [52].

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Fig. 5.18 Trans-stilbene. Top: a composite view of the disordered molecules at site B with 50% probability plots of thermal ellipsoids of the molecule with main-site occupancy in the disordered position (B1). Bottom: view of the molecule at site A with 50% probability plots of thermal ellipsoids [21, 47] (reproduced from Ref. [47], © 1984, with permission of the International Union of Crystallography) g

Table 5.3 Enthalpies of sublimation, s H(298.15 K), of trans-azobenzene and trans-stilbene and two mixed crystals: experimental data by Bouwstra et al. [48]; calculated data by Govers [53, 54], along with calculated excess enthalpy of the mixed solid phase (H E,sol ) Experimental

Lattice energy calculation

s Hexp (kJ mol−1 )

s Hcalc (kJ mol−1 )

H E,sol (kJ mol−1 )

94.65

g

Trans-azobenzene

g

96.1

0

X = 0.26

95.8

0.85

X = 0.56

96.5

0.78

98.2

0

Trans-stilbene

99.90

The two numerical values for the excess enthalpy in the fourth column of Table 5.3 allow one to calculate the two ‘unknowns’ A and B in Eq. (5.3). The calculated values are A = 3400 J mol−1 and B = 0.6. In Fig. 5.19, upper graph, the solid–liquid phase diagram of the system transazobenzene + trans-stilbene is shown. Bouwstra et al. [22] used adiabatic calorimetry to obtain the experimental solidus points (open circles) and liquidus points (filled

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Fig. 5.19 Phase diagrams of the systems {(1 − X) trans-azobenzene + X trans-stilbene} (upper graph) and {(1 − X) thiophene + X benzene} (lower graph). ◯: Experimental solidus points; ●: experimental liquidus points; —: calculated phase diagrams using liqfit. All experimental data were obtained by adiabatic calorimetry; samples of trans-azobenzene + trans-stilbene were prepared by zone levelling [22]; samples of thiophene + benzene were prepared by annealing [2, 55]

circles); see also Fig. 5.6. The liquidus points were subjected to a liqfit analysis, using the thermodynamic melting properties as obtained by Bouwstra et al. [22] and assuming ideal liquid mixtures. The outcome of the analysis is twofold: first the optimized phase diagram, and second the excess Gibbs energy of the solid state. The optimized phase diagram is represented by the solidus and liquidus curves in Fig. 5.19 (upper graph). As can be seen, the calculated solidus is in excellent agreement with the experimental solidus points, thus, giving evidence for the high degree of homogeneity of mixed crystalline material that can be obtained by zone levelling. The excess Gibbs energy, valid for the mean temperature of the diagram which is T m = 370 K, is given by the expression G E,sol (T = Tm , X ) = 530 · X · (1 − X ) · [1 + 0.7 · (1 − 2 · X )] J mol−1 .

(5.7)

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In terms of Eqs. (5.2) and (5.3), the compensation temperature Θ follows from the equality: 530 = 3400·(1 − 370 K/Θ); the result is Θ = 438 K.

5.12 The System Thiophene + Benzene Another system, whose change, from mixed crystalline solid to liquid, has been studied by adiabatic calorimetry, is the combination of thiophene and benzene: C4 H4 S + C6 H6 . Details of the investigation can be found in the papers by Okamoto et al. [55, 56] and Yamamuro et al. [2]. It is instructive to make a comparison between the systems thiophene + benzene (T + B) and trans-azobenzene + trans-stilbene (A + S), and so from the point of view of phenomenology and methodology. The two systems have in common that they form mixed crystals in all proportions, and that their change from solid to liquid goes together with a cigar type of phase diagram (type [0]; see Fig. 5.19). In the (A + S) case, the formation of mixed crystals is facilitated by the high degree of molecular homeomorphism. In the case of (T + B), mixed crystals are easily formed due to the high degree of rotational disorder; caused by the pseudo-fivefold symmetry and the sixfold symmetry of the molecules of T and B, respectively. The high degree of rotational disorder, which is most pronounced for thiophene, is reflected by the heats of melting: 5.0 kJ mol−1 for thiophene [57] and 9.9 kJ mol−1 for benzene [58], whereas the heats of melting of trans-azobenzene and trans-stilbene are 22.53 and 27.69 kJ mol−1 , respectively [22]. Another aspect of the rotational disorder is that homogeneous mixed crystals of (T + B) are easily formed. The mixed crystalline materials used in determining the solidus and liquidus temperatures as shown in Fig. 5.19 (lower graph) were obtained by annealing at a temperature just below the solidus point for 2–7 days. As can be seen, this relatively simple preparation method yields reasonable (although not superb) solidus temperatures for this system. In the case of (A + S), homogeneous mixed crystals had to be prepared by the laborious technique of zone levelling. As already mentioned, and shown in Fig. 5.19 (upper graph), this technique yields the best possible result.

5.13 The System Thianaphthene + Naphthalene The binary system thianaphthene + naphthalene is another example of a system showing crossed isodimorphism. Although the molecules of thianaphthene (benzo[b]thiophene, C8 H6 S) and naphthalene are similar in shape and form, the substances crystallize in different crystal systems: thianaphthene is orthorhombic (β); naphthalene is monoclinic (α). This gives rise to an interrupted series of mixed

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crystals. The solid–liquid phase diagram can be considered as the result of two interfering solid–liquid loops. The phase diagram presented in Fig. 5.20 is the result of an extensive thermodynamic analysis [59, 60] of the available experimental data by Klipp et al. [59], Mastrangelo and Dornte [61], and Kravchenko and Pastukhova [62]. The metastable melting points (α → liquid of thianaphthene, and β → liquid of naphthalene) and the corresponding transition enthalpies were derived by extrapolation of the available calorimetric data. For the α form of the system, the compensation temperature Θ in Eq. (5.2) was calculated as Θ = 420 K.

Fig. 5.20 Phase diagram of the system {(1 – X) thianaphthene + X naphthalene}. ●, ◯: Experimental liquidus and solidus data by Mastrangelo and Dornte [61]; , : experimental liquidus and solidus data by Kravchenko and Pastukhova [62]; —: calculated phase diagram; – – –: calculated metastable extensions (reproduced from Ref. [60], © 1995, with permission from Springer)

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5.14 Ternary Systems, as an Extra As already mentioned in Sect. 5.4, the change from liquid to solid in the ternary system p-dichlorobenzene + p-dibromobenzene + p-bromochlorobenzene was studied by Campbell and Prodan [17]. In 1983, Moerkens et al. [63] published the results of an investigation in which the ternary liquidus and solidus surfaces had been calculated, starting from the thermodynamic melting properties of the pure components and the thermodynamic mixing properties of the three binary subsystems; see Fig. 5.21 for a typical result [7, 63]. It was found that the mean absolute difference between calculated liquidus temperatures and corresponding experimental liquidus temperatures, measured by Campbell and Prodan and taken for 39 points inside the composition triangle, was as little as 0.125 K! Another investigation on a ternary system of isomorphic aromatic compounds was carried out by Teunissen, who studied the combination of p-dibromobenzene,

Fig. 5.21 Ternary solid–liquid phase diagram of the system p-dichlorobenzene + pdibromobenzene + p-bromochlorobenzene at 343 K (adapted from Ref. [7], © 2008, with permission from Springer)

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p-chloroiodobenzene, and p-bromoiodobenzene; see Oonk et al. [64]. Also worth mentioning is the study by Stolk et al. [65] on the ternary n-alkane system n-pentadecane + n-hexadecane + n-heptadecane.

References 1. Van der Linde PR (1992) Molecular mixed crystals from a thermodynamic point of view. Thesis, Utrecht University 2. Yamamuro O, Suga H, Usui Y, Kimura T, Takagi S (1997) Thermodynamic functions of the thiophene-benzene system in their liquid and solid solutions. J Phys Chem B 101:6541–6548 3. Calvet MT, Cuevas-Diarte MA, Haget Y, van der Linde PR, Oonk HAJ (1991) Binary pdihalobenzene systems – correlation of thermochemical and phase-diagram data. Calphad 15:225–234 4. Bouwstra JA, Brouwer N, van Genderen ACG, Oonk HAJ (1980) A thermodynamic method for the derivation of the solidus and liquidus curves from a set of experimental liquidus points. Thermochim Acta 38:97–107 5. Haget Y, Housty JR, Maiga A, Bonpunt L, Chanh NB, Cuevas M, Estop E (1984) Alliages moléculaires à forte chaleur latente: système paradichlorobenzène (pDCB)paradibromobenzène (pDBB). J Chim Phys 81:197–206 6. Cuevas-Diarte MA, Calvet T, Labrador M, Estop E, Oonk HAJ, Bonpunt L, Haget Y (1991) Coefficients of molecular homeomorphism and crystalline isomorphism in the series of paradisubstituted benzene derivatives. J Chim Phys 88:509–514 7. Oonk HAJ, Calvet MT (2008) Equilibrium between phases of matter, phenomenology and thermodynamics. Springer, Dordrecht 8. Oonk HAJ, Calvet T, Cuevas-Diarte MA, Tauler E, Labrador M, Haget Y (1995) Molecular alloys in the series of para-disubstituted benzene derivatives. Part 8. The para-dichlorobenzene + para-bromoiodobenzene system. Thermochim Acta 250:13–18 9. Van der Linde PR, van Miltenburg JC, van den Berg GJK, Oonk HAJ (2005) Lowtemperature heat capacities and derived thermodynamic functions of 1,4-dichlorobenzene, 1,4-dibromobenzene, 1,3,5-trichlorobenzene, and 1,3,5-tribromobenzene. J Chem Eng Data 50:164–172 10. Dworkin A, Figuière P, Ghelfenstein M, Szwarc H (1976) Heat capacities, enthalpies of transition, and thermodynamic properties of the three solid phases of p-dichlorobenzene from 20 to 330 K. J Chem Thermodyn 8:835–844 11. Deffet L (1938) Recherches piézométriques. IV. Influences des hautes pressions sur la courbe de fusion des mélanges binaries. Bull Soc Chim Belg 47:461–505 12. Küster FW (1895) Beitrage zur Molekulargewichtsbestimmung an „festen Lösungen”. 3. Mitteilung: Die isomorphen Mischungen von p-Dichlorbenzol mit p-Dibrombenzol und von s-Trichlorphenol mit s-Tribromphenol. Z Phys Chem 50U:65–80 13. Bruni G, Gorni F (1900) Sui fenomeni di equilibrio fisico nelle miscele de sostanze isomorfe. Gazz Chim Ital 30:127–140 14. Beck K, Ebbinghaus K (1906) Ueber Umwandlungspunkte und eine Methode zur Beobachtung derselben. Ber Dtsch Chem Ges 39:3870–3877 15. Beck K (1907) Beiträge zur relativen innern Reibung. Z Phys Chem 58:425–441 16. Kruyt HR (1912) Das Gleichgewicht Fest-Flüssig-Gas in binären Mischkristallsystemen. Z Phys Chem 79U:657–676 17. Campbell AN, Prodan LA (1948) An apparatus for refined thermal analysis exemplified by a study of the system p-dichlorobenzene–p-dibromobenzene–p-chlorobromobenzene. J Am Chem Soc 70:553–561

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18. Van Genderen ACG, de Kruif CG, Oonk HAJ (1977) Properties of mixed crystalline organic material prepared by zone leveling. I. Experimental determination of the EGC for the system p-dichlorobenzene + p-dibromobenzene. Z Phys Chem (N F) 107:167–173 19. Kolkert WJ (1974) Growth of homogeneous organic mixed crystals by repeated pass zone levelling. Thesis, Utrecht University 20. Kolkert WJ (1975) Growth of uniform solid solutions of naphthalene and 2-naphthol by repeated pass zone-leveling. J Cryst Growth 30:213–219 21. Bouwstra JA (1985) Thermodynamic and structural investigations of binary systems. Thesis, Utrecht University 22. Bouwstra JA, de Leeuw VV, van Miltenburg JC (1985) Properties of mixed-crystalline organic material prepared by zone levelling. IV. Melting properties and excess enthalpies of (transazobenzene + trans-stilbene). J Chem Thermodyn 17:685–695 23. Van der Linde PR, Bolech M, den Besten R, Verdonk ML, van Miltenburg JC, Oonk HAJ (2002) Melting behaviour of molecular mixed crystalline materials: measurement with adiabatic calorimetry and modelling using ultracal. J Chem Thermodyn 34:613–629 24. Mondieig D, Housty JR, Haget Y, Cuevas-Diarte MA, Oonk HAJ (1991) The system 1,2,4,5tetrachlorobenzene + 1,2,4,5-tetrabromobenzene. Part 1. Experimental phase diagram (93–460 K). Thermochim Acta 177:169–186 25. Van Genderen MJ, Mondieig D, Haget Y, Cuevas-Diarte MA, Oonk HAJ (1992) A special model of isodimorphism used for the thermodynamic description of the system 1,2,4,5tetrachlorobenzene + 1,2,4,5-tetrabromobenzene. Calphad 16:1–12 26. Haget Y, Bonpunt L, Michaud F, Negrier P, Cuevas-Diarte MA, Oonk HAJ (1990) Coefficients of molecular homeomorphism and crystalline isomorphism in the series of 2-R-naphthalene (R = H, F, Cl, CH3 , SH, Br). J Appl Crystallogr 23:492–496 27. Kitaigorodsky AI (1984) Mixed crystals. Springer-Verlag, Berlin-Heidelberg 28. Haget Y, Chanh NB, Meresse A, Bonpunt L, Michaud F, Negrier P, Cuevas-Diarte MA, Oonk HAJ (1999) Isomorphism and mixed crystals in 2-R-naphthalenes: evidence of structural subfamilies and prediction of metastable forms. J Appl Crystallogr 32:481–488 29. Oonk HAJ, van der Linde PR, Haget Y, Bonpunt L, Chanh NB, Cuevas-Diarte MA (1991) Molecular mixed crystals from a macroscopic thermodynamic point of view. J Chim Phys 88:329–341 30. Van Duijneveldt JS, Chanh NB, Oonk HAJ (1989) Binary mixtures of naphthalene and five of its 2-derivatives. Thermodynamic analysis of solid-liquid phase diagrams. Calphad 13:83–88 31. Oonk HAJ, Tamarit JLI (2005) Condensed phases of organic materials: solid-liquid and solidsolid equilibrium. In: Weir RD, De Loos ThW (eds) Measurement of the thermodynamic properties of multiple phases. Experimental thermodynamics, vol VII. Elsevier, Amsterdam 32. Küster FW (1895) Beitrage zur Molekulargewichtsbestimmung an „festen Lösungen”. 2. Mitteilung: Das Gleichgewicht zwischen Wasser, Naphtalin und β-Naphtol. Z Phys Chem 17U:357–373 33. Rudolfi E (1909) Über die Dielektrizitätskonstanten von Gemischen fester Körper. Z Phys Chem 66U:705–732 34. Vetter H, Rössler S, Schildknecht H (1963) Theorie des Zonenschmelzens. Ermittlung der Solidus- und Liquiduskurve von Schmelzdiagrammen organischer Zweistoffsysteme durch Mikro-Zonenschmelzen und Normales Erstarren. In: Schildknecht H (ed) Symposium über Zonenschmelzen und Kolonnenkristallisieren. Kernforschungsanstalt, Karlsruhe, pp 57–68 35. Oonk HAJ, Pleijsier HL (1971) The equilibrium distribution coefficient and its derivation from TX solid-liquid equilibrium diagrams. Sep Sci 6:685–697 36. Baumgarth F, Chanh NB, Gay R, Lascombe J, Le Calvé N (1969) Contribution à l’étude de la phase cristalline haute température du β-naphtol et des solutions solides β-naphtol/naphthalène. J Chim Phys 66:862–868 37. Robinson PM, Rossell HJ, Scott HG, Legge C (1970) Binary phase diagrams of some molecular compounds—II. Mol Cryst Liq Cryst 11:105–117 38. Robinson PM, Scott HG (1972) The naphthalene–β-naphthol system. Mol Cryst Liq Cryst 18:153–156

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39. Haget Y, Chezeau N, Meresse A, Housty J, Chanh NB (1979) Syncristallisation organique avec peritexie. Cas du système binaire fluoro-2 naphtalene – naphtol-2. Mol Cryst Liq Cryst 55:109–118 40. Meresse A (1981) Implications du polymorphisme dans la formation des alliages moléculaires. Syncristallisation en série naphtalénique β substituée. Thesis, Université de Bordeaux I 41. Michaud F, Negrier P, Haget Y, Alcobé X, Cuevas-Diarte MA, Oonk HAJ (1998) Is 2-naphtol isomorphous with 2-fluoronaphthalene or with naphthalene? J Chim Phys 95:2199–2213 42. Calvet T, Cuevas-Diarte MA, Haget Y, Mondieig D, Kok IC, Verdonk ML, van Miltenburg JC, Oonk HAJ (1999) Isomorphism of 2-methylnaphthalene and 2-halonaphthalenes as a revealer of a special interaction between methyl and halogen. J Chem Phys 110:4841–4846 43. Jaeger FM (1904) Ueber molekulare und krystallographische Symmetrie von stellungsisomeren Benzolabkömmlingen. Z Kristallog 38:555–601 44. Oonk HAJ (1992) Solid-liquid equilibria in the fifteen binary systems shared by the six isomeric tribromotoluenes. Homage to F.M. Jaeger. Calphad 16:37–46 45. Van Duijneveldt FB (2002) Personal communication 46. Bouwstra JA, Schouten A, Kroon J (1983) Structural studies of the system transazobenzene/trans-stilbene. I. A reinvestigation of the disorder in the crystal structure of trans-azobenzene, C12 H10 N2 . Acta Crystallogr C 39:1121–1123 47. Bouwstra JA, Schouten A, Kroon J (1984) Structural studies of the system transazobenzene/trans-stilbene. II. A reinvestigation of the disorder in the crystal structure of trans-stilbene, C14 H12 . Acta Crystallogr C 40:428–431 48. Bouwstra JA, Oonk HAJ, Blok JG, de Kruif CG (1984) Properties of mixed crystalline organic material prepared by zone levelling III. Vapour pressures of (trans-azobenzene + trans-stilbene). J Chem Thermodyn 16:403–409 49. Van Miltenburg JC, Bouwstra JA (1984) Thermodynamic properties of trans-azobenzene and trans-stilbene. J Chem Thermodyn 16:61–65 50. Bouwstra JA, Schouten A, Kroon J, Helmboldt RB (1985) Structural studies of the system transazobenzene/trans-stilbene. III. The structure of three mixed crystals of trans-azobenzene/transstilbene; determinations by X-ray and neutron diffraction. Acta Crystallogr C 41:420–426 51. Oonk HAJ, van Genderen ACG, Blok JG, van der Linde PR (2000) Vapour pressures of crystalline and liquid 1,4-dibromo- and 1,4-dichlorobenzene; lattice energies of 1,4dihalobenzenes. Phys Chem Chem Phys 2:5614–5618 52. Van Eijck BP (2002) Crystal structure predictions for disordered halobenzenes. Phys Chem Chem Phys 4:4789–4794 53. Govers HAJ (1974) Calculations of lattice energies of unary and binary molecular crystals. Thesis, Utrecht University 54. Govers HAJ (1977) A mean-field approximation for multicomponent solid solutions of molecular crystals with orientational and substitutional disorder. J Chem Phys 67:4199–4205 55. Okamoto N, Oguni M, Suga H (1990) Transition phenomena in the thiophene crystal. Thermochim Acta 169:133–149 56. Okamoto N, Oguni M, Suga H (1989) Calorimetric study of thiophene-benzene solutions (I). J Phys Chem Solids 50:1285–1295 57. Figuière P, Szwarc H, Oguni M, Suga H (1985) Calorimetric study of thiophene from 13 to 300 K. Emergence of two glassy crystalline states. J Chem Thermodyn 17:949–966 58. Domalski ES, Hearing ED (1996) Heat capacities and entropies of organic compounds in the condensed phase. Volume III. J Phys Chem Ref Data 25:1–525 59. Klipp N, van der Linde PR, Oonk HAJ (1991) The system thianaphthene + naphthalene. Solid-liquid equilibrium. A case of crossed isodimorphism. Calphad 15:235–242 60. Oonk HAJ (1995) The syncrystallization of thianaphthene and naphthalene—an exercise in thermodynamic phase diagram analysis. In: Van der Eerden JP, Bruinsma OSL (eds) Science and technology of crystal growth. Kluwer Academic Publishers, pp 27–38 61. Mastrangelo SVR, Dornte RW (1957) The system naphthalene–thianaphthene. Anal Chem 29:794–797

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62. Kravchenko VM, Pastukhova IS (1958) The equilibrium of condensed phases in the naphthalene – thionaphtene system. Dokl Akad Nauk SSSR 119:285–287 63. Moerkens R, Bouwstra JA, Oonk HAJ (1983) The solid-liquid equilibrium in the system pdichlorobenzene + p-bromochlorobenzene + p-dibromobenzene. Thermodynamic assessment of binary data and calculation of ternary equilibrium. Calphad 7:219–269 64. Oonk HAJ, Calvet MT, Cuevas-Diarte MA, Haget Y, van Miltenburg JC, Teunissen EH (1989) The system p-dibromobenzene + p-chloroiodobenzene + p-bromoiodobenzene–a methodological study on thermal analysis and thermodynamic phase-diagram analysis. Thermochim Acta 146:297–309 65. Stolk R, Rajabalee F, Jacobs MHG, Espeau P, Mondieig D, Oonk HAJ, Haget Y (1997) The RI-liquid equilibrium in the ternary system n-pentadecane + n-hexadecane + n-heptadecane. Calculation of liquidus surface and thermal windows comparison with experimental data. Calphad 21:401–410

Chapter 6

Chains D. Mondieig, E. Moreno-Calvo, and M. À. Cuevas-Diarte

Abstract Starting from a simple aliphatic hydrocarbon chain, we have studied the effect on structural properties caused by the incorporation of an increasing number of hydrogen bonds. In reality, this means that we have studied the structural characteristics—including polymorphism—and the phase behavior of binary mixtures in the following groups of substances: the n-alkanes; the 1-alkanols; the α, ω-alkanediols; the alkanoic acids; and the alkanedioic acids. The results that have been obtained clearly show that two “parameters” have a crucial influence on the structural and thermodynamic properties. These are (i) the parity (odd versus even) of the carbon chain; and (ii) the aim at realizing as many hydrogen bonds as possible.

6.1 Introduction During the past few years, the families of long-chain aliphatic n-alkanes and nalkanols, and n-carboxylic acids have been studied in the REALM group. The aim is to have a comprehensive idea of the general laws that govern the thermodynamics and the structural properties of each family rather than those of the individual members and the stability of molecular alloys in each family. The logic sequence of chain aliphatic families allows us to rationalize the importance and the implications of the diversification of the molecular interactions and the effect on the crystal packing of the different families. The three families dealt with in this chapter are molecular compounds with straight hydrocarbon chains in which by substitution of one or both methyl end groups by D. Mondieig UMR 5798, LOMA, Université de Bordeaux, Talence, France e-mail: [email protected] E. Moreno-Calvo (B) · M. À. Cuevas-Diarte Grup de Cristal·lografia Aplicada, Universitat de Barcelona, Barcelona, Spain e-mail: [email protected] M. À. Cuevas-Diarte e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. À. Cuevas-Diarte and H. A. J. Oonk (eds.), Molecular Mixed Crystals, Physical Chemistry in Action, https://doi.org/10.1007/978-3-030-68727-4_6

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a OH, or COOH give rise to the formation of n-alkanols, and n-carboxylic acids families (Fig. 6.1). Generally, the saturated hydrocarbon part (–CH2–)n of longchain organic compounds assumes the all-trans zigzag conformation as the most stable conformation in the crystalline phase. However, some defects can exist in the chains of the n-alkane derivatives families that adopt four different types of molecular conformations (all-trans, terminal gauche, terminal double gauche, and kink defect) on increasing chain length or temperature [1]. Likewise in n-alkanes, for those families having the same terminal groups at both ends of the chain, the plane containing the chain is always a plane of symmetry to the molecule, but according to the parity of the number of carbon atoms, the symmetry of the molecule is different (Fig. 6.2). Therefore, the molecules with a number even of atoms of carbon in the chain have an inversion center (symmetry 2/m) while molecules with a number odd of atoms of carbon in the chain have a second plane of symmetry perpendicular to the chain (symmetry mm2).

Fig. 6.1 Scheme of the general molecular formula of long-chain aliphatic compounds

Fig. 6.2 Symmetry of the molecules of n-alkanes

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6.2 n-Alkanes 6.2.1 The Family Normal alkanes Cn H2n+2 (hereafter denoted by Cn) and their mixtures are the basic products of petroleum waxes and the knowledge of the structural and thermodynamic behavior of binary, ternary and multicomponent mixed crystals of n-alkanes are of great interest to petroleum refiners. Furthermore, from a fundamental point of view, normal alkanes can be used as a model for other compounds like as liquid crystals [2], membranes [3], polymers, and others. Moreover, alkane alloys are pertinent molecular alloys as phase change materials (MAPCM) for energy storage and thermal protection applications [4–9]. In the case of alkanes, molecules are packaged in successive layers in which chains are parallel among them. Global intermolecular packaging can be considered as a result of three types of packaging: • Packaging side of CH2 groups, which represent the main body of the chains. This packaging governs the internal structure of a layer. • Packaging of group’s methyl CH3 , i.e., the relative position of the heads of the chain in the adjacent chains. These CH3 heads are flat interlayer. • Longitudinal packaging of successive chains, which governs the superposition of layers. Crystalline forms are referred to as “vertical” when they are perpendicular to the plane of the methyl groups and “oblique” in case contrary (Fig. 6.3). The distance between two layers or distance interlayer allows characterizing different crystalline forms. We signal that in both cases (vertical or oblique forms) chains bodings packaging can be described by the repetition of a same reason (dashed line in Fig. 6.3) called subcell. This notion of subcell was used by Bunn in 1939 [11] when he describes the structure of polyethylene and retaken by Vand in 1951 [12] which gives name to simplify the classification of paraffinic structures. Consider, for example, the orthorhombic structure with parameters a, b, and c, where c is taken according to the longitudinal axis of the chain and ab as the section perpendicular to the chains. We take a ≥ b (conventional space group Pca21 of the orthorhombic form of even alkanes) to continue normally adopted representation. The subcell is called a0 , b0, and c0 where a0 and b0 , in the present case, are simply a and b. As for c0 corresponds to the zigzag elementary CH2 –CH2 (c0 = 2. 54 Å). This subcell is used as a model for polymers. It is characterized by a0 = 7. 42 Å, b0 = 4. 96 Å, c0 = 2. 54 Å, containing two CH2 –CH2 motifs.

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Fig. 6.3 Schematic view of the intermolecular packing (adapted from [10]). a Right forms; b Tilted forms. Solid circles are CH2 , and open circles are CH3 groups. The small parallelepipeds indicate the subcell

6.2.2 Polymorphism Keller, in 1961 [10], studying the polymorphic behavior of the alkane C36 as a function of the temperature, notes at 60ºC the appearance of stretch marks on the surface of the crystal according to the largest diagonal, and from 70 °C the apparition of new stretch marks according to the minor diagonal. Supported in the work of Schoon in 1938 [13] that was evoked the existence of crystalline forms corresponding to glide chains parallel to c in a number integer of CH2 according to one or other of the axes a and b, Keller suggests that the appearance of these stretch marks is the result of these movements and shows, thanks to elegant crystallographic calculations which stretch marks correspond to tilted monoclinic structures with molecules gliding about on each other well according to a, but according to b, but always parallel to the axis c. Would be more rigorous to say is produce a glide of chains according to c in the plane (a, c) or (b, c). Keller proposes to note the planes containing the methyl groups with a notation (hkl) indexed in the referential of the orthorhombic subcell. This

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plane is therefore the plane (00l) for vertical structures, while there is no never h and k simultaneously equal a zero in the oblique structures. If the glide takes place according to a, will have methyl planes (h0l) and if it takes place according to b, the planes will be of the type (okl). So, are for example, we will note as M011 a monoclinic form derived from an orthorhombic subcell a0 , b0 , and c0 by a glide of molecules of a length of 2CH2 (which means a c0 ) according to c in the plane (b, c). Kitaigorodskii, using the theory of close packaging, confirms later, in 1955 [14], the potential existence of diverse forms derived from a same subcell: monoclinic and triclinic. There are also triclinic forms that derive from the orthorhombic subcell (so-called Thkl) originating from mixed glides, i.e., at the same time according to the plans (a, c) and (b, c). These forms are also observed in (n ≥ 44) long-chain alkanes. When the temperature or the length of the chain increases, the molecule of alkanes can present various types of defects of conformation. The concentration of defects also depends on the ordered or rotatoring nature of the crystalline form. Mean conformation defects are trans-gauche (tg), double gauche (gg), “kink” (gtg’) as shown in Fig. 6.4. In 1930, Müller [16] suggested that the increment of interlayer distance during the transition from low temperature to rotator form linked to an increase in the degree of liberty so that chains rotated around its longitudinal axe. Much later, Kim et al. in 1989 [17] come to the same conclusion. According to these authors, the transition is not due more than the appearance of an orientational disorder. More recently it has been shown that this transition involves also a disorder of conformation. After the work of Müller, we know alkanes can show, before the melting, one or more solid forms where the molecules are affected by a strong degree of freedom of rotation around their molecular axis. These forms, called rotator (R), have been studied by several authors [18–30].

Fig. 6.4 Mean types of conformational defects present in alkanes. a all-trans (tt); b trans-gauche (tg); c double gauche (gg); d “kink” (gtg’) (adapted from [15])

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Just as ordered forms of low temperature (ord), rotator form R presents structures in layers. A three-dimensional long-distance order persists in what concerns the position of molecules. The essential difference lies in a loss of the long-distance order in what concerns the degree of rotational freedom of molecules around the molecular axis. However, we can observe manifestations of diffuse diffusion, sign of the existence of a certain local order. On the other hand, there is another degree of freedom which is the longitudinal movements whose relatively weak amplitude increases with the number of carbon atoms and temperature. This longitudinal disorder conduces to a disturbance of the planes containing the methyl groups and as a result in a decrease in the intensity of the picks of diffraction 00l. N-Alkanes present a rich and complex polymorphism. In the range from C8 to C28 and in the “low temperature” domain, seven forms are identified [31]: • • • • • • •

triclinic Tp (P-1, Z = 1) in C8 to C24 [32], triclinic Ti (P-1, Z = 2) in C9 and C11 [33], monoclinic M011 (P21/a, Z = 2) in C26 and C28 [29, 34, 35], orthorhombic Op (Pca21 , Z = 4) in C28 [29, 35, 36], orthorhombic Oi (Pcam, Z = 4) in odd alkanes [37], orthorhombic Odci (Pnam, Z = 4) in C23 to C27 [29, 38], monoclinic Mdci (Aa, Z = 4) in C25 and C27 [23, 38].

We will use the following nomenclature for defining the different spatial symmetry: T for triclinic, O for orthorhombic, and M for monoclinic. In triclinic forms, we will also add a “p” to define even (“pair” in French) and “I” for odd (“impair” in French) members. Conformational defects in the alkanes chain will be denoted with “dci” (“défauts de conformation” in French). The notation Mhkl will be also used in monoclinic forms (e.g., M011). In this case, (hkl) refers to the plane formed by the methyl end groups (CH3) in the reference orthorhombic subcell. The five ordered forms may be described in two groups: • The forms in which the defects of conformation are non-existent: Tp, Oi, M011. These forms are not derived from the same subcell. Oi and M011 forms derived from the orthorhombic subcell while the Tp form (observed in even (n ≤ 24) short-chain alkanes) derived from the triclinic subcell. • Odci and Mdci forms are of a different nature. They are induced by defects of conformation. Hence, the “dc” indexes. The index “i” means that these two forms, which have the same orthorhombic subcell, are characteristic of the odd-numbered alkanes. Since the 1980s appears, a new batch of studies more profound on the polymorphic behavior of alkanes. In 1981, Doucet et al. [20] propose a structural model of the rotator phase that they observe in C17, C19, and C21, and that they call RI. They describe this as a three-dimensional form with space group Ccmm. In another article also 1981 Doucet et al. [21] they show the existence of a second rotator phase called RII, in alkanes C23 and C25, appearing during the rise in temperature after the RI form, some degrees prior to the melting. It identifies as perfectly hexagonal, but its crystalline structure is not determined. The same authors subsequently published an

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article on alkanes with an even number of carbon atoms, from C18 to C26 [24]. They highlight a transition from the ordered phase to RII for alkanes, C22 and C24, C26 and proposed space group P63 mc for form RII. In 1984, Doucet et al. expand your research to alkanes from C27 to C34. Diagrams of X-ray diffraction prior to the melting in alkanes with 27 ≤ n ≤ 34 cannot be indexed nor as orthorhombic (RI) or as hexagonal (RII). Two new forms, a triclinic RIII) and the other monoclinic (RIV), are proposed. In parallel, other authors are also studying the crystallographic behavior of alkanes. In 1983, Ungar [23] studied C11 to C25 alkanes with odd n by X-ray diffraction, differential scanning calorimetry, and infrared spectrometry. Propose an orthorhombic packing, centered in the faces with a spatial group Fmmm for the RI form. They agree with the hexagonal packing for RI, for which they propose the space group R-3 m. Besides, they announced the existence of a new rotator form, which precedes RI, for alkanes C23 and C25. On the other hand, some degrees before the transition that leads to a rotator phase detect two transitions from ordered to ordered phase. These transitions are also detected by infrared spectrometry [22, 39]. According to these authors, these transitions are present in alkanes C25 and C27 C29. In 1990, Heyding et al. [40] propose the cell parameters for the high-temperature forms of some alkanes using as a base the spatial group Fmmm with Z = 4 for RI and R-3 m with Z = 6 for RII. They do not take in account the new rotator phase proposed by Ungar, nor the RIII and RIV forms proposed by Doucet et al. in 1984 [25]. In 1991 [41] and 1992 [42], Gerson et al. using X-ray diffraction measurements establish a general proposal of cell parameters from C18 to C28, only for the lowtemperature forms. In 1993, Sirota el al. [26] using X-ray diffusion propose the five rotator forms that can be observed in alkanes with 20 ≤ n ≤ 33 (see Fig. 6.5). These rotator phases are characterized by disorder. They named as RV the form announced by Ungar. These authors were interested in the structural disorder characterization. They concluded that the five rotator forms never are present in a single alkane. The nomenclature proposed by Sirota et al. in 1993 [26] to differentiate the five rotator forms is used commonly today. • Orthorhombic RI (Fmmm, Z = 4) in C13 to C21, C22, C24, C23, and C25 characterized by a molecular oscillation around their average positions increasing

Fig. 6.5 Scheme of packaging layers proposed for the different rotator forms (adapted from [26])

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• • • •

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with the temperature. To be coherent with the mmm molecular symmetry, the rotation is suggested to be a swing between two or four equivalent molecular orientations. The packing is bilayer ABAB [43]. Monoclinic RV in C23 and C25 present before RI it is a tilted version of RI. The tilt angle diminishes with the temperature. The packing is also bilayer ABAB. Trigonal RII (R-3 m, Z = 3), from C22 to C26, is the most disordered rotator phase, with a total rotation of the molecules. Correspond to a packing of sticks (molecular envelops) perpendicular to layers. The packing is ABCABC. Triclinic RIII, in C26 and C27, where the cylindrical molecular envelops are tilted from the normal of layer planes. The packing is monolayer. Monoclinic RIV, in C27 and C28, where the molecules are also tilted like as RIII. The packing is monolayer.

The sequence of polymorphic transitions, as a recompilation of works made in the REALM group [30], is: Even Alkanes C8 to C20: C22 and C24: C26*: C28**: Op

Tp Tp M011 M011 M101

RI

L L L L L

RII RII RIV RIV Odd Alkanes

C9 and C11: Ti C13 to C21: C23: C25: C27:

Oi Oi Oi Oi Oi

Odci Odci Odci

Mdci Mdci

RV RV

RI RI RI RI

RII RII RIII

RIV

L L L L L

*In C26, the form Tp can coexist with the M011. **Two sequences of transition have been observed in C28, increasing the temperature: The first involves the stable forms. Until n = 21, the behavior is related to the parity. For n ≥ 21, the effect of parity is even sensitive for ord forms, while it disappears for rotator forms. Only the even alkanes from C8 to C20 melt in an ordered form (triclinic Tp). Elsewhere, with one or several ordered forms, the melting is preceded by the step by one or more rotator forms. Even alkanes have a unique stable orderly form. Until n = 24 is triclinic Tp (P − 1); (Z = 1). For n = 26 still observes the triclinic form but is metastable. The stable form is monoclinic M011 (P21 /a); (Z = 2). Odd alkanes have all an orthorhombic form Oi (Pcam; Z = 4) is the only ordered form between C13 and C21. When n increases, one (for n = 23) or two (for n = 25 and 27) ordered forms insert between Oi and rotator forms. It is Odci and Mdci (Aa; Z = 4).

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Only Tp, RI, RII and RIV forms melt directly. RI and Tp forms show a perfusion effect associated with a strong increase in heat specific. RV to RI and RIII to RIV transitions are second order. The rest are first order, but the energies involved are very different depending on the case. Transitions more energetic are, without doubt, the melting of the Tp form; the enthalpies of melting vary between 195 and 240 J g−1 from C8 to C20. After, the melting of the rotator form, from 160 to 165 J g−1 order, they are almost independent of the length of the chain. Although less energetic, the transition enthalpies from an ordered form to a rotator form can be also important, in the order from 50 to 80 J g−1 . On the contrary, ordered to ordered and RI to RII transitions are very weak energetically, on the order from 1 to 6 J g−1 . The cell parameters of stable forms calculated from powder X-ray diffraction data [29] are presented in Table 6.1. Infrared and Raman spectroscopies can be used to characterize the molecular conformation defects as well as particular intermolecular interactions of each polymorph. The vibrational mode assignments of the Raman and IR spectra of n-alkanes were made from both experimental and theoretical calculations [22, 44–47] and have serve the REALM as a basis to investigate the polymorphism and molecular conformations of the different families studied. The infrared spectra of the different long-chain families show the characteristic bands of the n-alkanes and the bands associated with the vibrations of the functional group and its interaction with the hydrocarbon chain. The modes of vibrations and the frequency at which they show up are described in Table 6.2 followed by a model IR spectra in Fig. 6.6. This spectra are characteristic for showing progression bands associated with the vibrations of the skeletal C–C bonds, vibrations of the methylene groups (CH2 ), and the vibrations of the terminal, in this case methyl group. The modes of vibration can be classified into stretching symmetric/asymmetric (due to C–H or C–C bonds) and bending vibrations (either rocking, scissoring, wagging, and twisting). These vibrations are very sensitive to the immediately adjacent neighboring, and their frequencies and intensity can notable vary. There are two types of bending vibrations associated with methylene groups (CH2 ): bending in plane of the type scissoring and rocking and bending out of plane of the type wagging and twisting. For long chains made up of molecules like CH3 –CH2 )n –CH3 in planar conformation (all-trans) like the alkanes, the interaction between adjacent CH2 groups results into progression of equidistant bands in the IR spectra and two characteristic bands of all the methylene groups of the chain associated with CH2 rocking and CH2 twisting in phase vibrations. The CH2 in phase rocking band is very intense in the IR spectra of n-alkanes (n > 3) but in the low-temperature-ordered phase, this bands split into a doublet likewise the CH2 scissoring mode. The rocking vibration frequency of CH2 is sensitive to the conformation of the immediately adjacent C–C bonds. Different local conformations give rise to distinguishable bands in the region 620–670 cm−1 , and the relative intensities of these

233

253

268

270

291

291

291

291

291

C12

C14

C16

C18

C20

C22

C24

C26*

298

C28

248

263

273

291

291

291

C13

C15

C17

C19

C21

C23

Orthorombic forms Oi (Pcam; Z = 4)

323,5

C26

Monoclinic forms M011 (P21 /a; Z = 2)

173

C10

T(K)

C8

Triclinic forms T p (P-1; Z = 1)

Cn

4,83 ± 0,01 4,843 ± 0,010 4,828 ± 0,007 4,831 ± 0,009 4,831 ± 0,004 4,824 ± 0,009

4,29 ± 0,01 4,334 ± 0,008 4,296 ± 0,008 4,282 ± 0,008 4,283 ± 0,010 4,28 ± 0,03

4,99 ± 0,01 4,99 ± 0,01 4,98 ± 0,01 4,989 ± 0,005 4,980 ± 0,005 4,983 ± 0,007

7,43 ± 0,01 7,43 ± 0,02 7,43 ± 0,01 7,482 ± 0,010 7,477 ± 0,009 7,467 ± 0,008

7,44 ± 0,05

4,84 ± 0,01

4,28 ± 0,01

7,56(1)

4,79 ± 0,01

4,30 ± 0,01

5,59 ± 0,04

4,81 ± 0,01

5,61(2)

4,76 ± 0,02

4,31 ± 0,01

b(Å)

4,16 ± 0,02

a(Å)

62,19 ± 0,04

57,09 ± 0,07

51,96 ± 0,08

46,6 ± 0,1

42,3 ± 0,1

36,8 ± 0,1

38,1 ± 0,3

35,75(4)

35,0 ± 0,2

32,48 ± 0,07

29,90 ± 0,05

27,5 ± 0,3

25,09 ± 0,07

22,49 ± 0,09

19,84 ± 0,06

16,42 ± 0,06

14,9 ± 0,2

12,4 ± 0,2

c(Å)

86,4 ± 0,4

86,28 ± 0,09

85,9 ± 0,3

86,0 ± 0,3

84,97 ± 0,07

84,2 ± 0,3

83,9 ± 0,2

81,3 ± 0,6

81,6 ± 0,2

80,1 ± 0,9

α(°)

Table 6.1 Cell parameters for n-alkanes ([33] for n < 21, [34] for n > 21, [35] for C28 and [29]) (*metastable form)

119,1 ± 0,6

119,2(1)

69,2 ± 0,7

69,0 ± 0,3

68,9 ± 0,2

68,6 ± 0,3

67,3 ± 0,2

68,2 ± 0,3

66,9 ± 0,2

65,6 ± 0,2

64,7 ± 0,2

62 ± 2

β(°)

(continued)

72,7 ± 0,2

72,70 ± 0,07

72,7 ± 0,1

72,8 ± 0,2

72,6 ± 0,1

74,0 ± 0,3

73,0 ± 0,2

73,8 ± 0,3

73,5 ± 0,1

75,4 ± 0,9

γ(°)

116 D. Mondieig et al.

291

291

C25

C27

317

319

C25

C27

323

C27

278

288

296

305

315

320

C15

C17

C19

C21

C23

C25

316.2

318

321

324

C22

C23

C24

C25

Hexagonal forms RII (R-3 m; Z = 3)

263

C13

Orthorombic forms RI (Fmmm; Z = 4)

319

C25

Monoclinic forms Mdci (Aa; Z = 4)

312

C23

Orthorombic forms Odci (Z = 4)

T(K)

Cn

Table 6.1 (continued)

4.988 ± 0.005

7.542 ± 0.008

5.05 ± 0.01 5.07 ± 0.01 5.06 ± 0.01

7.64 ± 0.02 7.77 ± 0.01 7.77 ± 0.01

4.79

4.79

4.80

4.80

8.02

4.93

4.94

5.13 ± 0,02

8.04

5.08 ± 0.02

7.69 ± 0.01

4.997

7.73 ± 0.03

7.589

4.993

4.980 ± 0.006

7.553

4.989 ± 0.007

7,456 ± 0,009

7.529 ± 0.010

4,974 ± 0,005

7,449 ± 0,010

7.546 ± 0.008

b(Å) 4,968 ± 0,005

a(Å)

c(Å)

102.7

98.6

94.7

90.9

67.37

63.03

57.7 ± 0.2

52.3 ± 0.2

46.7 ± 0.2

42.3 ± 0.2

36.9 ± 0.2

72.84

67.53

72.45 ± 0.07

67.33 ± 0.07

62.26 ± 0.04

72,37 ± 0,07

67,26 ± 0,07

α(°)

94.1

91.54

β(°)

γ(°)

(continued)

6 Chains 117

T(K)

327

Cn

C26

Table 6.1 (continued) 4.80

a(Å)

b(Å) 106.2

c(Å)

α(°)

β(°)

γ(°)

118 D. Mondieig et al.

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Table 6.2 Mean values and intensity of the characteristic vibration of n-alkanes Vibrations

Frequencies

IR

Raman

CH3 stretching asymmetric

2972–2952

Very strong

Very strong

CH3 stretching symmetric

2882–2862

Very strong

Very strong

CH3 bending asymmetric

1470–1440

Middle-strong

Middle-strong

CH3 bending symmetric

1380–1370

Middle

Weak-absent

CH2 stretching asymmetric

2936–2916

Very strong

Very strong

CH2 stretching symmetric

2863–2843

Very strong

Very strong

CH2 scissoring

1475–1445

Middle-strong

Middle-strong

−(CH2 )n –twisting in phase

1305–1295

–

Strong

−(CH2 )n –twisting in phase

726–720

Middle-strong

–

C–C stretching

1180–1120

Middle

Middle

1100–1040

–

Middle

900–800

–

Middle

Fig. 6.6 IR spectra of n-nonadecane (C19 H40 ) as an example of an ordered phase (Reproduced from [48] with permission of the author)

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bands provide a measure of the relative gauche and trans population at an specific chain position. In Raman spectra, the vibrations due to C–C stretching (between 885–1300 cm−1 ) and C–C–C bending (below 540 cm-1) are also typically observed, as well as the socalled longitudinal acoustic bands or LAM’s associated with contraction—expansion of the whole chain that gives rise to a series of bands below 800 cm−1 . At frequencies, below 200 cm−1 is the characteristic bands of the different lattices, that depend on the lattice type and the number of molecules contained in the lattice.

6.3 Solid-State Miscibility The REALM was performed a global overview of the phase relationships in the alkane family. The analysis relies on 19 experimental binary phase diagrams, for systems with n = 1 and n = 2, in the range from n = 8 to n = 28. To construct the binary phase diagrams, samples of different compositions were analyzed by differential scanning calorimetry and X-ray diffraction as a temperature function. The components, in adequate proportions, were melted until obtain a homogeneous dissolution and quenched to liquid nitrogen temperature. The studied systems are shown in Fig. 6.7 (n = 1, even + odd and odd + even systems), Fig. 6.8 (n = 2, odd + odd systems), and Fig. 6.9 (n = 2, even + even systems). The study of the 19 binary systems with n = 1 and n = 2 presented herein has evidenced several facts concerning the polymorphism of both pure components and binary system [50–53]. First fact to mention is the effect of intermolecular interactions. They bear the same characteristics; however, intermolecular interactions produce a complex polymorphic behavior of both pure components and mixed crystals. Beside the pure components forms, five different additional phased are stabilized upon mixing. At high-temperature RI rotator form is stabilized. At low temperature, Odci, Mdci, Mdcp, and Op forms are stabilized. The later are ordered forms not showing the rotation defect. Overall, binary phase diagrams present a high number of single and two phases regions. Total miscibility is showed in the rotator phases when the components have an identical stable phase, in contrast with ordered phases where only in n = 2 systems, we observe isomorphism between their components. Nevertheless, their miscibility is very limited, owing to the appearance of additional phases for high degrees of crystalline isomorphism εi m (Oi) and εi m (Tp) (>0.9 for n ≥ 20). The mismatch in size among the crystal molecules produces conformational defects that favor the formation of additional phases. It worth to note that some forms are stabilized—even at lower temperature, or for shorter chain lengths—upon mixing. The RI rotator form, for example, can be stabilized in mixed samples at lower temperatures than in pure components. In addition, this rotator form is formed already in the C14+C16 system, whereas in pure components, the RI form appears

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Fig. 6.7 Binary phase diagrams: even + odd and odd + even systems (n = 1). Solid single-phase regions: green for [Tp], red for [Oi], pink for [Odci], orange for [Op], blue for [Mdci], yellow for [RI] and [RV], and light yellow for [RII] (Reprinted from Mondieig D, Rajabalee F, Métivaud V, Oonk HAJ, Cuevas-Diarte MA, n-Alkane binary molecular alloys. Chem. Mater., 16:786–798. Copyright (2004) American chemical society [49])

from C22 onwards. Therefore, we can affirm that substitutional disorder catalyzes conformational and rotational disorder. The high degree of rotational freedom in long-chain compounds like n-Alkanes is responsible for their polymorphism, both in pure and mixed samples. Both temperature and chain length increase are responsible for the increase of intramolecular rotations and the occurrence of conformational defects. Short n-alkanes molecules adopt the ordered- defect-free trans configuration. However, a temperature or chain length increase produces conformational defects (called gauche, kink, double gauche, etc.) because molecules are not rigid. These molecules are no longer flat, which generates significant defects in their crystal packing. Consequently, the lamellar surface is affected. As a result, the packing between molecular layers or interlamellar packing (the zone where the defects are positioned) is affected. Above a certain threshold of temperature, the number of defects is so high that the interlayer spacing changes

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Fig. 6.8 Binary phase diagrams: odd + odd systems (n = 2). Solid single-phase regions: green for [Tp], red for [Oi], pink for [Odci], orange for [Op], blue for [Mdci], violet for [M011], yellow for [RI] and [RV], light yellow for [RII], and green yellow for [RIII] and [RIV] (Reprinted from Mondieig D, Rajabalee F, Métivaud V, Oonk HAJ, Cuevas-Diarte MA, n-Alkane binary molecular alloys. Chem. Mater., 16:786–798. Copyright (2004) American chemical society [49])

Fig. 6.9 Binary phase diagrams: even + even systems (n = 2). Solid single-phase regions: green for [Tp], orange for [Op], cyan for [Mdcp], violet for [M011], yellow for [RI] and [RV], light yellow for [RII], and green yellow for [RIII] and [RIV] (Reprinted from Mondieig D, Rajabalee F, Métivaud V, Oonk, HAJ, Cuevas-Diarte MA, n-Alkane binary molecular alloys. Chem. Mater., 16:786–798. Copyright (2004) American chemical society [49])

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discontinuously and produces a solid–solid transition (as it is the case with Oi to Odci to Mdci solid–solid transitions). The same sequence of phases occurs, for an odd alkane, when the concentration of second component molecules in the crystal increases. We have observed that the same sequence of solid ordered forms is obtained by increasing the intrinsic defects (x = 0 or 1, temperature increase) or the extrinsic ones (temperature, rise in composition). Instead, in even n-alkanes the Op phase is induced by the increasing concentration of second component molecules in the crystal. This form is also observed as metastable in even n-alkanes with more than 28 carbon atoms, the stable form being monoclinic M011. Crystallization from the melt and evaporation of high-boiling petroleum ether solutions rendered the orthorhombic phase. The orthorhombic Op form is also observed near the equimolecular composition in odd + odd systems. The statistical molecular entity is very similar to the even alkane C2p at the equimolecular composition. However, the geometrical mismatch between molecules with end-gauche defects produces a different symmetry. In this case, the orthorhombic Pca21 with Z = 4 also observed in even alkanes with longer chains.

6.4 A Perfect Family The interpretation of the various equilibria has been based on the concepts of isopolymorphism and crossed isopolymorphism. The thermodynamic assessments are carried out for a double reason: to verify the experimental data and to use the available data for calculating the thermodynamic mixing properties [54]. Thermodynamic assessments of phase equilibria in the binary systems were performed successfully, for numerous transitions, including ord to R. The thermodynamic analysis leads to correlations between characteristics of mixed crystals and properties of pure components. For the complete set of 19 binary systems, a uniform thermodynamic description is applied, referred to as the ABθ model. The model is such that the excess enthalpy and excess entropy are independent of temperature. The parameter A is a measure of the magnitude of the excess functions. Its value can be correlated to the mismatch in molar volume, or a related property, between the components of a given binary system. For the alkanes, the mismatch parameter we use is the relative difference in chain length n/n. The parameter B is a measure of the asymmetry of the excess functions; its value is about 0.2. The parameter θ has the property that, within a family of mixed crystals, it is system independent. In other words, it has a value that is characteristic for the family as a whole. Families of mixed crystals, therefore, are classes of similar systems in terms of enthalpy/entropy compensation: at the compensation temperature, θ excess enthalpy and excess entropy compensate one another, such that the excess Gibbs energy is zero (or rather goes through zero). The ord systems are characterized by θ = 335 K and the rotator ones by θ = 320 K. Figure 6.10 [55] shows excess enthalpy vs. excess entropy. The perfect straight line obtained indicates a uniform behavior. The low scatter of points at the beginning of the curve is due to having a compensation temperature close to the temperature

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Fig. 6.10 For the mixed crystalline rotator I form of binary n-alkane systems, equimolar excess enthalpy plotted against equimolar excess entropy. Open circles: n = 2 and n’ and n are odd; filled circles: n = 2 and n’ and n are even; half-filled circles: n = 1 (Reprinted from Oonk et al. [55], with the permission of AIP publishing)

for which the HE and GE data are valid. Even having such a minor difference, the evolution of the phase diagrams is also evidencing a uniform compensation temperature. The family of n-alkanes represents a coherent group of substances from a chemical perspective. They can be present in a series of solid forms depending on carbon parity and carbon content. The RI form is stabilized in the mixed systems subgroup studied herein. The members of this subgroup show the same structural characteristics and same type of intermolecular interactions between entities with similar structural and chemical characteristics. As structural characteristics, we consider space group, arrangement of structural units, conformation, degree of orientation disorder, and the like. We suggest the definition of “A perfect family of mixed crystals” to those groups of mixed crystals exhibiting the following characteristics: (1) the members are chemically coherent and have constant structural characteristics, (2) they all have mostly the same nature of interactions, and (3) the enthalpy–entropy compensation curve is characterized by a uniform temperature.

6.5 Thermal History of Alkane Alloys A bibliographic analysis of the binary phase diagrams of alkanes show that in some cases the single-phase field for the liquid state (L) is separated, by a narrow two-phase region, from a single-phase field in the central part of the diagram: the rotator state (R). It is particularly relevant that for a same phase diagram, one can observe notable differences with regard to the width of the [R + ord ] region. These differences are too significant to be ascribed to experimental errors, or to differences in the interpretation

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of thermograms. In other terms, the differences have to correspond to differences in experimental procedures, in combination with the absence of full thermodynamic equilibrium. The fact is that, for a mixed crystal, the equilibrium transition from ord to R requires a continuous redistribution of the component substances over the two different phases, ord and R, until the material is fully homogeneous again. In this respect, the transition of a mixed crystal differs from the transition of its pure components. It is possible to make mixed n-alkane samples to change isothermally from the ordered crystal state (ord ) to the rotator state R. This isothermal transition has been observed in DSC and X-ray powder diffraction experiments, for all compositions in various binary systems, by maintaining mixed samples for a long time in the R state [56]. Thermodynamic analysis of the isothermal data, which correspond to experimental EGC temperatures, results in a successful calculation of the [ord + R] equilibrium temperatures. The thermodynamic interpretation and X-ray diffraction observations show that the phenomenon corresponds to diffusionless, or martensitic transitions, in metal alloys. Three different types of thermal treatment were performed in mixed samples. Those samples where then analyzed by DSC in the temperature range from the ord to the R transition. The analysis of mixed samples performed corresponded to the following thermal treatments: (a) analysis of quenched samples; (b) analysis of quenched samples at the temperature corresponding to the ord state. With this experiment, we analyzed the effect of the isothermal arrest duration; (c) analysis of quenched samples that were previously kept at a temperature corresponding to the R state, thereafter, cooled to the ord state, and subsequently analyzed. With this experiment, we investigated the effect of the duration of the isothermal arrest in the R state. Experiments (a) and (b) produced similar DSC curves meaning the length of the thermal arrest in ord does not have a significant effect on the thermal behavior after the arrest. Type (c) experiments, however, evidenced significant influence when the arrest is performed in the rotator state. Mixed samples of C19+C21 held at 305 k for 16 months were evaluated by DSC. The DSC signals become increasingly sharp. The results showed that the value of δ = Tend − Tonset is invariably zero: The change from ord to R has become an isothermal event. The temperature of this isotherm is about halfway between the onset and end temperatures of the t = 0 signals. The heat effect of the transition, apparently, remains unchanged. The same effect was observed on mixed samples of the systems C18+C19, C19+C20, C20+C21, and C23+C25. The transition of a pure substance and an alloy is different. A transition of an alloy becomes in a strict thermodynamic equilibrium demanding a continuous redistribution of matter. The redistribution of matter is a process that takes place on a different timescale than the transition from one state to the other. The latter process is in addition dependent on external factors like sample handling, and the way the experiment is performed. For example, we can assume that those materials prepared by quenching a melted substance might exhibit lattice imperfections and voids which enhances molecular

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mobility. Consequently, we can conclude that experiments of long-term duration might bring the material to a state of perfection, characterized by homogeneity and compactness. The rotator state offers the molecular mobility necessary to reach the state of perfection. Therefore, an alloy that is kept in the rotator state over a long period of time will transform into a homogeneous and compact phase. The relevance of thermal history of mixed samples is well known. The onset and end temperatures of the ord to rotator state transition of n-alkanes mixed samples vary with the duration of the arrest in the R state. In addition, the width of the transition temperature decreases with the duration of the arrest making it possible to achieve and isothermal transition from the ord to the rotator state. This way we have obtained isothermal ord to R transitions for all compositions in C18+C19, C19+C20, C19+C21, C20+C21, and C23+C25, after keeping mixed samples for a long time in the rotator state before studying the ord to R transition. The isothermal transition curves T(x) correspond to the EGC (equal G curves) of the two phases. The EGC experimental data can be used for a thermodynamic analysis of the change from ord to R. The calculated results (lower solvus and upper solvus temperatures of the [ord + R] equilibrium) are in good agreement with the onset and end temperatures, obtained by DSC experiments, on samples that had not been kept for a long time in the R state. The thermodynamic interpretation of the isothermal temperatures and the X-ray diffraction observations show that the isothermal phenomenon in alkane alloys is similar to diffusionless, or martensitic, transformations in metal alloys.

6.6 n-Alkanols 6.6.1 The Family Normal alkanols Cn H2n+1 OH (hereafter denoted by CnOH) are among the simplest of the substituted hydrocarbons. A single -OH group replaces a hydrogen atom at one end of the aliphatic chain giving rise to formation of hydrogen bonds. Molecules bind longitudinally pairwise by hydrogen bonds forming dimers and laterally by Van der Waals bonds forming a structure in layers [57].

6.6.2 Polymorphism For the n-alkanols with a number of carbon atoms between 12 and 20 [58–61], two ordered monoclinic phases at room temperature, named γ and β, have been described and observed [62–70]. The principle of a common structural description is summed up elsewhere [57]. The γ -phase (A2/a, Z = 8) is more frequent in even n-alkanols, the β-phase (P21 /c, Z = 8) in odd n-alkanols. The β-phase can be observed also,

6 Chains

127

Fig. 6.11 Packing of β a and γ b forms (Reproduced from [59] with permission of the author)

although in a metastable state, in alkanols with an even number of carbon atoms as a result of a fusion-quenching process. In both γ and β forms, the molecules are tilted toward the plane ab (more tilted in the γ form) and are stacked forming alternating planes of pure CH3 and planes of pure OH groups (Fig. 6.11). At these planes, OH groups are forming infinite hydrogen bond networks. The more important difference between the two phases that determines the different packaging is the presence of a gauche defect at the end of the chain that affects the oxygen atom in 50% of the molecules forming the asymmetric unit of the β form [58]. The ordered phases undergo a transition into rotator phases at temperatures below the melting point. On heating, the molecules of the ordered phases start to rotate along their long axis generating various conformational defects [71]. These phases are called RII, RIV, and RV [22, 26, 33, 34, 55, 72, 73]. For a matter of clarity and to avoid confusion with n-alkanes, we will call them as R’II, R’IV, and R’V. Table 6.3 and Fig. 6.12 summarize the main studied alkanols features. From a general point of view, we can say that: 1.

2.

The transition from an ordered form to a rotator form takes place immediately before the melting in alkanols with an even number of carbon atoms (Tordered-rotator ≤ 1 K than the Trotator-liquid). While in alkanols with an odd number, although it is also close to the melting temperature, it takes place a few degrees before (Tordered-rotator ≤ 4 K than the Trotator-liquid). Therefore, a parity effect is observed in the n-alkanols family and even–odd alternation in the melting points is observed. Two polynomial relations of degree 2 are possible. One of them corresponds to solid–solid transition temperatures in β-form: β-form to R’II form and β-form to R’IV. The other related temperatures corresponding to transitions from the γ -form: transitions of γ -form to R’II and γ -form to R’IV. The intersection of

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D. Mondieig et al.

Table 6.3 Temperatures and enthalpies in normal alkanols with 12 ≤ n ≤ 20 (m: metastable) [58, 59] Transition Solid–solid

Solid–liquid

T(K)

Forms

T(K)

Forms

C12 H26 O

–

–

295.5 ± 0.5

β→L

C13 H28 O

–

–

302.9 ± 0.5

β→L

C14 H30 O

310.0 ± 0.5

γ → R’II

310.0 ± 0.5

R’II → L

309.0 ± 0.5

(β → L)m

C15 H32 O

314.5 ± 1.5

β → R’II

316.4 ± 0.5

R’II → L

C16 H34 O

321.1 ± 0.5

γ → R’II

321.6 ± 0.5

R’II → L

318

(β → R’II )m

C17 H36 O

323.3 ± 0.5

β → R’IV

326.6 ± 0.5

R’IV → L

C18 H38 O

329.5 ± 0.5

γ → R’IV

330.3 ± 0.5

R’IV → L

328

(β → R’IV )m

C19 H40 O

329.7 ± 0.9

β → R’IV

333.9 ± 0.5

R’IV → L

C20 H42 O

335.5 ± 0.5

γ → R’IV

336.6 ± 0.5

R’IV → L

332

(β → R’IV )m

Fig. 6.12 Solid–solid transitions in n-alkanols with 11 ≤ n ≤ 20 as a function of n (Reproduced from [59] with permission of the author)

6 Chains

3.

129

the two curves that mark the figure is a point where n is lower than 13. This point can tell us two things: (i) for n ≤ 13 stabilizes the β-form as a form of low temperature, regardless of the parity of the number of carbons in the chain; (ii) this point indicates that for chains with a number lower than 13 carbon atoms will not appear rotator forms before the melting. Significantly, it indicates that C13OH does not show a transition solid–solid during the rise in temperature, but on the descent appears the form R’II metastable, while the C12OH no longer displays rotator forms stable or metastable. This result is consistent with reticular energies calculated for β and γ forms for a same member of the family equates for n = 14. Theoretically the C14OH may have two stable forms, whereas for smaller n stable form is only β-form. The melting temperature, instead, seems to follow a single law of variation that can be described by a polynomial of degree 2, regardless of the nature of the melting form here, the parity effect is not observed.

A similar discussion is possible for the enthalpy of transition and melting. The enthalpies of melting of rotator forms also follow a single law of variation that can be described by a polynomial of degree 2, regardless of the form that melts (R’II or R’IV). Parity effect is not observed. The values of these enthalpies are sufficiently high to consider these substances directly as phase change materials (MCPs), or good candidates to provide materials with phase change based on molecular alloys (MAPCM of the acronym in English). On the other hand, the ordered to rotator form transition enthalpies also follow a single law of variation that can be described by a polynomial of degree 2, irrespective of whether the ordered phase is γ or β, and this transform to R’II or R’IV. Once again, the parity effect is not observed. In addition, for each member of the family, whatever the phases involved, the ordered to rotator form transition enthalpies are quite high if compared with the enthalpies of melting. The relationship between both shows a chain length effect. The influence of hydrogen bonding in the intermolecular interactions is more pronounced for shorter members but without reaching the behavior of plastic or liquid crystals. This feature makes n-alkanols differ from the n-alkane family, where the absence of hydrogen bonds is manifested in much smaller solid–solid transition enthalpies. Heat capacities and derived thermodynamic functions were determined for C12OH, C13OH, C18OH, C19OH, C20OH, and C22OH between 10 and 370 K and the C15OH and C17OH between 300 and 380 K [74, 75]. The cell parameters were calculated from powder X-ray diffraction data gather in Table 6.4 followed by an example of diffraction pattern for the C19OH alkanol (Fig. 6.13). From the crystallographic point of view, the n-alkanols presented a structure in layers. The molecules within a layer may be tilted (γ , β or R’IV) or vertical (R’II). We can observe that: • For a given phase, regardless of the parity of the carbon atoms in the chain, a linear behavior of d hkl as a function of n is denoted. The distance between layers increases by increasing the length of the chain.

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Table 6.4 Crystalline parameters of forms observed in normal alkanols with 12 ≤ n ≤ 20 (m: metastable) [58, 59] Form

a(Å)

b(Å)

c(Å)

α(º)

β(º)

γ(º)

C12 H26 O

β

5.083(4)

7.348(5)

69.44(8)

90

90.84(3)

90

C13 H28 O

β

5.030(5)

7.433(5)

74.61(3)

90

91.08(4)

90

C14 H30 O

(β)m γ R’II

5.088(3) 9.027(4) 4.857(3)

7.379(5) 4.953(2) 4.857(3)

79.44(7) 77.25(5) 116.8(2)

90 90 90

90.87(8) 121.38(3) 90

90 90 120

C15 H32 O

β R’II

5.029(4) 4.847(3)

7.341(2) 4.847(3)

84.75(4) 124.7(8)

90 90

91.12(2) 90

90 120

C16 H34 O

γ (β)m R’II

9.003(4) 5.04 4.905(4)

4.960(3) 7.35 4.905(4)

87.82(5) 89.92 133.3(3)

90 90 90

121.95 91.5 90

90 90 120

C17 H36 O

β R’IV

5.031(3) 8.414(6)

7.403(3) 4.854(3)

94.52(4) 46.61(3)

90 90

91.11(4) 94.18(5)

90 90

C18 H38 O

γ (β)m R’IV

9.031(3) 5.009 8.458(3)

4.959(3) 7.322 4.933(4)

98.19(5) 101.24 48.58(6)

90 90 90

122.41(5) 91.1 92.1(1)

90 90 90

C19 H40 O

β R’IV

5.023(3) 8.419(6)

7.335(3) 4.825(6)

104.77(3) 50.92(3)

90 90

91.11(2) 93.11(4)

90 90

C20 H42 O

γ (β)m R’IV

9.035(3) 5.046 8.450(5)

4.970(4) 7.378 4.940(7)

108.71(9) 107.41 53.24(8)

90 90 90

122.82(4) 91.35 93.7(2)

90 90 90

Fig. 6.13 X-ray powder diffraction pattern for the β and R’IV forms of C19 H40 O (Reproduced from [59] with permission of the author)

6 Chains

131

• In the phase R’II (vertical molecules), logically, d hkl is greater than in the phases β and R’IV (almost vertical molecules) and γ (more tilted molecules). At the same time, d hkl is similar in the forms β and R’IV, because the tilt angle is similar. • From the crystallographic point of view, crystalline parameters follow a trend for a same crystalline form and different members of this family. For example, parameters a and b of all the β forms remain virtually constant, regardless of the length of the chain and the parity of the number of carbons in the chain. Obviously, the c parameter is the one dependent on the length of the post string that corresponds to the longitudinal axis of the chain. • Ordered forms present a greater compactness than rotator forms as evidenced by their degree of packing η of a crystalline form, quantified by the coefficient of occupation or packing (the ratio between the volume of the molecule Vm and the volume occupied by this molecule in the crystalline unit cell V/Z being V the volume of the crystalline unit cell and Z the number of molecules containing the unit cell). This is due to the presence of more defects in the rotator forms. Also, it is appreciated as there is no difference between the forms β and γ . Equally, both RIV as the RII forms have a constant factor η. When the γ -phase and β-phase are heated in the Guinier–Simon camera, they follow a similar change. Both phases transform to a rotator phase, and it is this phase that melts. The thermal stability range comprises only a few degrees before the melting. The fact that the rotator forms of n-alkanols with odd number of carbon atoms present a range of thermal stability more large than n-alkanols with an even number of carbon atoms is probably due to the first present defects of type gauche in the ordered form of low temperature (β) and this facilitates the stabilization of the rotator form. The form characteristic of the n-alkanols with an even number of atoms of carbon and n ≥ 16 has, on the other hand, an “all-trans” conformation, conformation of departure with which it is more difficult to stabilize rotator forms that present more defects. The rotator phases were characterized by X-ray powder diffraction at different temperatures. The heating and cooling processes were also studied by DSC. On heating, the solid–solid transitions are overlapped with solid–liquid phenomena (mainly the γ to rotator transition in even alkanols), whereas on cooling the crystallization and transition are clearly separated. Due to the overlapping, the temperatures and enthalpies of melting were obtained in a second heating of the sample from a temperature above the solid–solid transition. A scheme of the DSC for C19OH and C20OH is shown in Fig. 6.14. From the thermal hysteresis and the volume change in the transition, it is concluded that the transition of β or γ forms to rotator form is a first-order transition. The enthalpy change involved in this solid–solid transition is about two-thirds of the melting enthalpy, whereas in the n-alkanes, where only van der Waals interactions between the molecules are present, one-third of the melting enthalpy is involved. Characterization of the molecular conformation defects was made by Raman scattering and infrared spectroscopy. The vibrational mode assignment was based on

132

D. Mondieig et al.

Fig. 6.14 DSC analysis of commercial C19 H40 O and C20 H42 O (Reproduced from [59] with permission of the author)

that of n-alkanes considering vibrations generated by the functional OH group [60], and it allows the identification of β and γ forms. The conformation of the ends of the chain is evidenced by Raman scattering using the CH3 rocking mode zone (700 < ν < 1000 cm−1 ). n-Alkanols in β-phase spectra display a band at 875 cm-1 assigned to CO gt- (gauche) mode while a band observed at 845 cm−1 is assigned to CO tt- (trans). Kink defects (CC–gtg’- at 850 cm−1 ) and end-gauche defects (CC gt- at 870 cm−1 ) are also manifested in this zone. Snyder also described the identification of structural disorder by identifying irregular frequencies and intensities in the C–C stretching mode zone (1000 < ν < 1150 cm−1 ) and CH2 wagging/twisting mode zone (1150 < ν < 1400 cm−1 ). The IR-spectra of β- and γ -phases of n-alkanols show three broad new bands compared to that of all-trans n-alkanes assigned to: (i) OH bending out of plane at 686 cm−1 (β-phase) and 606 cm−1 (γ -phase). (ii) OH bending in the plane at 1406 cm−1 (β-phase) and 1425 cm−1 (γ -phase). (iii) OH stretching, a broad band at 3300 cm−1 (β-phase) and two broad bands at 3230 and 3330 cm−1 (γ -phase).

6 Chains

133

Fig. 6.15 Rietveld refinement (left), and the determined crystal structure for the γ form of C20 H42 O (right). Projection in the plane (010) (Reproduced from [59] with permission of the author)

Significant differences between the spectra of low-temperature phases and the rotator phase are: (i) the disappearance of external mode bands and a general broadening of all bands. (ii) A decrease of frequency and intensity of longitudinal acoustic modes (LAM). (iii) The existence of the band at 875 cm−1 assigned to CO gt- mode and a new band near 1083 cm−1 . (iv) The change from four to two broader bands in the CH2 wagging/twisting zone. Crystal structures of n-alkanols were determined from powder X-ray diffraction methods combining molecular modeling and Rietveld refinement using as starting structural models the information of the single-crystal structures of some members of the family [59, 63, 65, 69, 70, 76–81]. Figure 6.15 shows an example of Rietveld refinement and the determined crystal structure in the case of the C20OH (γ form).

6.6.3 Solid-State Miscibility Some binary systems among n-alkanols have been studied before [82–85]. Phase diagrams between n-alkanols experimentally studied by us with n = 1 or 2: C19OH–C20OH [60], C16OH–C18OH [58], C18OH–C20OH [86], and theoretically modeled: C15OH–C17OH [59] and C17OH–C19OH [59], are characterized by narrow domains solid–liquid (not exceeding 1.5 K). These narrow domains of melting, along with the high values of the enthalpies of melting (on the order of 40 kJ/mole) guarantee the ability to use mixed samples as MAPCMs in conditions virtually isotherms. In other systems like the ones with n = 4, (i.e., C16OH–C20OH [58], the solid–liquid domains presented considerable widths (as in some compositions of 5 K). Solid-state miscibility in these types of substances is governed by a

134

D. Mondieig et al.

series of characteristic parameters like as volume and the shape of the molecule, as well as the interactions between them. A way to quantify the similarity between molecules consists in the use of the concept of crystalline isomorphism. In this case, as in other families of chain conformation, the difference in length of chains is revealed as a determining factor. The experimental binary phase diagrams of the systems C19OH–C20OH with n = 1, and C18OH–C20OH with n = 2, and C16OH–C20OH with n = 4 permit us to propose very narrow solid–liquid domains only for the systems with n = 1 and 2 (lower than 1.5 K). Miscibility favors stabilization in mixed samples of forms that only are metastable in both components of the binary systems. So, for example, the β-form that only is seen as stable form in alkanols with an odd number of carbons in the chain, or alkanols with an even number but short chain as the C12OH is the form that stabilizes at low temperature in the majority of compositions of the C19OH–C20OH and C18OH– C20OH systems. On the other hand, the form R’II is that presents a greater range of miscibility. Let us remember this form only is seen as a stable form in the C16. In addition, this form is the more disordered. Therefore, we can clearly associate miscibility in solid state with disorder. The disorder favors miscibility in solid state. By analogy with the alkane family, the experience of the REALM [55, 87] has highlighted the existence of compensation between enthalpy and entropy of excess in families of molecular substances with different configurations and interactions. The representation of the enthalpies of excess as a function of the entropies of excess, for a specific equimolar composition, has allowed to observe in the family of the n-alkanols once more the notion of family of binary mixed crystals [60]. Also, correlations between thermodynamic magnitudes (as Gibbs energy, enthalpy and entropy of excess) and crystallographic magnitudes have been able to establish [55].

6.7 Fatty Acids 6.7.1 The Family Normal saturated carboxylic acids, with general formula Cn H2n O2 (hereafter denoted by CnCOOH), are substituted hydrocarbons in which a single carboxyl group (– COOH), replaces a hydrogen atom at one end of the saturated aliphatic chain (Fig. 6.1). In general, the saturated hydrocarbon part (–CH2 –)n assumes the all-trans zigzag conformation as the most stable conformation in the crystalline phase. In some cases, conformational defects of the gauche type, usually involving the C2–C3 bonds necessary to stabilize the crystal packing, are observed.

6 Chains

135

6.8 Polymorphism The debate around the complex polymorphic behavior of carboxylic acids started in the thirties [88] and was emphasized again during the decade of the fifties [89–93]. The various reasons originating such a long debate are the multiple crystal forms in which carboxylic acids might crystallize, their structural similarity, and the common obtaining of mixtures of crystal forms. In addition, the growing of carboxylic acids single crystals has been challenging and even impossible for some of the forms, a fact that has difficult even more the proper identification and characterization of carboxylic acids polymorphism. The polymorphism of normal saturated carboxylic acids is complex. It mostly depends on parity and temperature. Additional factors influencing the obtaining of a given crystal form include the rate of crystallization, the type (polarity, evaporation rate, etc.) of solvent, the acid purity, and the length of the hydrocarbonated chain. Odd-numbered carboxylic acids crystallize into the so-called A’, B’, C’, C”, and D’ crystal forms [88–98] and even number carboxylic acids crystallize into the A1 , A2 , A3 , Asuper , Bo , Bm , Eo , Em , and C [94, 96, 99–106]. Crystal forms are usually classified as (1) high-temperature phases of the type C, C’, and C”, and (2) lowtemperature phases. The former are stable only during few degrees before the melting point, where the later are stable from room temperature to the some degrees before the melting. When heated, low-temperature phases undergo a solid–solid transition and transform into the high-temperature C-type forms. Crystal structure determination of the C type forms was performed using in-situ crystallization and conventional X-ray diffraction from single crystals [96, 107]. The high-temperature form of evennumbered acids is called C while the high-temperature form of acids with an odd number of carbon atoms in the chain is called C’ or C”. The high-temperature form of even-numbered acids can be obtained from solvent crystallization and from the melt. Once obtained, the C form remains metastable for at least 2 years if kept at room temperature. However, the high-temperature forms of acids with an odd parity convert into the low-temperature forms on lowering temperature. Much effort has been made in order to control the appearance of each crystal form. In general, pure crystalline forms of fatty acids are rarely obtained and concomitant crystallization of several phases is the common trend (see Fig. 6.16 as an example). The polymorphism tendency of the even subfamily and the occurrence of each crystal forms are represented in the next scheme and Table 6.5.

Acids from C10 to C14 Acid C16 Acids from C18 to C20 A1 , A2 , A3 , Asuper

C

B0 , Bm , E0 , Em

136

D. Mondieig et al.

Fig. 6.16 Powder X-ray diffraction patterns at increasing temperatures of a mixture of E, A2 and C forms in a sample of C16 H32 O2 (Reproduced from [108] with permission of the author)

The polymorphic behavior for the odd-numbered acids is rather different. The Ah’ and B’ forms are observed at room temperature for all the acids from C13COOH to C23COOH acid, being the Ah’ form the stable one for C13COOH acid and the B’ form the stable one for acids from C17COOH to C13COOH acid. C15COOH acid represents the crossing point between the two stability regions. On raising the temperature, the Ah’ and B’ forms undergo a reversible transition into the hightemperature form C” (acids C13COOH and C15COOH) or C’ (acids from C17COOH to C23COOH). Finally, the D’ form is obtained by grinding samples of the B’ forms but it rapidly transforms again into B’ on increasing the temperature. The temperatures and enthalpies of the solid–solid and solid–liquid phenomena’s analyzed by means of DSC directly after the preparation of the sample gathers in Tables 6.6, 6.7, and 6.8. The differences in the molecular conformation of the different polymorphs can be observed by means of infrared spectroscopy with polycrystalline samples. Thus, forms A and B—end chain defect and gauche conformation—can be identified and distinguished from E or C—all-trans conformation-. Representative FT-IR spectra of the different polymorphs and detailed assignations of IR bands are published by Moreno [109] and Gbabode [48]. For the matter of concern here, we describe the bands evidencing gauche defects in the molecules which allow identification of B and A forms of even acids. In general, the gauche defect causes strong interactions between the methylene and the carboxyl group making the appearance of new bands or shifting them. For the A form, the δ(CH2 ) (1472 cm−1 ) and the r(CH2 ) (718 cm−1 ) bands are single; otherwise, these bands are split. In the zone 1320–1180 cm−1 , the bands associated with the ω(CH2 ) coupled with the carboxyl vibration appears. This progression of bands is regularly spaced for the structures with all-trans molecular

C

C

Asuper C

C

C

C

C10 H20 O2

C12 H24 O2

C14 H28 O2

C16 H32 O2

C18 H36 O2

C20 H40 O2

Comercial sample

C

C

C

C

C

C

Melting & Quenching N2 (l), water, ice

Em

Em C

A2 C Em (Eo )

A2 Asuper C

–

Em (Eo )

Em (Eo )

Em (Eo ) C

C A2 Asuper

–

–

–

–

C

C (A2 )

–

–

Bo

Bo Bm A2

C A2

C A2

–

–

WPSa

–

Slow crystallization SPSc

WPSa

IPSb

Rapid crystallization

Bo Em Bm [Bo ]

Bo Bm A2 Em C

C Asuper Bm

Asuper (C)

–

–

IPSb

Table 6.5 Crystal forms obtained from diverse crystallization methodologies. Predominant forms are in bold and trace forms between parentheses

–

–

Bm

C

–

–

SPSc

6 Chains 137

138

D. Mondieig et al.

Table 6.6 Measured temperatures and enthalpies of the solid–solid transition [109]. The differences between authors can be attributed to the purity of the samples, the erroneous nature of the solid phases involved, and the method used to determine the transition temperatures Solid–Solid transitiona C12 H24 O2c C14 H28 O2

C16 H32 O2

C18 H36 O2

C20 H40 O2

a The

H(kJ mol−1 )

Phase

T(K)

Asuper

308.1

References

B

283.1

A2

315.0(1.5)

1.8(0.4)

This work

Asuper

325.3(0.4)

6.4(0.7)

This work

A

317.1

B

298.1

A2

324.7(0.6)

2.6(0.7)

This work

Asuper

331.0(0.5)

7.6(0.5)

This work

A

332.1

Em

316.7(0.7)

3.1(0.2)c

This work

Bm

317.5(0.6)

4.9(0.4)

This work

B

313.1

A2

331.6(0.2)

A

327.1

A

337.1

Von Sydow [110] Clarck [111]

Stenhagen and von Sydow [112] Clarck [111]

46

Clarck [111] 2.8(0.3)b

This work Stenhagen and von Sydow [112] Sato and Kobayashi [113]

4.3(0.3)c

Em

327.4(0.6)

E

316.6

This work

Bm

324.4(0.7)

5.4(0.3)

Bo

325.9(0.5)

5.7(0.3)c

This work

B

324.1

5.1

Sato and M. Kobayashi [113]

B

328.1

Clarck [111]

B

319.1

Stenhagen and Sydow [112]

Em

332.8(0.4)

4.1(0.3)

This work

Bo

333.3(0.6)

6.1(0.2)

This work

Bm

332.6(0.4)

5.9(0.2)c

This work

B

325.1

Sato and Kobayashi [113] This work

Stenhagen and Sydow [112]

solid–solid transition involves the transformation from the room temperature form to the C

form b Enthalpy

obtained from one analysis. The error bar is estimated by comparison with the rest of results c The enthalpy is underestimated since traces of other form (E o/m or Bo/m ) are present in the samples analysed

6 Chains

139

Table 6.7 Melting temperatures and enthalpies measured on heating at normal pressure. The solid– liquid transition involves always the transformation from the C form to liquid phase [109] Solid–Liquid transition C10 H20 O2

T(K)

H(kJ.mol−1 )

Reference

303.8(0.6)

28.3(0.7)

This work

305

C12 H24 O2

C14 H28 O2

[100]

304.5(0.1)

28.0(0.1)

Schaake et al. [114]

304.7(0.1)

29.4(1.2)

Adriaansen [115]

316.2(0.4)

36.1(0.8)

This work

318

[100]

317.1

Stenhagen and von Sydow [112]

316.9(0.1)

36.3(0.1)

Schaake et al. [114]

317.2(0.1)

36.7(1.5)

Adriaansen [115]

326.5(0.5)

45.0(1.3)

This work

328

[100]

327.3

C16 H32 O2

Stenhagen and von Sydow [112]

327.3(0.1)

45.1(0.1)

Schaake et al. [114]

327.4(0.1)

44.7(1.8)

Adriaansen [115]

334.7(0.5)

53.0(1.0)

This work

335.6(0.1)

53.7(0.1)

Schaake et al. [114]

335.8(0.1)

53.4(2.1)

Adriaansen [115]

342.4(0.3)

63.2(1.4)

This work

342.5(0.1)

61.2(0.2)

Schaake et al. [114]

342.6(0.1)

63.0(2.5)

Adriaansen [115]

347.6(0.3)

71.6(1.6)

This work

348.2(0.1)

69.2(0.4)

Schaake et al. [114]

348.4(0.1)

72.0(2.8)

Adriaansen [115]

336.0

C18 H36 O2

Stenhagen and von Sydow [112]

342.8

C20 H40 O2

Stenhagen and von Sydow [112]

348.2

Stenhagen and von Sydow [112]

conformation (E and C), and it is very irregular for the B and A forms. The δip (O–C = O) band is very useful to identify the B form as it appears at an intense single band at 645 cm−1 only for the B form. The band is shifted to 686 cm−1 for the E form. A double band is observed for the triclinic forms at 683 and 638 cm−1 in agreement with the two different molecular conformations in the triclinic structures. For the C form, the occurrence of a dynamic conversion between two configurations cis and trans is observed in the δip(O–C = O) region. A broad double band associated with the C = O in plane deformation, δip (C = O), appears at 688 for the cis and at 668 cm−1 for the trans configuration. Cis–trans interconversion is promoting all low temperatures.

A’

A’

–

–

–

–

–

C13H26O2

C15H30O2

C15H30O2

C17H34O2

C19H38O2

C21H42O2

C23H46O2

SIII

Solid–solid transition SIII → SII

T

–

–

–

–

–

295.5 (5)

287.7 (6)

H

–

–

–

–

–

0.27 (7)

0.06 (1)

S

–

–

–

–

–

0.9 (2)

0.21 (4)

B’

B’

B’

B’

B’

A’h

A’h

SII

Solid–solid transition SII → SI T

349.9 (4)

344.6 (4)

339.0 (4)

331.2 (5)

321.9 (4)

320.8 (3)

309.1 (4)

H

2.5 (6)

5 (1)

7.4 (6)

7.5 (9)

8.2 (6)

–

8.5 (3)

7 (2)

13 (3)

22 (1)

23 (3)

25 (2)

–

27 (1)

S

Melting SI → Liquid

C’

C’

C’

C’

C”

C”

C”

SI

352.0 (5)

346.7 (5)

340.4 (3)

333.5 (5)

325.5 (4)

325.5 (4)

314.6 (5)

T

75 (3)

63 (3)

57 (1)

46.5 (9)

40.4 (6)

40.4 (6)

33 (1)

H

212 (9)

183 (9)

167 (3)

139 (3)

124 (2)

124 (2)

105 (3)

S

Table 6.8 Temperatures (K), enthalpies (kJ mol−1 ), and entropies (J mol−1 K−1 ) of the various phase changes transitions in odd carboxylic acids [102]

140 D. Mondieig et al.

6 Chains

141

Single crystals of carboxylic acids of appropriate quality and size for singlecrystal X-ray diffraction are scarcely obtained. For this reason, the crystal structures of carboxylic acids were determined mainly from powder X-ray diffraction data by Gbabode et al. in [98] who reported the high-temperature crystal structures of oddnumbered acids from C13COOH to C23COOH and Moreno et al. [116] who reported the structures of even acids from C10COOH to C20COOH. In addition, the crystal structures of C’ and C” forms of acids from C6COOH to C15COOH were determined from single-crystal X-ray diffraction data using in-situ crystallization technique by Bond in [96] and the structure of the C form of C16COOH was determined from single-crystal X-ray diffraction data by Moreno et al. [107]. In general, the structures of carboxylic acids polymorphs consist of acid molecules stacked into bilayers, with the saturated carbon chains in the all-trans conformation (except for the B and some triclinic forms, where the molecules adopt a gauche conformation), and with their carboxyl groups forming dimers through a typical R2 2 (8) hydrogen bond system. Dimers arrange so as to exhibit monolayers of pure methyl and pure carboxyl groups, except for some triclinic forms (Asuper, A1) for which both terminal groups coexist in the same monolayer face. As evidenced by Fig. 6.17, the crystal structures of the different polymorphs are very similar. Consequently, for a given acid, the derived physicochemical attributes such as morphology and latent heat storage capabilities are also similar. The high-temperature forms are monoclinic. The C form of even carboxylic acids has space group P21 /a and Z = 4. The C’ and C” forms of odd carboxylic acids have P21 /c with Z = 4 and C2/c with Z = 8, respectively. The C” structure is adopted by C13COOH and C15COOH while the C’ structure is adopted by the acids with n = 7, 9, 11, 17, 19, 21, and 23. The molecules in the high-temperature forms adopt the extended all-trans conformation. The main difference among the three structural types is found at the methyl group interface. The cell parameters and important structural data of the high-temperature phases gather in Tables 6.9, 6.10, 6.11 and 6.12.

Fig. 6.17 From left to right: crystal packing of the C, Asuper and A1 forms of C12COOH and the Bm, and Bo forms of C18COOH (Reproduced from [117] with permission of the author)

298

C20 H40 O2

8.31

8.25

–

298

298

9.97

–

270

298

10.55

298

–

270

C18 H36 O2

C16 H32 O2

C14 H28 O2

9.05

–

298

170

C12 H24 O2

8.90

298

C10 H20 O2

Rwp

T [K]

Acid

P21 /c

P21 /c

P21 /c

P21 /c

P21 /c

P21 /c

P21 /c

P21 /c

P21 /c

P21 /c

Space group

4

4

4

4

4

4

4

4

4

4

Z

44.12(1)

39.99(1)

35.620(11)

35.72(1)

31.559(3)

31.63(1)

27.563(2)

27.54(1)

22.8440(16)

23.10 (1)

a [Å]

4.965(1)

4.960(1)

4.9487(16)

4.975(1)

4.9652(5)

4.967(1)

4.9627(3)

4.953(1)

4.9612(3)

4.940 (1)

b [Å]

9.318(1)

9.354(1)

9.406(3)

9.439(1)

9.4260(11)

9.492(1)

9.5266(6)

9.604(1)

9.3977(6)

9.834 (1)

c [Å]

93.69(1)

95.00(1)

90.447(5)

90.39(1)

94.432(4)

95.10(1)

98.006(2)

97.28(1)

93.559(4)

90.01(1)

β[º]

This work

This work

Moreno et al. [118]

This work

[100]

This work

[100]

This work

[100]

This work

Reference

Table 6.9 Crystallographic data determined for the high-temperature phases of even-numbered carboxylic acids from C10COOH to C20COOH determined from powder X-ray diffraction data compared with single-crystal results [116]

142 D. Mondieig et al.

57.8

C20 H40 O2

3.74

3.74

3.97

62.7

63.3

3.59

3.98

55.7

64.7

3.79

55.1

3.98

62.0

C18 H36 O2

C16 H32 O2

C14 H28 O2

3.80

3.95

62.9

60.9

C12 H24 O2

3.71

63.3

C10 H20 O2

4.13

4.09

4.16

4.01

4.16

4.16

4.18

4.22

4.07

4.08

4.13

4.09

4.16

4.01

4.15

4.16

4.18

4.22

4.07

4.08

4

4

4

2

3

2

2

1

3

3

171

173

173

177

178

175

170

175

168

174

C–C–C = O O–H… O torsion angle [º] angle [º]

(0)-(3)

(0)-(1)

(0)-(2)

Carboxyl group interface

Tilt towards the (b, c) plane [º]

Acid

C… C distances at the methyl interface [Å]

2.63

2.64

2.62

2.70

2.63

2.65

2.63

2.63

2.63

2.64

O… O distance [Å]

3.05

3.05

3.05

2.96

3.05

2.98

3.07

2.95

3.00

3.08

C1… O1 Distance [Å]

This work

This work

Moreno et al. [118]

This work

[100]

This work

[100]

This work

[100]

This work

Reference

Table 6.10 Selected structural data within (carboxyl interface) and between (methyl interfaces) the layers of the high-temperature phases of even carboxylic acids from C10COOH to C20COOH determined from powder X-ray diffraction data [116]

6 Chains 143

333

340

346.4

351.7

C21 H42 O2

C23 H46 O2

0.029

0.035

0.035

0.034

–

320

C19 H38 O2

0.033

–

310

324

0.031

–

300

313

–

Rwp

296

T [K]

C17 H34 O2

C15 H30 O2

C13 H26 O2

C11 H22 O2

P21 /a

P21 /a

P21 /a

P21 /a

A2/a

A2/a

A2/a

A2/a

P21 /a

P21 /a

Space group

4

4

4

4

8

8

8

8

4

4

Z

9.488(1)

9.526(1)

9.575(1)

9.634(2)

9.720(1)

9.724(2)

9.812(1)

9.854(1)

10.029(1)

9.62(3)

a [Å]

4.957(1)

4.957(1)

4.953(1)

4.953(1)

4.956(1)

4.952(1)

4.943(1)

4.940(1)

4.912(1)

4.92(1)

b [Å]

65.23(1)

60.18(1)

54.99(1)

50.03(1)

84.02(1)

84.02(3)

73.74(1)

73.66(1)

35.10(1)

34.2(1)

c [Å]

127.63(1)

128.10(1)

128.72(1)

129.44(1)

125.29(1)

125.30(1)

125.88(1)

125.95(1)

133.36(1)

131.3(3)

β [°]

This work

This work

This work

This work

[100]

This work

[100]

This work

[100]

von Sydow (1955) 191, 919, 551, 919, 551, 955

Reference

Table 6.11 Crystallographic data determined for the high-temperature solid forms of the odd-numbered fatty acids from C13COOH to C23COOH together with those found in the literature. The non-standard space groups settings A2/a and P21 /a have been chosen for convenience, so that the chain axis is almost parallel to the c axis [98]

144 D. Mondieig et al.

0.041

–

0.047

–

0.054

0.053

0.055

0.050

C13 H26 O2

C13 H26 O2 *

C15 H30 O2

C15 H30 O2 *

C17 H34 O2

C19 H38 O2

C21 H42 O2

C23 H46 O2

Rwp

124.6 (1)

124.6 (1)

124.9 (2)

125.2 (3)

–

125.2 (2)

–

125.8 (3)

89.62 (4)

89.30 (3)

89.49 (5)

89.91 (6)

–

90.71 (4)

–

88.92 (6)

Towards a axis Towards b axis

Angle of tilt of the alkyl chain [°]

4

9

13

7

179

4

179

3

C–C-C-O torsion [°]

159

152

153

161

156

164

177

165

O–H···O angle [°]

2.70 (7)

2.60 (5)

2.62 (7)

2.72 (9)

2.63

2.55 (4)

2.63

2.54 (4)

O···O length [Å]

Packing arrangement of the carboxyl groups

2.74

2.83

2.49

2.51

–

2.73

–

2.70

(0)-(1) and (0)-(2)

2.80

2.95

2.69

2.62

–

2.77

–

2.82

2.84

3.09

3.01

2.84

–

2.83

–

2.90

(0)-(3)

3.07

2.93

2.92

2.74

2.60

2.62

2.51

2.58

H···H lengths at the methyl group interface [Å]

Table 6.12 Selected structural data determined from the crystal structures of the C” [96, 98] and C’ forms of the six fatty acids [98]

6 Chains 145

146

D. Mondieig et al.

The low-temperature forms of the even subfamily are of three types: forms of the type E, B, and triclinic. On the one hand, the B and E forms show conformational polymorphism. The molecules in the E form adopt all-trans conformation while the molecules in the B form adopt gauche conformation around the C 2 -C 3 bond. On the other hand, the B and E crystal forms of even n-carboxylic acids exhibit polytypism, which arises from a different stacking sequence of the bilayers. The monoclinic polytype, Em or Bm , has a single bilayered structure, space group P21 /c and Z = 4. The orthorhombic polytype, Eo or Bo, has a double bilayered structure (Pbca, Z = 8). Four different types of triclinic forms named A1 , A2 , A3, and Asuper exist for even acids. The Asuper form was first described by Goto and coworkes, it has space group P 1¯ with Z = 6. Or according to the usual description—in terms of the chainpacking of dimers—Asuper form has A1¯ space group, with Z = 12. The 12 molecules are arranged such that, within a monolayer, three adjacent molecules have their carboxyl groups in one direction, and the next three molecules have their carboxyl groups in the opposite direction. In the asymmetric unit, molecules show the all-trans conformation and other type of conformation in which the C 1 –C 2 bond is rotated so that the carboxyl group plane and the hydrocarbonated chain plane are almost perpendicular. Lomer elucidated the structure of the form A1 for acid dodecanoic (C12). The A1 structure consists of two molecules in the unit cell with P 1¯ space group. The crystal structure idf characterized for gathering into the same monolayer the two terminal groups of the molecules: the carboxyl group of one molecule is next to the methyl terminal group of the other molecule. Having adjacent and successive carboxyl and methyl groups in the same monolayer is a common and unique characteristic of the ¯ Z = 2) and Asuper (A1, ¯ Z = 12) structures. A1 (P 1, The A2 form has a triclinic cell and follows the typical structure characteristic of the high-temperature form in which each monolayer contains only carboxylic or methyl groups. The peculiarity of the A2 lies in that molecules show two different types of conformations of the carboxyl group. The A2 form undergoes a reversible transition into the A3 form at temperatures below 140 k. The cell parameters of the low-temperature polymorphs of even saturated carboxylic acids are listed in Table 6.13. The low-temperature forms of the odd subfamily are the so-called A’, B’, and D’ forms, all of them show all-trans molecules packed in the typical bilayer stacked way. The A’ form of C15COOH acid was determined in the fifties by Von Sydow. Twenty-five years later, the structure of C13OOH acid was determined by Goto in 1984 [95], and in the late nineties, it was observed that the A’ form was different at temperatures below and above 293 K. They called this forms the A’h (A’ high) and A’l (A’ low) [117]. So, under certain conditions, the A’l phase undergoes a rapid reversible solid-state transition to the A’h phase on heating around 293 K. The Ah ’ form is triclinic P 1¯ with Z = 2. The B’ form was determined by Gbabode in 2006 [97]. B’ is triclinic with space group P 1¯ and Z = 4. The lateral packing of the hydrocarbon chains in the bilayer is described by the O⊥ subcell. The existence of the D’ form was revealed by F. Francis in the thirties. However, there is no literature

6 Chains

147

Table 6.13 Crystallographic data determined for the low-temperature phases of even-numbered carboxylic acids [118] a (Å)

b (Å)

c (Å)

α (°)

β (°)

γ (°)

Source

Polymorph C, P21/c Z = 4 C10 H20 O2

23.10(1) 4.940(1)

9.834(1)

90

90.01(1) 90

Powder

C12 H24 O2

27.54(1) 4.953(1)

9.604(1)

90

97.28(1) 90

Powder

C14 H28 O2

31.63(1) 4.967(1)

9.492(1)

90

95.10(1) 90

Powder

C16 H32 O2

35.72(1) 4.975(1)

9.439(1)

90

90.39(1) 90

Powder

C18 H36 O2

39.99(1) 4.960(1)

9.354(1)

90

95.00(1) 90

Powder

C20 H40 O2

44.12(1) 4.965(1)

9.318(1)

90

93.69(1) 90

Powder

Polymorph Bm, P21/c Z = 4 C16 H32 O2

39.44(1) 7.410(1)

5.589(1)

90

92.96(1) 90

Powder

C18 H36 O2

43.95(2) 7.397(1)

5.598(1)

90

90.31(1) 90

Powder

C20 H40 O2

48.43(1) 7.404(1)

5.582(1)

90

93.11(1) 90

Powder

Polymorph Em, P21/c Z = 4 C16 H32 O2

39.72(1) 7.374(1)

5.614(1)

90

90.13(1) 90

Powder

C18 H36 O2

44.26(1) 7.386(1)

5.608(1)

90

93.25(1) 90

Powder

C20 H40 O2

48.76(1) 7.377(1)

5.603(1)

90

90.88(1) 90

Powder

Polymorph Bo, Pbca Z = 8 C18 H36 O2

7.408(3) 5.587(3)

87.69(6)

90

90

90

Powder

C20 H40 O2

7.414(1) 5.592(1)

96.74(1)

90

90

90

Powder

88.41(5)

90

90

90

Single crystal

Polymorph Eo, Pbca Z = 8 C18 H36 O2

7.359(3) 5.609(1)

Polymorph Asuper, P1 Z = 6 C12 H24 O2

5.415(1) 17.900(4) 21.370(4) 111.10(3) 97.03(2) 90.63(4) Single crystal

Polymorph A1, P1 Z = 2 C12 H24 O2

5.400(2) 7.450(3)

15.989(5) 88.57(4)

86.73(3) 98.30(1) Single crystal

concerning this solid form since Gbabode [98] crystallized it again for acids from C15COOH to C23COOH on applying mechanical pressure into samples of the B’ form. He also found that the D’ form disappears upon heating the samples. The cell parameters and relevant structural data of the low-temperature polymorphs of even saturated carboxylic acids are listed in Tables 6.14 and 6.15. An analysis of the crystallographic data showed a linear evolution of the parameter d100 that represents the thickness of a double layer of molecules, upon increasing the chain length in even and odd subfamilies. In contract, the b and c parameters can be considered constant with mean values of b equal to 4.95(1) Å and c equal to 9.25(3) Å and 9.44(1) Å for even and odd-numbered acids, respectively. Consequently, the C, C’ and C” forms are isostructural for acids between C10 and C20. Exhaustive analysis of the cell parameters and structural features of the lowtemperature forms Em , Bm, and Bo of the even subfamily and the B’ forms of the odd

C’17

C”15

B’17

B’15

r. t 324 320 333

Gbabode et al. (2007) monoclinic A2/a Z = 8

Bond [100] monoclinic A2/a Z = 8

Gbabode et al. (2007) monoclinic P21 /a Z = 4

293

r. t

293

T(K)

Goto and Asada (1984) triclinic P1¯ Z = 4

Gbabode et al. [120] triclinic P1¯ Z = 4

Sydow (1954) triclinic P1¯ Z = 4

Gbabode et al. [120] triclinic P 1¯ Z = 4

Reference

0.974(1)

0.975(1)

0.975(1)

1.017(1)

1.018(1)

1.018(13)

1.020(1)

Density (g cm−3 )

9.634(2)

9.720(1)

9.723(2)

5.561(1)

5.547(1)

5.54(1)

5.552(1)

a(Å)

4.953(1)

4.956(1)

4.951(1)

8.018(1)

8.036(1)

8.06(3)

8.045(1)

b(Å)

50.03(1)

84.02(1)

84.01(3)

47.90(1)

47.89(1)

42.6(1)

42.84(1)

c(Å)

90

90

90

114.18(1)

114.37(2)

114.3(2)

114.78(1)

α(°)

Table 6.14 Crystallographic data of the solid forms exhibited by the C15COOH and the C17COOH [119] (r.t.: room temperature)

129.44(1)

125.29(1)

125.30(1)

114.96(1)

114.91(2)

114.2(2)

114.76(1)

β(°)

90

90

90

80.22(1)

80.18(1)

80.6(1)

80.05(1)

γ(°)

148 D. Mondieig et al.

6 Chains

149

Table 6.15 Structural data determined from the structures of the B’ form of C15COOH [119], C17COOH [95] and C19COOH [97] Reference C15 H30 O2 C17 H34 O2 C19 H38 O2

Gbabode et al. [120] Goto and Asada (1984) Gbabode et al. (2006)

C–C-C-O torsion angle(°)

O–H···O angle(°)

O···O length(Å)

(i)

179

173

2.63

(ii)

175

163

2.58

(i)

162

–

2.64

(ii)

177

–

2.62

(i)

177

174

2.65

(ii)

161

164

2.65

subfamily reported elsewhere [48, 108] evidenced a high degree of structural similarity among the different polymorphs. In consequence, the exceptional similarity observed in the crystal packings is translated into the exceptional similar morphologies observed for the different polymorphs (Fig. 6.18). The observed morphologies have been rationalized on the basis of their crystal packing and attachment energies upon crystal growing [118]. In summary, crystal forms C, Bo/m, and Eo/m, with plate-like morphology, have the carboxyl groups and methyl groups in different planes parallel to the basal plane. The plane containing the methyl groups correspond to the faces with the lowest attachment energy, since only weak van der Waals interactions exist. These faces develop slower and therefore have the most morphological importance. On the contrary, in the Asuper form each terminal plane contains both carboxyl and methyl terminal groups which is going to cause strong hydrogen bond interactions along a direction rather than in a plane given rise to long fibers. It has been found that crystal morphology in CA is not solvent dependent, probably because

Fig. 6.18 a SEM images of A2 and Asuper forms C form of C16 H31 O2 H, b a crystal of E form of C16 H31 O2 H, and c a crystal of B form of C18 H35 O2 H. Note A2 form has needle-like shape and dendrite growth. The Asuper form crystallizes as long fibers. C shows the plate-like crystals with prismatic shape and an acute angle of around 54º and similar crystals but with a larger acute angle (around 74º) are characteristic of the Eo/m and the Bo/m forms (Reproduced from [108] with permission of the author)

150

D. Mondieig et al.

Table 6.16 Results of the solvent-mediated transformation experiments between A2 , Em , Bm, and C forms of C16COOH [108] Initial forms

Experimental conditions

Final forms Ethanol

Pentane

Toluene

Mixture A2 , Em, C

12 days / 25 ºC

Asuper

Asuper

Asuper

Mixture Bm , Em

12 days / 25 ºC

Asuper

Evaporated

Asuper + C

Mixture Bm, C

12 days / 25 ºC

Asuper

Asuper

Solid dissolved

C

12 days / 25 ºC

Asuper

Evaporated

Asuper

as shown by the predicted morphologies, in this case there should be no large differences in the solvent–solute intermolecular interactions at the various crystal faces because all faces are formed by hydrophobic C-H sites. According to the empiric Burger and Rabemberg rules [121], in particular the density rule, for acids between C10COOH and C20COOH the C form is metastable at room temperature. This was also confirmed experimentally on the basis of the Ostwald law using the so-called solvent-mediated transformation experiments. This method is based on the relationship between solubility and stability of crystal forms: The less thermodynamically stable form will also be the most soluble at given conditions of temperature and pressure. If crystals of different forms are mixed with a saturated solution of the product, the most stable form will growth at expenses of the less stable one. Results of the experiments performed by Moreno [108] (Table 6.16) confirmed that the triclinic Asuper form represents the thermodynamically most stable form at room temperature for acid C16COOH. The rest of the forms being metastable and with a very slow kinetic of transformation in the solid state since they do not transform in the solid state at least for 5 years at 6ºC or two years at 25ºC. The C form being the stable form before melting. The results observed for C16COOH can be extrapolated shorter acids since they have shown very similar polymorphic behavior. For longer acids from C18COOH to C22COOH, however, Sato and co-workers [122] studied the relative stability of the crystal forms by means of solubility experiments and concluded that B is the stable form below a certain temperature and C form is the stable form above this temperature.

6.9 Solid-State Miscibility Some binary systems among n-carboxylic acids with n = 1 or 2 have been extensively discussed by Gbabode et al. [119, 123] for odd acids, and Fucks [124] and Feldman et al. [120, 125] for even n-carboxylic acids with n = 2: C10COOH– C12COOH, C16COOH–C18COOH, n = 4: C12COOH–C16COOH, and n = 6: C12COOH–C18COOH. The binary phase diagrams corresponding to even acids are characterized by the formation of a single eutectic and a region with an incongruently

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melting solid-state compound (Fig. 6.19). The incongruent compound transforms during melting in two phases; i.e., a higher melting solid phase and a liquid phase in a region containing an excess of the low molecular component. For even straightchain fatty acids which differ in chain length by two carbon atoms the eutectic occurs at approximately 73 mol % of the lower-melting component. The hump of the freezing curve, representing the component formation region, occurs at about 50 to 73 mol % of the lower-melting component. For even straight-chain fatty acids which differ by more than two carbon atoms in the chain length the eutectic location is shifted to a higher molar concentration of the lower-molecular weight component. For C12COOH–C16COOH binary system, the eutectic occurs at a concentration of 80 mol % C12COOH: 20 mol % C16COOH. For C12COOH–C18COOH binary system, the eutectic occurs at a concentration of 85 mol % C12COOH: 15 mol % C18COOH. Also, the compound formation region is displaced to a concentration range containing a higher molar concentration of the lower-molecular weight component. A complete study of the miscibility among odd n-carboxylic acids with n = 1 and n = 2 (C15COOH–C17COOH and C17COOH–C19COOH) is reported by Gbabode [48] and Gbabode et al. [119, 123]. In general terms, at low temperature the miscibility of the B’ form for the systems with n = 2 is very weak and do not exceed 10% in concentration. Two intermediate solid solutions are stabilized in

Fig. 6.19 General phase behavior of a binary system of even straight-chain acids. Reprinted from [125] Solar Energy Materials, 18:201–216 Feldman D, Shapiro MM, Banu D, Fucks CJ Fatty acids and their mixtures as phase change materials for thermal energy storage ©1989, with permission from Elsevier

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Fig. 6.20 Experimental phase diagram of the C15COOH–C17COOH (left) (Reprinted from [119], with permission from Elsevier) and C17COOH–C19COOH (right) binary system (Reproduced from [48] with permission of the author) Abscissa, mol fraction x of the acid with the shortest chain

narrow domains ( ξ A (ξ B + ξ A )

(7.1)

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Fig. 7.5 Equimolar excess enthalpy as a function of the mismatch parameter for the two-component systems involving the In, IIn, and IIIn series. The line represents the eq. H E, OD (X = 0.5)/J·mol−1 = 3526 m. (Reproduced with permission from ACS, Chem Mater 12:1108–1114) [17]

which contains the main characteristic of a mismatch parameter, i.e., it relates the relative difference between the values of a certain property, concerning the packing coefficient in this case, shown by the pure components of the system. The validity of the proposal mismatch parameter is shown in Fig. 7.5. It can be seen that although the mismatch correlates well for most of the two-component systems, it fails to describe some of them, in particular the system II2 + II3 , which concerns CI mixed crystals and the system I2 + I3 . To end with, it has been demonstrated that the group of systems the components of which pertain to the series I [(CH3 )4−n C(CH2 OH)n ] and series III [NO2 (CH3 )3−n C(CH2 OH)n ] showing face-centered cubic (CF ) mixed crystals, form a class of similar systems for which the excess enthalpy and excess entropy compensate at ca. 630 K. Moreover, it is also seen that the excess enthalpy and excess volume compensate at a pressure ca. 0.5 GPa. The relevant role played by the hydrogen bonds in the underlying systems makes that the mean number, n, of substituted –OH groups in the molecules of the series, a key parameter. Put into real numbers, such a role is expressed in terms of the packing coefficient which in turn allows the introduction of a mismatch parameter describing reasonably well the excess enthalpy of the involved two-component systems.

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7.3 Two-Component Systems Involving Components with Weak Intermolecular Forces Here we report on binary systems composed of globule-like molecules where the directional forces are far weaker than those due to excluded-volume interactions and therefore they constitute systems much closer to those able to be explored by theoretical models. The materials in question are the series of halogenomethanes, (CH3 )4−(m–n) CX m Y n (X, Y = F, Cl, Br, I, n, m = 0, …4). For a subgroup of these materials, the methylchloromethanes (CH3 )4−nCCln , referred as Cl:n, the rich polymorphic behavior at normal and high pressure has been extensively characterized and many of binary systems sharing them have been determined. In what follows we will briefly focus in such issues for the methylchloromethane compounds, although some systems concerning other halogenomethane (in particular containing bromide atom) will be enclosed. The compounds of Cl:n series, from Cl:0 (neopentane, C(CH3 )4 ) to Cl:4 (carbon tetrachloride, CCl4 ) exhibit solid–solid phase transitions. These thermally activated phase transitions are attributed to the ability of thermally activate rotational degrees of freedom within the crystalline state due to the weak intermolecular forces which enable the appearance of reorientational processes giving rise to the high-temperature OD (plastic) phase. As stated at the beginning of this chapter, CCl4 crystallizes to an OD FCC phase, and upon further cooling, it transforms to another OD rhombohedral (R) phase, which turns to a low-temperature monoclinic phase (C2/c, Z = 32) on further cooling. When cooling is limited such that an FCC phase is formed, this phase melts on heating without passing through the R phase. Similarly, when heated from phase R, a new melting point higher than that of phase FCC is obtained. This indicates that phase FCC is a metastable phase and, consequently, that the FCC-R transition has a monotropic character. On the contrary, FCC is the stable form of Cl:0 and Cl:1, while R is the stable form of Cl:2, Cl:3, and Cl:4. As for Cl:3 and Cl:4, FCC form is invariably metastable, but really appears for both compounds, as it does the SC (simple cubic) form for Cl:2. According to the symmetry of the stable and metastable OD phases, two types of phase diagrams (represented in Fig. 7.6) involving three phases α, β, and L can be distinguished. For the first case, Fig. 7.6a, the OD phase α is invariably metastable, for pure components and for mixed crystals, while the β phase is always the stable OD phase. Three two-component phase diagrams sharing Cl:2, Cl:3, and Cl:4 conform to Fig. 7.6a case are presented in Fig. 7.7 [14, 15, 18]. For the case represented in Fig. 7.6b, referred to as crossed isodimorphism, α form is stable for component A, whereas the other form β is metastable. It is worth mentioning that for component B β form is the stable whereas α form is metastable. As stated in a precedent chapter, the stable phase diagram can be considered as the stable result of two, each other crossing, solid–liquid loops and giving rise to eutectic or peritectic three-phase equilibrium. Due to the FCC lattice of the OD metastable phase for Cl:2, Cl:3, and Cl:4 and for the stable OD phase of Cl:1, two-component systems sharing them would appear as the general sketch represented in Fig. 7.6b [19–21].

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Fig. 7.6 Two representative different types of two-component systems involving two methyl chloromethane compounds. Solid and dashed lines represent stable and metastable equilibria, respectively

Examples of such binary diagrams are given in Fig. 7.8. It should be emphasized the relevant experimental finding concerning the extension of the [FCC+L] equilibrium in the Cl:1+Cl:4 binary system, which can be measured beyond the composition range where the FCC molecular alloys are stable. In particular, the equilibrium [FCC+L] was determined experimentally and continuously from X = 0 to X = 1. This finding means that the monotropic behavior of the CCl4 pure component FCC phase behaves similarly when molecular mixed crystals are formed. To illustrate the stability regions of FCC and R phases, Fig. 7.9a shows the Gibbs energy difference between those phases as a function of composition at 240.0 K: it nicely reveals that from X = 0 to X EGC = 0.633 FCC form is stable, while R form is metastable and, from X EGC = 0.662 to X = 1 R form is stable and FCC form metastable. The relative stability can be observed in Fig. 7.9b which represents the Gibbs energy functions for the R ∗,L and FCC mixed crystals, both with respect to (1 − X )μ∗,L A + X μ B , i.e., the linear contribution which does not change the relative stability described in Chap. 3. In order to apply the thermodynamic assessment to the two-phase equilibria, two kinds of procedure were undertaken. As far as the thermodynamic analysis of non-interfering two-phase equilibria experimentally determined, as those shown in Fig. 7.7 the thermodynamic properties, melting temperature and enthalpy change, are directly accessible from experiments. Nevertheless, for binary systems described by means of the crossed isodimorphism concept, as those depicted in Fig. 7.8, some of the pure component properties should be obtained when they are not experimentally accessible. These would concern, for example, the melting of the metastable R form

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Fig. 7.7 Stable (full symbols) and metastable (open symbols) equilibria for three phase diagrams sharing Cl:2, Cl:3, and Cl:4. Note that in the Cl:2+Cl:3 system (Fig. c) two kinds of metastable equilibria appear: SC mixed crystals for almost the whole range of composition and FCC, for a reduced range of composition. (Reproduced with permission from ACS, Chem Mater 17:6146– 6153) [22]

for Cl:1 (see Fig. 7.8 at the X = 0 side). Taking this metastable transition as a speaking example, the melting enthalpy change can be derived by combining the melting enthalpy changes from the R and FCC forms together with the transition enthalpy between them, related by the equation: H R→FCC + H FCC→L = H R→L

(7.2)

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Fig. 7.8 Three examples of phase diagrams showing crossed isodimorphism. Full and open symbols represent the stable and metastable equilibria, respectively. (Reproduced with permission from ACS, Chem Mater 17:6146–6153) [22]

Figure 7.10a illustrates the procedure which enables to get the melting enthalpy for the metastable R form of Cl:1. Such a procedure was used for different systems sharing Cl:1 in order to obtain coherent extrapolated values for the enthalpy change [22]. As far as the melting temperature of the metastable R form of Cl:1, Fig. 7.10b describes the thermodynamic coherence when the [R+L] two-phase equilibrium involving three different phase diagrams sharing Cl:1 component is considered. The melting temperatures and enthalpy changes are experimentally determined and those obtained by means of the described procedure are gathered in Table 7.3.

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Fig. 7.9 For the binary system Cl:1 + Cl:4: a Gibbs energy difference between R and FCC mixed crystals (at 240.0 K) and b Gibbs energy plus a linear contribution for R (continuous line) and FCC (dotted line) mixed crystals. The inset magnifies the circled region. (Reproduced with permission from RSC, Phys Chem Chem Phys 3:2644–2649) [19]

Fig. 7.10 a Experimental enthalpies of melting for the FCC (circles) and R (diamonds) mixed crystals and R to FCC transition (squares) for the Cl:1+Cl:3 binary system. Empty symbols correspond to calculated values according to Eq. 7.2. b Extrapolation of the metastable melting of R form for the Cl:1 through the [R+L] equilibrium for three different systems: Cl:1+Cl:2, Cl:1+Cl:3, and Cl:1+Cl:4. (Reproduced with permission from ACS, Chem Mater 17:6146–6153) [22]

It should be emphasized that the existence of isomorphism between two forms of two compounds must be demonstrated by means of the continuity of a physical parameter as a function of composition. Figure 7.11 provides examples for the lattice parameters as a function of composition of the OD FCC and R phases for two

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Table 7.3 Temperature (T) and enthalpy (H*) and entropy (S* ) changes of melting of the methylchloromethanes from their different polymorphs Substance

Melting form

T/K

H* /kJ·mol−1

S* /J·mol−1 ·K−1

Cl:0

FCCS

256.8a

3.09a

12.03a

Rm

220.0c

2.35c

10.68c

FCCS

248.1a

1.76a

7.09a

Rm

234.0b

2.60b

11.11b

RS

236.6a

2.30a

9.72a

SCm

230.4a

1.93a

8.25a

FCCm

224.0b

0.86b

3.84b

RS

241.9a

2.42a

10.01a

SCm

237.1b

2.35b

9.91b

FCCm

235.8a

1.55a

6.57a

RS

250.3a

2.52a

10.07a

SCm

245.8b

1.82b

7.40b

FCCm

244.0a

1.80a

7.38a

Cl:1 Cl:2

Cl:3

Cl:4

a Experimental

values. b Values obtained from the extrapolation of experimental values. c Values obtained from the thermodynamic assessment. The stable and metastable phases are denoted by superscript “s” or “m”, respectively (Reproduced with permission from ACS, Chem Mater 17:6146– 6153) [22])

Fig. 7.11 FCC (a and c) and rhombohedral (b and d) lattice parameters as a function of composition for the Cl:1+Cl:3 and Cl:2+Cl:4 binary systems. (Reproduced with permission from ACS, Chem Mater 17:6146–6153) [22]

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Table 7.4 Excess enthalpy and excess Gibbs energy changes at melting, expressed in the form of the two Redlich–Kister coefficients, obtained either experimentally or by thermodynamic phase diagram analysis, respectively ( Reproduced with permission from ACS, Chem Mater 17:6146–6153) [22]) Experimental System

Form

Phase diagram analysis

H1

H2

T EGC (X = 0.5)

G1

G2

/kJ·mol−1

/kJ·mol−1

/K

/J·mol−1

/J·mol−1

R

−1.43

0.66

230.0

−56

449

FCC

−2.70

0.40

237.0

−602

9

R

−0.20

−0.10

234.7

−6

−62

FCC

−0.77

1.62

232.6

−154

−40

Cl:1+Cl:3

R

−0.67

−0.67

237.9

28

−16

FCC

−1.75

0.19

232.6

−263

−62

Cl:1+Cl:4

R

−1.48

0.11

238.0

−100

−25

FCC

−0.68

−0.18

242.0

−141

−66

R

0.10

0.04

239.6

14

−9

FCC

0.00

0.00

234.4

−18

0

−0.40

0.34

242.0

−67

13

Cl:0+Cl:4 Cl:1+Cl:2

Cl:2+Cl:3 Cl:2+Cl:4

R

Cl:3+Cl:4

R

FCC FCC

0.70

0.05

238.6

14

8

−0.23

−0.13

246.2

−20

54

0.15

0.23

243.2

29

55

kind of systems: Cl:1+Cl:3 showing crossed isodimorphism and Cl:2+Cl:4 showing metastable ([FCC+L]) and stable ([R+L])-independent equilibria (see Fig. 7.8). The experimentally determined melting enthalpy changes as a function of composition (as those shown in Fig. 7.10) provide the values of the excess enthalpy differences between the involved OD phases and the liquid state. The so obtained values are collected in Table 7.4. The thermodynamic assessment of each individual twocomponent phase diagram conducted by means of the EGC procedure, enabled to determine the excess Gibbs energy at TEGC . For this proposal, excess Gibbs energies were described in terms of two coefficients of the Redlich–Kister polynomial, which in the absence of strong local anomalies is fairly adequate and has physically understandable coefficients. The values are gathered in Table 7.4. These set of values, together with those corresponding to the thermodynamic properties of the melting of the different forms for pure compounds compiled in Table 7.3 provide the required information to calculate the phase diagrams. For the collected systems in Table 7.4, temperature differences between experimental and calculated values are within the 1–2 K, which is certainly not far away from the experimental error limits. For many of the systems sharing methylchloromethane compounds, the excess properties in the liquid state have been measured and, thus the set of data offered an extraordinary opportunity to go deeper into the thermodynamic analysis of the OD mixed crystals. The available data concerning the excess enthalpy and excess Gibbs energy in the liquid state are included in Table 7.5. The excess properties of the liquid

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Table 7.5 Enthalpy and Gibbs energy excess properties of the liquid mixtures (at the given temperature) in the form of the two Redlich–Kister coefficients ( Reproduced with permission from ACS, Chem Mater 17:6146–6153) [22]) System

H1/J mol−1 H2/J mol−1 T/K

Cl:0+Cl:4 1550

0

Cl:1+Cl:2 44

Refs G1 /J mol−1 G2 /J mol−1 T/K

Ref.

293.15 [23] 1380

15

273.15 [24]

298.15 [25] 89

122

298.15 [26]

298.15 [25] 186

−198

298.15 [26]

298.15 [27] 812

363

298.15 [28]

−7

298.15 [25] 72

3

298.15 [26]

−317

298.15 [25] 547

132

298.15 [26]

−115

298.15 [27] 330

−143

298.15 [29]

Cl:1+Cl:3 10 Cl:1+Cl:4 210

−338

Cl:2+Cl:3 23 Cl:2+Cl:4 541 Cl:3+Cl:4 437

state enable the access to the particular excess properties of the individual systems, all of them collected in Table 7.6 as two-parameter Redlich—Kister expressions.

7.3.1 The ABθ Model 7.3.1.1

Compensation Temperature θ

Within the framework of the ABθ model, the compensation temperature θ is the ratio of the excess enthalpy to the excess entropy. Under the assumption that excess enthalpy and excess entropy are temperature independent, the aforementioned quotient is equal to the quotient of H1 OD and S1 OD . Figure 7.12 shows the values of H1 OD versus their corresponding S1 OD (the factor “4” in both magnitudes simply scales both magnitudes to the value of the excess property at the equimolar composition) reported in Table 7.6. Figure 7.12 nicely shows, within the error limits, that the excess properties, enthalpy and entropy, correlate in a linear form, so establishing the existence of a common compensation temperature (θ≈430 K) for the whole of binary systems. Because the mean of T EGC for this set of systems is 238 K, it can be easily seen that logθ/logTEGC is around 1.10, which means that the OD mixed crystals sharing methylchloromethane compounds are within the long-time phenomenologically stablished trend [30]. 7.3.1.2

The Magnitude of A: Finding a Mismatch Parameter

As stated before, methlychloromethane is composed by globule-like molecules where the directional forces are far weaker than those due to excluded-volume interactions and therefore they constitute systems mainly controlled by van der Waals interactions. Thus, stearic factors are known to play here the relevant key parameters. On these grounds, a mismatch parameter m, defined as m = (v B − v A )/vm , where vi is the molar volume of the component i = A, B, and vm , is their mean, has been used to

−181

−679

230

−128

−227

148

55

343

−398

703

0

0

37

110

−1475

101

−671

191

−673

−1749

−103

1618

−196

−774

656

396

−1428

−5.97

0.49

−0.84

2.89

−1.37

0.08

0.36

−2.22

−5.78

−6.39

−2.94

−2.66

−0.81

−8.86

0.90

0.72

−0.74

0.19

1.36

0.00

0.19

−0.48

0.57

1.08

−2.75

7.13

−0.18

1.63

437

541

23

210

10

44

1550

(Reproduced with permission from ACS, Chem Mater 17:6146–6153) [22])

FCC

Cl:3+Cl:4 R

FCC

Cl:2+Cl:4 R

FCC

Cl:2+Cl:3 R

FCC

Cl:1+Cl:4 R

FCC

Cl:1+Cl:3 R

FCC

Cl:1+Cl:2 R

FCC

−2703

Solid H1 L−OD H2 L−OD S1 L−OD S2 L−OD H1L H2L form /J·mol−1 −1 −1 −1 /J·mol ·K /J·mol

Cl:0+Cl:4 R

System

−0.57

−0.15

0.62

−0.16

−115 0.36

−317 −0.02

−7

−338 −2.02

0

0

0

/J·mol−1 ·K−1

S1 L

S2 L

683 1757

0.09

1.51

280

−345

13

−372

−162 664

−660

7

−44

−156

−477

−191

671

−16.8

103

−396

−656

H2OD

939

23

−0.06 −78

888

−2.35 1684

0.66

818

−0.44 240

4253

−0.05 2978

/J·mol−1

H1OD

−0.95

−0.13

1.20

−2.91

−0.63

0.83

1.70

−2.87

−0.03 1.35

−0.24

1.87

−2.91

−0.42

3.41

−7.57

−0.26

−1.68

−0.22

S2 OD

−0.52

0.2

3.76

5.80

2.35

2.51

0.66

9.48

6.59

/J·mol−1 ·K−1

S1 OD

Table 7.6 Excess enthalpy and excess entropy changes at melting of the R and FCC forms involved in the methylchloromethane compounds and excess enthalpy and entropy in the liquid phase and in the R and FCC forms, all of them in the form of two Redlich–Kister coefficients

180 J. Ll. Tamarit et al.

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Fig. 7.12 Excess enthalpies of the OD R (diamonds) and FCC (squares) mixed crystals as a function of the corresponding excess entropies for the systems involving methylchloromethane compounds at the equimolar composition. (Reproduced with permission from ACS, Chem Mater 17:6146–6153) [22]

parameterize the excess enthalpy of the individual systems. Similar mismatch parameters have been used for other families of systems for which, as in the present case, components of the binary phase diagrams belong to a chemically coherent group, as alkali halides or n-alkanes [31, 32]. For methylchloromethane mixed crystals, correlation between mismatch parameter m and the excess enthalpy can be represented, according to the line in Fig. 7.13, by Eq. 7.3 A(m)/kJ mol−1 = −0.7 + 26 m

(7.3)

Fig. 7.13 Excess enthalpies of the OD R (diamonds) and FCC (squares) mixed crystals at the equimolar concentration against the mismatch parameter. Line represents the eq. A(m)/kJ mol−1 = −0.7 + 26·m. Inset concerns exclusively the rhombohedral mixed crystals. (Reproduced with permission from ACS, Chem Mater 17:6146–6153) [22]

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The negative value of A for m = 0 shows up the attractive effect between methyl and halogen (Cl in the present case) of the different A and B molecules shared in the mixed crystal. By inspecting details in Fig. 7.13a fine tuning concerning the excess enthalpy of the OD rhombohedral mixed crystals can be achieved (see inset). It can be seen that systems involving “dipole dilution”, i.e., those sharing Cl:4 (devoid of dipole) and Cl:1, Cl:2, or Cl:3 have systematically higher values of the excess enthalpy. On the contrary, systems in which the molecules of both components show a dipole moment seem to lie below. Such behavior has been rationalized by dividing the excess enthalpy contribution in two terms, one coming from stearic effects and a second one accounting from the dipole dilution contribution. Details can be found in reference [22].

7.4 Pressure–temperature Phase Diagrams and Their Relation with Two-Component Systems at Normal Pressure Some of the binary systems sharing methylchloromethane compounds are nice examples of crossed isodimorphism. In systems like Cl:1+Cl:4, Cl:1+Cl:3, Cl:1+Cl:2,or Cl:2+Cl:4, it has been possible to measure not only the thermodynamic properties of the metastable phases (read FCC) for the pure components, but also those of the metastable mixed crystals. Such an experimental advantage is due to the extension of the monotropic behavior of the metastable OD FCC phases of the pure components to the mixed crystals, something that is quite exceptional. In some cases for which the metastable thermodynamic properties of pure components cannot be measured, the formation of large composition domains of mixed crystals enables to get the non-experimentally available properties with high accuracy (as, for example, the thermodynamic properties of the rhombohedral phase of Cl:1 compound, see Fig. 7.10. The set of thermodynamic properties for the different (stable and metastable) phases can be used to build up the topological pressure–temperature phase diagrams of pure compounds and compare them with the experimentally determined pressure–temperature phase diagrams. An explicit case to highlight this scenario appears in the two-component system sharing Cl:1 and CBrCl3 . As far as CBrCl3 is concerned, it is well known that it exhibits two stable OD phases at normal pressure, FCC and R. According to the experimental pressure–temperature phase diagram [33], it has been shown that the former OD phase exhibits a narrow (equilibrium) pressure range, as it is shown in Fig. 7.14c. The polymorphic behavior of some methylchloromethane and, in general, halogenomethane, is sketched in Fig. 7.14. Thus, for example, Fig. 7.14b, which represents the case of Cl:4 [34, 35], shows that the cubic phase is metastable and consequently behaves monotropically with respect to the R phase whatever the pressure is (i.e., displaying “overall monotropy”). For CBr2 Cl2 and CBr4 compounds,

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Fig. 7.14 Representative sketch of the temperature–pressure phase diagrams for the series of methylchloromethane compounds. Stable and metastable two-phase equilibria are denoted by continuous and dashed lines, respectively. a Cl:1, b Cl:2, Cl:3, and Cl:4 (for Cl:2, the symmetry of the metastable phase is SC), and for halogenomethane compounds c CBrCl3 and d CBr2 Cl2 and CBr4 . L represents the liquid phase and R and FCC the OD phases, respectively. T stands for the tetragonal phase (space group P4/nmm) whereas LT stands for the low-temperature ordered phases. (Reproduced with permission from Elsevier, Chem Phys 358:156–160) [38]

the OD R phase does not appear at normal pressure, but such a phase makes its real physical appearance at high-pressure, as it is shown in Fig. 7.14d [36, 37]. Figure 7.15 shows in the central panel the Cl:1+CBrCl3 binary phase diagram [38]. What is remarkable in this two-component system is the experimental emergence of a maximum and a minimum for the [R+FCC] two-phase equilibrium, since very few systems show a solid–solid equilibrium with those extremes. A similar behavior has been also found for the binary system Cl:4+CBr(CH3 )3 [39]. The left panel in Fig. 7.15 reports on the pressure–temperature phase diagram of Cl:1 [40]. Here, in addition to the OD FCC phase previously reported at normal pressure, an OD phase appears at high pressure, the symmetry of which was unknown at the time of the

Fig. 7.15 Two-component system Cl:1+CBrCl3 at normal pressure including the extrapolated equilibria from the thermodynamic assessment (central panel). The inset shows details for the lowtemperature extrapolated [R+FCC] two-phase equilibria at X = 0. Left and right panels show the corresponding pressure–temperature phase diagrams of the pure compounds Cl:1 (left) and CBrCl3 (right) compounds. Dotted line for the [R+FCC] transition line of the Cl:1 pressure–temperature phase diagram nicely corresponds to the extrapolation of the high-pressure two-phase equilibria at normal pressure given the metastable R to FCC phase transition for Cl:1. (Reproduced with permission from Elsevier, Chem Phys 358:156–160) [38]

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Fig. 7.16 Melting enthalpies for the FCC (open circles) and for the R to FCC transition (open squares) as a function of composition for the two-component system Cl:1+CBrCl3 obtained experimentally. The full square at X = 0 corresponds to the extrapolated enthalpy value for the R to FCC metastable transition of Cl:1 (1.07 ± 0.05 kJ mol−1 ). (Reproduced with permission from Elsevier, Chem Phys 358:156–160) [38]

p–T phase diagram determination. In addition, it can be seen that the OD FCC phase transforms, on decreasing temperature, to a tetragonal (P4/nmm) phase (called T). At lower temperature this tetragonal phase transforms to a monoclinic (P21 /m) phase. The right panel sketches the pressure–temperature phase diagram for CBrCl3 , with the low-temperature monoclinic (C2/c) phase. The equilibria involving OD R and FCC phases in the Cl:1+CBrCl3 phase diagram can be treated by means of the crossed isodimorphism in order to perform the thermodynamic assessment. The properties for the transition from the metastable OD R to the stable FCC phases for Cl:1 are needed. They can be obtained either by trial and error (as it was generally done for the systems sharing compounds of the series In, IIn, and IIIn) or by extrapolation when possible. Figure 7.16 shows that, due to the large concentration domain of OD mixed crystals, the enthalpy change from R to FCC for Cl:1 can be easily obtained with high accuracy. As for the transition temperature, TR→FCC , virtually the same value can be obtained by means of two different ways: (i) extrapolation of the EGC temperature of the [R+FCC] two-phase equilibria at X = 0 provides the value of 208.0 ± 1.0 K. Such a procedure is depicted as an inset in the central panel of Fig. 7.15, (ii) as can be seen in the pressure–temperature phase diagram of Cl:1 (left panel of Fig. 7.15), under the assumption that the high-pressure phase of Cl:1 corresponds to the OD R phase, the extrapolation of the [R+FCC] two-phase equilibrium at normal pressure provides the same value, thus making thermodynamically coherent the hypothesis. Additional arguments can be given for the assignment of the OD R character to the (initially) unknown high-pressure phase of Cl:1. On the one hand, the previously calculated topological phase diagram of Cl:1 [41] provides for the triple point [R+FCC+vapour] a temperature value of 208.7 K, which corresponds to the normal pressure transition point from R to FCC. On the second hand, Wilmers et al. [40] determined the volume changes for the R to FCC transition as a function of pressure.

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If these reported values (at normal pressure) are extrapolated at normal pressure, we v(0.1 MPa) = 1.95 cm3 ·mol−1 . With this extrapoobtain a volume change of  FCC R lated value into the Clapeyron equation applied to the R to FCC transition, together with the aforementioned corresponding temperature, TR→FCC = 208.7 K, and with the slope of the pressure–temperature phase diagram of the R to FCC transition, an enthalpy change at normal pressure of 1.05 ± 0.10 kJ mol−1 is obtained. This value perfectly matches the value obtained by extrapolation as a function of composition, 1.07 ± 0.05 kJ mol−1 (see Fig. 7.16). Moreover, and finally, it is worth highlighting that the orientationally disordered character of the high-pressure R phase of Cl:1 was clearly revealed through the high values of the static permittivity [40]. The exercise we have done with Cl:1 reveals the irrefutable connection between extrapolated phase transitions appearing as “virtual” phases within the two-phase equilibria emerging in the two-component systems with those obtained through the extrapolation of the two-phase equilibria (transition lines) present in the temperature– pressure phase diagrams. This is an important result because it implies that either from the two-component phase diagrams or from the temperature–pressure phase diagrams, thermodynamic information (seemingly hidden) can be obtained and new phases, which would make its physical appearance somewhere in the temperature– pressure space (at positive, as the present case, or negative pressure [42, 43]), can be predicted and characterized.

7.5 Concluding Remarks The existence of polymorphism in organic compounds considered in this chapter, i.e., compounds displaying at least one orientationally disordered phase, enables to analyze the syncrystallization conditions under two different types of order. The first corresponds to an ordered, low-temperature or solid phase and the second, the focus of the present chapter, an orientational, disordered, high temperature or plastic phase. These well-defined situations, where a long-range translational order exists, allow the factors ruling the miscibility to be distinguished. These factors are known to be supported, as sketched in Chap. 2, (i) by the size and shape of the molecules and the unit cell dimensions, which, on the whole, form the stearic or geometric conditions, and (ii) by the role of the molecular interactions in the mixed crystal structure. The symmetry of the crystallographic site versus the molecular symmetry appears to be of fundamental importance to our understanding of the formation of molecular alloys between organic compounds. All the phase diagrams reported in this chapter, whose components correspond to tetrahedral derivatives of neopentane (Cl:0), with strong interactions (as the In, IIn and IIIn, described in the first part) or with van der Waals interactions (as the methylchloromethane Cl:n), show similar features with regard to the extensions of the miscibility domains. Whatever the binary system considered, the low-temperature region shows a wide immiscibility zone with two solid forms. Two examples of phase diagrams [20, 44] showing the low-temperature region corresponding to the ordered

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Fig. 7.17 Binary phase diagrams for the systems I2 +II3 and Cl:1+Cl:3 from the liquid state down to the low-temperature ordered solid forms. (Reproduced with permission from RSC, J Mater Chem 5:431–439 [44], left panel, and from ACS, J Phys Chem B 105:10326–10324, right panel) [20]

solid phases are shown in Fig. 7.17. By increasing the temperature, such that ordered solid and OD phases are in equilibrium, the concentration range of the OD mixed crystals increases. On the contrary, the solubility boundaries of the mixed ordered crystals do not change noticeably. This means that a high solubility (large formation of mixed crystals) is observed when molecular substitution is performed within an orientational disordered, i.e., within the OD state. In order to explain the aforementioned experimental evidences (two-phase diagrams in Fig. 7.17 are just two representative examples), several arguments can be evoked: (i) The stearic factor and the (ii) lattice symmetry vs. molecular symmetry argument. As for the stearic factor, it is worth mentioning that the density of the ordered solid is higher than the density of the corresponding plastic phase (in general, the difference is around 6% and 11% for the compounds here collected). As far as the symmetry argument, if we assume that geometry makes possible the molecular substitution in the host crystal, it is evident that the guest molecule (within the host lattice) might not have the symmetry elements of the site. Thus, in order to comply with them, a “symmetry simulation” by the guest molecule should appear. We can look at it from the other side, i.e., when the host molecule is removed from the site, the guest molecule occupying its site can have different orientations, and thus the orientational disorder in the solid solution must emerge. One of the most typical examples of this symmetry simulation concerns planar molecules lacking a center of inversion which are oriented in opposite directions so that the crystal becomes centrosymmetric [45]. As a result, the guest molecule would acquire as many orientations as required by the crystallographic site symmetry. This argument is completely general for the formation of mixed crystals. For the case of systems with orientational disorder, as plastic phases (OD), it is obvious that a large but finite number of

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energetically equivalent orientations already exist in the “pure compound lattices”. Then, when the guest molecule replaces the host molecule, the number of possibilities for simulating the lattice symmetries is quite high and, thus, the miscibility largely increases when substitution in OD phases is performed. And, consequently, the solubility in the OD phases is much larger than in the low-temperature ordered solid forms of the counterpart compounds. Coming back to the geometrical factor, as reported in Chap. 1, this factor for mixed crystals involving organic compounds plays a decisive role. As we already mentioned, the greater the similarity in the shape and size of the molecules, the higher the solubility. The similarity concerning simultaneously shape and size can be accounted for the methods described in previous chapters, as degree of molecular homeomorphism coefficient εk , According to Kitaigorodsky, who first introduced this parameter, complete solubility in systems where isomorphism between the components is supposed to exist, can be achieved for values of εk around or higher than 0.8. Regarding derivatives of neopentane, the value of εk is really high, even when including systems that share compounds in which the substituted groups (–CH2 OH, –NO2 , CH3 , …) are quite different. Some representative examples are: I2 + II2 , εk = 0.942 ; I3 + II3 , εk = 0.946 ; or, for the case with the lower value, I2 + II3 , εk = 0.881 . As for methylchloromethane compounds, the values are even greater, which is not surprising due to the close size between the methyl group and the Cl atom. Then, εk parameter for plastic crystals has to be used as a parameter that provides just a clue on the degree of solubility that can be obtained a priori. There is also an issue that cannot be underestimated, the ubiquitous temperature-independent character of the εk , because for its calculation only the molecules in the mixed crystals are involved, without any reference to the polymorph to which they belong. Although the existence of continuous mixed crystals is impossible when the symmetry of the OD phases of the pure components is different (as in the present cases, FCC or BCC for neopentane derivatives, or FCC, BCC, SC and R for the methylchloromethane derivatives), the regions of mixed crystal formation are very wide even between phases for which isomorphism is not possible. In search of a similar criterium to εk , the ratio between the difference in the volume occupied by a molecule in the lattice with respect to the mean volume of the molecule unity (also called packing coefficient) or the ratio between the difference of the volume occupied by a molecule in the lattice with respect to its value for one of the pure compounds, are reasonable candidates to account for the mixed crystal formation and, thus possible parameters to rationalize the values of the excess thermodynamic properties of the mixed crystals in the OD phases. It is quite obvious, in addition, that the existence of orientational disorder gives rise to an “average” of the possible directional interactions as, for example, dipole– dipole or dipole-induced dipole interactions. It would mean then that such interactions can have a crucial role within the low-temperature (orientationally) ordered forms while they are “smeared out” within the high-temperature orientationally disordered phases. One speaking example which extremely reveals such behavior is the (CH3 )3 CBr+CBrCl3 binary system, depicted in Fig. 7.18.

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Fig. 7.18 Binary phase diagram of (CH3 )3 CBr+CBrCl3 showing the formation of two complexes in the low-temperature region as total or large miscibility in the high-temperature FCC and R OD phases, respectively. (Reproduced with permission from ACS, J Phys Chem B 358:115:1679–1688) [46]

This system [46] clearly shows the existence of large composition domains of OD mixed crystals (of both R and FCC forms), while that, for the low-temperature region, not only the miscibility is strongly reduced, but even the special dipole–dipole interactions between the molecules of the components gives rise to two complexes. This experimental fact was explained in terms of the attractive interaction between unlike molecules within the complex coming from the particular interaction between halogen atoms and the methyl group on the basis of the similarities found with the binary phase diagrams composed by 2-methylnaphtalene+2-chloronaphtalene and 2-methylnaphtalene+2-bromonaphtalene studied by Calvet et al. [47] some years ago.

References 1. Parsonage NG, Staveley LAK (1978) Disorder in Crystals. Clarendon Press, Oxford 2. Reuter J, Büsing D, Tamarit JLL, Würflinger A (1997) J Mater Chem 7:41–46 3. Urban S (1981) Dynamical and structural aspects of molecular reorientation in plastic crystals of the (CH3 )3 CX type. Adv Mol Relax Interact Proc 21:221–258 4. Rudman R, Post B (1968) Polymorphism of the crystalline methylchloromethane compounds. Mol Cryst 5:95–109 5. Silver L, Rudman R (1970) Polymorphism of the crystalline methylchloromethane compounds. III a differential scanning calorimetric study. J Phys Chem 74:3134–3139 6. Lee-Dadswell SE, Torrei BH, Binbrek OS, Powell BM (1998) Disorder in solid CBr2 Cl2 and CBrCl3 . Phys B 241–243:459–462 7. Silver L, Rudman R (1972) Polymorphism of the crystalline methylchloromethane compounds. IV. The crystal and molecular structure of methyl chloroform at −60° and −145°. J Chem Phys 57:210

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8. Rudman R (1966) Carbon tetrachloride a new crystalline modification science 154:1009–1010 9. Salud J, López DO, Barrio M, Tamarit JLL, Oonk HAJ (1999) Two-component systems of isomorphous orientationally disordered crystals. Part II: thermodynamic analysis of the mixed crystals. J Mater Chem 9(4):917–922 10. Salud J, López DO, Barrio M, Tamarit JLL (1999) Two-component systems of isomorphous orientationally disordered crystals. Part 1 packing of the mixed crystals. J Mater Chem 9:909– 916 11. Tamarit JLL, Barrio M, López DO, Haget Y (1997) Packing disordered molecular crystals and their molecular alloys. J Appl Cryst 30:118–122 12. Negrier Ph, Pardo LC, Salud J, Tamarit JLL, Barrio M, López DO, Würflinger A, Mondieig D (2002) Polymorphism of (CH3 )2 CCl2 : crystallographic characterization of the ordered and disordered phases. Chem Mater 14:1921–1929 13. Tamarit JLL, López DO, Alcobé X, Barrio M, Salud J, Pardo LC (2000) Thermal and structural characterization of (CH3 )3 CCl. Chem Mater 12:555–563 14. Pardo LC, Barrio M, Tamarit JLL, López DO, Salud J, Negrier P, Mondieig D (2000) Stable and metastable orientationally disordered mixed crystals of the two-component system (CH3 )2 CCl2 +CCl4 . Chem Phys Lett 321:438–444 15. Pardo LC, Barrio M, Tamarit JLL, López DO, Salud J, Negrier P, Mondieig D (1999) Miscibility study in stable and metastable orientationaly disordered phases in two-component system (CH3 )CCl3 +CCl4 . Chem Phys Lett 321:204–210 16. Salud J, López DO, Barrio M, Tamarit JLL, Oonk HAJ, Haget Y, Negrier P (1997) On the crystallography and thermodynamics in orientationally disordered phases. J Solid State Chem 133:536–544 17. López DO, Salud J, Barrio M, Tamarit JLL, Oonk HAJ (2000) Uniform thermodynamic description of the orientationally disordered mixed crystals of some neopentane. Chem Mater 12:1108–1114 18. Pardo LC, Barrio M, Tamarit JLL, Salud J, López DO, Negrier P, Mondieig D (2004) Multiple melting of orientationally disordered mixed crystals of the two component system (CH3 )2 CCl2 –(CH3 )CCl3 . Phys Chem Chem Phys 6(2):417–423 19. Pardo LC, Barrio M, Tamarit JLL, López DO, Salud J, Negrier P, Mondieig D (2001) First experimental demonstration of the crossed isodimorphism: (CH3 )3 CCl + CCl4 melting phase diagram. Phys Chem Chem Phys 3:2644–2649 20. Pardo LC, Barrio M, Tamarit JLL, López DO, Salud J, Negrier Ph, Mondieig D (2001) Stable and metastable phase diagram of the two-component system (CH3 )3 CCl –(CH3 )CCl3 : an example of crossed isodimorphism. J Phys Chem B 105(42):10326–10334 21. Pardo LC, Barrio M, Tamarit JLL, López DO, Salud J, Negrier P, Mondieig D (2002) Stable and metastable mixed crystals in the orientationally disordered state of the [(CH3 )3 CCl] + [(CH3 )2 CCl2 ] system. Chem Phys Lett 355:339–346 22. Pardo LC, Barrio M, Tamarit JLL, López DO, Salud J, Oonk HAJ (2005) Orientationally disordered mixed crystals sharing methylchloromethanes ((CH3 )4−n CCln , n = 0, 4). Chem Mater 17:6146–6153 23. Harsted BS, Thomsen EJ (1974) Excess enthalpies from flow microcalorimetry 2. Excess enthalpies for carbon tetrachloride + n-hexane, + n-heptane, + n-octane, + n-hexadecane, + isooctane, and + neopentane, and of octamethylcyclotetrasiloxane + isooctane and + cyclohexane. J Chem Thermodyn 6:557–563 24. Mathot V, Desmyter A (1953) Application of the cell method to the statistical thermodynamics of solutions. II. Experimental. J Chem Phys 21:782–789 25. Das SK, Diaz-Peña M, McGlashan ML (1961) Heats of mixing Pure. Appl Chem 2:141–146 26. Loras S, Barrio M (2000) Unpublished results 27. Van Miltemburg JC, Obbink JH, Meijer EL (1979) Excess enthalpies from displacement calorimetry excess enthalpies for 1,1,1- trichloroethane + carbon tetrachloride and 2-chloro-2-methylpropane + carbon tetrachloride at 298.15 K. J Chem Thermodyn 11:37–39 28. Minguez M, Chóliz G, Gutiérrez CR (1969) Real. Acad Cien Exact Fís Natur 6:533–563

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29. Meijer EL, Brouwer N, Van Miltemburg JC (1976) Vapour pressures and excess gibbs energy of the liquid mixture carbon tetrachloride + 1,1,1-trichloroethane. J Chem Thermodyn 8:703–706 30. López DO, Van Braak J, Tamarit JLL, Oonk HAJ (1994) Thermodynamic phase diagram analysis of three binary systems shared by five neopentane derivatives calphad 18(4):387–396 31. Oonk HAJ (2001) Solid-state solubility and its limits. The alkali halide case. Pure Appl Chem 73(5):807–823 32. Mondieig D, Espeau P, Robles L, Haget Y, Oonk HAJ, Cuevas-Diarte MA (1997) Mixed crystals of n-alkane pairs: a global view of the thermodynamic melting properties. J Chem Faraday Trans 93(18):3343–3346 33. Parat B, Pardo LC, Barrio M, Tamarit JLL, Negrier Ph, Salud J, López DO, Mondieig D (2005) Polymorphism of CBrCl3 . Chem Mat 17:3359–3365 34. Maruyama M, Kawabata K, Kuribayashi N (2000) Crystal morphologies and melting curves of CCl4 at pressures up to 330 MPa. J Cryst Growth 220:161–165 35. Baldelmeier U, Würflinger A (1989) High pressure differential thermal analysis of cyclohexanol-D11 (C6 D11 OH) and carbon tetrachloride (CCl4 ). Thermochim Acta 109–114 36. Barrio M, Tamarit JLL, Negrier Ph, Pardo LC, Veglio N, Mondieig D (2008) Polymorphism of CBr2 Cl2 New. J Chem 32(2):232–239 37. Levit R, Barrio M, Veglio N, Tamarit JLL, Negrier Ph, Pardo LC, Sanchez-Marcos J, Mondieig D (2008) From the two-component system CBrCl3 + CBr4 to the high-pressure properties of CBr4 . J Phys Chem B 112:13916–13922 38. Barrio M, Tamarit JLL, Céolin R, Pardo LC, Negrier Ph, Mondieig D (2009) Connecting the normal pressure equilibria of the two-component system CCl(CH3 )3 +CBrCl3 to the pressuretemperature phase diagrams of pure components. Chem Phys 358:156–160 39. Barrio M, Pardo LC, Tamarit JLL, Negrier Ph, Lopez DO, Salud J, Mondieig D (2006) Two-component System CCl4 + (CH3 )3 CBr: singularities in phases equilibria involving orientationally disordered phases. J Phys Chem B 110:12096–12103 40. Wilmers J, Briese M, Würflinger A (1984) Dielectric measurements at high pressures and low ternperatures V. Dielectric and pVT data of t-Butylchloride. Mol Cryst Liq Cryst 107:293–302 41. Barrio M, de Oliveira P, Céolin R, López DO, Tamarit JLL (2002) Polymorphism of 2-methyl-2chloro-propane and 2,2-dimethyl-propane (neopentane): thermodynamic evidence for a highpressure orientationally disordered rhombohedral phase through topological p-T diagrams. Chem Mater 14:851–857 42. Drozd-Rzoska A, Rzoska SJ, Imre AR (2004) Liquid–liquid phase equilibria in nitrobenzene– hexane critical mixture under negative pressure. Phys Chem Chem Phys 6:2291–2294 43. Imre AR, Drozd-Rzoska A, Kraska T, Rzoska SJ, Wojciechowski KW (2008) Spinodal strength of liquids, solids and glasses. J Phys Condens Matter 20:244104 44. Barrio M, López DO, Tamarit JLL, Negrier P, Haget Y (1995) Miscibility degree between non-isomorphous plastic phases: binary system NPG/TRIS. J Mater Chem 5(3):431–439 45. Kitaigorodsky A (1984) Mixed crystals. Springer, Berlin 46. Barrio M, Tamarit JLL, Negrier Ph, Mondieig D (2011) From high-temperature orientationally disordered mixed crystals to low- temperature complex formation in the two- component system (CH3 )3 CBr+CBrCl3 . J Phys Chem B 115:1679–1688 47. Calvet T, Cuevas-Diarte MA, Haget Y, Mondieig D, Kok IC, Verdonk ML, van Miltenburg JC, Oonk HAJ (1999) Isomorphism of 2-methylnaphthalene and 2-halonaphthalenes as a revealer of a special interaction between methyl and halogen. J Chem Phys 110:4841–4846

Chapter 8

Liquid Crystals J. Salud and D. O. López

Abstract The tricritical and the re-entrant nematic behaviour are two of the most relevant features of the Smectic A (SmA)-to-Nematic (N) phase transition in binary mixtures of liquid crystals. Both of these concepts are studied from a theoretical and an experimental point of view for some two-component systems whose members are calamitic liquid crystals belonging to the alkylcyanobiphenyl (nCB) or alkoxycyanobiphenyl (nOCB) series, n being the number of the carbon atoms in the alkyl or alkoxy chain, respectively.

8.1 Introduction The solid crystalline state of the ordinary matter is characterized by a uniform distribution of the atoms or molecules the centres of mass of which are located on a three-dimensional periodic lattice or what is the same, they have a long-range order. On the other hand, the isotropic liquid state does not present any kind of organization of the molecules, being randomly distributed from average distances greater than their characteristic length. Although these two states, along with the gas state, in which the type of disorder is similar to that of liquid but with much larger intermolecular distances, are typical of the majority of existing substances, there are, however, certain substances in which some other condensed states between the solid crystalline and the isotropic liquid are found. Two basic kinds of these intermediate phases may be distinguished: – Plastic crystalline states, in which the positional order is present, but the orientational order has disappeared or is strongly reduced. The compounds into which

J. Salud · D. O. López (B) Grup de Recerca de les Propietats Físiques dels Materials (GRPFM), Universitat Politècnica de Catalunya, Barcelona, Spain e-mail: [email protected] J. Salud e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. À. Cuevas-Diarte and H. A. J. Oonk (eds.), Molecular Mixed Crystals, Physical Chemistry in Action, https://doi.org/10.1007/978-3-030-68727-4_8

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these mesophases can be found are formed by pseudospherical or globular molecules and are known as plastic crystals or orientationally disordered crystals. – Liquid crystalline phases or mesomorphic phases, are characterized by orientational order and with the positional order strongly reduced or even completely disappeared. The compounds where those phases are present are formed by anisotropic molecules, usually rod-like or disk-like molecules, and are known as liquid crystals or mesogenic compounds. Liquid crystals form from organic compounds and are thought of as the phase of matter between the solid and liquid states. According to the way in which the liquid mesomorphic phases can be obtained, liquid crystals are classified as lyotropics or thermotropics. Lyotropic liquid crystals are always solutions of materials of unlike molecules in which one is a nonmesogenic liquid. Usually one of the components of these liquid crystals is formed by amphiphilic molecules and the other one is water. As the water content is increased, several mesophases are exhibited. The first documented liquid crystal, discovered by Reinitzer [1] and Lehmann [2], was a lyotropic-like. Nowadays, lyotropic liquid crystals have a great importance in the fields of biology, biotechnology and industrial oils, detergents or food. Thermotropic liquid crystals are compounds in which mesophases appear by changing temperature. The most important applications of such materials are based on the properties of fluidity and anisotropy, which makes them as preferred candidates for electro-optical devices, because their molecules can be reoriented by applying an electric field. According to the geometrical structure of their molecules, in the past, thermotropic liquid crystals were classified as calamitics and discotics, depending on the shape anisotropy of their molecules. In calamitic liquid crystals, the molecules are rodshaped, i.e. with one axis much longer than the other two (Fig. 8.1a) and are composed of a rigid core formed by polarized chemical bonds and flexible side chains. The core rigidity imposes the elongated molecular shape and helps the van der Waals anisotropic interactions. A certain flexibility is needed in order to make relatively lower melting points and stabilize the molecular alignment of the molecules. This

Fig. 8.1 Molecule of calamitic (a) and discotic liquid crystal (b)

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flexibility is accounted for the flexible side chains (R1 and R2 in Fig. 8.1a), which may be polar or non-polar. Discotic liquid crystals are formed by disc-shaped molecules, i.e. molecules with one molecular axis much shorter than the other two, and with a core containing several aromatic or alicyclic rings to which several flexible side chains are added (Fig. 8.1b). Apart from the above-mentioned liquid crystals, there are other substances whose molecular morphologies can lead to liquid crystal mesophases. In some of these substances, the molecules are formed by the repetition of certain mesogenic units linked by flexible chains, called monomers, and are known as polymer liquid crystals, whereas other ones are made of dimeric rigid molecules with a “V-like”, bent-shaped or banana-shaped geometry, which are usually called bananas. As a result of their special geometry, they exhibit mesomorphic states different from the other types of liquid crystals. This chapter will be devoted to binary mixtures whose components are thermotropic calamitic liquid crystals and in the next sections some of their physical properties, concerning basically the involved phase transitions, will be exposed.

8.2 Liquid Crystalline States Mesogenic compounds exhibit a rich variety of states (polymorphism), passing through more than one mesophase between solid and isotropic liquid. The transition between these different states corresponds to the breaking of some symmetry elements, which can be categorized in terms of their orientational and translational degrees of freedom. By reducing the temperature from the completely disordered liquid, isotropic (I) state, one can distinguish positional order and orientational order, which can be either short range or long range depending on if it extends only between molecules close to each other or to larger dimensions, respectively. One of the most common mesophases is the nematic (N) one, where the molecules have no positional order, but they have long-range orientational order. The molecules flow and their centre of mass positions are randomly distributed as in one liquid. In calamitic liquid crystals, the nematic phase is said to be uniaxial, because their molecules have one axis that is longer and preferred, whereas the other two are equivalent and shorter (cylindricallike molecules). In this phase, the molecules tend to align parallel to each other with their long axes all pointing roughly in the same direction. The average direction along which the molecules are arranged is called the director of the phase n (Fig. 8.2a). Nevertheless, some liquid crystals are biaxial nematics (Fig. 8.2b, c), because in addition to their long axis, their molecules also orient along a secondary axis [3]. The other most common kind of mesophases is the smectic (Sm) one, in which the molecules form well-defined layers or planes that can slide over one another and the characteristic length of which is of the order of molecular length. The Sm phases have thus not only the orientational N order, but also a certain positional order

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Fig. 8.2 Schematic arrangement of molecules in: uniaxial a, biaxial nematic phases b and c. The molecules are: cylindrical-like in (a), anisometric parallelepiped platelets in (b), and bent-shaped in (c) (Reprinted with permission from [3] © 2004 by the American physical society)

along one direction. This way of layer distribution in fact reflects the existence of a weak density modulation. One of the best known Sm mesophases is the smectic A (SmA), in which the molecules are oriented along the layer normal. In the smectic C (SmC) mesophase, the molecules are arranged as in the SmA phase, but they are tilted with respect to the layer normal. There is a large number of Sm mesophases, all characterized by different types and degrees of positional and orientational order [4, 5]. According to the type of positional order within the layers, they are called SmB (Hexatic B), SmF or SmI. If the positional order is long range compared to molecular dimensions, the phases are more “solid-like” (B, E, J, G, K, or H smectic phases). The different kinds of Sm mesophases are gathered in Fig. 8.3. There exists a special class of calamitic liquid crystals the molecules of which lack improper elements of symmetry and their congruence with their mirror image is not achieved by any conformational change. This kind of molecules is called chiral calamitic molecules and they give rise to specific mesophases: chiral nematic (N*) and chiral smectic (being SmC* the most important). On the other hand, in a narrow range of temperature between N* and I phases, there appear singular liquid crystal phases, called blue phases. These phases have a regular three-dimensional cubic structure of defects and the characteristic period of these defects is of the order of the wavelength of visible light, giving rise to vivid specular reflections that are controllable with external fields. Although the materials that exhibit blue phases are examples of tunable photonic crystals with many potential applications, the thermal stability of such phases is rather limited (not more than 2 °C).

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Fig. 8.3 Structure of calamitic Sm liquid crystal phases (From [6], page 5. 1998. © Wiley–VCH Verlag GmbH & Co. KGaA. Reproduced with permission)

8.3 Nematic-To-Isotropic and Smectic A-To-Nematic Phase Transitions The N-to-I and SmA-to-N phase transitions are the most frequently studied both from the experimental and theoretical points of view. In this section, some of their most remarkable characteristics are detailed. Regarding the N-to-I phase transition, it seems to be an order–disorder phase transition that, according to the Maier–Saupe theory [7–9], is viewed as a sort of competition between thermally excited forces aiming to eliminate the orientational order and “mean field” molecular forces aiming to restore it. The role played by fluctuations is such that it can profoundly alter the nature of the properties of the liquid crystal in the vicinity of the phase transition. The Landau-de Gennes theory [5] is the simplest theoretical approach of the N-to-I phase transition based on the free energy density expansion in powers of the nematic order parameter up to the fourth order with a cubic term responsible of the firstorder nature of the transition, being T NI the corresponding transition temperature. However, experimental results do not seem to go with these theoretical predictions, which have led to think that the effect of fluctuations about its mean value also plays a role in phase transitions. These fluctuations of local order are responsible for the pretransitional effects at temperatures immediately preceding and following the transition temperature. It is worth noting that critical fluctuations are important when strong external fields are applied. However, weak external fields applied to strongly polar mesogens around the N-I phase transition lead to some fluctuation effects in the linear static dielectric permittivity or in the heat capacity experiments or molar volume

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studies. Thus, the prenematic region of the isotropic liquids is characterized by an abrupt increase of physical properties such as specific heat, [10, 11], isobaric thermal expansion, [11, 12] and isothermal compressibility coefficients, [12] and electric [13] and magnetic [14] induced birefringence, so that the mean-field theory breaks down in the neighbourhood of the transitions and other theoretical models incorporating fluctuations at many length or time scales have been proposed [15]. Among the most recent of such theories, the fluid-like model [16] considers that T NI is a fluid-like critical temperature of a hypothetical coexistence binodal curve, being T * and T ** the so-called spinodal temperatures which are identified with the metastable limits of the I and N phases, respectively. Thus, in this hypothetical coexistence region, the Landau-de Gennes theory is not satisfied. The behaviour of pretransitional effects is usually expressed by means of potential expressions, in such a way that, C p (T ) ∝ (T − TC )−α for the specific heat, ρ(T ) ∝ (T − TC )1−α for the density or S ∝ (T − TC )β for the nematic order parameter, where T C is set equal to T * , T ** and α and β are critical exponents. The Landau-de Gennes theory proposes the same critical exponents α and β in the N phase (α N = β N = 0.5), being α I = 0 (in the isotropic phase), but these values are not consistent with the experimental results that seem to point out a nearly tricrititcal behaviour in which the exponent of the nematic order parameter is β = 0.25. This tricritical behaviour was firstly suggested by Keyes [17] and the Landau-de Gennes theory accounts for this behaviour when the free energy density is expanded up to the sixth-order term in the order parameter. The SmA-to-N phase transition has probably been the most extensively studied of all liquid crystal phase transitions. Theoretical studies of certain relevance started in the 1970s, by McMillan [18] and Kobayashi [19]. All of them worked in an independent way and developed a molecular theory for the SmA-to-N phase transition based on the mean-field approach. The theory predicts that the SmA-to-N phase transition should be second order or first order in nature, depending on the nematic range (T AN − T NI , where T AN and T NI , are, respectively, the temperatures of the SmA-to-N and N-to-I phase transitions). Nearly at the same time, de Gennes [5] studied the SmAto-N phase transition as being described by a Landau–Ginzburg functional belonging to the 3D-XY universality class. However, because of a coupling between both the smectic and nematic order parameters, being that stronger as the nematic temperature range decreases, a crossover from the strictly 3D-XY second-order to first-order behaviour via a tricritical point (TCP) is expected to be observed. That TCP was defined as the limit point at which the transition changes from first- to second-order phase transition. McMillan proposed the ratio T AN /T NI , called the McMillan’s ratio, which should be 0.87 at the TCP as a simple parameter to control the extension of the nematic range, and therefore the coupling of the order parameters. However, experimental evidences claims for values somewhat higher, ranging from 0.94 to 0.99, displaying a non-universal value and with a strongly dependence on the polarity of the material molecules. Experimental evidence for the TCP has been reported on temperature–pressure phase diagrams of pure materials [20, 21], and mostly on two-component systems [22–24], in which the TCP is readily attainable at ordinary pressure.

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On the other hand, Halperin, Lubenski and Ma [25, 26] published, also in the 1970s, a theory according to which the SmA-to-N phase transition should always be first order in nature, even when it is very weak and a certain confusion could exist form an experimental point of view. Thus, the conventional TCP behaviour is ruled out. Some accurate experimental studies in supporting this theory have been undertaken from the 1990s [27–29] but other studies on several liquid crystal binary systems ([28] and references therein) are not able to clarify on whether the SmA-to-N phase transition is weakly first order or essentially second order.

8.4 Re-entrant Nematic Behaviour Liquid crystals consisting of molecules with a strong polar end group, CN and NO2 , for example, exhibit some specific properties from the more classical liquid crystals. One of such properties is the re-entrant phenomenon, triggered by complex molecular geometrical factors or competing fluctuations. This kind of liquid crystals exhibits a SmA phase (named partial bilayer smectic A: SmAd ) in which the layer spacing d is such that l < d < 2 l, where l denotes the length of a molecule. Under certain pressure conditions (or for certain mixtures with selected liquid crystal materials), a non-standard mesophase sequence appears, when cooling the sample from the I phase: it is the sequence from N, to SmAd , and to N again—a behaviour referred to as re-entrant nematic (RN). The re-entrant nematic behaviour was observed for the first time by Cladis [30] in binary mixtures of polar para-cyanosubstituted liquid crystals, studied at atmospheric pressure. It should be observed that re-entrant nematic behaviour is also found in non-polar liquid crystals [31, 32]. Besides, re-entrant phase phenomena are also observed in other non-mesogenic materials, such as pure iron, superconductors, and vapour–liquid systems. The first theoretical studies about the re-entrance phenomenon in nematic liquid crystals were made by Cladis et al. [33, 34]. They proposed a microscopic model of a bilayer Smectic A and also gave a qualitative description of the change from SmAd to-N. Somewhat later, a phenomenological Landau treatment of the phenomenon [35, 36] and its refinement [37] were proposed. Several microscopic theories appeared and, among them, the frustrated spin-gas model [38–40] was the most useful in reproducing the experimental data. From the 1990s, this model has been substantially developed [41]; and other microscopic approaches have been proposed [42–44]. The theoretical advances about the re-entrant nematic behaviour were followed in a parallel way by thermodynamic calculations of phase boundaries. The first thermodynamic studies are those by Klug and Whalley [45] and Clark [46]. The thermodynamic approach applied in this work is the equal-G curve (EGC) method. The EGC method, with G for Gibbs energy, was detailed by Oonk [47]; and applied for the first time to liquid crystal systems by Van Hecke [48].

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8.5 The EGC Method Applied to Liquid Crystal Binary Systems This section is about the development of the main features of the EGC method—to be applied (i) to the estimation of tricritical points (TCP) [24] and (ii) to reproduce and quantify the re-entrant behaviour in binary systems. We also applied the Oonk-van Hecke approach to second-order re-entrant transitions, and so, by taking an almost zero entropy change [49, 50].

8.5.1 Tricritical Behaviour in Binary Systems As is well known, the thermodynamic properties of a mixture made of two components A and B (see Fig. 8.4a1 , b1 ), taken under isobaric conditions and for a given phase, can be determined if the molar Gibbs energy is known as a function of temperature and composition. For two different phases α and β, composed of X moles of B and (1-X) moles of A, the molar Gibbs energies, as a function of temperature T and mole fraction X, are given by the following expressions:

Fig. 8.4 (a1 ) Scheme of a generic two-component system A + B. Component A shows a second order β-to-α phase transition (at T A temperature) and component B displays a (weakly) first-order β-to-α phase transition (at T B temperature); (a2 ) and (a3 ) molar Gibbs functions of pure components A and B, respectively, versus temperature, compatible with their phase diagram showed in (a1 ). (b1 ) Qualitative two-component system A + B in which pure component B displays a re-entrant α phase, αR.; (b2 ) and (b3 ) molar Gibbs functions of pure components A and B, respectively, versus temperature and compatible with their phase diagram showed in (b1 ). T is the absolute temperature; in (a1 ) and (b1 ), X B is the mole fraction of pure component B, and EGC is the acronym of the Equal Gibbs composition curve. FEGC in (a1 ) accounts for to the first EGC composition curve. See the text for the meaning of X REN , T REN , X TCP and T TCP (Reproduced from Ref. [51] with permission of the author)

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∗,α G α (X, T ) = (1 − X )μ∗,α A (T ) + X μ B (T )

+ RT [(1 − X ) ln(1 − X ) + X ln X ] + G E,α (X, T ) ∗,β

(8.1)

∗,β

G β (X, T ) = (1 − X )μ A (T ) + X μ B (T ) + RT [(1 − X ) ln(1 − X ) + X ln X ] + G E,β (X, T )

(8.2)

where GE,α (X, T ) and GE,β (X, T ) are, respectively, the excess Gibbs energy functions of the α and β phases, μA *,α and μA *,β are, respectively, the molar Gibbs functions of pure component A in α and β phases, μB *,α and μB *,β are the same functions for the B compound in the same phases and R is the gas constant. Let us consider the arbitrary two-component system A + B of Fig. 8.4a1 , in which the α-to-β phase transition is weakly first order for pure B compound and second order for pure A compound. The molar Gibbs functions of pure compounds are plotted against temperature in a qualitative way, as shown in Fig. 8.4a2 , a3 . In Fig. 8.4a2 , for any temperature below T A , both phases have similar Gibbs energies, being their slopes the same up to T A . In other words, there are no differences between both phases. If temperatures above T A are considered, both molar Gibbs energy curves diverge in a way that the lower value corresponds to the α phase (μA *,α ) being the stable phase at these temperatures. It should be noted that, at T A there are no changes in the slope of both G-curves, a fact that in turn, it means the same molar entropy between α and β at T A . However, changes in specific heat between α and β at T A behave changes in the curvature of both Gibbs functions. On the other hand, in Fig. 8.4a3 we show the generic plot of two molar Gibbs energy curves crossing one to each other at the transition temperature T B for a first order phase transition. The Gibbs function slope change at the transition temperature T B is related with the typical change in molar entropy. A qualitative description of the Gibbs energy functions for the α and β phases (Gα and Gβ ), expressed by Eqs. (8.1) and (8.2), at a temperature T 1 in Fig. 8.4a1 at which their corresponding curves intersect, is shown in Fig. 8.5a1 . At this temperature, a first-order phase transition associated to two-phase equilibrium region between α and β takes place according to the well-known criterion of the double tangent line giving the equilibrium compositions (X α and X β ). In this case, a very narrow twophase composition region is obtained because both X α and X β are very close. A zoom window of this region is represented in the top left inset of Fig. 8.5a1 . Additionally, it is easy to observe how the composition of the crossing point of both Gibbs functions, named equal Gibbs composition (EGC), is very close to the equilibrium compositions (X α and X β ) and changes with temperature. That temperature change of the EGC defines the equal Gibbs composition curve (also named EGCC) [47] being representative of the temperature composition two-phase equilibrium (Fig. 8.4a1 ). In Fig. 8.5a2 , both Gibbs energy functions, to which the term RT [(1 − X) ln (1 − X) + X ln X] has been subtracted, denoted as G , are shown. This procedure lets the EGC curve unchanged and has the advantage of better observe small differences in very similar G curves. When such a situation takes place, as in the case of many

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Fig. 8.5 (a1 ) Qualitative representation of the Gibbs functions related to α and β phases of a generic two-component system A + B versus the mole fraction of the B component, X B , at a T 1 according to Fig. 8.4(a1 ). The inset is a zoom window of intersection of the Gibbs energy functions; (a2 ) the same plot of the Gibbs functions modified by the subtraction of the term RT [(1 − X) ln (1 − X) + X ln X] to the Gibbs energy functions given by (Eq. (8.1)) and (Eq. (8.2)), giving rise to (Gα ) and (Gβ ) functions; (b1 ) and (b2 ) are, respectively, the same representations as (a1 ) and (a2 ), but corresponding to the temperature T 2 of Fig. 8.4(a1 ). X B is the B mole fraction (Reproduced from Ref. [51] with permission of the author)

liquid crystal mesophase transitions, the corresponding entropies of pure compounds are very small. On the other hand, both X α and X β can be substituted by the EGC and Fig. 8.5a2 can be taken as representative of the phase stability. In this case, the two-phase equilibrium of the two-component system can be substituted by a single line, the EGC curve, which is a very good approximation. Let us consider, as an example, a typical situation in a two-component system (A + B) in which a second-order phase transition is observed for the mixture X B ≈0.25 at a temperature T 2 , according to Fig. 8.4a1 . The Gibbs energy functions of both involved phases (Gα and Gβ ), as described by Eqs. (8.1) and (8.2), are plotted as a function of the mole fraction of B component, X B , in Fig. 8.5b1 . From this figure, it is easy to observe how there is no crossing between both Gibbs functions. In order to clarify this situation, in Fig. 8.5b2 , the G α and G β functions, obtained by subtracting the term RT [(1 − X) ln (1 − X) + X ln X] from Gα and Gβ , respectively, are shown. Some aspects must be remarked: (i) There is a clear coincidence between both modified Gibbs functions (G α and G β ) in the composition range from 0 (A component) to X B ≈ 0.25, which also implies that both Gα and Gβ are coincident in the same composition range and, in turn, we have more than one equal Gibbs

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composition (EGC); (ii) As it can be clearly seen, the more stable phase from X B ≈ 0.25 up to X B = 1 (pure component B) is the β phase, since Gβ is lower than Gα in such a composition range. All these considerations not only provide a coherent thermodynamic description of the second-order transition for the mixtures, but also provide a simple methodology to calculate second-order transition points from a quantitatively point of view. This method states that: at fixed temperature, when the composition (in this case X B ), at each temperature, is going from X B = 1 (pure component B, with a first-order phase transition) to X B = 0 (pure component A, with a second-order phase transition), the second-order transition binary mixture is identified as the first EGC point (FEGC) at which both Gibbs energy functions do not cross each other but they are coincident. In the temperature–composition diagram, such as the diagram showed in Fig. 8.4a1 , there is a point, called the tricritical point (TCP: T TCP , X TCP ) identified with the limit point at which the α-to-β first-order phase transition becomes second order in nature. The corresponding temperature, T TCP , is the temperature at which an EGC point (coming from the crossing of both Gibbs energy functions of the implied phases (α and β)) changes to a set of EGC points (where the corresponding Gibbs energy functions are coincident throughout the composition range). In order to quantify the situations arising from Fig. 8.5 for whatever twocomponent system similar to that of Fig. 8.4a1 , the EGC curve must be obtained. For that purpose, making the subtraction of both Eqs. (8.1) and (8.2) set equal to zero, will give the EGC curve equation: G α (X, T ) − G β (X, T ) = (1 − X )μ∗A (T ) + X μ∗B (T ) + G E (X, T ) = 0

(8.3)

∗,β

where μi∗ (T ) = μi∗,α − μi (i = A, B). For μi∗ (T ), taking into account the transition temperature of the pure i-component as a reference point, we obtain     μi∗ = −Si∗ (T − Ti ) + C ∗pi T − Ti − T ln T Ti

(8.4)

where Si∗ and C ∗pi are the molar entropy change and the specific heat change, respectively, at the transition temperature of the pure i-compound T i . If the α-to-β phase transition is purely second order, S i * would set equal to zero and in this case, according to Eq. (8.4), μi * depends exclusively on the change in C pi * . As far as the change C pi * is concerned, it can be finite, infinite or zero, depending on the type of second-order phase transition. For GE (X, T ), a two-parameter form of the Redlich–Kister expansion is commonly used G E (X, T ) = X (1 − X )[A1 (T ) + A2 (T )(1 − 2X )]

(8.5)

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where A1 (T ) and A2 (T ) are usually taken either constant or as a function of temperature as H jE − T S Ej ( j = 1, 2). The EGC method is based on an iterative procedure by means of which from some experimental data concerning the phase transition of the mixtures, a reasonable EGC curve, as well as the GE (X, T ) along this curve are obtained. This procedure is automatically done using WINFIT software [52] in which, as a novelty, the option of zero entropy change for pure components has been implemented.

8.5.2 Re-entrant Nematic Behaviour in Binary Systems Let us consider an arbitrary two-component system A + B in which re-entrant αphase α R exists (see Fig. 8.4b1 ). As it can be seen, one of the pure components, in this case B compound, exhibits re-entrant behaviour: αR -to-β at T B L and β-to-α at T B H . Both phase transitions must be first order in nature. However, the A compound does not exhibit the β-to-α phase transition, but it will be considered as virtual in order to apply the Oonk’s method. To do so, the virtual temperature T A β−α , lower than T B L (see Fig. 8.4b1 ), has to be known. Taking into account these requirements, the molar Gibbs energy functions of pure A and B compounds are qualitatively represented in Fig. 8.4b2 , b3 respectively. For the A compound (Fig. 8.4b2 ), α is the only stable phase and therefore its molar Gibbs function in this phase, μ∗,α A , must be lower than ∗,β the corresponding Gibbs function in β phase, μ A , in the considered temperature range. As far as B compound is concerned (Fig. 8.4b3 ), its molar Gibbs functions, ∗,β L H μ∗,α B and μ B , must intersect at two temperatures: T B and T B . Obviously, this qualitative picture shows the alternation in stability of phases which is consequence of the temperature dependence of the specific heat of that compound. This can be easily attained following the procedure of van Hecke [48] in which nonzero value for C pi ∗ (see Eq. (8.4)) of pure B component is needed. Under the particularities cited above, by using Eqs. (8.3)–(8.5), via WINIFIT software, the calculated re-entrant boundary in the composition-temperature plane can be obtained.

8.6 The Heptyloxycyanobiphenil (7OCB) + Nonyloxycyanobiphenil (9OCB) Binary System: An Experimental Study The binary system 7OCB + 9OCB is an example of binary mixtures that exhibit not only RN behaviour, but also the existence of an experimental TCP in the SmA-to-N transition. The concurrence of both experimental facts makes this system as the most appropriate one for the treatment of the two essential features in liquid crystal binary mixtures to be studied in this issue: re-entrance and tricritical point phenomena.

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The experimental partial phase diagram of 7OCB + 9OCB, exhibiting a monotropic RN behaviour, is shown in Fig. 8.6. Experimental data come from two sources: modulated differential scanning calorimetry (MDSC) (filled and empty circles) and polarized optical microscopy (triangles). From an experimental point of view, binary first-order phase transitions correspond to a two-phase equilibrium region. In our case, the MDSC technique gives a narrow N-to-I region, of about 0.05 K. In Fig. 8.6 the region is represented by a single curve—and it means that the curve is also representative of the EGC. As regards second-order phase transitions (open symbols in Fig. 8.6), they are strictly represented by only one line because no phase coexistence region exists. This is the case for (i) the change SmAd -to-N for compositions between pure 7OCB and the TCP composition and (ii) for the change RN-to-SmAd , both represented by the dashed line in Fig. 8.6. In this figure, it also can be seen that the MDSC results perfectly are in line with those coming from polarized optical microscopy. Figure 8.7 is a representation of the outcome of a cooling experiment during which the specific heat was measured of a sample containing 26 mol % of 9OCB, the cooling rate being 1 K·min−1 . Also shown are optical texture photographs, for the sample being in inhomogeneous alignment. The small peak in the specific heat curve at 329 K is typical of a second-order transition—the transition N-to-SmAd . Additionally, at lower temperatures, the typical twined specific heat peak representative of

Fig. 8.6 Partial temperature–composition phase diagram for 7OCB + 9OCB. Specific heat measurements by means of MDSC technique are reflected as open and filled circles. Triangles come from optical measurements. Vertical dashed lines are three selected mixtures of microscopic measurements designed to study the re-entrant nematic phase. (see Fig. 8.7) (Reprinted with permission from [50]. © 2006 American Chemical Society)

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Fig. 8.7 Specific heat data as a function of temperature, along with some photographs showing optical textures of the different phases, for one of the selected mixture X 9OCB = 0.26, pointed out in Fig. 8.6. The experiments were executed on cooling mode, from the isotropic state at a slow cooling rate of 1 K·min−1 (Reprinted with permission from [50]. © 2006 American chemical society)

the crystallization of the sample can be observed. By means of optical textures (the upper one on the left side in Fig. 8.7), a like peak is interpreted as the transition of the supercooled SmAd -mesophase to crystal. For the 26 mol %, and other mixtures with a higher content of 7OCB the SmAd -to-RN specific heat peak could not be observed, whatever the cooling rate. The only experimental technique from those available to detect the RN-phase was polarized optical microscopy, in cooling experiments, starting from the isotropic phase, at a rate of 2 K·min−1 . This rate appeared to be high enough to avoid the crystallization of the supercooled SmAd -mesophase. Two examples of this procedure are shown in Fig. 8.8, where the optical textures for two selected mixtures (X 9OCB = 0.12 and X 9OCB = 0.23) are shown at a number of temperatures. The X 9OCB = 0.23 textures were obtained with EHC cells in an inhomogeneous alignment, while those of X 9OCB = 0.12 were obtained with Linkam cells in perfect planar alignment (for details, see [50]). As it was reported [11, 50, 53, 54], a weakly first-order phase transition is characterized by peaks in the δ-phase shift and specific heat data, both located at the same temperature. This coincidence is not casual because the δ-phase shift peak is attributed to the existence of latent heat in a way that the smaller the peak, the weaker first order is the transition and the lower the associated latent heat, as it can be seen in the insets of Fig. 8.9a. A δ-peak is observed for the mixture X 9OCB = 0.93, but this peak is attenuated more and more as the 9OCB content of the mixtures decreases in such a way that for the mixture X 9OCB = 0.67, no δ-peak is observed. This fact is an indication that the order of the SmAd -to-N transition changes for a composition close to 0.67 in mole fraction of 9OCB. A more accurate location of the TCP composition can be determined by means of the experimental determination of the latent heat H SmA-N for several mixtures. The total enthalpy change for a given experiment is defined as:

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Fig. 8.8 Sequence of optical textures obtained for mixtures X 9OCB = 0.12 (LC in planar alignment) and 0.23 (LC in inhomogeneous alignment) at several temperatures (Reprinted with permission from Ref. [50]; © 2006 American chemical society)

 T OT HSmA-N

= HSmA-N +

C p dT

(8.6)

where H SmA-N is the latent heat and C p is defined as C p −C p (background) where C p (background) corresponds to the specific heat far from the phase transition. The second term of the right hand of Eq. (8.6) is the pretransitional fluctuation contribution. In the case of a second-order phase transition, the latent heat H SmA-N vanishes. Both right terms in Eq. (8.6) can be obtained by means of the MDSC technique. It should be stressed how a linear relationship can be fitted when (H SmA-N )0.5 versus composition is plotted. This procedure has been revealed as a very useful experimental procedure to obtain the tricritical composition. This can be seen in Fig. 8.9a, where the selected (H SmA-N )0.5 data corresponding to 7OCB + 9OCB mixtures are shown as a function of composition. It is straightforward to observe how the linear fit extrapolation to the experimental points goes to a zero value for a X 9OCB ≈0.69. Taking into account the experimental uncertainty (the estimated error is about 0.05 in mole fraction), the latent heat for the mixture X 9OCB = 0.67 is extremely low and is impossible to give a value because is out of the confidence

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Fig. 8.9 (a) Latent heat square root ([HSmA-N]0.5 ) is plotted as a function of composition for several mixtures of the two-component system 7OCB + 9OCB. The insets provide enlarged details of the δ-phase shift data as a function of the (T-Tpeak) at the SmAd-to-N phase transition for some selected 7OCB + 9OCB mixtures. (b) Experimental specific heat data for some selected mixtures X9OCB = 0.67, 0.57 and 0.41 are plotted as a function of the (T-Tpeak) at the SmAd-to-N phase transition. All C p data have been shifted in relation to that corresponding to X9OCB = 0.41 in order to facilitate comparison (Both figures Reprinted with permission from Ref. [50]; © 2006 American chemical society)

limit of the calorimeter. The study of the second-order SmAd -to-N phase transition was undertaken by measuring the specific heat of some binary mixtures at very low heating rates (at about of 0.01 K·min−1 ). These specific heat data (see Fig. 8.9b) were analysed through the renormalization-group expression [55]   ± −α 1 + D ± |ε|0.5 C± p = B + Eε + A |ε|

(8.7)

where ± refers to the specific heat above and below the transition, respectively. The parameter ε (=(T − T c )/T c ) is the reduced temperature, and T c is the critical temperature at which divergence of both the smectic and nematic behaviour is observed. The parameters A± are the corresponding amplitudes above (+) and below (−) the phase transition. The constants B and E correspond to the specific heat background and α is the specific heat critical exponent, the same in the smectic and nematic phases. The term D± |ε|0.5 is the first-order correction to scaling term, above (+) and below, (−), the phase transition. The exponent of this term was fixed at the value 0.5 which is the 3D-XY value predicted by the model. The theory for the 3D-XY universality class predicts an amplitude ratio of (A− /A+ ) = 0.971, a critical exponent α = −0.007, and D− /D+ ≈1. Although around the TCP, D− /D is also predicted to be close to one, the predictions about the theoretical amplitude ratio (A− /A+ ) and the critical exponent α are different, being 1.6 and 0.5, respectively.

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Table 8.1 Results of fits to Eq. (8.7) for some 7OCB + 9OCB mixtures. The quoted errors are the statistical uncertainties X 9OCB

Nu

α

A− /A+

0.67

350

0.43 ± 0.04

1.1 ± 0.4

0.63

342

0.38 ± 0.03

0.6 ± 1.5

0.57

350

0.26 ± 0.08

1.2 ± 1.5

D− /D+

T c /K

χ 2 ·103

1.0 ± 0.1

342.81 ± 0.41

1

1.5 ± 1.9

342.43 ± 0.60

2

15.4 ± 15a

341.11 ± 0.35

5

a The obtained D− /D+

value is too high and is consequence of the fitting procedure considered with the purpose to obtain the same value of α below and above the transition

For some analyzed mixtures of the two-component system 7OCB + 9OCB, the values obtained for each of the fitted parameters of Eq. (8.7) (α, A− /A* , D− /D+ , T c ), over the range |ε| ≤ 5 · 10−3 , are presented in Table 8.1. Taking into account the corresponding χ 2 values of the table, all the parameter set are considered to reproduce well the measured specific heat data. It should be noticed that the more the content in 7OCB, the smaller the specific-heat peak is (see Fig. 8.9b). This fact, also observed on 7OCB + 8OCB mixtures [49], was explained by Kortan et al. [56] as an effect due to an increasing of the correlation volume for the mixtures as the reentrance approaches. For 7OCB + 9OCB mixtures, no specific heat peak is observed for X 9OCB < 0.2.

8.7 The Heptyloxycyanobiphenil (7OCB) + Nonyloxycyanobiphenil (9OCB) Binary System: A Thermodynamic Analysis The phase diagram of the 7OCB + 9OCB system, shown in Fig. 8.6, is an example of monotropic re-entrant behaviour. The EGC procedure to be applied is similar to the methodology cited above for Fig. 8.4b1 . It must be performed in two analytical steps: the first one concerns the first-order N-to-I phase transition and the second one is related to the SmAd -to-N transition in which a monotropic re-entrant nematic behaviour has been evidenced. This theoretical analysis needs one Gibbs energy function for I and N phases, in the same way as those previously expressed by Eqs. (8.1) and (8.2). Another required input are the experimental data read from Fig. 8.6, in addition to the experimental thermal properties of pure components concerning the first-order N-to-I transition: temperature, latent heat and the excess specific heat (see [50]). From these data, WINIFIT software, by using the EGC method, provides the excess Gibbs energy difference, GE , between N and I phases along the EGC curve in the form of an Redlich–Kister equation with two coefficients being temperature-independent as in Eq. (8.5)

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Fig. 8.10 Calculated monotropic re-entrant behaviour for the two-component system 7OCB + 9OCB. Filled and empty symbols are representative of first-order and second-order phase transitions, respectively. Triangles account for optical experiments (Reproduced from Ref. [51] with permission of the author)

G E (X ) = G E,I − G E,N = X (1 − X )[−8.8(±0.5) + 0.2(±1.0)(1 − 2X )]J mol−1

(8.8) Assuming I-phase to be strictly ideal, the coefficients of the GE,N are obtained from Eq. (8.8) for the N phase. This N-I equilibrium is very narrow and only the corresponding EGC curve has been drawn for simplicity, as it is shown in Fig. 8.10 (dotted line in the N-to-I phase transition). The SmAd -to-N phase transition has the peculiarity that is first order for 9OCB and does not exist for 7OCB, but it can also be calculated using the EGC method by means of the WINIFIT software and by comparing the Gibbs energy functions of the N and SmAd phases. This method, based on the applications of Eq. (8.3), needs a virtual second-order SmAd -to-N phase transition for 7OCB and therefore, the temperature, the entropy, and specific heat changes (Eq. (8.4)) for this phase transition of 7OCB are needed. Considering that the entropy change is zero, WINIFIT software provides an estimate of both the transition temperature and specific heat change, C p * from the experimental two-component phase diagram shown in Fig. 8.6 (see [51]). The estimated (virtual) transition temperature, T 7OCB SmA-N , has been ≈ 235 K. Likewise, for the 9OCB compound, both the entropy change and temperature of its SmAd -to-N phase transition are experimentally known and the specific heat change at this phase transition, Cp * , can also be estimated either by means of WINIFIT (in a similar procedure as in 7OCB) or from experimental data [11]. Both procedures lead to a comparable value of about 2.7 J·mol−1 ·K−1 . WINIFIT calculations lead to the following excess Gibbs energy for the SmA phase

8 Liquid Crystals

G E,Sm A (X ) = X (1 − X )[19.2(±0.2) + 1.2(±0.4)(1 − 2X )]J mol−1

209

(8.9)

and the calculated EGC curve, drawn in Fig. 8.10 (as a dotted line), matches the experimental re-entrant nematic data. The value of T 9OCB L is about 247 K and the limit values in composition and temperature of the re-entrance, X REN and T REN , are about 0.05 and 298 K, respectively (Fig. 8.10). The thermodynamic determination of the tricritical point deserves special attention. From the experimental point of view, in a previous section it has been evidenced the existence of a tricritical point in the SmAd -to-N phase transition. But since a tricritical point (TCP) has been found for the SmAd -to-N phase transition, on may wonder if it is possible to find another one for the RN-to-SmAd phase transition. The search for this second TCP is based on the Oonk’s equal Gibbs energy analysis [24] and is described in Fig. 8.11, where several sets of Gibbs energy functions (GSmA ) and (GN ) , one set for each chosen temperature in Fig. 8.11a, are shown. At 345 K, both Gibbs functions are crossing each other at the EGC (Fig. 8.11b), clearly exhibiting a first order SmAd -to-N phase transition. For temperatures higher than 345 K up to the transition temperature of pure 9OCB, the situation is similar to that described in Fig. 8.11b. On decreasing temperature from 345 K, the first temperature at which no crossing between both (GSmA ) and (GN ) but a coincidence in a composition range is observed, corresponds to the TCP temperature, as cited in Section tricritical behaviour in binary systems (see Fig. 8.11c). This situation, taking into account the FEGC, allows us to obtain the SmAd -to-N and RN-to-SmAd boundary. However, for temperatures low enough, for example, at 250 K (Fig. 8.11d), a crossing between both (GSmA ) and (GN ) is found again. This means that an additional TCP temperature (at about 254 K), which is not experimentally accessible as it has been previously commented, is found.

8.8 Other Binary Systems Exhibiting Re-entrant Nematic Behaviour As cited above, re-entrant nematic phases appear, among others, in binary mixtures between selected liquid crystals at atmospheric pressure. The first binary mixtures where the reentrant nematic behaviour was evidenced were those of {p-[p-(hexyloxy-benzylidene)-amino] benzonitrile}(HBAB) and [N-pcyanobenzylidene-p-n-octyloxyaniline] (CBOOA), obtained by Cladis in 1975 [30] and whose corresponding phase diagram is shown in Fig. 8.12. The discovery of the re-entrant behaviour in liquid crystals has resulted both in experimental and theoretical studies of this phenomenon. Thus, from the experimental point of view, an active research has been done concerning the re-entrancy in new systems, as well as the analysis of this phenomenon in their phase diagrams in which temperature, pressure and composition play an important role. As far as the

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Fig. 8.11 (a) Calculated RN behaviour in the 7OCB + 9OCB mixtures. First-order (continuous line) and second-order (dashed line) phase transitions are clearly distinguished. The horizontal dashed lines correspond to temperatures for which Gibbs energy functions, (G) , of the SmA and N phases have been calculated. (b), (c) and (d) correspond, respectively, to Gibbs energy functions as a function of the mixtures composition at temperatures of 345 K, 334 K and 250 K (Reproduced from Ref. [51] with permission of the author)

theoretical point of view is concerned, the main efforts have been addressed towards the understanding of the microscopic origin and nature of the re-entrancy. According to the Cladis model [34], the re-entrance appears only in compounds, exhibiting layered phases, of amphiphilic molecules having both a polar (aromatic rings with polar heads) and a non-polar (alkyl or alkoxy chains) parts. One of the most experimentally studied liquid crystal re-entrant nematic binary system is hexyloxycyanobiphenyl (6OCB) + octyloxycyanobiphenyl (8OCB). It has

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Fig. 8.12 Phase diagram for the binary system {p-[p-(hexyloxy-benzylidene)-amino] benzonitrile} (HBAB) and [N-p-cyanobenzylidene-p-n-octyloxyaniline] (CBOOA). (Reprinted with permission from [30]. © 1975 by the American physical society)

been characterized by means of calorimetry [57], birefringence studies [58], Xray experiments [36, 56], volumetric determinations [36, 59], dielectric relaxation studies [60, 61] and under pressure measurements [36]. The partial phase diagram, at ordinary pressure, is shown in Fig. 8.13. Light scattering intensity, birefringence and density measurements [58, 59] seem to prove the continuity of the N and RNmesophases, the latter having a stronger molecular packing. The RN-mesophase gives rise to the unusually slow nucleation of a crystalline phase some degrees below the normal melting temperature and this crystalline phase “melts” at the RN-to-SmA temperature. Another classical example of the re-entrant nematic phenomenon has been obtained as a result either of the discovery of several SmAd phases or the study of the N-to-SmAd critical behaviour in binary liquid crystal mixtures whose smectogenic components contain strongly polar (cyano or nitro) end groups. An example of the first case is the system pentylphenyl cyanobenzyloxy benzoate +

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Fig. 8.13 Partial phase diagram for the binary system hexyloxycyanobiphenyl (6OCB) and octyloxycyanobiphenyl (8OCB) where solid and open triangles indicate calorimetric and visual transitions, respectively [57]. Solid phase lines are from X-ray experiments [62] (Reprinted with permission from [57]. © 1980 by the American physical society)

4-n-octyloxybenzoyloxy-4’-cyano-stilbene (DB5 + T 8 , for short), for which an SmAd domain exists in the temperature range of 405–520 K, being surrounded by a nematic one for which the lower concentration of T8 is 86% for the reentrancy (Fig. 8.14a) and the binary phase diagram whose components are 4-nheptyloxybenzoyloxy-4 -cyano-stilbene (T 7 ) and T 8 constitutes an example of the second case (Fig. 8.14b). Finally, in Fig. 8.15 two examples of “multiple re-entrance” in binary polar liquid crystal mixtures are shown: paraazoxyanysol (PAA) + T 8 ), in which there

Fig. 8.14 Partial phase diagram for the binary systems: (a) DB5 + T 8 . Reprinted from [63]. © 1981 by the American physical society. (b) T 7 + T 8 . (Reprinted from [64]; © 1987 by the American physical society)

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213

SmAd

473

T/K

N

423

SmA d

373 0.5

1

X DB ONO 9

(a)

2

(b)

Fig. 8.15 (a) Isobaric phase diagram between paraazoxyanysol (PAA) and T8 . Reprinted from [65], with permission from Elsevier, © 1979. (b) Partial phase diagram between DB8 and DB9 (Adapted from [66])

appears not only a nematic but also a Sm-A re-entrant phase (Fig. 8.15a) and 4-oxyloxyphenyl-4 -nitrobenzoyloxybenzoate (DB8 ONO2 ) + 4-nonyloxyphenyl-4 nitrobenzoyloxybenzoate (DB9 ONO2 ), where two re-entrant nematic phases and a re-entrant SmA phase are evidenced (Fig. 8.15b).

8.9 Other Binary Systems Exhibiting Tricritical Behaviour There has been and there is still an enormous effort concerning the nature of the SmA-to-N phase transition both from theoretical and experimental points of view. According to what has been exposed in Section “nematic-to-isotropic and smectic A-to-nematic phase transitions”, the character of the SmA-to-N phase transition can be driven from second to first order depending on the strength of the coupling between the smectic and nematic order parameters. As cited in this section, that coupling depends on the nematic range which, inter alia, can be controlled by mixing two convenient liquid crystals. Some binary systems in which the SmA-to-N phase transition shows the existence of a TCP, at one temperature and composition, are presented in this section.

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Figure 8.16 shows several binary systems, built up from homologous compounds belonging to the nCB and nOCB series, in which experimental TCP points at the SmA-to-N phase transition have been evidenced [22, 53, 54, 67]. For such compounds, data concerning the critical exponent α for pure components and several binary mixtures are collected and for each one, the corresponding Mc Millan’s ratio (T AN /T NI ) has been found. In Fig. 8.17, the α-values are plotted against (1 − T AN /T NI = NR/T NI ). These values are located in a range that covers from the tricritical point (TCP-mixtures in the systems 8OCB + 9OCB and 8CB + 9OCB) to nearly 3D-XY mixtures (X 8OCB = 0.73 in the system 7OCB + 8OCB). All the data follow a common trend that can be adjusted by the curve

Fig. 8.16 Phase diagrams for the binary systems whose components belong to the nCB and nOCB series of liquid crystals and for which the experimental existence of TCP in the Sm-to-N phase transition has been evidenced (Reprinted with permission from [24]; © 2008 American chemical society)

Fig. 8.17 Effective critical exponent α against (NR/TNI ) for the nCB and nOCB series of liquid crystals compounds and their binary mixtures. ( ) 8CB + 9OCB [53], ( ) 7OCB + 8OCB [49], (✚) 7CB + 8CB [68], ( ) 8OCB + 9OCB [53], (●) 8OCB + 8CB [69], ( ) 8OCB + 10OCB [54], (⊕) 7OCB + 9OCB [50], ( ) 8CB + 10CB [22, 70]. The solid line corresponds to Eq. (8.10)

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Fig. 8.18 Phase diagram for the binary system 4O.8 + 6O.8

  2    α = 397 N R TN I − 40.5 N R TN I + 1.02

(8.10)

As a consequence of the goodness of the curve described by Eq. (8.10) in Fig. 8.17, the existence of a common trend for the set of nCB and nOCB-series of compounds is clearly stated. In spite of the well-known non-universality of the crossover behaviour for liquid crystals, Eq. (8.10) can be used to predict the α-critical exponent from the values of NR/ TNI in this particular series of liquid crystals and their binary mixtures. Another binary system for which the presence of a TCP has been experimentally determined is the butyloxybenzylidene octylaniline (4O.8) + hexyloxybenzylidene octylaniline (6O.8) (Fig. 8.18) for which high-resolution AC calorimetry heat capacity measurements by Salud et al. [71] provided a tricritical composition X TCP6O8 ≈ 0.1. Finally, in Fig. 8.19, two additional examples of binary liquid crystal mixtures where TCP occurs, both of them obtained by pyroelectric AC calorimetry, are shown. One concerns the system 7AB + 8AB (Fig. 8.19a). The other one is the 5CB + D55 system (see Fig. 8.19b) and is an example of double TCP with a so-called injected SmA phase, because neither 5CB nor D55 exhibit the SmA phase. Moreover, for this system, the SmA range exhibits a dome-like structure with rather strong concentration dependence on the SmA-to-N transition temperature. According to the theoretical statements, at the top, the nematic range is small and the corresponding SmA-to-N phase transition is expected to be first order in nature. As we move away from the concentration at the maximum of the dome, the nematic range increases, showing the possiblity of one TCP at the left side of the dome and another at right side, but extremely difficult to be measured.

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Fig. 8.19 Phase diagram for: (a) heptylazoxybenzene (7AB) + octylazoxybenzene (8AB) and (b) pentylcyanobiphenyl (5CB) + 4-n-propylcyclohexylcarboxylate (D55) binary system (Reprinted with permission from [72]. © 2001 Japan society for analytical chemistry)

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44. de Miguel E, Martín del Rio E (2005) Computer simulation of nematic re-entrance in a simple molecular model. Phys Rev Lett 95:217802 45. Klug DD, Whalley E (1979) Elliptic phase boundaries between smectic and nematic phases. J Chem Phys 71:1874–1877 46. Clark NA (1979) Thermodynamics of the re-entrant nematic-bilayer smectic-A transition. J Phys Colloq 40:C3-345 47. Oonk HAJ (1981) Phase theory. The thermodynamics of heterogeneous equilibria. Elsevier, Amsterdam 48. Van Hecke GR (1985) The equal-G analysis—a comprehensive thermodynamics treatment for the calculation of liquid-crystalline phase-diagrams. J Phys Chem 89:2058–2064 49. Sied MB, Salud J, López DO, Allouchi H, Diez S, Tamarit JLI (2003) Reentrant behaviour in binary mixtures of (octyloxy) cyanobiphenyl (8OCB) and the shorter chain homologue (heptyloxy) cyanobiphenyl (7OCB). J Phys Chem B 107:7820–7829 50. Cusmin P, Salud J, López DO, de la Fuente MR, Diez S, Pérez-Jubindo MA, Barrio M, Tamarit JLI (2006) Reentrant nematic behaviour in 7OCB+9OCB mixtures. Evidence for multiple nematic-smectic tricritial points. J Phys Chem B 110:26194–26203 51. Cusmin P (2009) Alkycyanobiphenyl mesogens: theoretical and experimental aspects for the SmA-to-N and N-to-I transitions, Ph thesis. Politècnica de Catalunya University, Barcelona 52. Daranas D, López R, López DO (2000) WINIFIT 2.0 computer program. Polythecnical University of Catalonia, Barcelona 53. Sied MB, Salud J, López DO, Barrio M, Tamarit JLI (2002) Binary mixtures of nCB and nOCB liquid crystals. Two experimental evidences for a smectic A-nematic tricritical point. Phys Chem Chem Phys 4:2587–2593 54. Sied MB, Diez S, Salud J, López DO, Cusmin P, Tamarit JLI, Barrio M (2005) Liquid Crystal Binary Mixtures of 8OCB + 10OCB. Evidence for a smectic A-to-nematic tricritical point. J Phys Chem B 109:16284–16289 55. Kumar S (2001) Liquid crystals: experimental study of physical properties and phase transitions. Cambridge University Press 56. Kortan AR, von Känel H, Birgenau RJ, Litster JD (1984) Nematic-smectic a reentrant nematic transitions in 8OCB-6OCB mixtures. J Physique 45:529–538 57. Lushington KJ, Kasting GB, Garland CW (1980) Calorimetric investigation of a re-entrantnematic liquid crystal mixture. Phys Rev B 22:2569–2572 58. Chen NR, Hark SK, Ho JT (1981) Birefringence study of re-entrant-nematic liquid crystal mixtures. Phys Rev A 24:2843–2846 59. Zywocinski A (1999) Volumetric study of the nematic-Smectic-Ad -reentrant nematic phase transitions in the 8OCB+6OCB mixture. J Phys Chem B 103:3087–3092 60. Nozaki R, Bose TK, Yagihara S (1992) Dielectric relaxation of a re-entrant nematic-liquidcrystal mixture by time-domain reflectometry. Phys Rev A 46:7733–7737 61. Urban S, Dabrowski R, Gestblom B, Kocot A (2000) Dielectric relaxation studies of 6OCB/8OCB mixtures with the nematic-smectic A-nematic re-entrant phase sequence. Liq Cryst 27:1675–1681 62. Guillon D, Cladis PE, Stamatoff J (1978) X-ray study and microscopic study of the re-entrant nematic phase. Phy Rev Lett 41:1598–1601 63. Levelut AM, Tarento RJ, Hardoauin F, Achard MF, Sigaud G (1981) Number of SA phases. Phys Rev A 24:2180–2186 64. Evans-Lutterodt KW, Chung JW, Ocko BM, Birgeneau RJ, Chiang C, Garland CW (1987) Critical behaviour at nematic-to-smectic-A phase transitions for smectic-A1 and re-entrant smectic-Ad phases. Phys Rev A 36:1387–1395 65. Hardouin F, Sigaud G, Achard MF, Gasparoux H (1979) Measurements in a pure compound exhibiting an enantiotropic reentrant nematic behaviour at atmospheric pressure. Solid State Commun 30:265–269 66. Hardouin F, Levelut AM, Achard MF, Sigaud G (1983) Polymorphisme des substances mesogènes à molecules polaires. I. Physico-Chimie Et Structure. J Chim Physique 80:53–64

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Chapter 9

Enantiomers H. A. J. Oonk and I. B. Rietveld

Abstract Only in exceptional cases, a pair of enantiomers do form a series of mixed crystals. The most well-known of the exceptions is the system laevorotatory carvoxime+dextrorotatory carvoxime; its properties are discussed in some detail. Also, attention is given to recent work on the polymorphism of optically active drugs.

9.1 Systems of Enantiomers This chapter is about binary systems whose components are a pair of optical antipodes, or a pair of enantiomers—the molecules of one of the antipodes being the mirror image of the molecules of the other. One of the antipodes is laevorotatory— rotates the plane of polarization of polarized light to the left—and the other dextrorotatory. Optically active substances are said to have chiral molecules—molecules that have at least one chirality centre and do not have a plane of symmetry. The arrangement of groups around the chirality centre is either R or S; one speaks of R configuration and S configuration. Among the optically active substances whose molecules have the R configuration, some are laevorotatory and some are dextrorotatory (see, e.g. Janoschek [1]). A forum devoted to chirality in its broadest meaning is the journal Chirality, which was founded in 1989. For the sake of simplicity, we will define a system whose components are a pair of enantiomers by [(1-X) mole of L + X mole of D], and use L for the laevorotatory enantiomer and D for the dextrorotatory one. The substances D and L have the same H. A. J. Oonk Universiteit Utrecht, Utrecht, The Netherlands e-mail: [email protected] I. B. Rietveld (B) Laboratoire Sciences Et Méthodes Séparatives, Université de Rouen Normandie, Mont Saint Aignan, France e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. À. Cuevas-Diarte and H. A. J. Oonk (eds.), Molecular Mixed Crystals, Physical Chemistry in Action, https://doi.org/10.1007/978-3-030-68727-4_9

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thermodynamic properties, such as melting point and heat of melting. The phase diagram of a binary D+L invariably is symmetrical with respect to the equimolar composition. The equimolar combination is optically inactive.

9.2 Racemates and Conglomerates An L-molecule is not superimposable on a D-molecule, and it means that the formation of mixed crystals between a pair of enantiomers is rather the exception than the rule. In the majority of cases, L+D systems show the presence of a 1:1 compound (compound in the sense of the theory of phase diagrams); and the name for it is racemate. An example for the formation of a racemate (R) is the combination of L- and D-isopropylsuccinic acid (IPSA, succinic acid being COOH. CH2 . CH2 . COOH; by substitution of an isopropyl group for one of the two H atoms on the second C atom, the second C atom turns into a chirality centre). The solid–liquid phase diagram for the combination is shown in Fig. 9.1. In that diagram, the racemate is represented by the vertical line at equimolar composition (the racemate is also referred to as a line compound). At the left hand side of the line, there are three two-phase regions: at the bottom for solid L+R; a small region for solid L+liquid; a large field for liquid+R. a

c

b

t / °C

120

80

+ MSA

IPSA

+ IPSA

+ MSA

Fig. 9.1 Determination of (solid + liquid) phase diagrams is a means to establish the configurative relationships between optically active substances. The compound formed by (−) and (+) IPSA is a racemate. A quasiracemate is formed between (−) IPSA and (+) MSA Adapted with permission from Elsevier, Fig. 8.5.3 H.A.J. Oonk, Phase Theory 1981 [2] with data from Fredga and Miettinen [3]

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The left hand phase diagram in Fig. 9.1 is for the combination of D-methylsuccinic acid (MSA) and L-isopropylsuccinic acid. In spite of the fact that, in comparison with the diagram for L-IPSA+D-IPSA, D-IPSA is replaced by D-MSA, again a compound has been formed. And it means that D-MSA and D-IPSA have the same configuration of groups around the chirality centre. The compound between D-MSA and L-IPSA is a quasiracemate. The right hand diagram in Fig. 9.1 demonstrates that IPSA and MSA, when having the same configuration, do neither combine into a compound nor form a series of mixed crystals. The phase diagram is a simple eutectic phase diagram. Quite often (see Jacques et al. [4]) a pair of enantiomers is giving rise to a simple eutectic phase diagram, which is symmetrical, of course. Crystals of L and D coexist as a mechanical mixture, referred to as a conglomerate.

9.3 Formation of Mixed Crystals, Pseudoracemates As remarked before, an L-molecule is not superimposable on its mirror image, the D-molecule. Does it mean that a pair of enantiomers never will form mixed crystals, knowing that (a high degree of) superimposability is a requirement per se for the formation of mixed crystals? The answer is “no”. A human being’s left hand and right hand are not superimposable. Yet, there are two extreme situations for which the left and the right hand are more superimposable than for situations between the two extremes. In the first of the two extremes, the hands are fully stretched and kept in a horizontal plane, such that the two of them have the thumb pointing to the right. The other extreme is for the two fists, clenched as much as possible. In terms of molecules, these two situations correspond to nearly planar molecules, and to nearly spherical molecules, respectively. The latter type of molecules normally gives rise to plastic crystals. Following Bakhuis Roozeboom [5], the general three types of phase diagram I, II, and III—for systems that show complete subsolidus miscibility, assuming that the A and B molecules are randomly mixed for all of the compositions—imply, for a pair of enantiomers, three types of solid–liquid phase diagrams that are symmetrical with respect to the equimolar composition. These types are: I II III

liquidus and solidus coincide on the straight line between the melting points of the enantiomers; liquidus and solidus have a common maximum; liquidus and solidus have a common minimum.

For all of these three cases, the equimolar composition was by Bakhuis Roozeboom referred to as a pseudoracemate—a term introduced by Kipping and Pope [6]. In the following, we will use the term racemate when the equimolar composition is a perfectly ordered 1:1 (or DL) compound, and the term pseudoracemate when the equimolar composition is a solid solution in which the D and L molecules are randomly distributed over the lattice sites (substitutional disorder).

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9.4 Adriani’s Investigation A key paper on systems of optical antipodes was published in 1900 by Bakhuis Roozeboom’s pupil Adriani [7]. Among the six systems studied, four systems showed the formation of racemic compounds, similar to the type of phase diagram in Fig. 9.1b. The remaining two systems, the camphoroxime and the carvoxime ones, are of special interest in the context of this chapter; they include the formation of mixed crystals. The outcome of Adriani’s experiments on the camphoroxime system is shown in Fig. 9.2. The diagram reveals the existence of a racemate, stable at room temperature. On heating, the racemate first changes into a pseudoracemate, form β, subsequently in a pseudoracemate, form α, and finally into the equimolar liquid mixture. From the diagram, as a whole, it follows that the change from α to liquid amounts to a type I behaviour—typical of the change from ideal plastic crystalline to ideal liquid mixture. The change from β to α is an analogue of type III behaviour. The existence of a racemate at room temperature also follows from solubility experiments; in particular from those carried out by Jacques and Gabard [4, 8].

Fig. 9.2 Camphoroxime system. Phase diagram according to Adriani [7]

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The carvoxime system is discussed hereafter in detail, making allowance for Adriani’s work and his ideas.

9.5 The Carvoxime System Following Adriani The optical antipodes D-, and L-, and the optically inactive DL-carvoxime were synthesized for the first time in the second half of the nineteenth century (Tilden and Shenstone [9], Goldschmidt and Zürrer [10], and Wallach [11, 12] all describe the synthesis starting from limonene). Carvoxime, Fig. 9.3, is the oxime of carvone. Carvone and limonene are components of a variety of essential oils extracted from plants (see e.g. Bauer and Grabe [13]). D-carvone is the main component of caraway oil (about 60%) and is found in dill seed oil; L-carvone is present in spearmint oil (up to 80%) and in kuromoji oil. D-limonene is found in the essential oils of orange, grape fruit, and lemon. A geometric description of the crystals of optically active and DL-carvoxime was given by Beyer [14], who emphasized the similarity between the two sets of axial ratios: D- and L-carvoxime DL-carvoxime

A:B: C = 0.87389: 1: 0.36669. A:B: C = 0.85241: 1: 0.35777.

The solid–liquid phase diagram of the combination of antipodes, which was measured for the first time by Adriani [7], is one with a maximum, just like Bakhuis Roozeboom’s type II. The data collected by Adriani are displayed in Table 9.1 and

Fig. 9.3 Carvoxime molecule; chirality centre at carbon atom 4

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Table 9.1 Carvoxime system. Melting and freezing points as a function of composition. Compositions in % of dextrorotatory component, and temperatures in °C %D

100

99

98

95

90

80

70

60

50

25

8

1

0

m.p

72.0

72.4

73.0

75.4

79.0

84.6

88.2

90.4

91.4

86.4

77.4

72.4

72.0

f.p

72.0

–

–

73.0

75.0

80.0

85.0

–

91.4

92.0

–

–

72.0

also in Fig. 9.4. The melting points (m.p.), which were defined as the temperature at which, on heating, the last trace of solid disappears, were observed in a visual manner. The freezing points (f.p.), defined as the temperature at which, on cooling, the last trace of liquid disappears, were determined in a cooling-curve like manner: by measuring the time needed for a decrease in temperature of one degree, degree by degree. In all of the cooling experiments, the samples had been liquid before the start. Adriani did a crystallization experiment and dissolved 0.68 g of D- and 0.23 g of L-carvoxime in methanol. On evaporation of the solvent at room temperature, the solution first yielded crystals having a melting point of 92 °C, and thereafter crystals having a melting point of 72 °C. The observations made Adriani to conclude that DL-carvoxime is a racemate rather than a pseudoracemate, in spite of the type of phase diagram. A racemate capable of forming mixed crystals with each of the two antipodes.

Fig. 9.4 Carvoxime system. Phase diagram according to Adriani [7]

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9.6 Intermezzo Ever since Adriani’s time, the carvoxime system has exercised the minds of many scientists, and so because of its maximum and the elusive nature of the 1:1 solid. Van Laar [15] in his influential work on binary solid–liquid equilibria devoted a section to “Die Schmelzkurven bei optischen Isomeren”—the melting curves in systems of optical isomers. On theoretical grounds—which in fact was a mixing model derived from the van der Waals equation of state—van Laar came to the conclusion that the only two possibilities were the type I diagram (shown by the camphoroxime system, Fig. 9.2) and the diagram with the racemic compound (Fig. 9.1b for the IPSA system). The type II diagram with the minimum, even more so the type II diagram with the maximum, and even the eutectic diagram in the case of a conglomerate: categorically impossible. Timmermans [16] devoted a chapter to “cases where the criteria of the phase rule must be applied with discrimination”. One can read: “An historic example is the study of mixtures of optical antipodes, whose interpretation perplexed even van’t Hoff”. In Timmermans’s view: “The phase rule characterizes substances only by their thermodynamic constants, and does not differentiate between two substances whose optical properties alone are opposed to each other. In such a case, thermodynamically speaking, there is present only one component, always identical with itself, whatever the value of its rotatory power. The mixture of the camphor oximes (m.p. 119°) fulfils these conditions. From the thermodynamic point of view, it behaves like a single substance, even though the rotatory power varies with the concentration of the liquid”. In line with this view, DL-carvoxime is a compound (substance) that gives a continuous series of mixed crystals with L- and D-carvoxime (another substance). And also, DL-IPSA is a substance that gives a eutectic phase diagram with L- and D-IPSA (another substance). Apparently, both van Laar and Timmermans rejected the possibility of the symmetrical eutectic phase diagram for the case that the antipodes do neither show solid-state miscibility nor the formation of a racemate.

9.7 The Carvoxime System, Tammann’s View In a 1914 paper, Tammann [17] reported heat-of-melting values for D-, L-, and DLcarvoxime derived from the times of arrest in cooling curves. The times of arrest were between 270 and 285 s, and were compared with the times of arrest of substances with known heats of melting. The temperatures at which the arrests took place were 344.5 K (D), 344 K (L), and 364 K (DL). Curiously enough, Tammann’s results come down to the observation that for all of the three cases the entropies of melting have the same value of 5.62 times the gas constant. Tammann’s conclusion was that DL-carvoxime is a mixed crystal (pseudoracemate) and not a 1:1 compound (racemate).

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Fig. 9.5 Crystals of carvoxime. Left: DL-carvoxime; right: crystal with an inner mole-fraction gradient [19]

Given the equality of the entropies of melting, Tammann’s view was adopted by Mauser [18], who pointed to the similarity between carvoxime’s melting diagram and the liquid–vapour TX phase diagram in the system chloroform+acetone (see this work Chap. 3, Fig. 9.5) and by Oonk [19] at the time of writing his thesis. At this point, it may be remarked that, apart from the entropies of melting, Oonk’s view was based on the results of X-ray diffraction and crystallization experiments, some of which are presented here. X-ray experiments revealed that the structures of the antipodes and inactive carvoxime must closely resemble one another. Apart from similarity in axial ratios, there is a pronounced similarity in the intensities of the various reflexions. The most typical features appeared from the h0 diffraction (Weissenberg) photographs. The h0 reflexions from the antipode crystal with odd  are markedly weak compared with h0 reflexions with even . The same holds, yet to a higher degree, for the h0 reflexions from the DL crystal. These properties point to the presence of a pseudo glide plane—to a structure close to space group P21 /c. The number of 4 molecules per unit cell supports this view. A mixture of unequal amounts of D- and L-carvoxime dissolved in methanol started to yield crystals of DL-carvoxime. After that, and before the crystallization of antipode crystals, well-developed crystals of an unknown shape separated out; see Fig. 9.5. The front and back planes (a) of the crystals are not parallel; the angle between c and a is between those of the antipodes (100.9°) and the inactive substance (105.4°). Pieces of a crumbled crystal, selected at random, show melting ranges of several degrees spread over an interval between 72 and 92 °C. Thin sheets, cut from the crystal parallel to c, show, towards the inside of the crystal, gradually rising melting

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temperatures, while outside fragments melt at 72 °C. The compositions of three single crystals, determined by means of a polarimeter, were 82 ± 4, 72 ± 2, and 75 ± 3% of one of the antipodes. These properties point to a mixed crystal with an inner mole-fraction gradient. This view was confirmed by Weissenberg photographs.

9.8 The Carvoxime System, More Recent Advances In the years after 1965, it became clear that the ideas about the carvoxime system had to be revised, principally because of the observation that the entropy of melting of D-carvoxime (m.p. 72 °C) and that of DL-carvoxime are in the ratio 2:3 [20]. Jacques’s observation was an incentive to carry out a more detailed study of the system by making use of advanced techniques: X-ray structure determination, microcalorimetry, and adiabatic calorimetry. The outcome of the study is summarized below starting with the crystal structure of DL-carvoxime.

9.9 X-ray Studies The DL-carvoxime melting at 92 °C proved to have an ordered structure with space group P21 /c instead of pseudo P21 /c as was assumed in 1965 [21]. The weak reflexions observed on Weissenberg photographs (mentioned above) presumably were Renninger reflexions. Positional disorder in the structure of the crystal is out of the question. Figure 9.6 shows the projection of the structure down b. The H atom of the oxime group is involved in hydrogen bonding: D- and L-molecules form dimers by hydrogen bonding across the centre of symmetry. The six-membered rings with the two hydrogen bonds are planar within experimental error. According to expectation, the crystals of D- and L-carvoxime (m.p. 72 °C), space group P21, are isostructural with those of DL-carvoxime. The similarity between the two structures [22] extends to the statistics of the X-ray reflexions. This is demonstrated by the comparison of the quotients of normalized intensities of h0 reflexions with odd  and those with even : for DL-carvoxime the quotient is zero as a consequence of space-group symmetry; for D- and L-carvoxime, the quotient is as low as 0.06. The most dominant feature of the crystal structure is the hydrogen-bond formation O–H… N, which, in L-carvoxime, is essentially the same as in DL-carvoxime, although, in the former, the six-membered ring has lost its centre of symmetry. An X-ray study of a crystal containing unequal amounts of D and L has been carried out by Baert et al. [23]. The crystal investigated had a composition not far from the 1:1 ratio; it was the first crystal obtained by evaporation at 20 °C of a solution of equal amounts of D-carvoxime and DL-carvoxime. As a result, from a crystal-structure point of view, the DL crystal is an ordered structure, in which the D-molecules are related to the L-molecules by space group symmetry; they are located on D- and L-sites, respectively. With an increasing amount

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H. A. J. Oonk and I. B. Rietveld

Fig. 9.6 Projection of DL-carvoxime’s structure along b Reproduced with permission from the IUCr, H.A.J. Oonk and J. Kroon, Acta Cryst. (1976) B32, 500 [21]

of L, the D-molecules on the D-sites are randomly replaced by L-molecules, whereas inversely, the L-sites are gradually filled by D-molecules.

9.10 Calorimetry Heat capacities between 160 and 385 K and heats of melting of DL-carvoxime and L-carvoxime were determined by adiabatic calorimetry [24]. The samples used for the study had masses of 3.1532 g and 3.4646 g, respectively. The heats of melting are displayed in Table 9.2. Table 9.2 Change from solid to liquid of DL-carvoxime and L-carvoxime. Melting points (m.p.) in K; heat of melting (H) expressed in kJ. mol−1 m.p

H

365.1 ± 0.2

22.70 ± 0.06

adiab. cal

364.9 ± 0.1

22.2 ± 0.4

DSC

346.5 ± 0.2

17.02 ± 0.02

adiab. cal

346.1 ± 0.2

17.2 ± 0.4

DSC

DL-carvoxime, metastable

353.7 ± 0.5

15.1 ± 0.4

DSC

L-carvoxime, metastable

339.6 ± 0.5

13.8 ± 0.4

DSC

DL-carvoxime, stable L-carvoxime, stable

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By differential scanning calorimetry (DSC), the change from solid to liquid was studied as a function of composition, using samples whose mass was typically 2 mg [25]. The results confirmed the phase diagram obtained by Adriani [7], and, in addition, revealed the existence of a continuous series of metastable mixed crystals, which showed an unusual, erratic melting behaviour of DL-carvoxime, and yielded the heats of melting of stable and metastable L-carvoxime and DL-carvoxime. The heats of melting measured by DSC are included in Table 9.2. The melting diagram of the metastable series of mixed crystals has a maximum. Possibly, metastable DL-carvoxime, unlike stable DL-carvoxime, has a structure in which the D- and L-molecules are randomly distributed over the lattice sites.

9.11 Thermodynamic Analysis Irrespective of the nature of the DL composition in stable and metastable carvoxime, the continuity shown in the melting diagrams can be accounted for by the ABΘ model introduced in Chap. 3. Clearly, the B parameter of the model vanishes for symmetrical systems. For the systems at hand, the Eqs. (3.11, 3.13, and 3.14) for the excess functions reduce to G E (T, X ) = A X (1 − X ) (1 − T /)

(9.1)

H E (X ) = A X (1 − X )

(9.2)

S E (X ) = (A/)X (1 − X )

(9.3)

On the lines of Chap. 3, and adopting its assumptions (ideal liquid mixing; heat capacities may be ignored), the excess enthalpy and with it the value of A follow from the heats of melting of DL and L. The phase diagram provides information about the excess Gibbs energy, and yields, in combination with the excess enthalpy, the value of Θ. The results of the analysis [26] are shown in Table 9.3. The phase diagrams are fully reproduced using the information in Tables 9.2 and 9.3. Table 9.3 Carvoxime system. Calculated values of the model parameters A and Θ for the stable and the metastable series of mixed crystals. Parameter A is expressed in kJ. mol−1 and Θ in K A

Θ

Stable series

− 22.72

435

− 20.0

449

DSC

Metastable series

− 5.2

632

DSC

adiab. cal

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H. A. J. Oonk and I. B. Rietveld

Fig. 9.7 Carvoxime system. Diagrams of enthalpy vs mole fraction (left) and entropy vs mole fraction (right), including, from top to bottom, liquid, metastable solid (II), and stable solid (I) [26] Reproduced with permission from Elsevier, T Calvet, HAJ Oonk, Calphad (1995), 19(1), 49

In Fig. 9.7, the outcome of the study is shown in the form of enthalpy and entropy diagrams. The strong, negative entropy of mixing around the equimolar mixture in the stable series is evidence of pronounced ordering. For the metastable series, in spite of the negative excess entropy, the entropy of mixing remains positive. The enthalpy of mixing is negative, for the stable as well as the metastable solid states. The calculated values of Θ are in line with the global trend (see this work Chap. 3, Fig. 3.6).

9.12 Carvoxime: Concluding Observations When comparing the carvoxime system with an average mixed crystals forming system A + B, where A and B are members of a chemically coherent group of substances (think of A = 2-fluoronaphthalene and B = 2-bromonaphthalene), one can note an interesting difference between the two cases. The replacement, in the perfect crystal lattice of pure A, of an A-molecule by a B-molecule corresponds to a disturbance of the perfect structure. Owing to the geometric mismatch between the A- and the B-molecule, the replacement goes together with a positive enthalpy effect. In other terms, owing to the mismatch,

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the mixed crystalline state between A and B is characterized by a positive excess enthalpy, which has its maximum in the vicinity of the equimolar composition. Or, in experimental terms, the heat of melting of the highly disordered equimolar mixed crystal is lower than the mean of the heats of melting of pure A and pure B. In the carvoxime system, it is the other way around: the perfect structure is at the equimolar composition—and this perfect structure is disturbed when D-molecules are replaced by L-molecules, or L-molecules by D-molecules. The disturbance is at its maximum for pure L and pure D. In this case, the pure antipodes have the lowest heat of melting. One could say that the solid antipode is a ‘would-be racemate’.

9.13 Carvone and Limonene The optical antipodes L-carvone and D-carvone are liquid at room temperature, and so are their mixtures, and so are L-limonene and D-limonene and their mixtures. When cooled in a (micro)calorimeter carvone and limonene fail to crystallize, and become vitreous. On heating from the vitreous state, the material first undergoes the glass transition, and then crystallizes most of the time. Carvone crystallizes more readily than limonene; in DSC experiments, undercooled liquid limonene (D- or L-) crystallizes only once every five times [27]. The binary systems of the carvones and the limonenes have been studied by DSC and by adiabatic calorimetry.

9.14 The Carvone System The carvone system has a rich polymorphic nature that can be revealed by adiabatic calorimetry. The behaviour displayed by the system in a DSC, represented in Fig. 9.8 l-carvone O

2

glass transition

1 6

undercooled liquid

liquid

5

3 4

spontaneous crystallization

monoclinic solid 150

200

liquid 250 K

Fig. 9.8 Laevorotatory carvone—its forms and its behaviour in a DSC [28] Reproduced with permission from Springer Nature, HAJ Oonk and MT Calvet, Equilibrium between phases of matter 2008

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melting

glass transition heat taken

crystallization

given off 165

T/K

215

265

Fig. 9.9 Thermogram of a sample of laevorotatory carvone which is vitreous at the start Adapted with permission from Elsevier from H.E. Gallis et al. Thermochim. Acta (1996), 274, 231 [29]

for L-carvone, is simple, and in that, the recordings give no evidence of the existence of more than one solid form. The thermogram in Fig. 9.9, which is a recording of the behaviour of L-carvone in a DSC, is representative of the behaviour of mixtures of L-carvone and D-carvone, irrespective of their composition. The events, registered in L- carvone’s thermogram in order of increasing temperature, are the so-called glass transition (~171 K); a crystallization process (~193 K to 208 K); followed by a recrystallization phenomenon (~210 K to 218 K); and, finally, the melting of the sample (~248 K). The thermograms are evidence of the existence of a continuous series of mixed crystals, along with a phase diagram having a minimum—type III. The melting point of L-carvone is measured as (247.7 ± 0.5) K; its heat of melting is (11.55 ± 0.05) kJ. mol−1 . The phase diagram’s minimum is at 231 K; at the minimum the heat of melting is 9.0 kJ. mol−1 . From the melting temperatures and heat effects, the model parameters calculated are A = 10.2 kJ. mol−1 and Θ = 332 K. The A and Θ values imply that the system has a miscibility gap with a critical temperature of 215 K; see Fig. 9.10. Sañé et al. [30], who carried out an X-ray study at 218 K, found that the molecular packing in DL-carvone is similar to that in L-carvone and such that the D- and Lmolecules are randomly distributed over the lattice sites. The minimum in Fig. 9.10, as a result, would correspond to the melting point of a pseudoracemate. A closer study of the system’s phase behaviour was made by van Miltenburg and Gallis, who carried out a variety of experiments by adiabatic calorimetry [31–34]. Here, a short account is given of their findings starting with the study of L-carvone’s polymorphic behaviour.

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Fig. 9.10 Carvone system. Filled circles and diamonds: onset and end of melting determined by DSC. The solid curves represent the calculated phase diagram Reproduced with permission from Elsevier, H.E. Gallis et al. Thermochim. Acta (1996), 274, 231 [29]

In addition to the solid form I (which is the form that crystallizes and melts in the DSC experiments), there is a metastable form II. Heating from the vitreous state in an adiabatic calorimeter of L-carvone, results in a glass transition at 171 K, followed by crystallization to form II. This polymorph, when cooled down and subsequently heated again, spontaneously changes into form I between 202 and 225 K. In addition, the authors report the existence of a reversible transition in form I, from Iβ to Iα, taking place at about 200 K, and involving a small heat effect. The material that crystallizes in the adiabatic calorimeter, starting from vitreous DL-carvone, shows two melting peaks. The first melting peak is at 231 K (the melting temperature found by DSC) and the second at 241 K. Apparently, during the first melting, a recrystallization occurs to a more stable form whose melting point is 241 K. If, after the appearance of the first peak, the sample is kept under adiabatic conditions, it takes about 5 days to recrystallize completely into the stable form. The stable form melts at (241.15 ± 0.05) K, and its heat of melting is (12,697 ± 5) J. mol−1 . Experiments carried out on samples with compositions between DL and L gave evidence of the existence of anomalous racemates in ratios 1:4 and 4:1. An anomalous racemate is a compound like a racemate; however, with a ratio other than 1:1 (for examples see Jacques et al. [4]). The phase diagram of the carvone system, based on the observations by DSC and adiabatic calorimetry, is represented in Fig. 9.11.

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Fig. 9.11 Carvone system. The stable phase diagram derived from measurements by DSC and adiabatic calorimetry Reproduced with permission from Elsevier, HE Gallis et al., Thermochim. Acta (1999) 326, 83 [34]

9.15 The Limonene System The phase behaviour of the limonene system, when studied by means of adiabatic calorimetry, is shown in Fig. 9.12 [27]. The stable phase behaviour corresponds to the formation of a racemate, whose melting point is lower than that of the antipodes. In addition, there is evidence of the existence of a series of mixed crystals, the (metastable) melting diagram of which has a maximum. In the following, a brief account is given of the composition-dependent behaviour of the system in the calorimeter. Seven samples with compositions 0.5 < X ≤ 0.8, when cooled after crystallization and subsequently heated, produced heat-capacity diagrams typical of the melting of mixed crystals (see van der Linde et al. [35]). One of the samples, the one having X = 0.59, was kept under adiabatic conditions after it had partially melted. Under these conditions, the sample recrystallized. After crystallization, the material was cooled and next heated again until it was fully liquid. The heating curve of the experiment shows two, each other partially overlapping events—the eutectic arrest and further melting according to the system’s stable behaviour. Two samples, having compositions X = 0.82 and X = 0.87, when cooled after crystallization and next heated, directly displayed the stable phase behaviour evidenced by the eutectic arrest and the characteristic peak of residual melting. The sample having the equimolar composition, after crystallization, cooling and reheating, displayed a melting peak at 184.6 K and a small satellite peak at about 187 K, the total heat effect being 9.4 kJ. mol−1 . In a following experiment, starting from the liquid state, the procedures were repeated until the sample was partially

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Fig. 9.12 Limonene system, survey of observations and calculations Reproduced from Gallis et al. PCCP (2000), 2, 5619 with permission from the PCCP Owner Societies [27]

liquid and the temperature was 184 K. In that situation, the sample was kept under adiabatic conditions for 4 days. After the adiabatic arrest, the sample was cooled down and subsequently reheated. It resulted in a single melting peak at 186.7 K and a heat effect of 10.2 kJ. mol−1 . For the metastable melting diagram, it was assumed that the metastable melting point of the antipodes is 176.55 K [34] (from [36]. The heat of melting of the antipodes was set at 8.4 kJ. mol−1 , by extrapolation of the values measured for the samples with 0.5 ≤ X ≤ 0.8. With these values and Θ = 300 K (Eq. 36 in Chap. 3), the value of the parameter A is calculated as A = − 4.0 kJ. mol−1 .

9.16 Racemic Mixtures and Pressure The racemate of IPSA in Fig. 9.1b cannot be resolved in an easy manner using temperature, because it is more stable than the separate enantiomers i.e. its congruent melting temperature is much higher than the temperature of fusion of the pure enantiomers. Figure 1c, although not a mixture of optical antipodes, represents a eutectic

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equilibrium, which in terms of an enantiomer mixture is called a conglomerate. For a conglomerate, the temperature-composition (T-x) phase diagram as depicted in Fig. 9.1 would be symmetrical with both enantiomers melting at the same temperature. Conglomerates can generally be separated into pure enantiomers by entrainment or direct crystallization [4]. The idea to use pressure to influence the relative positions of the melting points of the racemate in relation to those of the pure enantiomers has existed for a long time and has been mentioned, for example, in the book by Jacques, Collet, and Wilen [4]. Nonetheless, the authors spent only half a page on the subject, because no experimental data existed at that time. In the following, experimental observations of the effect of pressure on pairs of optical antipodes will be discussed starting with the mixed crystal system camphor. For comparison, because very little has been published on optical antipodes under pressure, the systems mandelic acid and ibuprofen are presented here too.

9.17 The Temperature-Composition Phase Behaviour of Camphor The temperature-composition (T-x) phase diagram of camphor is very similar to the one of camphoroxime (compare Figs. 9.2 and 9.13). The racemic mixture consists of a racemate, phase III, which is stable at low temperatures up to about 206 K, where it turns into a hexagonal plastic crystal (phase II). The racemate contains disorder and after annealing for very long times in an adiabatic calorimeter, the III–II transition appears to shift to 218 K [37]. It has an orthorhombic space group, like the low temperature phase of D-camphor, however, the cell parameters are different [38, 39], which confirms that it is a racemate. The phases II and I are plastic crystals with a hexagonal and a cubic geometry, respectively. These phases seem to be continuous throughout the phase diagram and must therefore be solid solutions [40] like in the cases discussed above. Due to their rotation, the camphor molecules look very much like the closed fists as mentioned in the beginning of this chapter, creating a situation, for which the difference between an ordered or disordered system becomes effectively a metaphysical question. The phase transitions between the phases I and II and that of the fusion of I, should, considering the slight decrease in temperature of the phase transition towards the racemic mixture, be of Bakhuis-Roozeboom’s type III; however, by DSC, no difference between the solidus and the liquidus (and for the equivalent transitions between the phases I and II) has been observed [40]. The measured curves have therefore been described by the equal Gibbs energy curve: TE GC (X ) =

(1 − X )αβ H A + X αβ H B (1 − X )αβ S A + X αβ S B

+

αβ G EE GC (X )

(1 − X )αβ S A + X αβ S B

(9.4)

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Fig. 9.13 Temperature-composition phase diagram of the camphor system obtained by DSC measurements. Phase III is orthorhombic for the pure enantiomer and the racemate, but with different cell parameters. Phase II is hexagonal and phase I is cubic; both are continuous solid solutions Reproduced with permission from Elsevier, Rietveld IB et al. (2010) Thermochim. Acta 511:43–50 [40] E E with αβ G EE GC (X ) = X (1 − X )αβ G rac , in which the parameter αβ G rac expresses the magnitude of the excess Gibbs energy difference at equimolar composition between phases α and β. It can be found that the difference in excess Gibbs energy between the racemic liquid phase and phase I equals −53 J mol−1 . By fitting the transition between plastic phases I and II, a difference in the excess Gibbs energy of −2.6 J mol−1 can be found. Hence, if one assumes that the excess Gibbs energy in the liquid phase for two enantiomers is approximately zero, the excess Gibbs energy for phase I must be in the order of 53 J mol−1 and that of phase II will be close to 56 J mol−1 .

9.18 The Camphor System Under Pressure Although the plastic phases will not facilitate the separation of D- or L-camphor, it is still interesting to investigate how the stabilities of pure phase I and of the racemic phase I evolve in relation to the liquid with increasing pressure. Therefore, D-camphor

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and the racemic mixture have been measured with high-pressure differential thermal analysis (HP-DTA), which has been introduced in Chap. 4. In Fig. 9.13, it can be seen that the III-II transition temperature for D-camphor is found at a higher temperature than the one for DL-camphor. In Fig. 9.14a, it can be seen that with increasing pressure, the transition temperature increases for both DL-camphor and for D-camphor. Moreover, the temperature difference between the two transitions slowly increases too, although this may be harder to see, the initial slope for DL-camphor is 5.5 MPa K−1 and the slope for D-camphor is 4.6 MPa K−1 [41]. It implies that the orthorhombic structure with the pure enantiomer increases in stability compared to the racemic compound. A similar conclusion can be reached for the melting transition of D-camphor and of DL-camphor. Although, the melting temperatures are virtually the same at 0 MPa, it can be seen in Fig. 9.14b that the melting temperature of DL camphor exhibits more curvature and climbs more steeply than D-camphor does. Hence, the melting temperature of D-camphor rises faster with increasing pressure. In terms of a P–Tx phase diagram for the melting of camphor, it can be seen in Fig. 9.14c that the temperature of melting increases for all compositions with pressure; however, the curvature of the transition line, which is the equal Gibbs energy curve in this case, increases too. Using Eq. 9.4, it can be calculated that the excess Gibbs energy of the racemic mixture at 250 MPa has become 130 J mol−1 compared to 53 J mol−1 at 0 MPa [41]. In the case of camphor, for which the two optical antipodes form a solid solution of plastic crystals, it will be difficult to separate the pure enantiomers by simple crystallization experiments; nonetheless, the system clearly demonstrates that the racemic mixture is affected differently by pressure than the pure enantiomers. It will be shown in the following for the case of mandelic acid that such pressure behaviour can lead to an inversion of the stability between the racemate and the pure enantiomer.

Fig. 9.14 a Pressure–temperature measurements by HP-DTA for the III-II transition of D-camphor (solid circles) and of DL-camphor (open circles); b idem for the melting transition of phase I; c The P-T-x phase diagram of the melting of camphor (equilibrium I-L) Reproduced with permission from ACS, Rietveld IB et al. (2011) J. Phys. Chem. B 115:1672–1678 [41]

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9.19 Mandelic Acid In the case of mandelic acid, D- and L-mandelic acid crystalize in the space group P21 with a density of 1.35 g cm−3 [42]. DL-mandelic acid exhibits two racemates, one (I) with a space group Pbca and a density of 1.31 g cm−3 and a second (II) crystallizing in the space group P21 /c with a density of 1.36 g cm−3 [43, 44]. The melting temperature of the pure enantiomer is 405 K, whereas the stable racemate, phase I, melts at about 394 K [45, 46]. Racemate II, the metastable form, melts around 381 K [45, 46]. A eutectic equilibrium between the racemate I and the pure enantiomer can be found at 387 K (Fujita et al.) [47]. Hence, the phase diagram that schematically represents the system of mandelic acid is the bottom one in Fig. 9.15a. The effect of the pressure on this system can be predicted using the densities of the different phases and the Le Chatelier principle. Considering that the enantiomers have a higher density than that of racemate I, increasing the pressure will lead to a stabilization of the conglomerate (the eutectic system) in relation to racemate I. However, racemate II has the highest density and it is this phase that will ultimately become the most stable phase under pressure (unless another unknown even more stable phase appears). Under normal DSC conditions and for a given sample of mandelic acid, one of the following three stable melting events can be observed: that of the pure enantiomer, that of the pure racemate or that of the eutectic between the racemate and the pure enantiomer (followed by the liquidus). The influence of the pressure on these three events is depicted in Fig. 9.15b. It should be realized that Fig. 9.15b is not a phase

Fig. 9.15 a Evolution of the melting temperatures with pressure of the racemate in relation to the pure enantiomers and the two resulting eutectic equilibria, one between the pure enantiomers (mainly metastable) and one between the pure enantiomer and the racemate. In the top T-x phase diagram, the eutectic temperature of the two enantiomers coincides with the congruent melting temperature of the racemate. b The pressure dependence of the three stable melting events in the mandelic acid system, pure enantiomer: open circles, pure racemate: solid circles, and the eutectic transition between the pure enantiomer and the racemate: open diamonds Reproduced with permission from ACS, Rietveld IB et al. (2011) J. Phys. Chem. B 115:14,698–14,703 [44]

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diagram for a particular mixture x. It simply provides the evolution of three events in the mandelic acid system projected on the P–T plane with the composition axis perpendicular to both the P and T axis pointing out of the page; the melting of the pure enantiomer (open circles) will only be observed for a sample of one of the pure enantiomers, the melting temperature of the racemate (solid circles) will only be observed for the pure racemate and the eutectic transition (open diamonds) will be observed for mixtures of pure enantiomer and racemate with a signal’s size depending on the composition. Two interesting observations can be made in Fig. 9.15b. First of all, it is clear that with increasing pressure, the melting temperatures of the pure enantiomer and of the racemate diverge; the melting temperature of the pure enantiomer is increasing much faster in temperature than that of the racemate. Secondly, the melting temperature of the racemate and the eutectic transition converge with pressure. It implies that from the point of view of the T-x diagram, the system is moving from a situation as depicted at the bottom of Fig. 9.15a to that at the top of Fig. 9.15a; thus, the melting temperature of the racemate will eventually, with a high enough pressure, coincide with that of the eutectic temperature. At that point, the eutectic will not only exist between an enantiomer and the racemate, but it will coincide with the eutectic between the two pure enantiomers. Hence, at the point where the melting temperature of the racemate crosses the eutectic temperature, four phases must be in equilibrium: two enantiomers, the racemate, and the racemic liquid. Unfortunately, the HP-DTA equipment could not reach a high enough pressure to observe the actual quadruple point. Nonetheless, the phase equilibria can be approximated by straight lines (see also Chap. 4) and those lines can be extrapolated to find the coordinates of the quadruple point. Schematically, this has been depicted for the racemic mixture of mandelic acid in Fig. 9.16. On the left from pressure PA to pressure PB , the congruent melting temperature of the racemate (solid line) can be seen converging with the eutectic temperature of the pure enantiomers. This metastable eutectic (broken line ε2 ) becomes stable above the quadruple point, even though the experimentally observed eutectic in the measurements is the one between the enantiomer and the racemate, which is absent at the racemic concentration. They all meet in point B in Fig. 9.16, which was calculated to have the following coordinates: T = 457 K and P = 642 MPa [44]. As has been explained in Chap. 4 on topological phase diagrams, in a triple point (here a quadruple point, but due to the projection on the P–T plane it is equivalent in its appearance to a triple point in a unary diagram), three phase equilibria meet. The right hand side of Fig. 9.16 depicts only the stable phases and phase transitions. Given that eutectic ε2 (enantiomer R + enantiomer S + liquid) becomes stable at pressure PB , it can be seen that the two enantiomers must exist as a physical mixture of pure enantiomer crystals on the left of the solid line representing the equilibrium ε2 . It implies that at point B, the third intersecting equilibrium must be that between the racemate and the pure enantiomer crystals, and the other two being the melting of the racemate and the eutectic ε2 . The equilibrium between the racemate and the enantiomers, line a-B, decreases in pressure with increasing temperature for reasons that can be found in the original article [44]; however, the position of line

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Fig. 9.16 Pressure–temperature phase diagrams of the racemic mixture (x = 0.5 mol%) of mandelic acid. Left hand side: stable (solid lines) and metastable extensions (broken lines). Right hand side: stable domains for the racemic mixture as a function of pressure and temperature Reproduced with permission from ACS, Rietveld IB et al. (2011) J. Phys. Chem. B 115:14,698–14,703 [44]

a-B is only approximately known. The right hand side of Fig. 9.16 can be considered the topologically obtained P–T phase diagram of the racemic mixture of mandelic acid. Now, if the racemate of mandelic acid is brought above 642 MPa and slightly below 457 K (to avoid melting, but at the same time to be as close to the known coordinate, as the slope of the line a-B is not well defined by the experiments), the racemate may transform into the pure enantiomer crystals. This is not guaranteed, however, because phase diagrams provide thermodynamic stabilities, whereas for transformations, activation energies are also important. The latter are not necessarily linked to the thermodynamic stability of a phase. The phase diagram of mandelic acid under pressure was discussed by Weizhao et al. [48]. They carried out crystallization experiments under pressure, in which one may assume that the most stable compound will crystallize out. They found that, at room temperature, racemate I crystallize from a saturated solution under pressure up to 0.65 GPa, thus perfectly in line with the observations mentioned above. However, at around 0.7 GPa, they observed racemate II crystallizing out and not the pure enantiomers. This result merits a few remarks. First of all, it is unfortunate that it has not been possible experimentally to resolve (for now) mandelic acid by pressure. Nonetheless, the result found by Weizhao et al. [48] is not in contradiction with the abovementioned topological diagram and the crystallization result was to be expected, because racemate II is the densest known crystal structure of mandelic acid. The thermodynamic analysis leading to Fig. 9.16, gives rise to the conclusion that the solid state of the pure enantiomer becomes more stable than racemate I. It is a relative stability between those two phases disregarding any other phase that may exist. If at the same

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time another phase becomes more stable, than that phase will be the most stable of the three, but this does not alter the fact that also the pure enantiomers are more stable than the racemate I. Secondly, racemate I was observed crystalizing at 0.65 GPa, whereas racemate II appeared around 0.7 GPa; however, these findings were obtained at room temperature. It does not exclude the existence of a pressure–temperature domain where the pure enantiomers are stable; all depends on the positions and the slopes of the phase equilibria involved as a function of pressure and temperature: RacI—RacII, RacI— enantiomer and RacII—enantiomer.

9.20 Ibuprofen Under Pressure Ibuprofen is a nonsteroidal anti-inflammatory analgesic and antipyretic compound. It is the dextrorotatory (+)-ibuprofen that is pharmaceutically active; thus, it would make pharmaceutical sense to remove (−)-ibuprofen. The melting point of the pure enantiomer is found at 323.8 K, whereas the racemate melts at 347.9 K. The enantiomer has a specific volume at room temperature of 0.915 cm3 g−1 and the racemate has a specific volume of 0.893 cm3 g−1 . Hence, the density of the racemate is higher and it is therefore to be expected that pressure increases the stability of the racemate. This is corroborated by the HP-DTA data, which demonstrate that the melting equilibria slowly diverge with increasing pressure and thus the gap between the melting temperatures becomes larger (Fig. 9.17). Although in this case, the use of pressure does not promote a possible resolution of the enantiomers, the data can be used to illustrate that for a racemic mixture the same approach can be used to establish a topological phase diagram as for a “simple” case of polymorphism (simple in the sense that only one chemical compound is involved instead of two enantiomers). The topological approach has been explained in Chap. 4 and the details of how the phase diagram of ibuprofen can be determined can be found in Rietveld et al. [49]. The phase diagram is presented in Fig. 9.18 and it has exactly the same appearance as a unary phase diagram for dimorphism with overall monotropism, indicating that only the racemate has a stable domain over the entire pressure range for the racemic mixture.

9.21 Wallach’s Rule One of the arguments often used to disregard pressure for the separation of enantiomers is Wallach’s rule [50], which states that racemates tend to be denser than their chiral counterparts [51] and according to the logic of the Le Chatelier principle, this would mean that pressure stabilizes racemates. It appears to be true that the statistical average of the densities of racemates is higher than that of pure enantiomers; however, this observation may be due to a sampling bias [51]. In any case, whether Wallach’s

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Fig. 9.17 High-pressure differential thermal analysis of ibuprofen enantiomer (circles) and ibuprofen racemate (diamonds) pressures of fusion as a function temperature Reproduced with permission from ACS, Rietveld IB et al. (2012) J. Phys. Chem. B 116:5568–5574 [49]

Fig. 9.18 Pressure–temperature phase diagram of the ibuprofen system at the racemic mixture (left) and a typical T–x phase diagram (right) Reproduced with permission from ACS, Rietveld IB et al. (2012) J. Phys. Chem. B 116:5568–5574 [49]

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rule is valid or not is beside the point, as for enantiomer resolution one will always be dealing with a specific system of enantiomers and their racemate. Considering the data assembled by Brock et al. [51] from the Cambridge Structural Database, it can be seen that at that time, they found 65 enantiomers with racemates. Of those 65, 21 pure enantiomer structures have a higher density than their racemates. This means that following the Le Chatelier principle about one third of the enantiomer systems could potentially be resolved by using pressure. In addition, one should not forget that phase stability is defined as a function of pressure and temperature. Thus both variables together may also create stability domains for the conglomerate at high temperature and high pressure in analogy to the polymorphism phase diagram of bicalutamide [52], for example. It implies that a racemate that is denser than the conglomerate and melts at a higher temperature under ordinary conditions may still form a stable conglomerate under high pressure and high temperature.

9.22 Concluding Remarks The phase behaviour, including polymorphism, of mixtures of optical antipodes— also referred to as scalemic mixtures—is of vital importance when it comes to the development and subsequent registration of chiral drugs. A chiral drug is either the laevorotatory or the dextrorotatory member of a pair of enantiomers, such as the example of ibuprofen mentioned above. The optical antipode of a chiral drug, as a rule, does not have the desired effect; or worse, can be responsible for a harmful side effect. In many cases, a chiral drug is obtained by organic synthesis, accompanied by the same amount of its antipode (racemic mixture). An important issue in the pharmaceutical industry, therefore, is whether chiral drugs should be marketed as single enantiomers or as racemic mixtures (see e.g. Lennard [53]). Meanwhile, however, the guidelines of governmental authorities focus more and more on the necessity of developing chiral drugs as single enantiomers. Owing to the fact that the separation of the optical antipodes from a racemic mixture corresponds to a technically complex and at the same time costly operation, there is an increased interest in the establishment of practical stereoselective syntheses [54, 55], which may be facilitated by biocatalysis [56]. Polymorphs of a given drug differ in physical properties, such as solubility and stability (see, e.g. Grunenberg et al. [57]). Any difference in physical properties may have far-reaching consequences for pharmaceutical processing (see, e.g. Roberts and Rowe [58]). Recent examples are ritonavir [59, 60] and rotigotine [61, 62]. More information on polymorphism can be found in Chap. 4. If stereoselective synthesis is not possible or expensive, the desired enantiomer may also be obtained by a so-called resolution of the racemic mixture. In the case that two enantiomers form a conglomerate, resolution can be relatively simple by using direct crystallization methods or entrainment [4, 63]. Even if the enantiomer

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system does not form a conglomerate, it has to be kept in mind that metastable equilibria may be present just below the stable system as for example in the limonene system discussed above. A judicious choice of the solvent may stabilize a possible metastable conglomerate to be used for resolution. A particular interesting example of such a case is diprophylline, which exhibits complex polymorphic behaviour for the racemate as well as for the pure enantiomer [64, 65]. Enantiomer resolution can be obtained too by chemical modification followed by separation [4], but nowadays chromatographic methods are also very capable of separating enantiomers [66]. One recent discussion in the literature, which is of interest, is symmetry breaking of racemic mixtures due to ripening effects. It has been proposed as a reason for the observation of single chirality in living systems. Viedma observed symmetry breaking for chiral crystals of non-chiral molecules free to crystallize in one or the other chiral crystal when in solution. Using slurry-grinding techniques only a single chiral crystal was eventually obtained. Similar behaviour was observed in boiling solutions, where convection causes symmetry breaking. The phenomenon was called Viedma ripening [67, 68]. Noorduin et al. extended the idea of Viedma ripening to amino acid derivatives with an asymmetric carbon atom that quickly racemized in solution by chemical equilibrium. Using similar techniques as Viedma, they could cause complete symmetry breaking in their solutions ending up with a single enantiomer in the crystalline state [69]. This method was adapted by Suwanassang et al. [70] using heating and cooling cycles for a compound that racemizes in solution and crystallizes as a conglomerate. They ascribed the observed symmetry breaking to the small fluctuations in thermodynamic stability between the crystals of the two enantiomers. The stability differences are caused by the size differences between the crystals due to the stochastic nature of nucleation. Temperature cycling amplifies this effect resulting eventually in crystals consisting of a single enantiomer [70].

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35. Van der Linde PR, Bolech M, den Besten R, Verdonk ML, van Miltenburg JC, Oonk HAJ (2002) Melting behaviour of molecular mixed crystalline materials: measurement with adiabatic calorimetry and modelling using ULTRACAL. J Chem Thermodyn 34:613–629 36. Timmermans J (1914) Chem Zentralbl I:618 37. Nagumo T, Matsuo T, Suga H (1989) Thermodynamic study on camphor crystals. Thermochim Acta 139:121–132 38. Mora AJ, Fitch AN (1997) The low temperature crystal structure of RS-camphor. J Solid State Chem 134:211–214 39. Brunelli M, Fitch AN, Mora AJ (2002) Low-temperature crystal structure of S-camphor solved from powder synchrotron X-ray diffraction data by simulated annealing. J Solid State Chem 163:253–258 40. Rietveld IB, Barrio M, Veglio N, Espeau P, Tamarit J-L, Céolin R (2010) Temperature and composition-dependent properties of the two-component system d- and l-camphor at ‘ordinary’ pressure. Thermochim Acta 511:43–50 41. Rietveld IB, Barrio M, Espeau P, Tamarit J-L, Céolin R (2011) Topological and experimental approach to the pressure-temperature-composition phase diagram of the binary enantiomer system d- and l-camphor. J Phys Chem B 115:1672–1678 42. Patil AO, Pennington WT, Paul IC, Curtin DY, Dijkstra CE (1987) Reactions of crystalline (R)(-)-and mandelic acid with amines. Crystal structure and dipole moment of (S)-mandelic acid. A method of determining absolute configuration of chiral crystals. J Am Chem Soc 109:1529 43. Fischer A, Profir VM (2003) A metastable modification of (RS)-mandelic acid. Acta Crystallogr E Struct Rep 59:O1113 44. Rietveld IB, Barrio M, Tamarit J-L, Do B, Céolin R (2011) Enantiomer resolution by pressure increase: inferences from experimental and topological results for the binary enantiomer system (R)- and (S)-mandelic acid. J Phys Chem B 115:14698–14703 45. Lorenz H, Sapoundjiev D, Seidel-Morgenstern A (2002) Eantiomer mandelic acid system melting point phase diagram and solubility in water. J Chem Eng Data 47:1280 46. Lorenz H, Seidel-Morgenstern A (2004) A contribution to the mandelic acid phase diagram. Thermochim Acta 415:55 47. Fujita Y, Fujishir R, Baba Y, Kagemoto A (1972) Thermal properties of optically active compounds. 1. Study on thermal properties of mandelic acid using differential thermal-analysis method. Nippon Kagaku Kaishi 8(9):1563–1567 48. Weizhao C, Marciniak J, Andrzejewski M, Katrusiak A (2013) Pressure effect on D, L-mandelic acid racemate crystallization. J Phys Chem C 117:7279–7285 49. Rietveld IB, Barrio M, Do B, Tamarit J-L, Céolin R (2012) Overall stability for the ibuprofen racemate: experimental and topological results leading to the pressure-temperature phase relationships between its racemate and conglomerate. J Phys Chem B 116:5568–5574 50. Wallach O (1895) Zur kenntniss der terpene und der ätherischen öle. Liebigs Ann Chem 286:90– 143 51. Pratt-Brock C, Schweizer WB, Dunitz JD (1991) On the validity of Wallach’s rule: on the density and stability of racemic crystals compared with their chiral counterparts. J Am Chem Soc 113:9811–9820 52. Gana I, Ceolin R, Rietveld IB (2012) Bicalutamide polymorphs I and II: a monotropic phase relationship under ordinary conditions turning enantiotropic at high pressure. J Therm Anal Calorim 112:223–228 53. Lennard MS (1991) Clinical pharmacology through the looking glass: reflections on the racemate versus enatiomer debate. Br J Clin Pharmacol 31:623–625 54. Shinkai I (1997) Design and development of practical asymmetric syntheses of drug candidates. Pure Appl Chem 69:453–458 55. Zask A, Ellestad GA (2015) Recent advances in stereoselective drug targeting. Chirality 27:589–597 56. Patel RN (2011) Biocatalysis: synthesis of key intermediates for development of pharmaceuticals. ACS Catal 1:1056–1074

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57. Grunenberg A, Keil B, Henck JO (1995) Polymorphism in binary mixtures, as exemplified by nimodipine. Int J Pharm 118:11–21 58. Roberts RJ, Rowe RC (1996) Influence of polymorphism on the Young’s modulus and yield stress of carbmazepine, sulfathiazole and sulfanilamide. Int J Pharm 129:79–94 59. Bauer J, Spanton S, Henry R, Quick J, Dziki W, Porter W, Morris J (2001) Ritonavir: an extraordinary example of conformational polymorphism. Pharm Res 18:859–866 60. Chemburkar SR, Bauer J, Deming K, Spiwek H, Patel K, Morris J, Henry R, Spanton S, Dziki W, Porter W, Quick J, Bauer P, Donaubauer J, Narayanan BA, Soldani M, Riley D, McFarland K (2000) Dealing with the impact of ritonavir polymorphs on the late stages of bulk drug process development. Org Process Res Dev 4:413–417 61. Chaudhuri KR (2008) Crystallisation within transdermal rotigotine patch: is there cause for concern? Expert Opin Drug Del 5:1169–1171 62. Rietveld IB, Céolin R (2015) Rotigotine: unexpected polymorphism with predictable overall monotropic behavior. J Pharm Sci 104:4117–4122 63. Coquerel G (2015) Solubility of chiral species as function of the enantiomeric excess. J Pharm Pharmacol 67:869–878 64. Brandel C, Amharar Y, Rollinger JM, Griesser UJ, Cartigny Y, Petit S, Coquerel G (2013) Impact of molecular flexibility on double polymorphism, solid solutions and chiral discrimination during crystallization of diprophylline enantiomers. Mol Pharm 10:3850–3861 65. Brandel C, Cartigny Y, Coquerel G, ter Horst JH, Petit S (2016) Prenucleation self-assembly and chiral discrimination mechanisms during solution crystallisation of racemic diprophylline. Chem Eur J 22:16103–16112 66. Fanali S (2017) Nano-liquid chromatography applied to enantiomers separation. J Chromatogr A 1486: 20–34 67. Viedma C (2005) Chiral symmetry breaking during crystallization: complete chiral purity induced by nonlinear autocatalysis and recycling. Phys Rev Lett 94:065504/1–065504/4 68. Viedma C, Cintas P (2011) Homochirality beyond grinding: deracemizing chiral crystals by temperature gradient under boiling. Chem Commun (Cambridge, U. K.) 47:12786–12788 69. Noorduin WL, Izumi T, Millemaggi A, Leeman M, Meekes H, Van Enckevort WJP, Kellogg RM, Kaptein B, Vlieg E, Blackmond DG (2008) Emergence of a single solid chiral state from a nearly racemic amino acid derivative. J Am Chem Soc 130:1158–1159 70. Suwannasang K, Flood AE, Rougeot C, Coquerel G (2013) Using programmed heating-cooling cycles with racemization in solution for complete symmetry breaking of a conglomerate forming system. Cryst Growth Des 13:3498–3504

Chapter 10

Complexes A. Marbeuf and D. Mikaïlitchenko

Abstract In certain cases, two molecular substances A and B, having a high degree of molecular homeomorphism, give rise to the formation of a complex AB, rather than producing a series of mixed crystals. We demonstrate that complexes are formed when short-range van der Waals forces are overruled by long-range coulomb forces or by hydrogen bonds. Two groups of binary systems have been studied: (i) the group of benzene and benzene derivatives, and (ii) the group of naphthalene and naphthalene derivatives.

10.1 From Random to Ordered Alloys In Chap. 9, distinction was made in binary enantiomeric systems between random alloys (conglomerates) and ordered alloys (racemates) acting thermodynamically as line compounds. Such monophasic and biphasic solid phases belong to the generic family of co-crystals which was first introduced in the pharmaceutical research and which is now extended to other chemical systems [1]. We shall focus here on more common (1-X) A + X B systems in which one or more line compounds Am :Bn may occur: these molecular line compounds are called complexes. Complexes are found, either in gas phase—the so-called van der Waals complexes—such complexes between ozone and rare gases of environmental interest [2], either in liquid phase or in solid phase. In aromatic systems, in aliphatic systems with TAGs (see Chap. 11), m = n=1 mainly: therefore, we will restrict to A:B compounds. Because no composition range is detected, these congruent or uncongruent melting point lie exactly at the solid composition x = X = 1/2. In the first case, the liquid composition is also x and the phase diagram is of the doubleeutectic type if the complex is unique, whereas a non-congruent melting corresponds A. Marbeuf (B) · D. Mikaïlitchenko LOMA, UMR 5798, Université de Bordeaux, Talence, France e-mail: [email protected] D. Mikaïlitchenko e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. À. Cuevas-Diarte and H. A. J. Oonk (eds.), Molecular Mixed Crystals, Physical Chemistry in Action, https://doi.org/10.1007/978-3-030-68727-4_10

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to the existence of a peritectic transformation. When binary systems involve many complex compounds with m = n, combinations between eutectics or peritectics are possible. Because lattice stabilities of molecular alloys E lat are known to be related to intermolecular interactions V ij in terms of van der Waals forces, the purpose of this chapter is to define energetic characteristics which yield to random or ordered molecular alloys. Finally, examples in the benzene and naphthalene families will be presented in which structural determinations (X-ray diffraction, neutron scattering) and thermodynamic analysis (calorimetry, data assessments) yield correlations between complex solid phase and melt. Systems with many polyaromatic hydrocarbons are of biological interest because of their highly carcinogenic properties and their ability to intercalate between the bases in DNA [3].

10.2 Mean-Field Approach in Molecular Alloys In A1-x Bx alloys, a mean-field description may be used:molecular i-j interactions are separated into iA -jA , iB -jB and iA -jB (or iB -jA ) kinds and summarized into homomolecular A…A and B…B intermolecular interaction energies (εAA , εBB ) and heteromolecular A…B interaction energy (εAB ) [4]. Therefore, a pairwise-interaction model of the Borelius–Gorsky–Bragg–Williams type (BGBW approach [5] may be derived which takes into account short-range van der Waals potentials and long-range Coulombic ones or hydrogen bonds. Then, the lattice choice for alloys between random or ordered distribution appears as the consequence of a balance between these two kinds of interactions [6]. This mean-field approach is complementary of the macroscopic crystalline isomorphism degree εmi1 and of the molecular isomorphism concept εk discussed in Chap. 2: multiple examples of applications have been given for aromatic (Chap. 5) and aliphatic families (Chap. 6). By analogy with the BGBW approach for metal or inorganic alloys in which species are distributed on two sites α and β, the homomolecular interaction energies (εAA , εBB ) are assumed to be linearly composition-dependent: 0 (T ) − 2xγAB (T ), εAA (T, x) ∼ = εAA

(10.1)

0 εBB (T, x) ∼ (T ) + 2(1 − x)γAB (T ) = εBB

(10.2)

whereas heteropolar interaction energies vary with a long-range order parameter σ which describes the molecular distribution on the two sites. A simple way is to restrict εAB to be a quartic σ-function: 0 (T ) − φAB σ 2 + φAB σ 4 /2 εAB (T, σ ) ∼ = εAB

(10.3)

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0 0 where εAA (T ) and εBB (T ) correspond respectively to the internal energy of the species in their pure crystal. The crystal lattice is built by putting A and B molecules onto two equivalent sublattices, each distribution being a function of the alloy mol fraction and of the long-range parameter σ. Therefore, mixing entropy contains an excess term due to the non-ideal distribution. If species have z as mean coordination value, the BGBW model is able to express the Gibbs free energy S G(T, x, σ ) related to the pure compounds where both enthalpy and entropy term will be function of σ. In the peculiar case of interest (x = 1/2), S G(T,x,σ ) writes as: S

G(T, 1/2, σ ) = AB (T, 1/2)/4 + [AB (T, 1/2) − AB ]σ 2 /4 − AB σ 4 /8 + RT /2[(1 + σ ) ln(1 + σ ) + (1 − σ ) ln(1 − σ ) − 2 ln 2] (10.4)

where AB = z N φ AB is a positive term and AB (T,1/2) is the so-called interaction parameter coming from Eqs. (10.1–10.3):  0  0  0 (T ) − 1/2 εAA (T ) + εBB (T ) + (1 − 2x) AB (T ) (10.5) AB (T, x) = z N εAB The asymmetric term with AB (T ) = z N γAB (T ) term vanishes for x = 1/2. Without long-range order (σ =0), the BGBW model yields random alloys A1-x Bx with sub-regular excess properties given by eq. (10.5): S

G (T, 1/2, 0) = S G E (T, 1/2) = AB (T, 1/2)/4 − RT ln 2

(10.6)

When complete ordering (σ =1), the Gibbs free energy, deduced from eq. (10.4), corresponds to a line-compound A:B, i.e. a complex: S

G(T, 1/2, 1) = AB (T, 1/2)/2 − 3 AB /8

(10.7)

AB acts as a stabilizing quantity in the formation of a complex phase. It probably includes quadrupole–quadrupole interactions which have been found to stabilize complex lattice, as demonstrated by Hernandez-Trujillo et al. [7], for the C 6 H 6 :C 6 F 6 phase involving molecules with quadrupole moments of opposite sign. In other systems, the stabilizing effect may be related to charge-transfer or to hydrogen bonding appearing in the crystal lattice. For a given temperature, parameters are evaluated in the following manner: 0 0 (T ) and εBB (T ) are deduced from the sublimation homomolecular quantities εAA enthalpy of the corresponding components; for random alloys, excess functions give the quantity AB (298K , 1/2), and therefore the heteromolecular interaction energy 0 (T ); for ordered phase, both Gibbs free energy of formation s G (T,1/2,1) and the εAB 0 (T ) [6, 8]. equilibrium condition d(S G (T,1/2,1)/d = 0 lead to εAB Figure 10.1 gives the evolution of AB (298K , 1/2) versus the negative of the 0 0 homomolecular quantity −z N [εAA (298K )+εBB (298K )]/2 in normal alkane family,

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7

0

6 1

ΩAB (kJ/mol)

-2 -4

5

9

3

10

-6 -8

2

-10

8 4

-12 -14

80

120

160

200

240

280

320

360

-zN (ε°AA + ε°BB ) / 2 (kJ/mol) Fig. 10.1 Evolution of the interaction parameter, AB , versus the negative of the homomolecular interaction energy quantity, –zN (εoAA + εoBB ) /2, at 298 K [8]: - benzene complexes ():1 (C 6 H 6 :C 6 F 6 ), 2 (C 6 H 5 CH 3 :C 6 F 6 ), 3 (1,4 - C 6 H 4 (CH 3 )2 :C 6 F 6 ), 4 (1,3,5 - C 6 H 3 (CH 3 )3 :C 6 F 6 ), 5 (C 6 H 5 Cl:C 6 F 6 ), 6 (C 6 H 5 F:C 6 F 6 ) and 7 (1,4 C 6 H 4 CH 3 Cl:C 6 F 6 ), - naphthalene complexes: (●): 8 (C 10 H 8 :C 10 F 8 ), 9 (2 - C 10 H 7 Cl:C 10 F 8 ) and 10 (2 C 10 H 7 CH 3 :C 10 F 8 ), - benzene random alloys: () (C 6 H 5 Cl - C 6 H 5 Br, 1,4 - C 6 H 4 Cl 2 - 1,4 - C 6 H 4 Br 2 and 1,3,5 C 6 H 3 Cl 3 - 1,3,5 - C 6 H 3 Br 3 ), - naphthalene random alloys: (o) (2 - C 10 H 7 Cl-2 - C 10 H 7 Br, 2 - C 10 H 7 Cl-2 - C 10 H 7 CH 3 and 2 C 10 H 7 Br-2- C 10 H 7 CH 3 ), - random alloys between normal alkanes C n H 2n+2 : (Δ)Δn=1 (n=11,…,21), (♦) Δn=2 even (n=14,…, 20), (∇) Δn=2 odd (n=13,…,21) and (×) C19 H40 -C23 H48 Reproduced from [8] with permission of the author

in benzene and naphthalene families. The aliphatic family’s points are near the zero line for the n-alkane binary systems C n H 2n+2 -C n+1 H 2n+4 (Δn = 1) with |AB (298 K,1/2)| ≤ 0.6 kJmol−1 ; for the systems C n H 2n+2 -C n+2 H 2n+6 (Δn = 2), either n is even or n is odd, AB (298 K,1/2) is positive and maximal for the shortest chain length (C 14 H 30 -C 16 H 34 , AB (298 K,1/2) = 1.2 kJmol−1 ); and finally, for the system C 19 H 23 -C 23 H 48 (Δn = 4), AB (298 K,1/2) takes the highest positive value (2.5 kJ mol−1 ). In the aromatic families, some random alloys lie again near the zero line and correspond to the case of halogen-substituted ring (Cl or Br); when the number of halogen increases, AB (298 K,1/2) becomes more positive (AB (298 K,1/2) ≈ 2 kJ mol−1 ); negative values (AB (298 K,1/2) ≈ −2 kJ mol−1 ) are present for naphthalene binaries 2-C 10 H 7 X—2-C 10 H 7 CH 3 (X = Cl, Br). Complexes can correspond either to strong negative values of the interaction parameter (AB (298 K,1/2) = −11.7 kJ mol−1 for 1,3,5-C 6 H 3 (CH 3 )3 :C 6 F 6 , AB (298 K,1/2) = −10.87 kJ mol−1 for the complex C 10 H 8 :C 10 F 8 ) or positive

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ΔεAB / ε oAB (%) at 298K

3

7

6 5

1

2 3 1 2

0

4 3

4

5

6

7

8

9

-ΔfGcom (kJ/mol) at 298K

10

11

12

Fig. 10.2 Evolution of the relative A-B stabilization ΔεAB/ ε°AB versus the negative Gibbs free f energy of complexation -Gcom , at 298 K in the benzene family (Symbols have the same meaning as in Fig. 10.1)

values (AB (298 K,1/2) = 1.2 kJ mol−1 for the complex 1,4-C 6 H 4 CH 3 Cl:C 6 F 6 ). Finally, it is not surprising that Φ AB has the highest value for this complex (6.5 kJ mol-1) and a zero one for complex 1,3,5-C 6 H 3 (CH 3 )3 :C 6 F 6 as the consequence of the balance between AB (298 K,1/2) and AB in the preceding equation. Therefore, existence of complexes and random alloys in the same family cannot be understood by AB (298 K,1/2) only, as shown by the dispersive values of this parameter, but can be explained in connection with AB (see eq. (10.7)): the physical meaning of this parameter lies in the global stabilization of “bonds” A-B when long-range ordering. Figure 10.2 shows the correlation between the Gibbs free energy of complexf ation (Gcom = 2 S G (T,1/2,1)) for aromatic complexes and the coefficient 0 0 )298K ) which is equal to the relative quantity—( εAB /εAB )298 K . ( AB /(2z N εAB For alloys with negative mixing enthalpy, short-range ordering near the equimolar composition may be present. By increasing temperature, an order → disorder transition of the second kind may occur. According to Landau’s theory [9], the description of transitions of the first kind would require at least a sixth-order σ-dependence of εAB in order to reproduce the discontinuity of σ between the trivial zero value and the finite one. As seen in binary enantiomeric systems disordered conglomerate alloy and racemate ordered phase may coexist (see Chap. 9).

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10.3 Liquid Complexes Strong complexing means a significant emergence of the melting point of the A:B phase in the liquidus line and, therefore, to a complex with congruent melting, whereas complex phase melts non-congruently when weak complexing. Besides, interactions in the liquid phase are important for a good understanding of both complexing and phase diagram-type in binary systems. Literature contains many examples of heterocomplexes in various solvents, which are characterized by their spectroscopic properties in UV or visible range. Their chemical nature covers large area from charge-transfer complexes, complexes with Lewis acids or hydrogen bonding, addition compounds and transition metal complexes [10]. This is quite different from liquid complexes, quoted thereafter (AB), which are detected in a binary melt in equilibrium with a solid A:B phase: by neutron and Xray scattering experiments, [11, 12] have found (C 6 D6 C 6 F 6 ) liquid complexes with D…F bonds and arrangement similar to the ones present in the solid phase with an average intermolecular distance of 3.7 Å. Andrews et al. [13] measured the mixing enthalpy of several (1-X) C6 H6-n -Rn + X C6 F6 liquid: S-shaped curves are found, L E H (T,X) being negative except in regions of low X-values with a minimum lying at X = 1/2 (see Fig. 10.3 for the (1-X) C 6 H 6 + X C 6 F 6 melt). Moreover, near the melting point, a small departure of liquidus from a horizontal line appears. Thermodynamics are then similar to the ones of intermetallic systems involving III-V and II-VI semiconductors, magnetic compounds (GaN, MnTe, ZnO, CdSe, HgTe, etc. … [14–16]. By analogy with III-V and II-VI systems, a good description of the melt (1-X) A + X B will be the associated solution model (ASM) [8]. Therein, the liquid phase contains three species AL , BL and the associate (AB) with respective mol fractions y1 , y2 , y3 , in chemical equilibrium and interacting in a “sub-regular” way: A L + B L ↔ (AB)

(10.8)

yielding the following expression of the mixing free energy: L

  

(1) (1) (2) (2) + H12 (y1 − y2 ) G M (T, X ) = y1 y2 H12 − T S12 − T S12   (1) (1) (1) + y H + y1 y3 H13 − TS(1) y − TS 2 3 13 23 23 + RT(y1 ln y1  ◦ + y2 ln y2 + y3 ln y3 ) − y3 G diss /(1 + y3 ) (10.9)

In a ASM liquid model, the mol fraction of (AB) at the melting point depends on the dissociation energy of this liquid species G°diss and decreases with T. Near the melting point of the A:B complex, liquidus line follows the van’t Hoff law with a discontinuity due to the T-dependence of both excess functions L H E (T,X) (named thereafter Δmix H L (T,X)) and L S E (T,X), as a result of Eq. (10.9). Model will be able to

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Fig. 10.3 Mixing enthalpy of liquid (1-X) C 6 H 6 + X C 6 F 6 versus X for T above the melting point of the solid phase: experimental curves (__ 298.2 K, — 313.2 K, … 328.2 K,.-.-.- 343.2 K are adapted from Andrews et al. [13]), calculated values with ASM parameters (see Table 10.1) come from [8] (● 298.2 K,  313.2 K, 328.2 K, ◯ 343.2 K)

explain both Δmix H L (T,X) curve with negative values in a large X-composition range and possible liquid miscibility gaps due to repulsive AL -(AB) or BL -(AB) interactions as in Mn-Te binary(see Chevalier et al. [16]).

10.4 The Benzene-Hexafluorobenzene System The C 6 H 6 —C 6 F 6 binary is one of the most studied systems in the benzene family, showing a A:B solid phase with congruent melting (T fus = 297 K): calorimetric measurements by Andrews et al. [13], liquidus determination by Duncan et al. [17] through differential scanning calorimetry, liquid neutron scattering performed by Bartsch et al. [11], solid–solid transitions studied by Duncan et al. [17]), Brennan et al. [18], Ripmeester et al. [19]. Crystal structures determined by Overell et al. [20],

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Fig. 10.4 Cell volume of solid complex phases C 6 H 6 :C 6 F 6 versus temperature showing the first order type of the three solid–solid transitions Reproduced from [8] with permission of the author

Williams et al. [21–24] show three solid–solid transitions of the first kind. Mikaïlitchenko [8], Mikaïlitchenko et al. [25] have measured the cell volume expansion during heating of the complex phase (Fig. 10.4). The high-temperature form (C I ) is trigonal (space group R 3¯ m) with 3 complex entities per cell nearly perpendicular to the c–axis and turning around this. The behaviour of this plastic or rotator phase is certainly similar to the ones for normal alkanes as explained by molecular dynamics with particular corkscrew-like jumps of the molecules [26]. By decreasing temperature, C I transforms at 274 K into C II of monoclinic symmetry (space group I2/m) with 2 C 6 H 6 and 2 C 6 F 6 molecules where rings turn now with a precession movement around the c-axis. Below 249 K, crystal lattice ¯ always with 2 C 6 H 6 and distorts and leads to a triclinic symmetry (space group P 1), 2 C 6 F 6 . In this C III phase, the rotation of C 6 F 6 rings is frozen. Finally, the movement of C 6 H 6 molecules slows down, leading to the C IV phase with an unexpected increasing of symmetry (space group P21 /a). Optimization of the complete phase diagram was achieved by using ASM model in the liquid phase (Fig. 10.5 and Table 10.1): at the melting point of C I phase,

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Fig. 10.5 Benzene + Hexafluorobenzene binary system [8] including: DSC and X-ray measurements (▲ [17],  [19], x [8, 25]) and—calculated phase diagram Reproduced from [8] with permission of the author

the liquid associate (C 6 H 6 C 6 F 6 ) mol fraction is y3 fusT = 0.19 [8]; according to G°diss , associate becomes less stable when T increases (y3 decreases) and minimum of Δmix H L (T,X) less pronounced.

10.5 The Halogen-Substituted Benzene-Hexafluorobenzene Systems With one halogen atom (F, Cl) on the benzene rings, complexation strength decreases. A complex occurs with fluorobenzene and chlorobenzene, in C 6 H 5 F-C 6 F 6 system (T fus = 258 K) and in C 6 H 5 Cl-C 6 F 6 system (T fus = 259 K) [27], but with less intense f f Gibbs free energy at melting point (Gcom = −2.9 kJ/mol and Gcom = −2.6 kJ/mol, respectively). At the same time, data assessments lead to classical sub-regular melts

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Table 10.1 Thermodynamic data in the C 6 H 6 —C 6 F 6 system (SI units) Experiments

Optimization (1)

Liquid

(1)

H12 = −760(5) S12 = −3.2(1) (2)

(2)

H12 = 1095(2) S12 = 3.4(1) (1)

(1)

(1)

(1)

H13 = 4870(10) S13 = −3.5(5) H23 = 1950(10) S23 = −5.3(5) G°diss = 5080(10)-9.4(10.8)T Complex

Tfus = 297.6

fus H = 20500 ± 280 Ttrs,II → I = 273.7 ± 0.5

trs,II → I H = 950 ± 50 Ttrs,III → I I = 249.2 ± 0.7

trs,III → II H = 600 Ttrs,IV → II I = 199

trs,IV → III H < 200

Tfus = 297.2(2)

fus H = 20200(500) f

G Com = -21120(20) + 57.3(10.5)T Ttrs,II → I = 273.7(2)

Eutectic E1

TE1 = 270.9 xE1 = 0.12

fus HE1 = 9700 ± 110

TE1 = 269.8(5) xE1 = 0.122(6)

fus HE1 = 10190(10)

Metatectic M

TM = 273.7 ± 0.5 xM ≈ 0.14

TM = 273.7(2) xM = 0.15(1)

Metatectic M’

TM = 273.5 ± 0.5 xM’ ≈ 0.83

TM = 273.7(2) xM’ = 0.83(1)

Eutectic E2

TE2 = 268.8 xE2 = 0.85

fus HE2 = 10630 ± 110

TE2 = 268.6(1) xE2 = 0.849(6)

fus HE2 = 10850(5)

Reproduced from [8] with permission of the author

without associate. This result set means that electrophile halogen atoms have destabilizing effect on complexation. This is confirmed by inspecting the 1,4-C 6 H 4 Cl 2 C 6 F 6 system whose phase diagram is of the simple-eutectic type: p-dichlorobenzene does not give any complex phase [28]. One may notice that before melting C 6 H 5 Cl:C 6 F 6 presents the following poly¯ → C I (R 3¯ m): the high-temperature phase being morph sequence C II (P 1) isomorphous of the benzene homologous is of the rotator-type.

10.6 Influence of the Methyl Group on the Complexation in the Benzenic Family With benzene rings containing one or several CH 3 groups (toluene: C 6 H 5 (CH 3 ):C 6 F 6 ; p-xylene: 1,4-C 6 H 4 (CH 3 )2 ; mesitylene: C 6 H 3 (CH 3 )3 :C 6 F 6 ), all binaries built with hexafluorobenzene are of the double-eutectic type. Phase

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diagrams and mixing enthalpies were experimentally investigated and optimized [8, 13, 17, 25]. Corresponding complex phases (T fus = 283 K, T fus = 300 K, T fus = 309 K, respectively), referred to the C 6 H 6 :C 6 F 6 phase, show an increasing stability with number f of H-atoms substituted by CH 3 groups in the benzene ring (Gcom = −3.5 kJ/mol, f f Gcom = −4.6 kJ/mol and Gcom = −7.0 kJ/mol, respectively, at the medium temperature 313 K). In each case, liquid mixing enthalpy is S-shaped and melt is associated: y3 >0.34 may be compared to the corresponding value y3 = 0.16 in C 6 H 6 -C 6 F 6 system or in HgTe system at HgTe melting point (y3 fus T = 0.41) [15]. Dahl [29–33] has shown that the increasing of the number of methyl groups— up to six—leads both to a ring “alignment” and to a decreasing of the interplanar distance, that which stabilize more and more the complex phases. All these complexes ¯ → CI present a rich polymorphism (Table 10.2): the polymorph sequence C II (P 1) ¯ (R 3 m) is found again for toluene and hexamethylbenzene (C 6 H 5 (CH 3 ):C 6 F 6 , C 6 (CH 3 )6 :C 6 F 6 ) [8, 30, 31]). Such properties in C 6 H 6 -n Rn -C 6 F 6 systems are found again when C 6 F 6 is replaced by another molecule, such as fluoranil (C 6 F 4 O2 ) [34, 35], or C 6 H 6 by C 6 D6 [20]), and may be easily extended to other systems in the benzene family. Table 10.2 Polymorphism in benzene complexes: crystallographic data [8, 30–33] Complex with C6 F6

a (Å)

b (Å)

c (Å)

α (°)

β (°)

γ (°)

C6 H6

11.976 6.638 7.389 9.597

11.976 12.325 12.917 7.477

7.232 7.300 7.305 7.584

90 90 97.68 90

90 99.82 111.91 96.10

120 90 110.70 90

C6 H5 CH3

12.547 6.280 6.151

12.547 7.575 14.894

7.124 7.323 7.301

90 108.33 108.99

90 95.54 94.70

1,4C6 H4 (CH3 )2

8.883 6.736 6.173

11.033 7.401 7.745

7.311 7.314 7.302

90 103.42 107.46

1,3,5C6 H3 (CH3 )3

13.498 13.242 13.896

15.590 15.312 15.180

7.188 7.086 13.132

1,2,4,5C6 H2 (CH3 )4

9.478

15.771

1,2,3,5C6 H2 (CH3 )4

8.277

C6 (CH3 )6 C6 H5 Cl

Z

T (K)

3 2 2 2

278 263 233 113

120 107.91 107.11

Space Group R 3¯ m I2/m P 1¯ P21 /a R 3¯ m P 1¯ P 1¯

3 1 2

253 233 183

106.56 97.61 95.37

90 102.26 101.97

P21 /a P 1¯ P 1¯

2 1 1

298 290 203

90 90 90

90 99.25 90

90 90 90

Pnma P21 /n Pamb

4 4 8

293 165 103

7.232

90

133.36

90

C2/m

2

298

14.138

7.119

90

98.94

90

2

298

14.596 8.740

14.596 8.228

7.124 7.149

90 99.83

90 107.34

120 114.58

P21 /m ou P21 R 3¯ m P 1¯

3 1

278 233

12.510 9.436

12.510 9.986

7.100 7.723

90 92.13

90 110.95

120 116.17

R 3¯ m P 1¯

3 2

258 213

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These authors shown that mean interplanar distances between hexafluorobenzene molecules and partner ones are relatively large compared to those observed for ordinary π-π* complexes with charge-transfer and greater than the common van der Waals distance between aromatic rings (3.40 Å). In the same manner, binary systems belonging to the naphthalene family, and more generally to the aromatic series, are known to have a 1:1 complex phase.

10.7 The Naphthalene Parent System: C10 H8 -C10 F8 Even with more heavy aromatic molecules, complexing occurs. The parent system between octafluoronaphthalene and naphthalene (C 10 H 8 -C 10 F 8 ) and the derived ones by H-substitution—such as chloronaphthalene (2-C 10 H 7 Cl-C 10 F 8 ) or methylnaphthalene (2-C 10 H 7 (CH 3 )-C 10 F 8 )—, behave a 1:1 complex phase. After melting of theses solid phases, complex arrangement is maintained in the liquid phase where the associate mol fraction y3 fus T has the following values: 0.28 (C 10 H 8 -C 10 F 8 ), 0.23 (2-C 10 H 7 Cl-C 10 F 8 ) and 0.16 (2-C 10 H 7 (CH 3 )-C 10 F 8 ) [8, 36, 37]. The high-temperature range of the C 10 H 8 -C 10 F 8 phase diagram is nearly symmetric, as for the benzenic homologous (Fig. 10.6). Its complex phase (T fus = 406 K, [36]) is of interest by its high-temperature polymorph (C I ) with the R 3¯ m symmetry. That means naphthalenic rings, as benzenic rings, are able to turn themselves in their planes: C 10 H 8 :C 10 F 8 seems to be a rotator phase as C 6 H 6 :C 6 F 6 in its C I phase. The intermediate phase (C II ), stable between 293 K and 375 K, has the same symmetry than the room-temperature polymorph (C III , space group P21 /a with 2 complex molecules [38]) and C 6 H 6 :C 6 F 6 in its C IV form: in such a lattice, the two partner molecules correspond to each other in the same manner as shown by two layers in the hexagonal graphite structure (Fig. 10.7). Potenza et al. [38] found relatively short C…F and C…C contacts between the virtually eclipsed pairs and suggested a specific C…F interaction which might help to stabilize the complex via dipole-induced interactions. This molecular packing with stacking interactions is general in aromatic complexes, as discussed by Dahl in his review article [39]. Ternary systems, such as (1-X-Y) (2-C 10 H 7 Cl) + X (2-C 10 H 7 (CH 3 )) + Y (C 10 F 8 ) [37]), show the difference i) between systems—where random alloys may lead to a phase diagram of the II-type in the Bakhuis Roozeboom’s classification [40] with a maximum as in (1-X) (2-C 10 H 7 Cl) + X (2-C 10 H 7 (CH 3 ))—and ii) systems— where complex formation with congruent melting yields a double-eutectic type phase diagram as in (1-Y) (2-C 10 H 7 Cl) + Y (C 10 F 8 ). The presence of a maximum is the consequence of strong heteromolecular interactions in the lattice between 2C 10 H 7 Cl and 2-C 10 H 7 (CH 3 ) molecules, which overcompensate the natural effect of the geometric mismatch, whereas when one of the molecular partners is C 8 F 8 , specific interactions, such as quadrupole–quadrupole interactions, enhance heteromolecular interactions and allow ordering. Besides, the lack of isomorphism between C 8 F 8 and 2-C 10 H 7 Cl or 2-C 10 H 7 (CH 3 ) allows only ordering phases.

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Fig. 10.6 Naphthalene—Octafluoronaphthalene binary system: DSC and X-ray measurements (x [36, 37] and [8]) and—calculated phase diagram) Reproduced from [8] with permission of the author

The BGBW model applied to the complex solid phases (Table 10.3) shows that, even if homomolecular energies (zNε°AA , zNε°BB ) energies and heteromolecular ones (zNε°AB ) are of the same order, stabilizing effect due to ordering can explain the difference between random and ordered alloys as shown in eq. (10.7): this is the case of 1,4-C 6 H 4 CH 3 Cl:C 6 F 6 where the tendency to random alloys with a positive value of AB (298 K,1/2) (1100 J/.mol) is overcome by the strong influence of AB (6500 J/mol). Nevertheless, this quantity never exceeds 6% of heteromolecular energies in all studied systems.

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Fig. 10.7 Naphthalene-Octafluoronaphthalene complex: view along the normal to the naphthalene (full lines) plane (left) and along the c-axis (right); ◯ C atom, ● F atom, intermolecular distances come from [38]

10.8 Solid Complex and Associated Melt As seen in previous sections, C 6 H 6-n Rn :C 6 F 6 complexes, where R = CH 3 and n =0, 1, …,3 with high Gibbs free energy of formation melt congruently, are in chemical equilibrium with an associate liquid, whereas no melt association is present for low Gibbs free energy values when R = Cl, n = 1. Such a correlation between the f two phases types may be established by inspecting the value of Gcom (T) and the one of equimolar mix H L (T,1/2) at a given temperature (here 313.2 K): a linear relationship between these two functions is found by Mikaïlitchenko [8] (Fig. 10.8). f It may be noticed that the positive Gcom (T) value of the hypothetical complex 1,4f C 6 H 4 Cl 2 :C 6 F 6 (Gcom (313.2 K) ≈ 2 kJ/mol) agrees with its instability. In the same manner, the stability of CH 5 F:C 6 F 6 appears to be very weak; its existence is probably the consequence of H…F hydrogen bonding between parallel rings.

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Table 10.3 Intermolecular interactions and long-range stabilizing energies in benzene and in naphthalene binary systems (SI units) [8] zNεoAB

zNεoAA or zNεoBB

Ω AB (298 K,1/2)

Φ AB

−102500

C6 H6 CH5 F

−102000

C6 H5 Cl

−110900 −99940

C6 H5 CH3 C6 H4 CH3 Cl

−119150

1,4-C6 H4 (CH3 )2

−130600

1,3,5-C6 H3 (CH3 )3

−125630 −99800

C6 F6 C6 H6 : C6 F6

−102900

−1800

5000

CH5 F: C6 F6

−101400

−500

5700

C6 H5 Cl: C6 F6

−106200

−900

5500

C6 H5 CH3 : C6 F6

−108000

−8100

1800

1,4-C6 H4 CH3 Cl:C6 F6

−108300

1100

6500

1,4-C6 H4 (CH3 )2 : C6 F6

−119400

−4200

3800

1,3,5-C6 H3 (CH3 )3 : C6 F6

−124500

−11700

0

C10 H8

−154880

2-C10 H7 Cl

−160200

2-C10 H7 CH3

−128000 −125200

C10 F8 C10 H8 : C10 F8

−150800

−10800

400

2-C10 H7 Cl: C10 F8

−143900

−1200

5300

2-C10 H7 CH3: C10 F8

−130700

−4000

3900

10.9 Concluding Remarks We have seen how low molecular isomorphism may be accommodated by strong specific interactions which overcome geometric mismatch. Descriptions of both solid (pairwise-interaction model) and liquid phases (associate liquid model) have been presented. Using BGBW and ASM approaches as guides, and coupling them with the isomorphism concept, chemists should be able: (i) to predict complex formation in binary systems; (ii) to explain the resulting crystallographic properties of the solid phase and the thermodynamic behaviour of the melt. In some systems, hydrogen bonding is sufficient to stabilize interactions between molecules of different geometries: Chap. 9 has shown that such interactions occur in the racemate lattice, at the equimolar composition of the DL carvoxime system [41]. Molecular simulations will be also useful for predicting properties in systems with insufficient informations, specially for dynamics during solid–solid transitions: by modelling its vibration dynamics, Williams [24] has been able to interpret the

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Cl

Cl

1000 F

0

Cl

ΔfGoC (J/mol)

-1000

CH3

Cl

-2000 -3000 CH3

-4000 CH3

-5000

CH3

-6000 -7000

CH3 CH3 CH3

-8000 -3000

-2000

-1000

0

1000

2000

3000

ΔmixHL (J/mol) Fig. 10.8 Benzene systems C 6 H 6-n Rn —C 6 F 6 : correlation between enthalpy of mixing in the liquid state and Gibbs free energy of formation of the complex at 313.2 K, C6 H6-n Rn rings are represented by O Reproduced from [8] with permission of the author

phase transition C IV → C III of the complex C 6 H 6 :C 6 F 6 as an increasing amount of molecular diffusion arising from the large amplitude rotations in the C IV phase. Finally, the reader must have in mind that complex formation is not restricted to molecular partners belonging to the same family and to the 1:1 composition: picric acid (1,3,5-C 6 H 2 (NO2 )3 OH) is known to give 1:1 complex phase with naphthalene (C 10 H 8 :C 6 H 3 N 3 O7 [42]), or with phenanthrene (C 14 H 10 :C 6 H 3 N 3 O7 [43]), but a 2:1 complex with anthranilic acid (2 C 7 H 7 NO2 :C 6 H 3 N 3 O7 [44]).

References 1. Zaworotko MJ (2007) Molecules to crystals, crystals to molecules… and back again. Cryst Growth Des 7:4–9 2. Borges E, Ferreira GG, Oliveira JM, Braga JP (2009) A molecular dynamics simulation of Ar n O3 (n = 1-21) van der Waals complexes: size evolution of stable structures. Chem Phys Lett 472:194–199

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3. Hopfinger AJ (1977) Intermolecular interactions and biomolecular organization. Wiley, NewYork 4. Marbeuf A, Ll Casas, Estop G, Mikaïlitchenko D (2003) From all-atom vision to molecular pairwise-interactions: the mean-field approach in molecular alloys. J Phys Chem Solids 64:827– 832 5. Bragg WL, Williams EJ (1934) The effect of thermal agitation on atomic ar-rangement in alloys. Proc Royal Soc. Lond. 145:699–730 6. Marbeuf A, Mikaïlitchenko D, Würger A, Oonk HAJ, Cuevas-Diarte MA (2000) Unified stability concept of mixed molecular lattices: random alloys or complexes. Phys Chem Chem Phys 2:261–268 7. Hernandez-Trujillo J, Costas M, Vela A (1993) Quadrupole interactions in pure non-dipolar fluorinated or methylated benzenes and their binary mixtures. J Chem Soc, Faraday Trans 89:2441–2443 8. Mikaïlitchenko D (1999) Interactions intermoléculaires dans les alliages aromatiques: stabilité. Thèse, Université Bordeaux I, Ordre et Désordre 9. Landau LD, Lifshitz EM (1980) Statistical Physics. Pergamon Press, Oxford 10. Forster R (1969) Organic transfer complexes. Academic Press London, New York 11. Bartsch E, Bertagnolli H, Chieux P (1986) A neutron and x-ray diffraction study of the binary liquid aromatic system benzene-hexafluorobenzene ii. the mixtures. ber. bunsenges. Phys Chem 90:34–46 12. Cabaço MI, Danten Y, Besnard M, Guissani Y, Guillot B (1998) Structural investigations of liquid binary mixtures: neutron diffraction and molecular dynamics studies of benzene, hexaflorobenzene and 1,3,5-trifluorobenzene. J Phys Chem B 102:10712–10723 13. Andrews A, Morcom KW, Duncan WA, Swinton FL, Pollock JM (1970) The thermodynamic properties of fluorocarbon + hydrocarbon mixtures 2. Excess enthalpies of mixing. J Chem Thermodyn 2:95–103 14. Brebrick RF (1988) Thermodynamic modelling of the Hg-Cd-Te and Hg-Zn-Te systems. J Cryst Growth 86:39–48 15. Marbeuf A, Druihle R, Triboulet R, Patriarche G (1992) Thermodynamic analysis of Zn-Cd-Te, Zn-Hg-Te and Cd-Hg-Te: phase separation in Znx Cd1-x Te & in Znx Hg1-x Te. J Cryst Growth 117:10–15 16. Chevalier PY, Fischer E, Marbeuf A (1993) A thermodynamic evaluation of the Mn-Te binary sytem. Thermodynamica Acta 223:51–63 17. Duncan WA, Swinton FL (1966) Thermodynamics properties of binary systems containing hexafluorobenzene. part1. Phase diagrams. Trans Faraday Soc 62:1082–1089 18. Brennan JS, Brown MD, Swinton FL (1974) Thermodynamic study of the 1:1 solid compound, hexafluorobenzene-benzene. J Chem Soc, Faraday Trans 70:1965–1970 19. Ripmeester JA, Wright DA, Fype CA, Boyd RK (1978) Molecular motion and phase transitions in solid hexafluorobenzene + benzene complex by nuclear magnetic resonance and heat capacity measurements. J Chem Soc Faraday Trans II 74:1164–1178 20. Overell JS, Pawley GS (1982) An X-ray single crystal study of the molecular system C6 F6 C6 D6 . Acta Cryst B38:1966–1972 21. Williams JH (1991) An inelastic neutron scattering study of the dynamics of the van der Waals complex C6 H6 :C6 F6 . Mol Phys 73:99–112 22. Williams JH (1991) A quasielastic neutron scattering investigation of the van der Waals complex of benzene and hexafluorobenzene. Mol Phys 73:113–1225 23. Williams JH, Cockcroft JK, Fitch AN (1992) Structure of the lowest temperature phase of the solid benzene-hexafluorobenzene adduct. Angew Chem Int Ed Engl 31:1655–1657 24. Williams JH (1993) Modeling the vibrational dynamics of solid benzene: hexafluorobenzene. The anatomy of a phase transition. Chem Phys 172:171–186 25. Mikaïlitchenko D, Marbeuf A, Haget Y, Oonk HAJ (1998) Influence of the methyl-group on the complexation in the benzenic family: the toluene-hexafluorobenzene system and the p-xylene-hexafluorobenzene system. Mol Cryst Liq Cryst 319:291–305

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26. Marbeuf A, Brown R (2006) Molecular dynamics in n-alkanes: Premelting phenomena and rotator phases. J Chem Phys 124:054901 27. Marbeuf A, Mondieig D, Métivaud V, Négrier P, Cuevas-Diarte MA, Haget Y (1997) Binary system between chlorobenzene and hexafluorobenzene: phase diagram and molecular complex. Mol Cryst Liq Cryst 293:309–323 28. Mikaïlitchenko D, Marbeuf A, Oonk HAJ (1999) Destabilizing effects of chlorine on complexes belonging to the benzene family: p-C6 H4 Cl2 -C6 F6 and p-C6 H4 CH3 Cl-C6 F6 Systems. Chem Mater 11:2866–2871 29. Dahl T (1971) Crystal structure of the 1:1 complex between mesitylene and hexafluorobenzene. Acta Chem Scand 25:1031–1039 30. Dahl T (1972) Crystal structure of the trigonal form of the 1:1 complex between hexamethylbenzene and hexafluorobenzene. Acta Chem Scand 26:1569–1575 31. Dahl T (1973) Crystal Structure of the Triclinic Form of the 1:1 Complex between Hexamethylbenzene and Hexafluorobenzene at -40°C. Acta Chem Scand 27:995–1003 32. Dahl T (1975) Crystal Structure of the 1:1 Addition Compound between p-Xylene and Hexafluorobenzene. Acta Chem Scand A29:170–174 33. Dahl T (1975) Crystal Structure of the 1:1 Addition Compound between Durene and Hexafluorobenzene. Acta Chem Scand A29:699–705 34. Dahl T, Sorensen B (1985) Solid addition complexes of tetramethylbenzenes with fluoranil and hexafluorobenzene. crystal structure of the 1:1 complex between durene and fluoranil. Acta Chem Scand B39:423–428 35. Dahl T (1988) Crystallographic Studies of Molecular Complexes containing Hexafluorobenzene. Acta Chem. Scand A42:1–7 36. Michaud F, Négrier P, Mikaïlitchenko D, Marbeuf A, Haget Y, Cuevas-Diarte MA, Oonk HAJ (1999) Measurement and analysis of the naphthalene-octafluoronaphthalene phase diagram: complex formation in solid and liquid. Mol Cryst Liq Cryst 326:409–424 37. Marbeuf A, Mikaïlitchenko D, Négrier P, Cuevas-Diarte MA, Calvet-Pallas T (2000) 2substituted-naphthalene binary and ternary systems: from random alloys to complexes. Chem Mater 12:3280–3287 38. Potenza J, Mastropaolo D (1975) Naphthalene-octafluoronaphthalene, 1:1 solid compound. Acta Cryst. B31:2527–2529 39. Dahl T (1994) The nature of stacking interactions between organic molecules elucidated by analysis of crystal structures. Acta Chem Scand 48:95–106 40. Bakhuis Roozeboom HW (1899) Löslichkeit und Schmelzpunkt als Kriterien für racemische Verbindungen, pseudoracemische Mischkrystalle und inaktive Konglomerate. Z Physik Chem 28:494–517 41. Oonk HAJ, Kroon J (1976) The carvoxime system. I. X-ray study of dl-carvoxime (m.p. 92°C). Acta Cryst B32:500–504 42. Banerjee A, Brown CJ (1985) Picric Acid-Naphthalene 1/1 π Complex, C6 H3 N3 O7: C10 H8 . A Disordered Struct Acta Cryst C41:82–84 43. Yamaguchi SI, Goto M, Takayanagi H, Ogura H (1988) The crystal structure of phenanthrene: picric acid molecular complex. Bull Chem Soc Jpn 61:1026–1028 44. In Y, Nagata H, Doi M, Ishida T, Wakahara A (1997) Anthranilic acid-picric acid (2/1) complex. Acta Cryst. C53:646–648

Chapter 11

Triacylglycerols L. Bayés-García, M. À. Cuevas-Diarte, and T. Calvet

Abstract Triacylglycerols (TAGs), the main components of edible fats and oils, are widely employed in cosmetics and pharmaceutical formulations. This type of lipids exhibits a complex pattern of polymorphism, which determines the physicochemical properties of end products. In this chapter, an account is given of their polymorphic crystallization and transformation behaviour—and so from pure TAG components to more complex lipid systems. Special attention is given to the effects caused by the application of dynamic thermal treatment. These effects are the key to the design of end products that have the physical properties required for them.

11.1 Introduction Triacylglycerols (TAGs) are the tri-esters of 1,2,3-propanetriol (glycerol) and fatty acids. Being the main components of natural fats and oils, TAGs constitute an important class of nutrients, but they are also widely employed in cosmetic and pharmaceutical industries [1]. Natural fats and oils become very complex systems, as they are usually composed of several types of TAGs, including different fatty acid parts. For the design of applications, the properties of pure component TAGs may be understood, but also those of their mixtures, as a function of temperature and composition [2, 3]. In our work, this comes down to the determination of the structural and thermodynamic characteristics of single components and their mixed systems.

L. Bayés-García (B) · M. À. Cuevas-Diarte · T. Calvet Grup de Cristal·lografia Aplicada, Universitat de Barcelona, Barcelona, Spain e-mail: [email protected] M. À. Cuevas-Diarte e-mail: [email protected] T. Calvet e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. À. Cuevas-Diarte and H. A. J. Oonk (eds.), Molecular Mixed Crystals, Physical Chemistry in Action, https://doi.org/10.1007/978-3-030-68727-4_11

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Fat structures, compositions and polymorphism determine the physical properties of lipid systems, such as rheology, morphology and texture [4, 5]. Multiple polymorphic forms occur in almost all TAGs, which may be modified according to the fatty acid compositions included in the molecule. Among the most common polymorphic forms of TAGs, which are α, β’ and β, and in order to highlight the influence of lipids polymorphism with some examples, it should be mentioned that the desired polymorphic form of cocoa butter in chocolate is β, whereas β’ is preferred in margarine and shortenings. Referring to their binary mixing behaviour, three different phases have been obtained: (i) solid solution phase, (ii) eutectic phase and (iii) molecular compound (MC) formation phase. When a structural similarity and affinitive molecular interactions occur between the two TAG molecules, a solid solution phase is formed. By contrast, if they are immiscible due to steric hindrance, then eutectic equilibrium is obtained [6, 7]. Another immiscible behaviour is the MC formation, which occurs due to specific molecular interactions between the two component molecules. In this chapter, TAGs polymorphism and its relation with fatty acid compositions (saturated/unsaturated, chain length structure and so on) are discussed, by paying special attention to the influence of some kinetic factors, such as the application of dynamic thermal treatments, on its behaviour. Moreover, some examples of binary mixing properties are highlighted, and the possibilities given by cutting-edge techniques, such as synchrotron radiation microbeam X-ray diffraction (SR-μ-XRD), for studying the crystal microstructure of these mixtures, are also described.

11.2 Basic Terms on TAGs Polymorphism Physicochemical properties of TAG molecules are determined by the nature and compositions of the three fatty acid moieties, labeled R1 , R2 and R3 in Fig. 11.1a). The three main polymorphs of TAGs are α, β’ and β [8]. They are characterized by their corresponding subcell structures (hexagonal, orthorhombic ⊥ and triclinic//, respectively), which are based on the packing modes of the zigzag aliphatic chains, as shown in Fig. 11.1b. However, these three basic polymorphic forms may be modified depending on the fatty acid compositions and some of them can be absent or new forms can occur. The chain length structure generates a repetitive sequence of the acyl chains involved in a unit cell lamella along the long-chain axis, as depicted in Fig. 11.2. As an example, when the chemical properties of the three fatty acid moieties are the same or very similar, a double chain length structure is obtained. By contrast, a triple chain length structure is formed when the chemical properties of one or two fatty acid moieties are largely different from the others, due to chain sorting [9]. The major interactions which are thought to be most influential in exhibiting the polymorphic structures of TAGs are the following: (i) aliphatic chain packing due to hydrocarbon chain-chain interactions, (ii) glycerol conformation whose influences may act through dipole-dipole interactions of the glycerol groups, (iii) methyl end

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Fig. 11.1 a Chemical structure of a TAG molecule. b Typical subcell structures of TAG polymorphs

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Fig. 11.2 TAG molecules packing in bilayer (2L) and trilayer (3L) structures

stacking which may mostly determine the organization of different chain length structures, and (iv) olefinic interactions which may predominate in mixed-acid TAGs that contain unsaturated fatty acid moieties. In mixed-acid TAGs, the molecular interactions through the main body of the aliphatic chains, methyl end packing and glycerol groups are modified compared to those of mono-acid TAGs. Therefore, multiplicity and relative stability of polymorphic forms and their lattice energies are also modified (Sato et al. [10]). Referring to the relative stability of the polymorphic forms described above, the β form is the most stable, while β’ is metastable and α is the least stable and, thus, having the lowest melting temperature (Tm ). As the TAGs polymorphism becomes monotropic, the Gibbs free energy (G) values are largest for α, intermediate for β’ and smallest for β, as shown in Fig. 11.3. Therefore, it is possible to crystallize the three polymorphic forms from the liquid state (e.g. by changing the rates of cooling), where kinetic factors play an important role (see Section influence of kinetic factors on the polymorphic behaviour of TAGs). Furthermore, transformations from one polymorph to another can occur in the solid state or through melt mediation (melting of a less stable form followed by the recrystallization of a more stable form).

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Fig. 11.3 Gibbs energy (G)–Temperature (T) relationship of the three basic polymorphs of TAGs

11.3 Polymorphic Behaviour of Mixed-Acid TAGs Mono-acid and mixed-acid TAGs are defined, respectively, on whether the fatty acid chains are of the same fatty acid molecules or not. In this sense, it is important to consider the chain length and parity (odd or even number of carbon atoms) for both saturated and unsaturated fatty acids, but also the position and configuration (cis or trans) of double bonds in unsaturated fatty acids. As for mixed-acid TAGs, which are the most common in nature, polymorphic diversity is superimposed over that of the mono-acid TAGs in the form of sn (stereo-specific numbered) position. As an example of the diversified polymorphic forms that occur in TAGs with different fatty acid compositions, Table 11.1 summarizes the polymorphic forms described for tristearoyl glycerol (SSS), trioleoyl glycerol (OOO) and mixed-acid TAGs containing stearic and oleic fatty acids (1,3-distearoyl-2-oleoyl glycerol or SOS, 1,3-dioleoyl-2-stearoyl glycerol or OSO, 1,2-distearoyl-3-oleoyl glycerol or SSO and 1,2-dioleoyl-3-stearoyl glycerol or OOS). In the data shown, the chain length structure of each polymorphic form is specified. As already stated above, mixed-acid Table 11.1 Occurrence of polymorphic forms of triacylglycerols containing stearic and oleic fatty acids SSS

SOS

OSO

SSO

OOS

OOO

α-2L

α-2L

α-2L

α-3L

α-n.d.

α-2L

β’-2L

β’-3L

β’-2L

β’-3L

β’2 -3L

β’-2L

γ-3L β’1 -3L β-2L

β2 -3L β1 -3L

n.d. Not determined

β-3L

β-2L

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TAGs often exhibit a more complex polymorphic behaviour, with multiple β’ or β forms or additional γ and δ forms. Chapman [11] reported the polymorphism of some trisaturated, triunsaturated and mixed-acid TAGs. The author described three different polymorphic forms (α, β’ and β) for SSS and other trisaturated TAGs, such as tripalmitoyl glycerol (PPP) or trimyristoyl glycerol (MMM), all having double chain length structure (2L). The same polymorphic forms were described for triunsaturated TAG OOO, also having a double chain length structure [11, 12]. However, Hagemann et al. [13] described three different β’ forms for OOO. The polymorphic behaviour of OOO has been recently characterized by applying dynamic thermal treatments based on varying cooling and heating rates [14]. Under such experimental conditions, OOO exhibited five different polymorphic forms (α, two β’ forms and two β forms). As for mono-acid TAGs, both SSS and OOO exhibited 2L structure polymorphic forms due to identical chemical properties of the three fatty acid moieties. In mixed-acid TAGs containing stearic and oleic fatty acids, triple chain length structures were also detected (see Table 11.1). SOS exhibited five different polymorphic forms: α-2L form, an additional γ-3L form, β’-3L, and most stable β2 -3L and β1 -3L forms [15]. Regarding chain length structures, that of least stable α form was described as double, whereas γ, β’ and two β forms exhibited triple chain length structure. By contrast, OSO only showed the three main polymorphs α, β’ (both having a double chain length structure) and β (with a triple chain length structure) [16]. In asymmetric SSO and OOS TAGs, diunsaturated SSO was defined by two polymorphic forms with triple chain length structure (α-3L and β’-3L) [17], whereas three polymorphs occurred in OOS (α, β’2 -3L and β’1-3 L) [18]. Thus, no β forms were present in SSO and OOS, and the most stable polymorph was β’. This result was also observed in other asymmetric mixed-acid TAGs, such as 1,2-dipalmitoyl-3-myristoyl-sn-glycerol (PPM) (Kodali et al. [19]), 1,2-dipalmitoyl3-oleoyl-rac-glycerol (PPO) (Minato et al. [20]) and 1,2-dioleoyl-3-palmitoyl glycerol (OOP) [21]. Controlling polymorphism of TAGs becomes an important challenge for industrial fields, such as the pharmaceutical, biomedical or food areas, to obtain the desired product characteristics. Therefore, many studies have focused on the influence of external factors [22, 23] on the crystallization, transformation mechanisms and polymorphic behaviour of TAGs and more complex lipid samples, such as the use of dynamic thermal treatments [24–28], the use of additives [29], shear [30], emulsification [31] and sonication [32–34].

11.4 Influence of Kinetic Factors on the Polymorphic Behaviour of TAGs Many studies have focused on the influence of kinetic factors on the polymorphism exhibited by TAGs [35, 36]. As an example, the use of specific dynamic temperature

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variations may lead to the formulation of desired end food products properties for diversified industrial applications. However, the crystallization behaviour of TAGs under non-isothermal conditions is significantly complex owing to the occurrence of multiple polymorphic forms. Recent research was based on the influence of cooling rates variation on the particle size in lipid crystals [37, 38] and the polymorphic behaviour, thermal properties and microstructure of TAGs with different molecular structures [39–41]. A systematic series of research work was recently performed on the influence of the rates of cooling and heating on the polymorphic nucleation and transformation of main TAGs having different saturated-unsaturated fatty acid moieties in their molecular structures, which are present in vegetable and animal fats and oils. Hence, unsaturated-saturated-unsaturated OPO (1,3-dioleoyl-2-palmitoyl glycerol) [42], saturated-unsaturated-saturated POP (1,3-dipalmitoyl-2-oleoyl glycerol) [43], triunsaturated OOO (trioleoyl glycerol) and OOL (1,2-dioleoyl-3-linoleoyl glycerol) [14], and saturated-unsaturated-unsaturated POO (1-palmitoyl-2,3-dioleoyl glycerol), SOO (1-stearoyl-2,3-dioleoyl glycerol) and POL (1-palmitoyl-2-oleoyl3-linoleoyl glycerol) [44] were studied with a combination of Differential Scanning Calorimetry (DSC) and Synchrotron Radiation X-ray Diffraction (SR-XRD). Coupling these two techniques enables the monitoring of polymorphic crystallization and subsequent transformation at dynamic conditions of varying cooling and heating rates, providing rapid thermal programs and highly accurate structural information. In all cases, as the cooling rate applied to neat liquid samples decreased, the competitive polymorphic crystallization was directed to more stable forms (those with higher melting temperature, Tm ), whereas less stable polymorphic forms (with lower melting temperature) occurred when higher cooling rates were applied. Nevertheless, concurrent crystallization processes often occurred, increasing significantly the complexity of the phenomena. In more detail, Bayés-García et al. [42] reported that when OPO was cooled at 15 °C/min, α form crystallized; at 2 °C/min, α and β’ formed; at 1 °C/min, concurrent crystallization of α, β’ and β1 occurred; and when the controlled cooling rate was 0.5 °C/min, β’ and β1 crystallized. The Ostwald step rule for polymorphic nucleation [45] states that the firstly-crystallized solid from a melt or a solution may be the least stable form that is produced by spontaneous crystallization. However, some factors, like the cooling rate, affect crystallization and polymorphic nucleation. Therefore, this becomes an example in which kinetics, thermodynamics and structural factors are competing. As to the heating rates variation, more stable polymorphic forms also predominated when low heating rates were applied. As an example, Fig. 11.4 depicts the complex polymorphic pathways observed when the mixed-acid TAG POP was subjected to varying cooling and heating conditions [43]. The effects of heating rates appeared either in solid-state or melt mediated transformations, whose kinetics is determined by activation energies for the transformation and the heating rates applied. The results revealed that the transformation pathways clearly tended to change from solid-state to melt-mediated transformation when the heating rate increased. Figure 11.5 depicts the DSC cooling and heating thermo-

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Fig. 11.4 Polymorphic behaviour of POP under different cooling and heating conditions Reproduced from Ref. [43] with permission; © 2013 The Royal Society of Chemistry

grams and SR-XRD data (SAXD and WAXD patterns) when POP was cooled from the melt at 2 °C/min and subsequently heated at 2 and 1 °C/min. According to the SR-XRD, by applying a cooling rate of 2 °C/min to the molten POP sample, metastable α form occurred and, when heated, it transformed to γ and β, which finally melted. However, the transformation from α to γ differed between the heating rates of 2 and 1 °C/min. When the heating rate used was 2 °C/min, α crystals transformed into β’ through the liquid state (melt-mediated transformation), whereas at 1 °C/min, the same transition occurred in the solid state. This different behaviour may be understood by taking into account the activation free-energy diagrams related to each kind of transformation. Figure 11.6 illustrates these two typical transformations which occur from a less stable form (A) to a more stable form (B). The significance of the activation free energy, G# , involved in each process may determine the type of transformation pathway. That is, the transition rate of a solid-sate transformation based on structural modifications, like modifications in the subcell structure or chain length structure, is determined by G#ss . In the case of melt-mediated transformations, the transition rate is determined by the activation free energy corresponding to the melting process of a less stable form (G#m ) and following crystallization of a more stable form (G#c ). One may assume that activation energies associated to solid-state transformations and crystallization processes may increase for more stable forms, due to more stabilized crystal packing compared to that of less stable forms.

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Fig. 11.5 Polymorphic behaviour of POP when cooled at 2 °C/min and a heated at 2 °C/min; b heated at 1 °C/min. A. DSC thermogram. B. SR-SAXD pattern. C. SR-WAXD pattern Reproduced from Ref. [43] with permission; © 2013 The royal society of chemistry

A particular transformation pathway through a liquid crystal phase was observed in polymorphic transformations of POO, SOO and POL, when intermediate rates of heating between solid-state and melt-mediated transformation were applied. In such cases, the G# involved in transformation through the liquid crystal phase may be lower than that of solid-state transformation but higher than that of melt-mediation. The occurrence of transient liquid crystal phases in these saturated-unsaturatedunsaturated TAGs may be caused by lateral disorder due to some disparity in the molecule packing. This phenomenon may become more significant in asymmetric TAGs than in symmetric TAGs. The same general trends were basically observed for POP, OPO, OOO and OOL, POO, SOO and POL, although some slight differences may be described by considering their molecular structures. In general, POP (saturated-unsaturated-saturated TAG) exhibited more difficulties to obtain stable forms, such as β’ and β, compared to other TAGs (OPO, OOO and OOL, POO, SOO and POL) when lower cooling

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Fig. 11.6 Typical transformation pathways from less stable form A to more stable form B. (1) solid-state transformation, (2) melt-mediated transformation and (3) simple melting Reproduced from Ref. [43] with permission; © 2013 The royal society of chemistry

and heating rates were used. This may be explained by considering that activation energies associated to solid-state transformations based on structural changes from loosely packed to more closely packed subcell structures (e.g. transformation from hexagonal α form to orthorhombic β’ form) are higher than those which do not involve such structural modification (e.g. transformation from orthorhombic β’ form to triclinic β form). In the same line, activation energies of solid-state transformations between polymorphs having double chain length structures may be lower than that of transformation from a double to a triple chain length structure. By considering this, one may understand that crystallization and polymorphic transformations in triunsaturated OOO and OOL occurred easily, as all the polymorphic forms exhibited a double chain length structure. By contrast, POP showed a higher difficulty to transform to more stable polymorphic forms, probably due to the presence of a saturated palmitic acid in its structure, which may increase the corresponding activation energy compared with that associated to unsaturated oleic and linoleic acid moieties.

11.5 Phase Behaviour of TAGs Binary Mixtures As natural fats present in real systems in biotissues or food materials become mixtures of different types of TAGs, one may pay attention to the complicated behaviour of melting, crystallization and transformations of natural fats by examining the mixing behaviour of binary, ternary or more complex systems of specific TAG components [3]. In other words, the physicochemical properties of lipids must be studied not only in their pure systems but also in mixed systems. In particular, studies on binary mixture systems provide valuable information about molecular interactions among

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different lipid materials. Moreover, the phase behaviour of TAG mixtures has critical implications in fat blending and separation of component TAGs from natural fats and oil resources. Generally speaking, there are three types of mixing behaviour in TAGs binary systems: (i) the formation of eutectic mixtures, (ii) the formation of mixed crystals, and (iii) the formation of molecular compounds (MC). The most common mixing behaviour observed are those corresponding to (i) and (iii) types, which will be described in the present chapter. Primary factors determining the phase behaviour are the differences in chain length and chemical structures of the fatty acid moieties [46, 47]. Further complexity is caused by both polymorphism and acyl chain compositions attached to the glycerol group. Recent systematic examination of model fat systems of a series of binary mixtures of symmetric–asymmetric structure TAGs has been carried out [48–51]. The authors experimentally determined the kinetic phase diagram after applying different cooling rates to the following systems: (i) 1,3-dimyristoyl-2-stearoyl-sn-glycerol (MSM) and 1,2-dimyristoyl-3-stearoyl-sn-glycerol (MMS); (ii) 1,3-dicaproyl-2stearoyl-sn-glycerol (CSC) and 1,2-dicaproyl-3-stearoyl-sn-glycerol (CCS); (iii) 1,3dipalmitoyl-2-stearoyl-sn-glycerol (PSP) and 1,2-dipalmitoyl-3-stearoyl-sn-glycerol (PPS); and (iv) 1,3-dilauroyl-2-stearoyl-sn-glycerol (LSL) and 1,2-dilauroyl-3stearoyl-sn-glycerol (LLS). Monotectic behaviour was observed for MSM-MMS at the cooling rates used; whereas CSC-CCS mixtures exhibited eutectic behaviour, in which the eutectic appeared at different molar ratios when TAG mixtures were cooled at the different rates. The kinetic phase diagram of the LSL-LLS binary system displayed a singularity at the 0.5 molar fraction which delimited two distinct behaviours: eutectic behaviour in LLS-rich region and monotectic behaviour in the LSL-rich region. Also in this case, the eutectic position was shifted as a function of the cooling rate applied before the heating process. Finally, the PSP-PPS system also showed different behaviour at the two regions of the kinetic phase diagram, as it exhibited eutectic and monotectic in the PPS and PSP rich region, respectively. This singularity was attributed to the formation of a molecular compound at the 50-50 composition for the two cooling rates used. Authors described some general tendencies related to the mixing behaviour of these series of symmetric–asymmetric TAGs binary mixtures according to the neighbouring chain length mismatch (CLM), such as that eutectic compositions increased with increasing CLM. As typical examples of binary mixture systems, in this section, we present TAGs containing palmitic and oleic acid moieties: PPP, OOO, POP, OOP, PPO and OPO. Mixtures of PPP:POP [52], PPP:OOO [47], POP:OOP [21] and PPO:OPO [53] exhibited eutectic behaviour, whereas mixtures of POP:PPO [20] and POP:OPO [54] were MC-forming at ratios of 50:50. The mixing behaviour of POP:PPO and POP:OPO binary systems was examined not only in neat liquid system, but also in n-dodecane solution [55, 56], in which MC-forming mixtures were also confirmed. Recent work have also showed MC formation in OOP:OPO and PPO:OOP mixtures [53]. However, these MCs were only detected under metastable conditions, as long incubation periods forced the systems to evolve to the eutectic behaviour, as will be discussed further on. An equivalent behaviour was detected in binary systems

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Fig. 11.7 Diagram of binary mixture systems of palmitic-oleic mixed-acid TAGs POP, OPO, PPO and OOP

of SOS (1,3-distearoyl-2-oleoyl glycerol), OOS (1,2-dioleoyl-3-stearoyl glycerol), SSO (1,2-distearoyl-3-oleoyl glycerol), and OSO (1,3-dioleoyl-2-stearoyl glycerol), in which eutectic (SOS:OOS, Zhang et al. [18]) and MC formation (SOS:SSO [57] and SOS:OSO [58]) were determined. Figure 11.7 shows a diagram summarizing the mixing behaviour of mixed-acid TAGs containing palmitic and oleic fatty acid moieties. The combined usage of SR-XRD and DSC has clarified the thermodynamic (most stable phases) and kinetic (metastable phases) properties of the phase behaviour in the MC-forming systems. Furthermore, the molecular structures of the MC of POP:PPO and POP:OPO were assessed with FT-IR and the results obtained indicated that the steric hindrance and the glycerol interactions are operating in the MC formation [59]. Structural models of MCs of TAGs containing palmitic and oleic fatty acids are depicted in Fig. 11.8. The two typical glycerol conformations (tuning fork and chair conformation) of the palmitic-oleic mixed-acid TAGs are also shown. A chair conformation is typically adopted by asymmetric TAGs, whereas the tuning fork conformation predominates in symmetric TAGs [60]. Then, tuning fork becomes more favourable in POP and OPO as, therefore, oleoyl and palmitoyl may be located in separate leaflets. By contrast, the chair conformation may be more stable for PPO and OOP. In all cases, MCs have a double chain length structure, even if the stable forms of the component TAGs are all of triple chain length. This conversion in the chain length is basically caused by molecular interactions through the oleic acid chains, which are packed in the same leaflets in the double layers. In the POP:OPO molecular compound structure model (Fig. 11.8), aligned neighbouring glycerol groups are located along the chain axis with an opposite turn. For POP:PPO molecular compound formation, we may assume that both TAGs adopt the tuning fork conformation, so that juxtaposed glycerol groups may be set similarly to those in the POP:OPO mixture. Then, palmitoyl and oleoyl chains may be necessarily situated in one leaflet, whereas palmitoyl chains may fully occupy the

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Fig. 11.8 Structure models of a palmitic-oleic mixed-acid TAGs showing tuning fork and chair conformations, and b molecular compounds of palmitic-oleic mixed-acid TAGs Reproduced and adapted from Ref. [53] with permission; © 2015 American chemical society

other leaflet [20]. Nevertheless, the steric hindrance caused in the chain packing of the palmitoyl-oleoyl leaflet does not destabilize the MC molecule. Molecular compounds of POP:OPO and POP:PPO were, therefore, stable both in metastable and stable states. However, recent work [53] demonstrated the formation of molecular compounds at the 50:50 composition of OPO:OOP and PPO:OOP systems. The thermal stabilization experiments at fixed temperatures indicated that these molecular compounds were metastable and tended to separate into component TAGs such that the corresponding binary system evolved to eutectic. In the proposed OPO:OOP MC structural model (see Fig. 11.8), the glycerol groups directions are aligned with the lamellar plane. Moreover, the glycerol conformation of OPO and OOP are chair-type. Further stabilization may occur due to chain packing in the oleoyl-oleoyl leaflet, although the packing coefficient corresponding to the palmitoyl-oleoyl leaflet may be lower. Despite the non-destabilization of the MC structure in its metastable state, long incubation process forced binary mixtures to evolve into a eutectic behaviour. One hypothesis is that, during stabilization, the

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conformation of the glycerol group in OPO may have turned from chair-type, which is less stable, to tuning fork-type, resulting in a higher steric hindrance. Regarding the metastability of PPO:OOP molecular compound, the proposed model assumes a chair-type conformation and chain packing of palmitoyl and oleoyl chains in separate leaflets, involving a high packing coefficient. However, at this point, the racemicity of component TAGs PPO and OOP may be considered. The destabilization observed may be probably caused by predominant eutectic mixtures of R-PPO:S-PPO [61] and R-OOP:S-OOP. However, further research may focus on the study of the mixing behavior of enantiopure R- and S-TAGS. The use of cutting-edge techniques such as synchrotron radiation microbeam Xray diffraction (SR-μ-XRD) may allow the understanding on the mixing behaviour and microstructural characterization of TAGs mixtures. A systematic study was performed on the microstructure of POP:OPO spherulites, exhibiting molecular compound formation. The purpose was to examine the competitive polymorphic nucleation of mixture components at specific experimental conditions [62]. The crystallization behaviour of POP:OPO mixtures becomes significant for specific applications, such as the fractionation of palm oil to produce high-melting, medium-melting and low-melting fractions. This property is related to kinetic aspects of the binary mixing systems. However, few studies of kinetic properties of TAG mixtures have been conducted. Figure 11.9 shows the kinetic and stable phase diagrams in neat liquid (Fig. 11.9a and b, [54]) and n-dodecane (50%) solution (Fig. 11.9c and d [56]). The behaviour of stable forms of mixtures of POP and OPO were studied by SRμ-XRD (see Fig. 11.9b and d). This technique permitted to perform a mapping in the two dimensions of spherulites grown from neat liquid and n-dodecane solution, with steps of the order of the beam size (5 μm x 5 μm). Previous studies by Ueno [63] on spherulites of trilaurin (LLL) showed that the lamellar planes were set parallel to their radial direction from their inner to their outer areas. In addition, it was shown that, during the solid-state transition from β’ form to β, the lamellar directions were not randomized due to template effect. These crystalline aggregates become the most common in TAGs when crystallized from neat liquid and solution states [64]. Crystallization kinetics determine spherulites morphology and microstructures, as in mixed samples they permit to clarify the relative rates of nucleation of component materials. This is because the central region of such crystal aggregates is formed by many small crystals nucleation, and posterior crystal growth takes place toward the external region. Mixed spherulites formed by POP:OPO were characterized at ratios at which pure POP or OPO and MC crystals of POP:OPO were formed competitively. As MC was formed at the ratio 50POP:50OPO, the concentrations studied were 75POP:25OPO and 25POP:75OPO. The determination of the distribution of mixture components within each spherulite permitted to determine their relative crystallization rates. Small angle X-ray diffraction data was sufficient to distinguish the constituents, as molecular structures of most stable forms of pure POP and OPO have a triple chain length structure, whereas MC (POP:OPO) is defined by a double chain length structure. As an example, Fig. 11.10 shows SR-μ-XRD (small angle data) 2D

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Fig. 11.9 Phase behaviour of POP:OPO mixtures of a metastable forms (after cooling at a rate of 5 °C·min−1 ) in the neat liquid system b stable forms in the neat liquid system c metastable forms (after cooling at 5 °C·min−1 ) in solution system (50% n-dodecane) d stable forms in solution system (50% n-dodecane) Reproduced from Ref. [56] with permission; © 2010 American Chemical Society

patterns and related 2θ and χ extensions corresponding to three positions (central, intermediate and peripheral) of a spherulite of the 75POP:25OPO mixture without solvent. Two kinds of information were extracted from these patterns: identification of mixture component (long spacing value, 2θ extension) and lamellar direction of the crystals (azimuthal angle, χ extension). From Fig. 11.10, it follows that most stable β form of MC crystals, having a random lamellar orientation (see corresponding χ extension), was found in the central part of the spherulite. On the other hand, β form of POP, having an arrangement parallel to the radial direction, was detected at the periphery. In the middle zone β forms of MC and POP coexisted, such that their lamellar directions were virtually parallel. Contrarily,

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Fig. 11.10 Enlarged patterns taken at three positions within a spherulite of 75POP:25OPO in the neat liquid system. 2D diffraction patterns (left), 2θ extension (middle) and χ extension (right) Reproduced from Ref. [62] with permission; © 2011 The Royal Society of Chemistry

and for the growth temperatures applied, a predominance of β form of OPO was determined at 25POP:75OPO. The addition of n-dodecane was found to have no influence on the behaviour during the crystal aggregates growth. From the results obtained, it follows that the rate of crystallization of MC (POP:OPO) was higher than the one of POP, at the growing temperatures examined. Nevertheless, because the entire area of 25POP:75OPO was occupied by OPO, or OPO coexisting with MC, the rate of nucleation of OPO may be higher than the one of MC under the experimental circumstances applied. This heterogeneous composition could also be observed in the sherulite morphology, as shown in Fig 11.11. Different textures were observed within every 75POP:25OPO large spherulite: a compact pattern predominated in the central area, whereas the periphery had a clear needle-like morphology. By contrast, uniform patterns, without separated inner textures, were detected in 25POP:75OPO spherulites.

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Fig. 11.11 Schematic figure showing heterogeneous microstructure at the 75POP:25OPO and 25POP:75OPO compositions Reproduced from [62] with permission; © 2011 The Royal Society of Chemistry

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29. Smith KW, Bhaggan K, Talbot G, van Malssen KF (2011) Crystallization of fats: influence of minor components and additives. J Am Oil Chem Soc 88:1085–1101 30. Mazzanti G, Li M, Marangoni AG, Idziak SHJ (2011) Effects of shear rate variation on the nanostructure of crystallizing triglycerides. Cryst Growth Des 11:4544–4550 31. Wassell P, Okamura A, Young NWG, Bonwick G, Smith C, Sato K, Ueno S (2012) Synchrotron radiation macrobeam and microbeam x-ray diffraction studies of interfacial crystallization of fats in water-in-oil emulsions. Langmuir 28:5539–5547 32. Ueno S, Ristic RI, Higaki K, Sato K (2003) In situ studies of ultrasound-stimulated fat crystallization using synchrotron radiation. J Phys Chem B 107:4927–4935 33. Chen F, Zhang H, Sun X, Wang X, Xu X (2013) Effects of ultrasonic parameters on the crystallization behaviour of palm oil. J Am Oil Chem Soc 90:941–949 34. Ye Y, Martini S (2015) Application of high-intensity ultrasound to palm oil in a continuous system. J Agric Food Chem 63:319–327 35. Sato K, Ueno S, Yano K (1999) Molecular interactions and kinetic properties of fats. Prog Lipid Res 38:91–116 36. Himawan C, Starov VM, Stapley AGF (2006) Thermodynamic and kinetic aspects of fat crystallization. Adv Colloid Interface Sci 122:3–33 37. Acevedo NC, Marangoni AG (2010) Characterization of the nanoscale in triacylglycerol crystal networks. Cryst Growth Des 10:3327–3333 38. Acevedo NC, Marangoni AG (2010) Towards nanoscale engineering of triacylglycerol crystal networks. Cryst Growth Des 10:3334–3339 39. Bouzidi L, Narine SS (2012) Relationships between molecular structure and kinetic and thermodynamic controls in lipid systems. Part II: Phase behaviour and transformation paths of SSS, PSS and PPS saturated triacylglycerols-Effect of chain length mismatch. Chem Phys Lipids 165:77–88 40. Baker M, Bouzidi L, Garti N, Narine SS (2014) Multi-length-scale elucidation of kinetic and symmetry effects on the behaviour of stearic and oleic TAG. I. SOS and SSO. J Am Oil Chem Soc 91:559–570 41. Baker MR, Bouzidi L, Garti N, Narine SS (2014) Multi-length-Scale Elucidation of Kinetic and Symmetry Effects on the Behaviour of Stearic and Oleic TAG. II. OSO and SOO. J Am Oil Chem Soc 91:1685–1694 42. Bayés-García L, Calvet T, Cuevas-Diarte MA, Ueno S, Sato K (2011) In situ synchrotron radiation X-ray diffraction study of crystallization kinetics of polymorphs of 1,3-dioleoyl-2palmitoyl glycerol (OPO). Cryst Eng Comm 13:3592–3599 43. Bayés-García L, Calvet T, Cuevas-Diarte MA, Ueno S, Sato K (2013) In situ observation of transformation pathways of polymorphic forms of 1,3-dipalmitoyl-2-oleoyl glycerol (POP) examined with synchrotron radiation X-ray diffraction and DSC. Cryst Eng Comm 15:302–314 44. Bayés-García L, Calvet T, Cuevas-Diarte MA, Ueno S (2016) In situ crystallization and transformation kinetics of polymorphic forms of saturated-unsaturated-unsaturated triacylglycerols: 1-palmitoyl-2,3-dioleoyl glycerol (POO), 1-stearoyl-2,3-dioleoyl glycerol (SOO) and 1-palmitoyl-2-oleoyl-3-linoleoyl glycerol (POL). Food Res Int 85:244–258 45. Ostwald W (1897) Studien über die Bildung und Umwandlung fester Körper. Z Phys Chem Stoechiom Verwandtschaftsl 22:289–330 46. Rossel JB (1967) In: Paoletti R, Krichevsky D (ed) Advances in lipid research, vol V. Academic Press, New York, pp 353–408 47. Small DM (1986) In: Hanahan DJ (ed) The physical chemistry of lipids, from Alkanes to Phospholipids, Handbook of Lipid Research Series, vol 4. Plenum Press, New York 48. Boodhoo MV, Kutek T, Filip V, Narine SS (2008) The binary phase behaviour of 1,3dimyristoyl-2-stearoyl-sn-glycerol and 1,2-dimyristoyl-3-stearoyl-sn-glycerol. Chem Phys Lipids 154:7–18 49. Boodhoo MV, Bouzidi L, Narine SS (2009) The binary phase behaviour of 1,3-dipalmitoyl-2stearoyl-sn-glycerol and 1,2-dipalmitoyl-3-stearoyl-sn-glycerol. Chem Phys Lipids 160:11–32 50. Boodhoo MV, Bouzidi L, Narine SS (2009) The binary phase behaviour of 1,3-dicaproyl-2stearoyl-sn-glycerol and 1,2-dicaproyl-3-stearoyl-sn-glycerol. Chem Phys Lipids 157:21–39

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51. Bouzidi L, Boodhoo MV, Kutek T, Filip V, Narine SS (2010) The binary phase behaviour of 1,3-dilauroyl-2-stearoyl-sn-glycerol and 1,2-dilauroyl-3-stearoyl-sn-glycerol. Chem Phys Lipids 163:607–629 52. Minato A, Ueno S, Yano J, Wang ZH, Seto H, Amemiya Y, Sato K (1996) Synchrotron radiation X-ray diffraction study on phase behaviour of PPP-POP binary mixtures. J. Amer. Oil Chem. Soc. 73(11):1567–1572 53. Bayés-García L, Calvet T, Cuevas-Diarte MA, Ueno S, Sato K (2015) Phase behaviour of binary mixture systems of saturated-unsaturated mixed-acid triacylglycerols: effects of glycerol structures and chain-chain interactions. J Phys Chem B 119:4417–4427 54. Minato A, Ueno S, Smith K, Amemiya Y, Sato K (1997) Thermodinamic and kinetic study on phase behaviour of binary mixtures of pop and ppo forming molecular compound systems. J Phys Chem B 101(18):3498–3505 55. Ikeda-Naito E, Hondoh H, Ueno S, Sato K (2014) Mixing phase behaviour of 1,3-dipalmitoyl2-oleoyl-sn-glycerol (pop) and 1,2-dipalmitoyl-3-oleoyl-rac-glycerol (ppo) in n-dodecane solution. J Am Oil Chem Soc 91:1837–1848 56. Ikeda E, Ueno S, Miyamoto R, Sato K (2010) Phase behaviour of a binary mixture of 1,3-dipalmitoyl-2-oleoyl-sn-glycerol and 1,3-dioleoyl-2-palmitoyl-sn-glycerol in n- dodecane solution. J Phys Chem B 114(34):10961–10969 57. Engström LJ (1992) Triglyceride systems forming molecular compounds. Fat Sci Technol 94:173–181 58. Koyano T, Hachiya I, Sato K (1992) Phase behaviour of mixed systems of SOS and OSO. J Phys Chem 96:10514–10520 59. Minato A, Yano J, Ueno S, Smith K, Sato K (1997) FT-IR study on microscopic structures and conformations of POP-PPO and POP-OPO molecular compounds. Chem Phys Lipids 88:63–71 60. Craven RJ, Lencki RW (2013) Polymorphism of acylglycerols: a stereochemical perspective. Chem Rev 113:7402–7420 61. Mizobe H, Tanaka T, Hatakeyama N, Nagai T, Ichioka K, Hondoh H, Ueno S, Sato K (2013) Structures and binary mixing characteristics of enantiomers of 1-oleoyl-2,3-dipalmitoyl-snglycerol (S-OPP) and 1,2-dipalmitoyl-3-oleoyl-sn-glycerol (R-PPO). J Am Oil Chem Soc 90:1809–1817 62. Bayés-García L, Calvet T, Cuevas-Diarte MA, Ueno S, Sato K (2011) Heterogeneous microstructures of spherulites of lipid mixtures characterized with synchrotron radiation microbeam X-ray diffraction. Cryst Eng Comm 13:6694–6705 63. Ueno S, Nishida T, Sato K (2008) Synchrotron radiation microbeam x-ray analysis of microstructures and the polymorphic transformation of spherulite crystals of trilaurin. Cryst Growth Des 8(3):751–754 64. Marangoni AG (2004) Fat crystal networks. Marcel Dekker, New York

Part III

Applications

Chapter 12

Phase Change Materials M. À. Cuevas-Diarte and D. Mondieig

Abstract Apart from a purely scientific interest in molecular mixed crystals, the REALM continuously has been interested in finding applications—especially in the field of phase change materials for thermal protection and thermal energy storage. The key parameters are the heat of melting of the material and the thermal window, which is the temperature range in which the change from solid to liquid takes place. Applications are possible in the range of temperature from −50 °C to +200 °C. The composition of the material is one of the parameters that can be used to tune the thermal window to the required temperature.

12.1 Introduction The use of energy storage is a many-sided matter, both from a modern and an historical point of view. In the Middle Ages, hot stones were laid in beds to face the rigours of the cold winter nights—an example of the use of sensible heat. And in the early part of the twentieth century, the latent heat of salt hydrates contributed to the comfort of the sleeping cars of trains. “Energy storage is, in one way or another, part of all events both in nature and in man-made processes. There are many different kinds of energy storage systems, some involving large amounts of energy, and others very little. Some are part of energy transfer processes and others are part of information transfer systems. Such a variety of possible applications clearly means that several key parameters must be considered and that they differ from one application to another. A classification of energy storage systems therefore always tends to be very complex. In most cases, two features of the systems are crucial: the amount of energy to be stored and the length of time for maintaining the storage. Energy density and storage time are the key M. À. Cuevas-Diarte Grup de Cristal.lografia Aplicada, Universitat de Barcelona, Barcelona, Spain e-mail: [email protected] D. Mondieig (B) LOMA, UMR 5798, Université de Bordeaux, Talence, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. À. Cuevas-Diarte and H. A. J. Oonk (eds.), Molecular Mixed Crystals, Physical Chemistry in Action, https://doi.org/10.1007/978-3-030-68727-4_12

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parameters to be considered when discussing storage systems”. This is the beginning of the introduction chapter written by Jensen [1] in his book Energy Storage. “Energy storage is essential whenever the supply or consumption of energy varies independently with time. Traditionally, such imbalances have been handled mostly by storage of fuel, rather than of thermal or electrical energy itself. In the case of wood, coal, petroleum, and natural gas, fuel storage has been convenient and economical. Stored fuel has a high energy density, and there are established methods to deliver it to the consumer. Conversion systems at or near the ultimate consumer respond readily to his demands for heat, electricity, motive power, etc.”. Lane [2] wrote in the introduction and history chapter of his book Solar Heat Storage: Latent Heat Material. The work of Lane is a benchmark for the scientific basis and the storage of thermal energy. Lane refers to the patents of Douglas [3] and Newton [4] as the precursors of phase change materials (PCM), and to Maria Telkes [5] for the first practical application. Important, recent publications on PCM are the paper by Zalba et al. [6], and the book by Mehling and Cabeza [7]. Within the context of this chapter, we make a distinction between the following three types of storage of thermal energy: • sensible heat storage, which is based on the heat capacity of the medium; • latent heat storage, which is based on the energy of a change of phase of the medium (melting, evaporation, change of structure); • heat of a chemical reaction. Other kinds of energy storage (like mechanical, electrical or magnetic) are beyond the scope of our work. In this chapter, we concentrate on latent heat storage—in particular by means of the molecular alloys (mixed crystals) designed by the REALM (see Chap. 1). We speak of Phase Change Materials, and this chapter is about Phase Change Materials based on Molecular Alloys (MAPCM). In contrast to sensible heat storage, latent storage takes place without a (significant) change in temperature, taking advantage of a change of phase of the material. For a pure substance, as PCM, the change of phase is isothermal. For a molecular alloy, and for the change from solid to liquid, there is a small range of temperature, which is related to solidus and liquidus temperatures (see below).

12.2 Some Basic Matters The storage by latent heat has a number of advantages [8, 9]; the most important are: • it is “compact”: the energy that can be stored per unit of mass is considerable; • it allows to store energy in a small interval of temperature, which is of great importance for a number of applications;

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Fig. 12.1 Comparison of the storage capacities for water, solid concrete and hexadecane (Adapted from [10])

• and most importantly, the stored energy can be released at a virtually constant temperature. In Fig. 12.1, a comparison is made between the amounts of heat (in MJ), that can be stored in 1 m3 of material, as a function of temperature, for three commonly used materials. Sensible storage of heat takes place in the case of water and concrete. In the case of hexadecane there is, in addition to sensible storage, latent heat storage at 17.5 °C (normal melting point). While water can store 82 MJ.m−3 in a margin of 20 degrees (between 0 °C and 20 °C), hexadecane can store 240 MJ.m−3 at its melting point. For comparison, water, at its melting point of 0 °C, can store 306 MJ.m−3 ; its latent heat of melting. The changes of phase used for the storage of thermal energy are first-order transitions, and more specifically the solid–solid transition and the solid–liquid transition. The liquid–vapour transition involves volume changes that are too elevated to be convenient from a practical point of view. The liquid–liquid transition goes together with a heat effect which is too small for an application. Solid–solid transitions are not frequently used for storage of energy, and so for several reasons: they proceed slowly; the heat effect is often weak; or the temperature is too elevated to prevent the formation of vapour. Notwithstanding these features, the solid–solid transition has one important advantage: the advantage of proceeding in the absence of liquid. For a range of practical applications, the absence of liquid is a necessary condition. Promising solid–solid transitions are found in the change from ordered solid to plastic crystalline solid. A good bibliographic summary on

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these transitions was made by Lane [2]. In conclusion, the solid–liquid transition is the most frequently used—it is the focus of our work. The two most important characteristics of a solid–liquid PCM are the melting temperature of the material and the heat of melting. For a practical application, the temperature of melting is the first criterion of selection. For a given material, the temperature of melting is affected by impurities and their nature; read, by the source of the material. For economic reasons, and for most of the applications, the purity of the material is of secondary importance. For a given substance, the temperatures of melting and solidification are equal: the characteristic melting temperature of a substance is the temperature at which the solid and liquid phases are in equilibrium. We know that delay in melting does not exist. However, crystallization does not always occur at the same temperature as the transformation of solid to liquid. The kinetics of crystallization, which is governed by the phenomena of nucleation and crystalline growth, is totally different from the kinetics of fusion. This fact can become apparent in a certain delay or hysteresis of crystallization, as a result of which the material remains liquid at a temperature below the melting point. This is the metastable phenomenon of supercooling—related to the processes just mentioned: nucleation and (speed) of growth. Nucleation is related to the probability of the formation of a germ (or nucleus). The “birth” of a germ happens through the creation of a solid–liquid interphase that requires a certain amount of energy. Next the germ created has to reach a radius superior to a critical radius, so that it does not return to/dissolve in the fused mass. The critical radius is function of the free energy difference between solid and liquid and also of the superficial energy in the interphase. In the case of supercooling, it can happen that the supercooled liquid quickly returns to the temperature of fusion and subsequently dissolves the crystal seeds. The growth rate is related to the change in crystal size. We know that this rate is null at the melting point and that it quickly increases with decreasing temperature, until it reaches a maximum value. Beyond this maximum, the crystallization depends on the mobility of the molecules, which is related to viscosity, which itself increases with decreasing temperature. As a result, the velocity of growth diminishes with an increase of the degree of supercooling. The degree of supercooling depends on the material and the environment in which it is placed. The presence of certain impurities tends to diminish the phenomenon (in general the impurities act as a catalyst for germination; a particular case is the one of epitaxy, when the crystalline network of the impurity has similarities with that of the material that is wanted to crystallize). Supercooling is hardly to foresee. For a given material, it can vary according to its history [11]. For example, water can display a supercooling of 1–2 °C in its “normal” state; of 5–10 °C with a degree of 10−3 impurity; and until 40 °C in extremely pure water (10−12 ). The degree of supercooling also varies with the mass of the considered material. Indeed, it usually decreases when the mass of material increases [11–13]. For that reason, reported values on supercooling should be considered with caution because they generally are obtained from calorimetric measurements that are carried on small quantities of sample (milligram). In conclusion, supercooling may represent an important problem when it comes to applications; and, through its random nature, it requires to find solutions to reduce it, or even to eliminate it.

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The melting enthalpy is the heat that is absorbed at the solid–liquid transition and is fully released at the inverse transition. It can be determined experimentally. It is normally expressed in J.mol−1 , or J.g−1 , but the units more common from an economic perspective are kJ.kg−1 or rather MJ.m−3 . Because of the fact that the materials used in PCM applications do not have an absolute purity, the actual heat of melting can be somewhat smaller, since the presence of impurities usually has a lowering effect on the enthalpy of melting. The heat capacity C p (J.g−1 .K−1 ), which varies with temperature, has a discontinuity at the melting point. In many applications, the storage of heat takes place in a temperature range, from T initial below - to T final above the melting temperature. Apart from the storage of latent heat, there are two regions where sensible heat is stored (the case of hexadecane in Fig. 12.1). The heat which is stored is given by the following expression: T solidus

Q st = (m

Tfinal

C p(solid) (T ) dT ) + m H f + (m Tinitial

C p(liquid) (T ) dT )

(12.1)

Tliquidus

The expansion of volume V /V is another significant characteristic to be considered. First-order transitions are accompanied by a change in volume, and, for most of materials, the expansion of volume during the melting is positive (except for ice, gallium and its alloys…). It can vary from the 1 to 50%, depending on the material. In practice a PCM is put into containers (because of its transition to the liquid state, and also to protect it from the outside). It means that the parameter of expansion of volume is of importance when it comes to the selection of the characteristics and dimensions of the container. In addition, some PCM have a strong vapour pressure and can, therefore, produce constraints on the container. The thermal conductivity λ (W.m−1 .K−1 ) represents the aptitude of the material to transmit heat as a result of a thermal gradient. It varies with temperature. For many materials, the thermal conductivity in the liquid state is weaker than in the solid state. The thermal conductivity of the commonly used PCM is quite weak. Because of the fact that the parameter conditions the thermal interchanges, its weak value gives rise to temperature differences inside the material, hindering the release of heat. Besides, near the melting point thermal conductivity is not easy to measure. The main criteria for the selection of a good PCM are: • • • • • • • •

Temperature of interchange of energy appropriate for the application. A high enthalpy of transition. Chemical stability. Thermal stability, that is to say that thermal cycling does not involve a degradation of the material. A high heat capacity. Absence of, or minimal supercooling. A high thermal conductivity. A weak expansion of volume.

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Fig. 12.2 Protocol for the election of the PCM and the development of an application ( Adapted from [14])

• • • •

Not to be corrosive. Not to be toxic. A temperature of auto-ignition quite superior to the temperature of transition. Economic.

Abhat, in an excellent paper of 1983 [14], established a protocol for the selection of the PCM and the development of an application (Fig. 12.2). The REALM has attempted to provide solutions for the energy storage by latent heat by means of molecular alloys [15, 8]. Based on fundamental studies of molecular alloys of alkanes, alkanols, fatty acids, dicarboxylic acids and sugars, we have developed potential phase change materials (molecular alloy phase change materials or MAPCMs) for energy storage and thermal protection. When the component substances of an alloy have high latent heats of transition, then molecular alloys made from them may exhibit, in turn, very interesting properties for thermal energy storage. Let us now consider phase change materials constituted by molecular alloys (MAPCM). MAPCM are materials having the fundamental property of being thermo-adjustable (Fig. 12.3). The originality of the MAPCM concept lies in the fact that the choice of its composition is a means to tune the alloy to the optimal performance for a desired application—in terms of stored energy and/or the operating temperature. The last parameter, temperature, is essential for the selection of a suitable material for energy storage: its transition temperature must correspond exactly to that required for a given application [16].

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Fig. 12.3 Schematically representation of thermo-adjustable property and thermal window (δ) of a MAPCM

Available molecular alloys currently allow to cover a range of temperatures from −50 °C to + 200 °C. The melting of an alloy, as a rule, is not an isothermal phenomenon, in that it takes place in a certain range of temperature. Ideally, when an alloy is heated, its fusion begins at the solidus temperature and goes on continuously until the liquidus temperature is reached. The reverse transition, from liquid to solid, occurs on cooling and the heat stored on heating is released. The temperature range between solidus and liquidus is the “efficiency thermal window”, symbol δ. The efficiency thermal window is a property that can be rather elevated, say more than ten degrees. Whatever the case may, by the research carried out by the REALM, and highlighted in this book, MAPCMs can be designed that operate at a desired temperature, with a low thermal window (even below 1 °C). Materials and applications are the basis of a French patent, FR91/08695, deposited by the “Centre National de la Recherche Scientifique” (CNRS) in July 1991, with extension granted to Europe, USA, Canada and Japan [17]. A second patent was deposited in July 1992 jointly by the CNRS and the University of Barcelona [18]. In these patents, a MAPCM is characterized by its melting temperature, its “thermal window of effectiveness” and its latent heat of transition (Fig. 12.4). For example, a 45.31.6 MAPCM with H = 160 kJ.kg−1 is a material that stores this energy between 43.7°C and 45.3°C. As shown in Fig. 12.4, when heating the MAPCM its temperature increases gradually until the beginning of melting. While it melts (between T solidus

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Fig. 12.4 Mean characteristics of a MAPCM Reproduced with permission from Ref. [19]; © 2003 Elsevier

and T liquidus ), the MAPCM remains at a nearly constant temperature, if its thermal window is narrow, and stores the energy supplied in the form heat. Once all the MAPCM is liquid, its temperature increases again in a gradual manner. As a result, energy storage is carried out in three steps: the most important is the storage of latent heat and the other two steps correspond to the storage of sensible heat—in the solid state before the start of melting (below the solidus temperature) and in the liquid state above the liquidus temperature. When the liquid MAPCM is cooled it gives off the stored energy—then the restitution stage begins. The temperature of the MAPCM decreases by releasing part of the sensitive heat until it solidifies

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liberating the latent heat practically isothermally. When cooling continues, the solid state MAPCM releases the last part of sensible heat. If the MAPCM is properly chosen, these processes can be performed in a cyclic form, producing a series of storage and release events—that can be adapted at a frequency appropriate to the application in question (day–night in some cases). The family of alkane MAPCM (see Chap. 6) is of particular interest—from all points of view. Alloys based on alkanes, which we name Alcal ®, is a trademark registered by the CNRS. Among the many and diverse applications that can be envisaged with this type of materials, we have chosen some examples that have been developed within the REALM. Their level of development is variable. Laboratory-scale or full-scale feasibility studies on prototypes have been carried out in a number of cases. In one case it has reached the final product. The applications cover a variety of sectors: heating, transport and conservation of solid foods and liquids, transportation and conservation of biomedical and pharmaceutical products, thermal protection of electronic circuits, etc. All of them are based on the same principle of storage and release of thermal energy by latent heat. Many of the applications have been developed in collaboration with companies of the industrial sectors concerned.

12.3 An Example An example of the application of molecular alloys is found in the thermal protection of temperature-sensitive biomedical products, in particular during transport [19]. All steps from basic research to marketing have been addressed in collaboration with the French company ISOS. We used a double-wall bag containing MAPCM (see Fig. 12.5 (2)). This bag contained the blood bag and was placed in a transport box (see Fig. 12.5(3)). A cool box containing six of these units was used (see Fig. 12.5(1)). The bag containing the MAPCM is put in a freezer to complete its solidification. From that moment, and during the transport of blood in the box, external heat is blocked by the MAPCM. When the temperature reaches the solidus temperature, the MAPCM starts to melt. During melting of the MAPCM, the temperature remains virtually constant: the blood in the box is protected against an increase of temperature. Obviously, the duration of the melting process is dependent on the latent heat of melting and the amount of the MAPCM used, and also on the external temperature. On the scale of application, three temperatures were selected: −30, +4 and + 22 °C. Here we describe the design of the application at 4 ± 2 °C; meant to protect blood—during transportation from a hospital to its destination. The melting and solidification of the desired MAPCM should take place within the temperature range 2 °C ≤ T ≤ 6 °C. For the design of the MAPCM, and guided by the results of our research, we started from five pure n-alkanes: C12 , C13 , C14 , C15 and C16 . The polymorphism of these materials and their main melting parameters

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Fig. 12.5 Blood thermal protection device: (1) cool box; (2) double-wall bag containing MAPCM; (3) transport box containing the double-wall bag ( Adapted from [19])

were previously determined ([20, 21]; see also Chap. 6). Their main properties are shown in Table 12.1. It is important to observe that the properties of pure alkanes are needed to assess the properties of their mixtures, and that, on the contrary, commercially available mixtures must be used for the practical realization of the MAPCM. The reason is that the pure substances are very expensive, because they have to be obtained by synthesis (isolation from an oil fraction by distillation is impracticable). This Table 12.1 Main properties of alkanes form C12 to C16 liq

M δ liq (g/ml) C p (298.2 K) T melt (°C) H melt (J/g) T boil (°C) Melting Purity (g/mol) (J/g·°C) phase % C12 170.34

0.749

2.2

−10.0

210

216

Tp

99.10

C13 184.36

0.756

2.2

−5.4

157

234

RI

99.80

C14 198.39

0.763

2.2

5.2

215

254

Tp

99.79

C15 212.42

0.769

2.2

9.9

161

271

RI

99.37

C16 226.45

0.773

2.2

17.6

234

287

Tp

99.40

δ liq

M molecular weight; density; enthalpy; T boil boiling temperature

liq Cp

heat capacity; T melt melting temperature; H melt meeting

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observation explains why a series of five n-alkanes is taken as the starting point of our calculations. From our catalogue of binary phase diagrams, we selected the following five systems: C12 –C14 , C13 –C14 , C13 –C15 , C14 –C15 and C14 –C16 . As an example, three of these binary diagrams are shown in Fig. 12.6. In all cases, there is a composition domain in which molecular alloys melt in the desired temperature range, along with a narrow thermal window. These domains of composition permitted us to select and test some potential MAPCMs. From these three binary phase diagrams taken as starting point, five potential ternary systems were explored: C12 –C13 –C14 , C13 – C14 –C15 , C14 –C15 –C16 , C12 –C14 – C16 and C13 –C15 –C17 (see Fig. 12.7). To select the alloys with melting temperature

Fig. 12.6 Binary phase diagrams for C13-C15, C14-C15 [20] and C14-C16 [22]

Fig. 12.7 Domain of ternary alloys melting in the temperature range from +2°C to +6°C in the ternary systems C12-C13-C14, C13-C14-C15, C14-C15-C16, C12-C14-C16 and C13-C15-C17. We show the melting domain of disordered rotator phase I (in yellow) and some ordered low temperature phases (green and blue)

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and thermal window adapted to the application, some ternary molecular alloys were studied, and their melting range was calculated by using the Txy-CALC software [23, 24]. A good agreement was found between the computational results and the experimental data for a number of alloys prepared and studied. Finally, a number of compositions were selected as a potential MAPCM—with properties close to those required for the application. With this basis, a search was made for cheaper commercially available materials with properties similar to those of the molecular alloys studied. A multicomponent material was tested, and results indicated that its melting behaviour was comparable to that of the characterized binary and ternary alloys: a liquidus temperature close to the one required, a large melting enthalpy and a thermal window smaller than 2 °C. Furthermore, its behaviour during cycles of melting and solidification was constant. This material was tested at prototype scale in different situations. Six blood bags were set in double wall boxes containing the MAPCM and placed in a cool box. Blood and external temperatures were registered as a function of time. The results showed that the effect of the MAPCM was highly significant: the system was able to maintain a blood bag at a temperature below 10 °C during 6 h, when the outside temperature was 22 °C. Eight times longer than without the use of the MAPCM. Obviously, the protection time will be longer when the outside temperature is lower, if the cool box is protected by a thermal insulant, and if the double-wall box contains a larger quantity of MAPCM.

12.4 Other Examples To end this chapter, we present a list of applications of molecular alloy phase change materials developed by the REALM: the properties of the alloys; the industrial parties involved; and the object of the application. • An “active” double-wall thermos flask for cold drink protection. Developed in collaboration with SOFRIGAM (application temperature of +10°C). The container with 350 ml of MAPCM is able to keep the drink inside between 6 and 13º C for more than 3 h at an external temperature of 25 °C. The flask gives a protection 3 times longer than when no MAPCM is used [25]. • An MAPCM for thermal protection of electronic components (application temperature of +70 °C). This project was developed in collaboration with THOMSOM [26]. • An MAPCM for thermal protection of telecommunication components (application temperature of +35 °C). Developed in collaboration with FRANCETELECOM [22]. • A radiant floor with energy storage in MAPCM (application temperature close to +22 °C) during the night and restitution during the day, for a heat system. In collaboration with DUCASA [27].

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• A thermally controlled transport device using MAPCM (application temperature between 60 and 70 °C) in order to avoid contamination by microorganisms of hot food. Pizza is taken as the example, and it is shown that the delivering time can be increased by a factor of three [28]. • A plastic conical double-wall container with 800 ml of MAPCM, for ice-cream thermal protection during transport (application temperature of −11 °C). This thermal device is able to keep ice cream at a temperature less than −8 °C (temperature where the ice cream starts to melt) longer than 4 h 30’, at an external temperature of 20 °C. The protection is eight times longer than when the ice cream is maintained in only its pasteboard container [29]. • A container for thermal protection of single-crystal growth under microgravity conditions. Developed in collaboration with several research groups, in projects with the European Space Agency (ESA) and space corporation “Energy” (RKK“ENERGY”) (application temperature of +20 °C). A double-wall extruded polystyrene and aluminium box with 5 litres of MAPCM is able to keep the single-crystals at 20±2 °C for more than 6 days at an external temperature of 25 °C [30].

References 1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16.

Jensen J (1980) Energy storage. Butterworth Scientific and Ann Arbor Science, London Lane GA (1983) Solar energy storage: latent heat materials, vols I and II. CRC Press, Florida Douglas AAH (1933) British patent 398,927 Newton AB (1944) U.S. patent 2,342,211 Telkes M (1946) U.S. patent 2,595,905 Zalba Nonay B, Marín Herrero JM, Cuevas-Diarte MA, Calvet Pallàs T, Cabeza Fabra L (2002) Almacenamiento térmico mediante cambio de fase. Ingenieria Química 394:472–483 Mehling H, Cabeza LF (2008) Heat and cold storage with PCM. Springer, Berlin Mondieig D, Haget Y, Labrador M, Cuevas-Diarte MA, van der Linde PR, Oonk HAJ (1991) Molecular alloys as phase change materials (MAPCM) for the storage of thermal energy. Mat Res Bull 26:1091–1099 Labrador M (1990) Emmagatzemament d’energia per calor latent en aliatges moleculars. European Ph.D. Universitat de Barcelona Chevalier JL, Delcambre B, Martorana S, Sallee H (1982) Stockage thermique par changement de phase aux températures de confort Cahiers du CSTB nº229, pp 1–21 Lane GA, Glew DN, Clarke EC, Rossow HE, Quigley SW, Drake SS and Best JS (1975) Proc Work Sol Ener Stor. Charlottesville 43 Abhat A and Malatidis NA (1981) New energy conservation technologies and their commercialization. P Milhone-Springer, p 847 Cantor S (1978) Applications of differential scanning calorimetry to the study of the thermal energy storage. Termochim Acta 26:39 Abhat A (1983) Low temperature latent heat thermal energy storage: heat storage materials. Sol Energy 30:313–332 Cuevas-Diarte MA, Chanh NB, Haget Y (1987) Molecular alloys: status and opportunities. Mat Res Bull 22:985–994 Cuevas-Diarte MA, Calvet T, Tamarit JL, Oonk HAJ, Mondieig D, Haget Y (2000) Nuevos materiales termoajustables. Mundo Científico 213:45–49

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17. Haget Y, Mondieig D, Cuevas MA (1991) Compositions utiles notamment comme matériaux à changement de phase pour le stockage et la restitution de l’énergie. Patent FR 91/08695, EP 92915596.8, JP 04/3543179, US 07/988, 949, Can 07/2091350 18. Cuevas- Diarte MA, Labrador M, Mondieig D, Haget Y (1992) Alliages moléculaires pour le stockage de l’énergie par chaleur latente. Patent FR/ESP 92-08553 19. Mondieig D, Rajabalee F, Laprie A, Oonk HAJ, Calvet T, Cuevas-Diarte MA (2003) Protection of temperature sensitive biomedical products using molecular alloys as phase change material. Transfus Apheresis Sci 28:143–148 20. Espeau P, Robles L, Mondieig D, Haget Y, Cuevas-Diarte MA (1996) Mise au point sur le comportement énergétique et cristallographique des n-alcanes. I. Série de C8 H18 à C21 H44 . J Phys Chim 93:1217–1238 21. Robles L, Mondieig, Haget Y, Cuevas-Diarte MA (1998) Mise au point sur le comportement énergétique et cristallographique des n-alcanes. II. Série de C22 H46 à C27 H56 . J Chim Phys 95:92–111 22. Métivaud V (1999) Systèmes multicomposants d’alcanes normaux dans la gamme C14 H30 C25 H52 : Alliances structurales et stabilité des échantillons mixtes. Applications pour la protection thermique d’installations de télécommunications et de circuits électroniques. European Ph.D. Université Bordeaux I 23. Jacobs MHG, Oonk HAJ. (1990) Txy-CALC program. Utrecht University 24. Métivaud V, Rajabalee F, Oonk HAJ, Mondieig D, Haget Y (1999) Complete determination of the solid (RI)-liquid equilibria of four consecutive n-alkane ternary systems in the range C14 H30 -C21 H44 using only binary data. Can J Chem 77:332–339 25. Espeau P, Mondieig D, Haget Y, Cuevas-Diarte MA (1997) ’Active’ package for thermal protection of food products. Packag Technol Sci 10:253–260 26. Grignon R, Girardet C, Haget Y (1994) Intégration d’un MCPAM dans une structure composite pour réaliser un dispositif de protection thermique. XXémes Journées d’Étude des Equilibres entre Phases Bordeaux, pp 113–114 27. Cuevas-Diarte MA, Calvet T, Aguilar M, Arjona F Suelo radiante con almacenamiento de energía por calor latente en aleaciones moleculares. Technical rapport, Barcelona 28. Arjona F, Calvet T, Cuevas-Diarte MA, Métivaud V, Mondieig D (2000) Application of the N-Alkane molecular alloys to thermally protected containers for catering. Bol Soc Esp Cerám Vidrio 39:548–551 29. Ventolà L, Calvet T, Cuevas-Diarte MA, Métivaud V, Mondieig D, Oonk HAJ (2002) From concept to application. a new phase change material for thermal protection at −11 °C. Mat Res Innovat 6:284–290 30. Ventolà L, Cuevas-Diarte MA, Calvet T, Mondieig D (2005) Molecular alloys as phase change materials (MAPCM): thermal protection of macromolecule monocrystals at controlled temperature. Mater Res Innovations Online 9–1:204–208

Chapter 13

Crystallization H. P. C. Schaftenaar, M. Matovi´c, and J. H. Los

Abstract The crystallization of mixed crystals from a liquid mixture of the components is a complex event. Mass-transport- and heat-transport limitations prevent the crystallizing system from adopting through and through thermodynamic equilibrium: Equilibrium phase diagrams are making place for kinetic phase diagrams. The theoretical background of non-equilibrium crystallization is the main subject of the chapter.

13.1 Introduction The growth of molecular mixed crystals [1], often called solid solutions or alloys, of two or more components plays an important role in many modern industrial applications, like the food and pharmaceutical industry. Mixed crystals are particularly interesting due to the fact that their properties can be modified and optimized by changing their composition [2]. Examples are metal alloys, ceramic materials, organic compounds, superconductors and semiconductor compounds. Yet another possible, interesting application is to design mixed crystals for storage of thermal energy or for the purpose of thermal protection [2]. One can make a distinction between two types of molecular mixed crystals: the substitutional and the interstitial type. The substitutional solid solutions are solid phases in which the molecules or atoms in case of atomic solid solutions are replaced at random by those of the other component. It typically occurs when the components H. P. C. Schaftenaar (B) Universiteit Utrecht, Utrecht, The Netherlands e-mail: [email protected] M. Matovi´c Openbaar Lyceum Zeist, Zeist, The Netherlands e-mail: [email protected] J. H. Los Ecole Normale Supérieure, Paris-Saclay, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. À. Cuevas-Diarte and H. A. J. Oonk (eds.), Molecular Mixed Crystals, Physical Chemistry in Action, https://doi.org/10.1007/978-3-030-68727-4_13

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are isomorphous. The solute component in the interstitial solid solutions is usually relatively small and occupies interstitial positions in the crystal structure of the solvent [3]. Understanding the microscopic processes during crystallization of solid solutions is essential. In this chapter, we will give the outlines of various approaches to the problem of mixed crystals growth, without the aim of being exhaustive. The approaches are based on classical thermodynamics (equilibrium phase diagrams) and on linear non-equilibrium thermodynamics ((effective) kinetic phase diagrams). An important aspect is the segregation, which occurs at the solidification front during crystal growth [4]. At conditions close to equilibrium, one speaks of equilibrium segregation and the growth composition of the growing solid solution is given by the equilibrium phase diagram. However, at conditions far from equilibrium, often the case in industrial crystallization, one speaks of kinetic segregation which can be represented by kinetic phase diagrams (see, e.g. [5] for a comprehensive exposé of kinetic phase diagrams). For systems with a mixed crystalline phase, metastable situations can easily occur with a non-homogeneous solid phase of which only the surface is in equilibrium with the liquid phase, due to the fact that the diffusion rate in solids is usually very slow [3]. We will discuss several analytical models. Firstly, we will give the outline of a multicomponent equilibrium model adopting the formalism as used by Los et al. [3, 4] based on activities and activity coefficients quantifying the deviation from ideal mixing behavior. Secondly, several analytical kinetic models for mixed crystal growth developed by Los et al. are discussed. These include the linear kinetic segregation (LKS) model [4, 6, 7] to determine the kinetic segregation as a function of the undercooling, and the so-called linear effective kinetic segregation (LEKS) model [7, 8], an extension of the LKS model that takes into account the effects of mass and heat transport limitations in order to determine the effective segregation. Next, two models beyond the LKS model will be discussed, the mean field kink site kinetic segregation (MFKKS) model [6, 7] and the combined LKS-MFKKS model [6]. The last subject describes the determination of an equilibrium phase diagram from a kinetic analysis [7].

13.2 Multicomponent Equilibrium Model of Crystal Growth In this section, we will describe a multicomponent solid–liquid equilibrium model in a formulation as used in Refs. [3, 4, 9] which is based on the minimization of the Gibbs free energy G for a system consisting of one liquid phase with Nc crystallizing components, optionally a solvent and Ns different solid phases. The equilibrium condition in a multicomponent system with one liquid and N s solid phases reads, (see e.g. [10, 11]):

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307 liq

sol, j

μi = μi

, for(i = 1, . . . , Nc , j = 1 . . . , Ns )

(13.1)

In other words: The chemical potentials of each component i in each phase are equal. The superscripts “liq” and “sol” refer to the liquid and the solid phase, respectively. In this model, the deviation from ideality is described by activities or activity coefficients that enter in the expression for the chemical potential as follows:   μiP = μiP,∗ + RT ln aiP = μiP,∗ + RT ln γi P X iP

(13.2)

where aiP = γi P X iP is the activity of component i in solid phase P, with γi P the corresponding activity coefficient. The asterisk superscript refers to the pure component property. Now we can substitute eq. (13.2) into eq. (13.1) to find the equilibrium condition:  sol, j sol, j  liq,∗ sol, j,∗ γ Xi − μi μ (13.3) = i ln i liq liq RT γi,eq X i,eq liq

Note that we have put “eq.” as a subscript in the activity coefficient γi,eq and mole liq

fraction X i,eq of the liquid phase. It refers to the equilibrium liquid phase with respect sol, j

to a solid phase with given composition X i . This formulation will also be used for the description of the kinetic models, discussed in the next sections. Integrating from the temperature T to the pure component melting temperature sol, j Ti  of the  component i in the solid phase P = sol,j one finds for the Gibbs energy sol, j

G Ti  G

sol, j Ti

[11]: 

= G(T ) − S(T )



sol, j Ti





− T − c P (T )



sol, j

sol, j Ti

ln

Ti

T

−

sol, j Ti

+T (13.4)

where cp is the heat capacity at constant pressure , Eqs. (13.2) and (13.4) combined with the standard thermodyUsing μi = ∂∂G Ni namic relationship [12]: 

∂(G/T ) ∂T

=− P

H T2

(13.5)

leads to:  sol, j sol, j   sol, j  sol, j,∗ sol, j sol, j γi T Xi Ti Hi c P,i Ti c P,i ln = − + ln i liq liq sol, j RT R T RTi T γi,eq X i,eq (13.6)

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where Hi is the melting enthalpy for the pure component i and the polymorph liq sol, j sol, j sol, j = Ti − T and c P,i = c p,i −c p,i is the difference of the j-th solid phase, Ti between the heat capacities of the liquid and solid phase. Equation (13.6) can be simplified, because usually the terms involving the differences in heat capacities are small and can be neglected, which leads to the following expression for the equilibrium activity:  liq ai,eq

=

liq liq γi,eq X i,eq

=

sol, j sol, j γi Xi

exp −

sol, j,∗

Hi

sol, j

Ti

sol, j

RTi



T

(i = 1, . . . , Nc , j = 1, . . . , Ns )

(13.7)

We formulate the total Gibbs energy of phase P in terms of the chemical potentials P,E , as: and the mixing excess Gibbs energy, G mix P

G = P

Nc

i=1

P

NiP μiP,∗

+ RT

Nc

  P,E NiP ln X iP + G mix

(13.8)

i=1

Then, since ∂G p /∂ NiP = μi , consistency with eq. 13.2 implies the following P,E relation between the activity coefficient of component i in phase P, γi P , and G mix P,E   ∂G mix RT ln γi P = ∂ NiP

(13.9)

where NiP is the amount of component i in phase P. P,E and an entropy The excess Gibbs energy is composed of an enthalpy part Hmix P,E part Smix : P,E P,E P,E G mix = Hmix − T Smix

(13.10)

P,E The molar mixing excess Gibbs energy gmix for a binary mixture can be described by the empirical, so-called three-suffix Margules equation [4, 13, 14]: P,E = (A21 X 1 + A12 X 2 )X 1 X 2 gmix

(13.11)

where A21 and A12 are interaction parameters. For the activity coefficient, it follows from eq. (13.9) that: RT ln γ1 = X 22 [A12 + 2(A21 − A12 )X 1 ]

(13.12)

For a multicomponent system, including an excess entropy, this expression can be generalized to:

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Nc  Nc

P,E

T G mix P,E 1− = gmix = N θii  i=1 i  =i+1  X iP X iP P,E P,E ˜  X iP X iP × R Tii φii  + φi  i X iP + X iP X iP + X iP

(13.13)

where θii is the compensation   temperature for the binary mixture of components i and i  and T˜ii = Ti P + Ti P /2 denotes the average melting temperature of the pure  and φiP,E are dimensionless excess energy components i and i in phase P. φiiP,E  i parameters, defined by: φiiP,E = 

iiP,E  R T˜ P

(13.14)

ii

is proportional to the excess bond energy per ii bond in a dilute solution where iiP,E  of component i into the phase P of component i  . Finally, we can use eq. (13.9) to find: ⎛ NcP  P,E

g T T˜ii  mix P + γi = exp⎝− 1− RT θii  T i  =1,i  =i   ⎛ P,E  2  P 2 ⎞ ⎞ φi  i X iP + 2X iP X iP + φiiP,E Xi  ⎠ X iP ⎠ ×⎝ (13.15)  P  2 X i + X iP Solving the above equations to find the state of minimal Gibbs free energy is not an easy task but can be done by a so-called flash calculation for multicomponent systems [14, 15]. This includes a stability analysis, based on the Gibbs’ tangent plane criterion. Note that it is beforehand not known how many solid phases are present in the state of minimal Gibbs energy, which complicates the calculations. Although the equilibrium state provides an important reference, it is actually not obvious at all whether equilibrium is actually reached within a reasonable time scale in practice, because of the very slow diffusion rate in the solid phase, as opposed to that in liquid or vapor systems. Typically, one can have a metastable situation with a non-homogeneous solid phase where only the composition at the surface is in equilibrium with the liquid phase. So surface kinetic factors during the growth process (non-equilibrium situation) play an important role, sometimes overruling thermodynamics. We will discuss in the next sections several published analytical kinetic models, based on linear non-equilibrium thermodynamics [3, 4, 6–8, 16]. In these models, diffusion in the solid phase is completely neglected and by this they describe a

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situation that is in a sense the opposite to that of the equilibrium description which assumes no mass diffusion limitation at all in the solid phase. In general, experimental reality will be somewhere in between, depending on the type of system and on the time scale.

13.3 Analytical Kinetic Models 13.3.1 Linear Kinetic Segregation Model We will first treat a relatively simple kinetic model, known as the linear kinetic segregation (LKS) model, applied in Ref. [3] to crystal growth in multicomponent solid–liquid systems of fat mixtures (triglycerides or triacylglycerols (TAGs)). A fat molecule is an esterification of glycerol and three fatty acids. The LKS model has also been applied to other systems, such as the binary system 1,4-dichlorobenzene + 1,4-dibromobenzene, as discussed in Ref. [7]. In the LKS model, three main simplifications are adopted to overcome the complexity of a kinetic modeling: 1. The temperature is homogeneous, and crystallization is considered at constant temperature. 2. The composition of the liquid phase is homogeneous. 3. Details of the growth kinetics at the surface are neglected. During crystal growth atoms or molecules attach and detach from the interface with certain probabilities. In other words, they transform from the liquid to the solid phase and vice versa. According to linear non-equilibrium thermodynamics, the ratio of the attachment flux, Ji+ , and the detachment flux, Ji+ of component i is given by:  Ji+ μi = exp RT Ji−

(13.16)

liq

where μi = μi − μisol is the difference of the chemical potential of component i in the liquid phase and the growing solid phase. Applying eq. (13.2) of the previous section, we can write:  μi = RT ln liq

liq

liq

γi X i liq

 (13.17)

liq

γi,eq X i,eq

liq

in which the product γi,eq X i,eq is given by eq. (13.7). In accordance with chemical reaction rate theory, the attachment flux can be written as: + γi X i Ji+ = K i+ X i = K i,0 liq

liq

liq

(13.18)

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+ where K i+ = K i,0 γi , which is an attachment flux, i.e. number particles adsorbed + per unit of time per attachment site, with K i,0 the corresponding flux in a system of pure i. The net flux for component i, Ri , can be found by substituting Eqs. (13.17) and (13.18) into eq. (13.16): liq

  liq liq liq,gr liq,gr + + γi X i − γi,eq X i,eq = Nk K i,0 σi Ri = Ji+ − Ji− = Nk K i,0 liq

liq

liq,gr

(13.19)

liq,gr

where Nk is the 2D kink site density and σi = γi X i − γi,eq X i,eq is the absolute supersaturation for component i. The superscript “gr” is added to indicate that liq,gr liq,gr X i,eq and γi,eq are the equilibrium fractions and activity coefficient, respectively, of component i in the liquid phase with respect to the composition of the solid phase being formed at the solidication front. Subsequently, the total growth rate R reads: R=

Nc

Ri = N k

i=1

Nc

  liq liq liq,gr liq,gr + γi X i − γi,eq X i,eq K i,0

(13.20)

i=1

where Nc is the number of crystallizing components. In the steady-state regime, the composition of the solid phase at the solidification front should satisfy the Nc − 1 equations:

sol,gr

Xi

+ γ1 X 1 − γ1,eq X 1,eq K 1,0 σ1 R1 = + = κ1i,0 liq liq (i = 2, . . . , Nc ) (13.21) liq,gr liq,gr Ri K i,0 σi γi X i − γi,eq X i,eq liq

sol,gr

X1

=

liq

liq,gr

liq,gr

+ + where the kinetic constant ratio κ1i,0 is defined as κ1i,0 ≡ K 1,0 /K i,0 . This set of equations, combined with eq. (13.7), can be solved numerically for  sol,gr sol,gr (i = 1, . . . , Nc ) with Xi = 1. Xi i

The above set of Nc −1 equations is also the solutions of a set of coupled first-order differential equations (see [4]), describing the increase of the amount of component i that has crystallized, Nisol , with time. In real circumstances, in order to determine the effective kinetic segregation, i.e., the actual segregation between the bulk phases, it is important to keep in mind that the properties of the liquid phase at the interface and in the bulk can be different. This topic will be considered in the next section, dealing with mass and heat transport limitations, combined with the here discussed growth segregation at the interface.

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13.3.2 Linear Effective Kinetic Segregation Model Mass Transport Limitation In order to couple mass and heat transport limitations to the LKS model, an extended LKS model has been developed [8] and successfully applied to experiments [7], called the linear effective kinetic segregation (LEKS) model. In general, the segregation during mixed crystal growth is accompanied by concentration and temperature gradients near the solid-liquid interface, within mass and heat transport boundary layers. As a result, the properties of the liquid at the solidification front and in the bulk are different. The widths of the boundary layers are related to the width of the convective boundary layer, which depends on the geometry, scale of the system and hydrodynamic conditions of the liquid phase [17]. To determine the actual segregation at the interface, called effective segregation, the composition and temperature of the liquid at the interface have to be determined. This can be done by using hydrodynamic equations. The crucial hydrodynamic parameters are the liq so-called q-parameters, qm for mass transport and qT for heat transport, as will be discussed in this section. These q-parameters determine the extent of the mass and heat transport limitations at given crystallization conditions. Figure 13.1 illustrates mass transport limitation for a binary mixture. Clearly, in the depicted situation, the components 1 and 2 will be accumulated and depleted, respectively, in the liquid phase near the solid–liquid interface. Transport within the liquid phase consists of two fundamental processes: convection (natural or forced) and diffusion. In case of stirring (forced convection), the width of the convective boundary layer (c ) is given by [17]: c = 2.4

 v 1/2 ω

(13.22)

Fig. 13.1 Mass transport limitation schematically illustrated for a binary mixture; accumulation and depletion of components 1 and 2 (Reproduced from [8] with permission; © 2005 J. Phys. Chem. B) [8]

13 Crystallization

313

in which ω is the angular velocity of the rotating disk and v = η/ρ is the kinematic viscosity, with η the viscosity and ρ the mass density. For the steady-state situation, the width of the mass transport boundary layer (m ) in the liquid phase reads: m ∼ =



Dm v

1/3

 c =

1 NSc

1/3 c

(13.23)

where NSc is the so-called Schmidt number, defined by the quotient v/Dm with Dm the mass diffusion coefficient. To derive the earlier mentioned linear effective kinetic segregation (LEKS) model, first a relation between the liquid composition at the surface and the liquid bulk composition has to be found. Adopting a one-dimensional geometry, appropriate for a flat surface, and using a boundary layer approach as described in e.g. Ref. [17], this relation is found to be:     liq,surf sol,gr sol,gr liq,bulk liq,bulk sol,gr sol,gr exp qm σ˜ surf = gd gliq,surf X i + gliq,surf X i − gd gliq,surf X i Xi (13.24) in which qm is the key parameter for mass transport limitation, defined as: qm =

vˆ m Dm

(13.25)

where vˆ is an average crystal growth velocity constant. In case of a binary mixture, it is given by: 1

sol,gr K i Vi 2 i=1 2

vˆ =

(13.26)

sol,gr

where Vi is the volume per particle of component i in the growing solid phase. Next, the weighted supersaturation at the surface, σ˜ surf , is defined in terms of the net flux or growth rate Ri and the activity coefficients as: σ˜ surf =

2 2 sol,gr sol,gr  



Vi K i Vi liq,surf liq,surf liq,gr liq,grliq,gr Ri = γi (13.27) Xi − γi,eq X i,eq vˆ vˆ i=1 i=1

with   liq,surf liq,surf lig,gr lig,gr Xi − γi,eq X i,eq Ri = K i γi

(13.28)

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Finally, the factor gd in eq. (13.24) is defined as:   liq,bulk gliq,surf exp qm σ˜ surf − 1 gd = sol,gr     gliq,surf exp qm σ˜ surf − 1

(13.29)

csol,gr cliq,bulk liq,bulk and g ≡ liq,surf cliq,surf cliq,surf

(13.30a,b)

with: sol,gr

gliq,surf ≡

where the symbol c denotes the molar concentration. This parameter corrects for the difference in molar densities between the liquid and solid phases, and equals 1 when the densities are equal. Usually, the molar densities of the liquid and solid phase are similar. When we neglect the difference, eq. (13.24) simplifies to: liq,surf

Xi

sol,gr

= Xi

    liq,bulk sol,gr exp qm σ˜ surf + Xi − Xi

(13.31)

For small values of qm or qm = 0, there is no mass transport limitation, and the composition of the liquid phase at the surface becomes equal to that of the liquid bulk. This is the case for slow crystal growth, a thin mass transport boundary layer, and/or fast diffusion in the liquid phase. The other way round, for high values of qm implying a large exponential term, the composition of the growing solid phase will approach the composition of the liquid bulk, giving the reduced segregation. Note liq,surf liq,bulk sol,gr has to be within the interval [0, 1], the factor X i − Xi in that since X i front of the exponent has to be necessarily small in that case. In the beginning of the crystallization, there is no steady state, i.e. the mass boundary layer builds up with time and finally reaches a steady-state width. Since mass conservation has to be fulfilled at each time in the process, the average concentration of component i in the grown solid phase and in the boundary layer near the solid–liquid phase interface has to be equal to its concentration in the liquid bulk. Heat Transport Limitation For heat transport limitation, two crucial q-parameters can be identified, related to the fact that heat can be transported into both liquid and solid phase, in contrast to mass transport: liq

liq

qT = liq

vˆ T

liq DT

and qTsol =

vˆ sol T DTsol

(13.32a,b)

where DT and DTsol are the thermal diffusion coefficients for the liquid and solid liq phase, respectively. The thermal boundary layer thickness in the liquid T is given

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315

by:  liq T

∼ =

liq

DT v

1/3

 c =

1 N Pr



1/3 c =

liq

DT Dm

liq

1/3 m

(13.33)

liq

liq

in which NPr = ν/DT is the so-called Prandtl number and DT = k liq /(cliq c p ) is the liq thermal diffusivity in the liquid phase, with k liq , cliq and c p the thermal conductivity, the (molecular) concentration and the heat capacity, respectively, of the liquid bulk phase. The heat flux equation required to derive the temperature at the interface, T surf , is derived as: cliq cliq p

∂T liq liq liq liq 2 = −cliq cliq p ∇(vc T ) − ∇ Jq = −c c p ∇(vc T ) + k ∇ T ∂t

(13.34)

liq

in which the heat flux is given by Jq = −k liq ∇T according to Fourier’s law, and where vc is the position and time dependent convective flow velocity. For the general case, when the temperatures of the liquid and the solid bulk are different, T surf is derived as:       liq  liq surf cliq c p exp qTsol σ˜ surf − 1 T liq,bulk + csol csol T sol,bulk ˜ p 1 − exp −q T σ    T surf =    liq  liq surf cliq c p exp qTsol σ˜ surf − 1 + csol csol ˜ p 1 − exp −q T σ       liq csol,gr H sol,gr 1 − exp −qT σ˜ surf exp qTsol σ˜ surf − 1    + (13.35)    liq  liq surf cliq c p exp qTsol σ˜ surf − 1 + csol csol ˜ p 1 − exp −q T σ where cliq and csol are the concentrations of the liquid and solid bulk phases, liq sol,gr is the respectively, c p and csol p denote their heat capacities, whereas H melting enthalpy of the growing solid phase, which is composition and temperature dependent, defined by: H

sol,gr



sol,gr Xi , T surf



=

i

 sol,gr Xi

Hi∗

+

HiE

T surf

+ ∫ c p,i dT

 (13.36)

Ti

in which Hi∗ is the melting enthalpy of the pure component i at Ti and HiE = liq,E − Hisol,E is the difference in the excess enthalpies of component i in the liquid Hi and solid phase.

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liq

Fig. 13.2 Heat transport limitation schematically illustrated. Temperature profiles for qT = 0.1, 1.0 and 5.0 ( Reproduced from [8] with permission; © 2005 J. Phys. Chem. B) liq

Figure 13.2 shows heat transport limitation, for the cases qT = 0.1, 1.0 and 5.0, liq liq with qTsol = 4qT and sol T = 4T . As can be seen, the temperature profiles are convex at the liquid side and concave at the solid side. This implies a larger gradient or heat flux at the liquid side. The effective kinetic segregation, i.e. the composition of the growing solid as a function of the liquid composition and of the undercooling well away from equilibrium, can be represented by (effective) kinetic phase diagrams. Examples of such diagrams calculated by the here described LEKS model are shown in Fig. 13.3. Crystallization takes place at a certain degree of undercooling, which is defined in Fig. 13.5 by θ = θ − θeq , where θeq = Teq /T2 is the dimensionless equilibrium temperature with T2 the melting temperature of the component with the highest melting temperature. The  relative liquid bulk undercooling is defined as  θ = T /T2 = Teq − T liq,bulk /T2 . Figure 13.3 illustrates three situations of (effective) kinetic phase diagrams compared to equilibrium phase diagrams: (a) includes no transport limitations liq liq (qm = qT = 0), (b) includes only mass transport limitation (qm = 5, qT = 0), liq and (c) includes both mass and heat transport limitations (qm = 5, qT = 0.1). The kinetic liquidi are constructed simply by a downward shift of the equilibrium liquidi over a given temperature interval θ . The magnitude of the effective segregation is indicated by the horizontal arrow from the kinetic liquidus (taken at X 2 = 0.3 as an example in Fig. 13.3) to the kinetic solidus. In Fig. 13.3c, a help line is added (dashed-dotted line), giving the actual temperature at the solidification front. The point at the kinetic liquidus is first moved vertically upward to the help line and then horizontally to the kinetic solidus. Figure 13.3a, b and c can be summarized as follows. With increasing the bulk undercooling, the segregation decreases (a). Mass transport limitation reduces the effective segregation further (b). Heat transport limitation on the other hand enhances the effective segregation, for it reduces the effective undercooling at the interface tempering the effects of interfacial undercooling and mass transport (c).

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Fig. 13.3 (Effective) kinetic phase diagrams (full lines) for an arbitrary binary mixture which forms mixed crystals at a relative liquid bulk undercooling equal to θ = 0.05, compared to the equilibrium phase diagrams (dotted lines). The kinetic liquidus is just a copy of the equilibrium liquidus but shifted down by θ. In panel c, starting from a point on the kinetic liquidus (full circle), the temperature at the liquid–solid surface is given by the intersection of the vertical line through this point and the dashed-dotted line. The intersection of the horizontal line at the surface temperature with the other full line (kinetic solidus) gives the solid growth composition (indicated by the arrow), while its intersection with the dashed line gives the liquid composition at the surface ( Reproduced from [7] with permission from the author)

13.3.3 Mean Field Kink Site Kinetic Segregation (MFKKS) Model The mean field kink site kinetic segregation (MFKKS) model, developed in Ref. [6] and used in for the interpretation of experimental data in Ref. [7], will be discussed in

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Fig. 13.4 Kossel model for crystal growth schematically represented. Reactive particles at the surface may have one to five bonds (indicated by numbers 1 to 5). A particle with three bonds is called a kink particle. Kink sites play a crucial role in the growth process ( Reproduced from [6] with permission; © 2006 Elsevier)

this section. It is more kinetic than the LKS model, because it takes into account the growth kinetics at the surface. In this model, a key role is played by so-called kink sites. A kink site is an absorption site at the crystal surface for which the absorption energy of a particle (growth unit) is (approximately) equal to the crystallization energy per particle. Figure 13.4 illustrates a kink site for the so-called Kossel model, a traditional model for crystal growth simulation. For this model, representing a crystal with cubic symmetry, a growth unit (cube) at a kink site is bonded to three neighbors. To detach such a growth unit from the surface involves the breaking of these three bonds. The energy cost associated with such a detachment is indeed equal to the melting energy (i.e. minus the crystallization energy), invoking the fact that the energy of a growth unit in the crystal bulk is equal to one half times the bond energy with its six neighbors, the energy contribution of each bond being shared by the two particles involved. So, particles at kink sites start to become well bonded to the crystal, as opposed to particles at sites with 1 or 2 bonds which can easily dissolve again. In binary crystal growth, the composition at the kink site can be different from that of the bulk solid phase. Referring to a binary Kossel model, the net incorporation of component i can be approximated by:   liq Ri = Nk K i+ X i − K˜ i− X isol,k

(13.37)

where Nk is the mean 2D kink site density at the surface, X isol,k is the probability of finding component i at a kink site and K˜ i− is the mean rate of detachment of a particle i from a kink site. For a binary system, the ratio of the fractions of components 1 and 2 in the growing solid phase is given by:

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319 liq X 1sol K 1+ X 1 − K˜ 1− X 1sol,k = liq X 2sol K 2+ X 2 − K˜ 2− X 2sol,k

(13.38)

similar as in the LKS model. The mean detachment rate of a particle i from a kink site reads: K˜ i− =

2

sol sol − X sol j X k X l K ijkl

(13.39)

j,k,l=1 − in which K ijkl is the detachment rate of a particle i at a kink site surrounded by neighbours j, k and l. Within a mean field approximation, the fractions X sol,i , X sol j and X ksol at these three neighbour sites are equivalent to those in the bulk for the two components, while the mean fraction of the kink site, X isol,k , may be different. The detachment rate for a kink particle can be written as:



− K ijkl

φij + φik + φil − T  S˜i,ev + = K i,0 exp − kB T

 (13.40)

+ K i,0 is a kinetic constant for particle i, φij , φik and φil are the bond energy changes, k B is the Boltzmann constant, whereas  S˜i,ev is the change in the vibrational entropy for a selected event, which can be assumed to be equal to the melting entropy Si∗ of the pure component i:

 S˜i,ev = Si∗ =

Hi∗ 3φii = Ti Ti

(13.41)

where Hi∗ is the melting enthalpy of the pure component i, and Ti its melting temperature. For the variation in time of X 1sol,k we can write: dX 1sol,k liq liq − sol,k sol − sol,k sol = K 1+ X 1 X 2sol,k − K 2+ X 2 X 1sol,k + K˜ 21 X 2 X 1 − K˜ 12 X1 X2 dt (13.42) Note that the removal of a kink site particle along a straight step generates another kink site particle. The average detachment probability for a kink particle i with a neighbour j along the step obeys, within the mean field approximation: K˜ ij− =

2

k,l=1

− X ksol X lsol K ijkl

(13.43)

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The neighbor particle is assumed to become a kink particle itself after removing its neighbor at the kink site. For steady-state growth, in other words: dX isol,k /dt = 0 it holds that: X 1sol,k =

liq − sol X1 K 1+ X 1 + K˜ 21 liq liq + + − sol − sol ˜ K 1 X 1 + K 2 X 2 + K 21 X 1 + K˜ 12 X2

(13.44)

where we have used that X 2sol,k = 1 − X 1sol,k . Combining Eqs. (13.38), (13.39), (13.43), and (13.44) leads to an expression with one variable, X 2sol , keeping in mind that X 1sol = 1 − X 2sol , which can be solved numerically.

13.3.4 Combined Analytical LKS-MFKKS Model It turns out that the kinetic phase diagram resulting from the MFKKS model at vanishing undercooling does not necessarily coincide with the equilibrium phase diagram, unlike the situation for the LKS model. This artifact of the MFKKS model can be corrected by combining the two models in the so-called LKS-MFKKS model discussed in this section [6]. Monte Carlo (MC) simulations, based on the binary Kossel model, have been used to validate the analytical models. For binary systems, in which the two components have similar pure component melting enthalpies, the results from the MC simulations roughly agree with those of the LKS model at conditions close to equilibrium. At temperatures well below the equilibrium point, the LKS model still is a good approximation for systems with small melting energy (i.e. small bond energies), such as metal systems, and for systems with small excess energy, i.e. systems that mix easily. In other cases, however, the results from the LKS model can deviate considerably from those of the MC model. In these cases, the MFKKS model agrees much better with the MC results. But since at vanishing undercooling the results from MFKKS model do not coincide the equilibrium phase diagram, a fourth model was constructed, the combined LKS-MFKKS model. By construction, this model becomes equivalent to the LKS model at vanishing undercooling but rapidly tends to the MFKKS model for increasing undercooling. An alternative, advanced analytical model, the so-called extended kink site kinetic segregation (EKKS) model, can be found in Ref. [16]. In Ref. [7], the focus lies mainly on the binary mixture 1,4-dichlorobenzene + 1,4-dibromobenzene, as already mentioned, and on binary fat mixtures (TAGs). The combined segregation LKS-MFKKS model is coupled with the LEKS model and solved for time dependent boundary conditions. The composition of the growing

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solid and that of the liquid at the interface is calculated, as well as the temperature at the surface, for the given experimental conditions.

13.4 Determination of an Equilibrium Phase Diagram from a Kinetic Analysis In this part, we will summarize the kinetic analysis in Ref. [7] of the equilibrium phase diagram of the binary mixture 1,4-dichlorobenzene + 1,4-dibromobenzene. A kinetic modeling of the crystallization process is introduced to calculate the enthalpy path for a given set of excess parameters, as determined in [2]. Figure 13.5 shows the isobaric equilibrium solid–liquid TX phase diagram (Bakhuis Roozeboom’s TypeI) and illustrates the crystallization path for the mixture 1,4-dichlorobenzene + 1,4-dibromobenzene of overall composition z, which forms mixed crystals with a continuously varying composition. z is defined as the total amount of component 2 (in mole fraction). When the temperature is decreased continuously, nucleation occurs at point 0 (T0 ), which in general is below the solidus of the equilibrium phase diagram. The temperature rises from this point 0 (T0 ) to 1 (T1 ), due to the heat of crystallization. In this part, crystallization occurs fast. At point 1, the undercooling is vanishing and crystallization slows down. Upon continued cooling the temperature starts decreasing again. In this second part of the crystallization process, proceeding slowly, the liquid

Fig. 13.5 Crystallization path of the mixture 1,4-dichlorobenzene + 1,4-dibromobenzene of overall composition z, schematically depicted in the isobaric equilibrium solid–liquid TX phase diagram ( Reproduced from [7] with permission from the author)

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stays in (near) equilibrium with the solid at the surface and the amount of solid s formed between T and T − T is given by the lever rule: liq

s = (1 − s)

liq

X eq (T ) − X eq (T − T )

(13.45)

liq

sol (T − T ) − X (T − T ) X eq eq

liq

sol where X eq and X eq are the equilibrium mole fractions of component 2 in the liquid and the solid phase at the corresponding temperature. In case that diffusion in the solid phase is neglected, the following expression follows from integration of eq. (13.45):





liq

T

dX eq /dT

T1

sol (T ) − X (T ) X eq eq

s(T ) = 1 − (1 − s0 ) exp ∫

liq

dT

(13.46)

where s0 is the total initial amount of solid phase at T1 , which is in fact inhomogeneous. It is assumed that the surface of the solid at T1 is of equilibrium composition. In order to calculate s(T ) from eq. (13.46), one has to calculate the initial amount sol sol = X 2,av . of solid formed between T0 and T1 , s0 , and its average composition X av This average composition of the second component must satisfy the mass balance equation: liq sol + s0 X av =z (1 − s0 )X eq

(13.47)

“Next, the theoretical enthalpy path has been derived as: theor Hmix (T )

T



= ∫ c p,mix − (1 − (s(T )) T1

liq

dX eq /dT liq

sol (T ) − X (T ) X eq eq

 Hfus dT

(13.48)

in which Hfus is the composition-dependent enthalpy of fusion, written as:     sol sol sol Hfus,1 + X eq Hfus,2 + H E X eq Hfus = 1 − X eq

(13.49)

Here Hfus,1 and Hfus,2 are the temperature-dependent enthalpies of melting of the pure components, whereas H E is the difference between the excess enthalpy in the liquid and solid phases. Equation (13.48) can be computed numerically, after inserting eq. (13.47) into it. It was demonstrated that the kinetic model describes the crystallization process well, when the excess parameters are known. The other way round, the kinetic model can be used to determine the excess quantities (excess enthalpy, entropy and Gibbs energy of the solid phase), by fitting the theoretical enthalpy path eq. (13.48) to the experimental one, which is measured in the adiabatic calorimeter during slow

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cooling. The phase diagram is also calculated, which follows closely the equilibrium phase diagram of Fig. 13.5. An advantage is that both excess enthalpy and entropy are obtained without the assumption of complete equilibrium between totally homogeneous phases, which is often not the case.

13.5 Some Additional Information and Observations on Melting and Crystallization One of the systems that has played an important part in the field of crystallization and melting of mixed systems is the 1,4-dichlorobenzene + 1,4-dibromobenzene mixture, whose phase diagram is shown above in Fig. 13.5 (see also Chap. 5, Aromatic Compounds). A meaningful experimental study of this system has been reported by van Genderen [18], who had prepared homogeneous mixed crystals by means of zone leveling, using the experimental setup that had been designed by Kolkert [19, 20]. The most important outcome of the investigations by van Genderen et al. and Kolkert comes down to the facts that (i) when heated at high speed, the melting of the samples prepared by zone leveling is quasi-isothermal, and (ii) the melting points of those samples are situated on the Equal-G Curve (EGC). NB. Remember that the EGC is the locus of the temperature at which, for a given mole fraction, the liquid mixture has the same molar Gibbs energy as the homogeneous solid mixture—here, the mixture prepared by zone leveling (see Chap. 3, Thermodynamics). For the equimolar mixture in the system 1,4-dichlorobenzene + 1,4-dibromobenzene the EGC temperature is about 337 K. When studied under equilibrium conditions—read, when heated at very low speed in an adiabatic calorimeter—a homogeneous sample prepared by zone leveling starts to melt when the temperature of the solidus is reached and goes on melting, until the temperature of the liquidus is reached (melting starts at the temperature of the solidus, and is complete at the temperature of the liquidus). For the equimolar mixture in 1,4dichlorobenzene + 1,4-dibromobenzene, the temperatures of solidus and liquidus are about 333 K and 341 K, respectively; NB. An example of a so-called heat capacity curve—reflecting the results obtained by adiabatic calorimetry—can be found in Chap. 5; for a zone-leveled, equimolar sample of trans-azobenzene + trans-stilbene [21, 22]. A comparison of the results obtained by adiabatic calorimetry and by DSC on zone-leveled samples of 1,4-dichlorobenzene + 1,4-dibromobenzene reveals that in experiments of high speed (DSC), the equilibrium melting range of 8 K is reduced to a quasi-isothermal event at 337 K. In common language, and for the whole range of compositions, one could say that at high speed the wide melting loop is compressed to a narrow melting curve.

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As a conclusion, and returning to the main subject of this chapter, which is crystallization, one can observe that the influence of speed on crystallization (cooling) is similar to the influence of speed on melting (heating): In both cases, the effect of high speed comes down to a reduction of the temperature range, in which liquid and solid are present together.

References 1. Kitaigorodskii AI (1984) Mixed crystals. Springer, Berlin 2. van der Linde PR (1992) Molecular mixed crystals from a thermodynamic point of view. Thesis, Utrecht University 3. Los JH, Flöter E (1999) Construction of kinetic phase diagrams. Phys Chem Chem Phys 1:4251–4257 4. Los JH, van Enckevort WJP, Vlieg E, Flöter E (2002) Metastable states in multicomponent liquid-solid systems I: A kinetic crystallization. J Phys Chem B 106:7321–7330 5. Chvoj Z, Šesták J (1991). In: Tˇríska A(ed.) Kinetic phase diagrams; Nonequilibrium phase transitions. Elsevier, Amsterdam 6. Los JH, van den Heuvel M, van Enckevort WJP, Vlieg E, Oonk HAJ, Matovi´c M, Van Miltenburg JC (2006) Models for the determination of kinetic phase diagrams and kinetic phase separation domains. Calphad 30:216–224 7. Matovi´c M (2007) Investigation of binary solid phases by calorimetry and kinetic modeling. Thesis, Utrecht University 8. Los JH, Matovi´c M (2005) Effective kinetic phase diagrams. J Phys Chem B 109:14632–14641 9. Wesdorp LH (1990) Liquid-multiple solid phase equilibria in fats. Delft University, Thesis 10. Gibbs JW (1876) On the equilibrium of heterogeneous substances. Trans Connect Acad III, 108–248, (1878) 343–524 11. Bakhuis Roozeboom HW (1904) Die heterogene gleichgewichte, II, part 1. Systeme aus zwei Komponente, Vieweg, Braunschweig 12. Oonk HAJ, Calvet MT (2008) Equilibrium between phases of matter. Phenomenology and Thermodynamics, Springer, Dordrecht 13. Prausnitz JM, Lichtenthaler RN, Gomes de Azevedo E (1986) Molecular thermodynamics of fluid phase equilibria. Prentice Hall, Englewood Cliffs, N.J 14. Michelsen ML (1982) The isothermal flash problem. Part I. Stability. Fluid phase equilibria, 9:1–19 (1982) The isothermal flash problem. Part II. Phase-split calculation 9:21–40 15. Los JH, van Enckevort WJP, Vlieg E, Flöter E, Gandolfo FG (2002) Metastable states in multicomponent liquid-solid systems II: Kinetic phase separation. J Phys Chhem B 106:7331– 7339 16. Los JH, van Enckevort WJP, Meekes H, Vlieg E (2007) On the definition of a Monte Carlo model for binary crystal growth. J Phys Chem B 111:782–791 17. Rosenberger F (1979) Fundamentals of crystal growth I. Springer, Berlin 18. Van Genderen ACG, de Kruif CG, Oonk HAJ (1977) Properties of mixed crystalline organic material prepared by zone leveling. I. Experimental determination of the EGC for the system pdichlorobenzene + p-dibromobenzene. Z Physik Chem Neue Folge 107:167–173 19. Kolkert WJ (1974) Growth of homogeneous organic mixed crystals by repeated pass zone leveling. Thesis, Utrecht University 20. Kolkert WJ (1975) Growth of uniform solid solutions of naphthalene and 2-naphthol by repeated pass zone-leveling. J Crystal Growth 30:213–219

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21. Bouwstra JA (1985) Thermodynamic and structural investigations of binary systems. Thesis, Utrecht University 22. Bouwstra JA, de Leeuw VV, van Miltenburg JC (1985) Properties of mixed-crystalline organic material prepared by zone levelling. IV. Melting properties and excess enthalpies of (transazobenzene + trans-stilbene). J Chem Thermodyn 17:685–695

Subject Index

A AB model, 27, 80, 123, 165, 179, 231 Addition compounds, 256 Adiabatic calorimetry, 84, 86, 87, 99, 100, 230 Aliphatic chains, 270 Alloys, 305 All-trans zigzag conformation, 108 Alternation rule, 54 Analogous chemistry, 13 Analogous form, 13 Anomalous racemates, 235 A perfect family of mixed crystals, 124 Associated Solution Model (ASM), 256, 258 Associate species, 256

B Bananas, 193 BGBW model, 252, 253, 263 Biaxial nematics, 193 Binary mixing behaviour, 270 Binodal, 32 Blue phases, 194 Boiling point, 54

C Calamitics, 192 Calorimetry, 252 Cambridge Structural Database, 246 Chain-chain interactions, 270 Chain length structure, 270 Chair conformation, 280, 281 Charge-transfer, 253 Chemically coherent groups, 79, 80, 95

Chemical nature, 94 Chemical reaction, 292 Chiral calamitic molecules, 194 Chiral drugs, 246 Chiral molecules, 221 Chiral nematic, 194 Chiral smectic, 194 Chirality center, 221–223, 225 Chocolate, 270 Clapeyron equation, 46, 47, 52, 53, 55, 58 Class of similar systems, 37, 81 Clausius-Clapeyron equation, 53 Cocoa butter, 270 Cocrystals, 63, 65, 69, 251 Coefficient of crystalline isomorphism, 79, 83, 94 Coefficient of molecular homeomorphism, 80 Cold drink protection, 302 Compensation pressure, 168 Compensation temperature θ, 30, 37, 80, 82, 100, 165, 179 Complexes, 251 1:1 compound, 222 Computer software, 28, 38 Concurrent crystallization, 275 Condensation, 49 Conglomerate, 223, 251 Congruent melting, 251, 256, 257, 262 Congruent melting temperature, 237 Continuity of all properties, 15 Cooling curves, 36, 227 Coulomb force, 252 Critical point, 32, 49 Crossed isodimorphic, 70

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. À. Cuevas-Diarte and H. A. J. Oonk (eds.), Molecular Mixed Crystals, Physical Chemistry in Action, https://doi.org/10.1007/978-3-030-68727-4

327

328 Crossed isodimorphism, 70, 89, 92, 96, 97, 101, 172 Crossed isopolymorphism, 123 Crystalline isomorphism, 252 Crystalline polymorphism, 41, 42 Crystallization, 49 Crystallization experiment, 226, 228 Crystal structures, 44, 257 Cysteamine hydrochloride, 64 D Data assessments, 252 Degree of crystalline isomorphism, 120, 252 Degree of molecular homeomorphism, 21, 187 Density, 241 Difference properties, 81 Differential Scanning Calorimetry (DSC), 81, 85–87, 92, 93, 231, 257, 275 Dimorphic, 58, 61 Dimorphism, 14, 43, 49, 56–58, 63, 65, 244 Dipole-induced interactions, 262 Discotics, 192 Dissociation energy, 256 Distribution coefficients, 91 Double-eutectic, 251 Dynamic temperature variations, 275 E (Effective) kinetic phase diagrams, 306, 317 (Effective) kinetic segregation, 311, 316 EGC temperature, 37 Empirical relationships, 37 Enantiomers, 63, 69, 221 Enantiotropic, 48, 57, 58, 60–63 Enantiotropic relationship, 48 Enantiotropy, 48, 57, 58, 60, 61 Energy storage, 291 Enthalpy, 28, 55 Enthalpy changes, 53, 56 Enthalpy-entropy compensation, 37, 82, 123 Enthalpy of melting, 35 Enthalpy of mixing, 232 Entropy, 28, 53 Entropy change, 47 Entropy change on melting, 34 Entropy of mixing, 232 Equal-G curve, 32, 85, 86, 126 Equal Gibbs Composition, 199 Equal Gibbs-energy curve, 238 Equilibrium phase diagrams, 306 Equilibrium segregation, 306

Subject Index Equimolar mixture, 29 Essential oils, 225 Eutectic, 63–65, 69, 270 Eutectic arrest, 236 Eutectic mixtures, 63 Eutectic phase diagram, 223 Eutectic systems, 16 Excess enthalpy, 30, 80–83, 98, 99, 231, 233 Excess entropy, 30, 232 Excess functions, 253 Excess Gibbs energy, 27, 29, 65, 80, 81, 83, 94, 99, 100, 199, 231, 239 Excess heat capacities, 30 Expansion of volume, 295 External factors, 274

F Face-centered cubic (FCC) or body-centered cubic (BCC), 164 Fat blending, 279 Fats, 269, 275, 278, 279 Fatty acid, 269, 270, 273–275, 279, 280 2-fluoronaphthalene, 96 Food, 274, 278 Fractionation, 282 FT-IR, 280 Fusion, 49 Fusion enthalpy, 322

G Gas constant, 29, 227 Geometric mismatch, 232, 265 Gibbs energy, 28, 50, 54, 68 Gibbs energy change on mixing, 29 Gibbs free energy, 44, 54, 58, 253 Gibbs free energy of complexation, 255 Glass transition, 234 Glycerol conformation, 270

H Halomethane compounds, 62 Heat capacity, 44, 230, 295 Heat-capacity diagrams, 236 Heat-capacity melting curves, 87, 88 Heating rates, 274–276, 278 Heat-of-melting, 34, 227, 233 Heat of mixing, 80 Heat of vaporization (sublimation), 54 Heats of melting, 230 Heat system, 302 Heat transport limitation, 314

Subject Index Heteromolecular interaction energy, 252 Heteropolar interaction, 252 High-pressure differential thermal analysis (HP-DTA), 240 Homeomorphism, 18 Homeomorphism by atomic substitution, 20 Homeomorphism by radical substitution, 20 Homogeneous mixture, 29 Homologous homeomorphism, 21 Homomolecular interaction energy, 252, 254 Hot food, 303 Hydrates, 63, 65–67 Hydrogen bonding, 229, 253, 256, 264, 265 Hydrogen-bonds, 164, 252 Hydrostatic PVT system, 44

I Ice-cream thermal protection, 303 Ideal mixture, 29 Interaction energies, 252 Interaction parameter, 253 Intermolecular contacts, 94 Internal energy, 28, 253 Interstitial solid solutions, 306 Isodimorphism, 14, 69, 70 Isomorphic, 63, 67, 69–71 Isomorphism, 10, 42, 44, 62, 68, 69, 71 Isomorphous, 71 Isopolymorphism, 123 Isotrimorphism, 69

K Kinetic factors, 270 Kinetics, 275, 279, 280, 282 Kink defect, 108

L Lamella, 270 Lamellar direction, 282, 283 Latent heat, 291 Latent heat storage, 292 Lattice energy, 97, 98 Le Chatelier principle, 53, 241 Linear Effective Kinetic Segregation (LEKS) model, 312 Linear Kinetic Segregation (LKS) model, 310 Linear variation of the crystal parameters, 18 Line compounds, 222, 251 Lipids, 269, 270, 274, 275, 278

329 LIQFIT, 36, 81, 84–86, 99 Liquid crystalline phases or mesomorphic phases, 192 Liquid crystals, 192 Liquid neutron scattering, 257 Liquidus, 31, 223, 257 Liquidus temperature, 297 Long-range order parameter, 252 Lyotropics, 192

M Margarine, 270 Mass Transport Limitation, 312 McMillan’s ratio, 196 Mean-field approach, 252 Mean Field Kink Site Kinetic Segregation (MFKKS) model, 317 Melting enthalpy, 295 Melting point, 48 Melting point alternation, 153 Melting temperature, 60 Melt mediation, 272 Mesogenic compounds, 192 Metastable, 48 Metastable melting points, 84, 96, 101 Microcalorimetry, 36 Microstructure, 270 Miscibility gap, 32, 81, 83, 84, 234 Mismatch parameter, 38, 168, 179 Mixed-acid TAGs, 272 Mixed crystal growth, 306, 312 Mixed crystals, 10 Mixing enthalpy, 255 Mixing entropy, 253 Mixing free energy, 256 Mixing model, 227 Molar Gibbs energy, 28 Molecular alloys, 63, 69 Molecular Compound (MC), 270 Molecular homeomorphism, 100 Molecular interactions, 270, 272, 278, 280 Molecular isomorphism, 252 Molecular mixed crystals, 305 Monomers, 193 Monotropic, 48, 57, 58, 61–63, 66, 272 Monotropically, 182 Monotropic behaviour, 182 Monotropic character, 172 Monotropism, 272 Monotropy, 48, 57, 58, 60, 61 Morphotropism, 14

330 N Nematic range, 196 Neutron, 256 Neutron scattering, 252 Non-congruent melting, 66 Non-homogeneous solid phase, 306 O Optical antipodes, 69, 221 Order-disorder, 44 Order—disorder transition, 255 Ordered alloys, 251 Ordered forms, 112 Orientational disorder, 97, 186, 187 Orientationally disordered, 185 Orientationally disordered crystals, 192 Orientationally disordered (OD) phases, 163, 185, 187 Orientational order, 193 Overall enantiotropic, 59, 60 Overall enantiotropy, 57, 59 Overall monotropic, 58, 59, 62 Overall monotropy, 57, 59 P Packing coefficient, 20 Pairwise-interaction model, 252 Palm oil, 282 Partial bilayer smectic A, 197 Peritectic, 252 Phase Change Materials based on Molecular Alloys (MAPCM), 292 Phase Change Materials (PCM), 292 Phase diagram, 31, 48–50, 54, 57, 58, 60, 63–70 Phase equilibria, 50 Phase transitions, 44, 52, 53, 64 Plastic crystalline states, 191 Plastic crystals, 163, 192 Plastic phase, 185 Polymer liquid crystals, 193 Polymorph, 58, 71 Polymorphic, 65, 66 Polymorphic nucleation, 275 Polymorphic phase diagram, 58 Polymorphism, 42, 44, 49, 58, 63, 65, 67, 81, 80–82, 244, 270 Polymorphs, 48–50, 58, 66, 71 Positional order, 193 Pressure, 28, 237 Pressure-temperature phase diagrams, 51, 52, 54, 55, 57, 62

Subject Index Pseudoracemate, 223 P-T plane, 242 P-T-x phase diagram, 240 Pure substance, 28

Q Quadrupole-quadrupole interactions, 253 Quasiracemate, 223

R Racemates, 63, 70, 222, 251 Random alloys, 251 Rates of cooling, 272 Rate of nucleation, 284 R configuration, 221 Recrystallization, 66, 234 Redlich-Kister, 65 Re-entrant nematic, 197 Re-entrant phenomenon, 197 Region of demixing, 32 Relative stability, 272 Renninger reflexions, 229 Resolution, 246 Rotational disorder, 100 Rotator, 111 Rotator I form, 37 Rotator phase, 258

S Scalemic mixtures, 246 S configuration, 221 Sensible heat, 291 Sensible heat storage, 292 Shortenings, 270 Software package, 38 Solid-solid transitions, 50, 52, 58, 60 Solid solution, 16, 63, 65, 67, 69, 72, 69–73, 223, 270, 305 Solidus temperature, 31, 223, 297 Space group, 228 Specific entropy, 45 Specific volumes, 44, 45, 244 Spherulites, 282 Spinodal, 32 S-shaped mixing enthalpy, 261 Stabilizing effect, 253 Stable, 49 Stable melting point, 93, 96 Stacking interactions, 262 Stereoselective syntheses, 246 Steric hindrance, 270

Subject Index Structural subfamilies, 80, 90 Subcell, 109 Subcell structures, 270 Sublimation, 49, 52, 253 Sub-regular excess properties, 253 Substitutional disorder, 97, 223 Substitution solid solutions, 305 Superstructure, 18 Symmetry breaking, 247 Synchrotron radiation microbeam X-ray diffraction, 270 Synchrotron Radiation X-ray Diffraction (SR-XRD), 275 Syncrystallisation, 13 System-dependent parameters, 29 T Telecommunication components, 302 Temperature, 28 Temperature-composition (T-x) phase diagram, 238 Terminal double gauche, 108 Terminal gauche, 108 Thermal conductivity, 295 Thermal history, 124 Thermal protection, 299 Thermal protection of electronic components, 302 Thermal protection of single-crystal growth under microgravity conditions, 303 Thermo-adjustable, 296 Thermogram, 234 Thermotropics, 192 Topological method, 49, 55 Topological pressure-temperature phase diagram, 59–62 Transition of the second kind, 255 Transitions of the first kind, 255 Tricritical point, 196 Trimorphism, 56, 58

331 Triple points, 47, 49, 52, 54–57, 60, 59–62 Tunable photonic crystals, 194 Tuning fork conformation, 280 Two-phase equilibria, 57 Two-phase region, 31 Types of phase diagram, 223

U Ultracal, 87–89 Uncongruent melting, 251 Unit cell, 49, 50, 270

V Van der Waals complexes, 251 Van der Waals equation of state, 227 Van der Waals interactions, 164, 252 Van’t Hoff law, 256 Vaporization, 49 Vapor pressure, 54 Viedma ripening, 247 Volume, 28 Volume changes, 47, 53, 55, 56

W Wallach’s rule, 244 Weissenberg photographs, 228 Would-be racemate, 233

X X-ray diffraction, 228, 252, 256

Z Zero line, 35 Zone levelling, 85–87, 99, 100 Zone melting, 91

Substances Index

A Alkanes, 296 Alkanols, 296 2-aminonaphthalene, 92 Anthranilic acid, 266 Apatite, 41 Aragonite, 41, 42 Argon, 37 Arsenates, 12 Azobenzene, 97

B Benfluorex hydrochloride, 50, 51, 60, 61 Benzamide, 42 Benzene, 12, 100, 252 Benzo[b]thiophene, 101 Bicalutamide, 246 2-bromonaphthalene, 73, 74 Butyloxybenzylidene octylaniline, 215

C Calcite, 41, 42 Calcium carbonate, 42 Calcium oxide, 37 Camphor, 69, 70 Camphoroxime, 224, 227, 238 Carbonates, 12, 18 Carbon tetrachloride, 164, 172 Carvone, 225, 233–236 Carvoxime, 265 CBr2 Cl2 , 62 CBr4 , 62 CBrCl3 , 62, 67–69 CCl3 Br, 69, 70

CCl4 , 62 CCS (1,2-dicaproyl-3-stearoyl-sn-glycerol), 279 (CH3 )3 CBr, 68–70 (CH3 )3 CCl, 67–69 Chlorobenzene, 259 2-chloronaphthalene, 71–74 Chloronaphthalene, 262 CO3 Ca, 18 Copper sulphate, 10 Copper sulphate hydrate, 11 CSC (1,3-dicaproyl-2-stearoyl-sn-glycerol), 279 Cysteamine hydrochloride, 55, 59, 60, 63– 65

D 1,4-dibromobenzene, 321, 323. See also pdibromobenzene Dicarboxylic acids, 296 1,4-dichlorobenzene, 321, 323. See also pdichlorobenzene 1,2-dipalmitoyl-3-oleoyl-rac-glycerol, 274 Diprophylline, 247 D-methylsuccinic acid, 223

F Fatty acids, 19, 296 Ferrous and calcium carbonates, 12 Ferrous carbonate, 12 Ferrous sulphate, 10, 11 Ferrous sulphate hydrate, 11 Fluoranil, 261 Fluorobenzene, 259

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. À. Cuevas-Diarte and H. A. J. Oonk (eds.), Molecular Mixed Crystals, Physical Chemistry in Action, https://doi.org/10.1007/978-3-030-68727-4

333

334 2-fluoronaphthalene, 80, 90

H Halogenomethane, 172, 182 Heptyloxycyanobiphenil, 202 Hexadecane, 293 Hexafluorobenzene, 257–260 Hexamethylbenzene, 259, 261 Hexyloxybenzylidene octylaniline, 215 Hexyloxycyanobiphenyl, 210 Hydrate, 11

I Ibuprofen, 238, 244–246 Isopropylsuccinic acid, 222, 223

K KBr-KCl, 18 KCl-NH4 Cl, 18 Krypton, 37

L l-citrulline, 66, 67 Lewis acid, 256 Lime carbonate, 42 Limonene, 225, 233, 236, 237, 247 Lipids, 269, 270, 274, 275, 278 L-isopropylsuccinic acid, 222, 223 LLL (trilaurin), 282 LLS (1,2-dilauroyl-3-stearoyl-sn-glycerol), 279 LSL (1,3-dilauroyl-2-stearoyl-sn-glycerol), 279

M Magnesium oxide, 37 Magnetic compounds, 256 Mandelic acid, 238, 240–243 Margaric acid, 19 Mesitylene, 260 1--methyl-2,3,5-tribromobenzene, 95 1--methyl-2,4,6-tribromobenzene, 95 Methylchloromethane, 172, 182, 185, 187 Methylchloromethane compounds, 164, 182 Methylchloromethanes, 172 2-methyl-naphtalene, 70, 72–74, 80 Methylnaphthalene, 262 Methylsuccinic acid, 222 MMM (trimyristoyl glycerol), 274

Substances Index MMS

(1,2-dimyristoyl-3-stearoyl-snglycerol), 279 MSM (1,3-dimyristoyl-2-stearoyl-snglycerol), 279

N NaCl, 18 NaH2 AsO4 ·H2 O, 12 NaH2 PO4 ·H2 O, 12 n-alkanes, 107 n-alkanols, 107 Naphthalene, 70, 71, 80, 90–92, 96, 101, 252, 262, 266 2-naphthalene, 74 2-R-naphthalenes, 70, 71, 73 2--naphthol, 70, 80, 91–93 n-carboxylic acids, 107 n-dodecane, 282 Neopentane, 164, 172, 185, 187 4-n-octyloxybenzoyloxy-4’-cyano-stilbene, 211 NO3 Na, 18 Nonyloxycyanobiphenil, 202 4-nonyloxyphenyl-4 -nitrobenzoyloxybenzoate, 213 Normal alkanes, 253, 258 N-p-cyanobenzylidene-p-n-octyloxyaniline, 209

O Octafluoronaphthalene, 262 Octyloxycyanobiphenyl, 210 Oleic acid, 19 OOL (1,2-dioleoyl-3-linoleoyl glycerol), 275 OOO (trioleoyl glycerol), 273, 275 OOP (1,2-dioleoyl-3-palmitoyl glycerol), 274 OOS (1,2-dioleoyl-3-stearoyl glycerol), 273 OPO (1,3-dioleoyl-2-palmitoyl glycerol), 275 OSO (1,3-dioleoyl-2-stearoyl-glycerol), 273 Oxides, 18 4-oxyloxyphenyl-4 -nitrobenzoyloxybenzoate, 213 Ozone, 251

P 1-palmitoyl-2,3-dioleoyl glycerol, 275 1-palmitoyl-2-oleoyl-3-linoleoyl glycerol, 275

Substances Index Paraazoxyanysol, 212 Para-cyanosubstituted liquid crystals, 197 Para-dihalogen derivatives, 19 p-bromochlorobenzene, 81 p-bromoiodobenzene, 81, 103 p-chloroiodobenzene, 81, 103 p-dibromobenzene, 81, 84, 86–88, 103. See also 1,4-dibromobenzene p-dichlorobenzene, 80–82, 84, 86–88, 260. See also 1,4-dichlorobenzene Pentylphenyl cyanobenzyloxy benzoate, 212 Phenanthrene, 266 Phosphates, 12 Picric acid, 266 Piracetam, benzocaine, 55 POP (1,3-dipalmitoyl-2-oleoyl glycerol), 275 Potassium arsenate, 12 Potassium chloride, 30 Potassium phosphate, 12 p-[p-hexyloxy-benzylidene)-amino] benzonitrile, 209 PPM (1,2-dipalmitoyl-3-myristoyl-snglycerol), 274 PPO (1,2-palmitoyl-3-oleoyl glycerol), 274, 279–282 PPP (tripalmitoyl glycerol), 274 PPS (1,2-dipalmitoyl-3-stearoyl-snglycerol), 279 PSP (1,3-dipalmitoyl-2-stearoyl-snglycerol), 279 p-xylene, 260

R Rare gases, 251 RbCl, 18 Rimonabant, 58, 59 Ritonavir, 50, 61, 62, 246 Rotigotine, 246

335 S Semiconductors, 256 Silicates, 18 Sodium chloride, 18, 30 SOS (1,3-distearoyl-2-oleoyl glycerol), 273 SSO (1,2-distearoyl-3-oleoyl glycerol), 273 SSS (tristearoyl glycerol), 273 1-stearoyl-2,3-dioleoyl glycerol, 275 Stilbene, 97 Sugars, 296 Sulfur, 42, 43 Sulphates, 12

T Tert-butyl compounds, 164 Thianaphthene, 101 2-thionaphthalene, 92 2-thionaphthol, 73, 74 Thiophene, 100 Thiophenebenzene, 80, 89, 100 Toluene, 260, 261 trans-azobenzene, 97, 98, 100, 323 Transition metal complexes, 256 trans-stilbene, 97, 98, 100, 323 Triacylglycerols (TAGs), 251, 269, 273, 310 1,3,5-tribromobenzene, 80, 87, 88 1,3,5-trichlorobenzene, 80, 87, 89 Triethylenetetramine dihydrochloride, 66

U Urea, 19

W Water, 293

Z Zinc sulphate hydrate, 11