Chemical Physics of Molecular Condensed Matter [1st ed.] 9789811590221, 9789811590238

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Lecture Notes in Chemistry 104

Kazuya Saito

Chemical Physics of Molecular Condensed Matter

Lecture Notes in Chemistry Volume 104

Series Editors Barry Carpenter, Cardiff, UK Paola Ceroni, Bologna, Italy Katharina Landfester, Mainz, Germany Jerzy Leszczynski, Jackson, USA Tien-Yau Luh, Taipei, Taiwan Eva Perlt, Bonn, Germany Nicolas C. Polfer, Gainesville, USA Reiner Salzer, Dresden, Germany

The series Lecture Notes in Chemistry (LNC), reports new developments in chemistry and molecular science - quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge for teaching and training purposes. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research. They will serve the following purposes: • provide an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas, • provide a source of advanced teaching material for specialized seminars, courses and schools, and • be readily accessible in print and online. The series covers all established fields of chemistry such as analytical chemistry, organic chemistry, inorganic chemistry, physical chemistry including electrochemistry, theoretical and computational chemistry, industrial chemistry, and catalysis. It is also a particularly suitable forum for volumes addressing the interfaces of chemistry with other disciplines, such as biology, medicine, physics, engineering, materials science including polymer and nanoscience, or earth and environmental science. Both authored and edited volumes will be considered for publication. Edited volumes should however consist of a very limited number of contributions only. Proceedings will not be considered for LNC. The year 2010 marks the relaunch of LNC.

More information about this series at http://www.springer.com/series/632

Kazuya Saito

Chemical Physics of Molecular Condensed Matter

123

Kazuya Saito Department of Chemistry Faculty of Pure and Applied Sciences University of Tsukuba Tsukuba, Japan

ISSN 0342-4901 ISSN 2192-6603 (electronic) Lecture Notes in Chemistry ISBN 978-981-15-9022-1 ISBN 978-981-15-9023-8 (eBook) https://doi.org/10.1007/978-981-15-9023-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

I started my research activity as an undergraduate student in a chemical school under the supervision of Profs. Hideaki Chihara and Tooru Atake. My research object was a phase transition exhibited by the crystal of an organic compound, and our experimental setup was a laboratory-made calorimeter constructed by Tooru. The measurement itself of the heat capacity was not difficult (but very time-consuming) because of the state-of-the-art calorimeter. On the other hand, my knowledge was deficient in analyzing the results and even catching their meaning. The situation forced me to read textbooks of statistical and solid-state physics. I found the necessity to generalize the content if we intended to utilize them for molecular crystals. Indeed, the inclusion of rotational and even deformational degrees of freedom into the lattice-dynamical treatment was necessary. After graduation with my doctoral degree, I joined the group headed by Prof. Isao Ikemoto. The group was active in synthesizing and characterizing organic conductors, including some superconductors. My knowledge in solid-state physics was undoubtedly useful in this period of ca. 10 years. The important lesson I learned was that most of the interesting properties were well (although only partially) described without molecular languages. Although crystal structures were essential, the properties were analyzed using an effective Hamiltonian, which did not retain “molecular” properties. On the other hand, the complete oblivion of the molecular nature sometimes irritated the research community. After returning to the university of my graduation as a member of Prof. Michio Sorai’s group, I started studying liquid crystals and have continued it besides other issues at the other place till now. Through the whole period of my research, I have been more convinced that the essential feature of molecular systems is the molecular shape. The other belief I have acquired is that many attractive and challenging puzzles are in molecular systems as critical issues of condensed matter physics. The three preceding paragraphs give the background of my idea of writing this book. I intended to fill a gap in the knowledge about the properties of macroscopic (bulk) material between chemistry- and physics-trained persons and to illustrate the appeal of molecular systems without stepping into every detail. Although many v

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Preface

good textbooks are available on solid-state (or condensed matter) physics, they generally treat simple systems such as metals and crystals consisting of atoms. On the other hand, in solid-state chemistry, textbooks give a diverse variety of fascinating examples but often avoid descriptions of theoretical background even at the simplest level. This book gives a coherent description starting from intermolecular interaction up to properties of condensed matter ranging from the isotropic liquid to molecular crystals in terms of molecular language. The contents should serve as a starting point based on the current status for further study. It may be useful to give some comments about the adoption of subjects. The book highlights the effects of molecular properties, i.e., the presence of the shape and its deformation, on structure and properties. However, I omitted details for specific subjects for which a comprehensive monograph is available. Linear and branched polymers correspond to the case though they well fit the category. Many specialized textbooks are available. Besides, they need specific treatment as a string or ribbon. This coarse-grained treatment mostly throws the identity of molecules away. It is a kind of idealization of a higher order. On the other hand, even if the subject remains, the topic non-relevant to molecular nature is excluded. The continuum theory of liquid crystals is an example. Other subjects omitted from the content are systems, for which effective theories usually do not incorporate molecular natures. These include optical properties, molecular conductor and magnets, and properties of liquid solutions. Although the optical properties are directly related to spectroscopies of molecular systems, their effective theories, i.e., quantum-mechanical models, treat molecules only indirectly. Molecular conductors and magnets are spreading research fields, but their arguments rely on anisotropic and exotic crystal structures and the resulting effective Hamiltonians. For solutions, it seems difficult to give a coherent molecular description at present. When the researches in these fields begin to involve molecular natures of systems, a new version should be necessary. I assumed the pieces of knowledge of classical and quantum mechanics, thermodynamics, and statistical mechanics at the elementary level, for which undergraduate classes are available in chemical school. Mathematics behind them is implicitly assumed. It contains analysis including partial derivatives of multivariable functions, linear algebra, and group theory. However, the manipulation in the text does not go far beyond the very elementary level. I explicitly included step-by-step derivations for many issues because such examples seem valuable for practical applications to specific problems by readers. This necessity for particular treatments to case by case reflects the diverse nature of molecular systems. Concerning chemistry, on the other hand, I only assumed the meaning of chemical formulas and the fact that the bulk substance consists of a vast number of molecules! In total, this book should be readable for graduate students majoring in either chemistry or physics. However, I am grateful if the book is enjoyable to established researchers in physical chemistry and condensed matter physics. This book stands on my research/teaching activity with many colleagues. Drs. H. Chihara, T. Atake, Ichiro Hatta, and M. Sorai were my professors, and they guided directly or indirectly with their profound insights and broad perspective on

Preface

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condensed matter science. Dr. Yasuhisa Yamamura was a student at first and has been the most effective collaborator. Encouragement by Prof. Mary Anne White, a respected friend who has worked in a similar field, in an early stage of writing convinced me of some merit of books of this kind. My friends, Prof. Tadeus Wasiutyński, Maria Massalska-Arodź, Robert Pełka, and Kiku Ishii, read some parts of the draft and gave me valuable comments and suggestions for correcting faults and improving writing. Especially, Robert checked most of the math. Prof. Takehiko Mori provided information on polymorphism of organic conductors. Still, I declare to be solely responsible for all of the mistakes and faults in this book. Without people irrespective of the mentioned above or not, I could not prepare this book. It is my great pleasure to express my sincere thanks to all of them. Tsukuba, Japan June 2020

Kazuya Saito

Contents

1

Molecules and Intermolecular Interactions . . . . . . . . 1.1 Hierarchy of Materials . . . . . . . . . . . . . . . . . . . 1.1.1 Energetic Overview of Nature . . . . . . . . 1.1.2 Molecular World . . . . . . . . . . . . . . . . . 1.2 Intermolecular Interaction . . . . . . . . . . . . . . . . . 1.2.1 Charge Distribution in a Molecule . . . . . 1.2.2 Electrostatic Interaction . . . . . . . . . . . . . 1.2.3 Polarizability and Dispersion Interaction 1.2.4 Repulsion: Pauli’s Exclusion Principle . . 1.2.5 Practical Representation . . . . . . . . . . . . 1.2.6 Other Interactions . . . . . . . . . . . . . . . . . 1.3 What the Interaction Brings . . . . . . . . . . . . . . . . 1.3.1 Equation of State of Gas . . . . . . . . . . . . 1.3.2 Liquefaction . . . . . . . . . . . . . . . . . . . . . 1.3.3 Crystallization . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Thermodynamic Aspects . . . . . . . . . . . . . . . . . 2.1.2 Entropy and Boltzmann’s Principle . . . . . . . . . 2.1.3 Interface and Phase Growth . . . . . . . . . . . . . . 2.2 Landau’s Phenomenology . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Power Expansion of Thermodynamic Potential . 2.2.2 Order Parameter . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Simplest Case: Continuous Transition . . . . . . . 2.2.4 Discontinuous Transition . . . . . . . . . . . . . . . . 2.2.5 Plural Equivalent Order Parameters . . . . . . . . . 2.2.6 Implication and Outcome . . . . . . . . . . . . . . . .

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2.3 Critical Phenomena and Universality . . . . . . . . . . . . . . . . . . . . 2.4 Formation Versus Collapse of Order . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48 50 51

3

Molecular Liquids . . . . . . . . . . . . . . . . . . . . . . 3.1 Liquid Structure . . . . . . . . . . . . . . . . . . . 3.1.1 Scattering from Molecular Liquid 3.1.2 Case Study . . . . . . . . . . . . . . . . 3.2 Molecular Association . . . . . . . . . . . . . . . 3.3 Ionic Liquids . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4

Molecular Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Crystallization . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Alder Transition . . . . . . . . . . . . . . . . . . 4.1.2 Landau Theory of Weak Crystallization . 4.2 Diffraction and Modern Definition of Crystals . . 4.3 Molecular Shape and Crystal Structure . . . . . . . . 4.3.1 Within Landau Theory . . . . . . . . . . . . . 4.3.2 Close Packing . . . . . . . . . . . . . . . . . . . 4.3.3 Polymorphism . . . . . . . . . . . . . . . . . . . 4.4 Cohesive Energy . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Experimental Information . . . . . . . . . . . 4.4.2 Lattice Sum . . . . . . . . . . . . . . . . . . . . . 4.4.3 Crystal Engineering . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5

Lattice Dynamics of Molecular Crystals . . . . . . . . . . . . . . . . . 5.1 Scope of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Atomic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Atomic Array . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Atomic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Crystals of Rigid Molecules . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Properties of Problems . . . . . . . . . . . . . . . . . . . . 5.4 Crystals of Deformable Molecules . . . . . . . . . . . . . . . . . . 5.5 Related Issues and Examples . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Heat Capacity and Debye Temperature . . . . . . . . 5.5.2 Debye–Waller Factor . . . . . . . . . . . . . . . . . . . . . 5.5.3 Lattice Instability and Structural Phase Transition 5.5.4 Anharmonicity and Thermal Expansion . . . . . . . . 5.5.5 Large Amplitude Motion . . . . . . . . . . . . . . . . . . 5.5.6 Degrees of Freedom for Membrane Dynamics . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6

Melting of Molecular Crystals . . . . . . . . . . . . . . . . . . . . . . 6.1 Melting in Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Plastic Crystals and Liquid Crystals . . . . . . . . . . . . . . 6.2.1 Mean-Field Theory of Ising Model . . . . . . . . 6.2.2 Simple Theory of Melting of Atomic Crystal . 6.2.3 Translational and Orientational Melting . . . . . 6.3 Molecular Deformation . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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119 119 124 124 129 133 136 139

7

Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Thermotropic Versus Lyotropic . . . . . . . . . . . . . . 7.1.2 Various Thermotropics . . . . . . . . . . . . . . . . . . . . 7.2 Effects of Molecular Anisotropy . . . . . . . . . . . . . . . . . . . 7.2.1 Liquid Crystal of Hard Particles: Onsager Theory 7.2.2 Maier–Saupe Theory . . . . . . . . . . . . . . . . . . . . . 7.3 Effects of Molecular Shape . . . . . . . . . . . . . . . . . . . . . . . 7.4 Molecular Shape and Aggregation in Lyotropics . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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143 143 143 144 149 149 152 155 157 160

8

Molecular Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Glass as Frozen-in State . . . . . . . . . . . . . . . . . . . 8.1.1 Simple View on Glass Transitions . . . . . . 8.1.2 Structural Chemistry of Glass Transitions 8.2 Properties of Glasses . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Fragility . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Step in Heat Capacity at Tg . . . . . . . . . . 8.2.3 Residual Entropy . . . . . . . . . . . . . . . . . . 8.2.4 Effects of Structural Inhomogeneity . . . . . 8.3 Possibility of Ideal Glass Transitions . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9

Molecular Flexibility and Material Properties . . . . . . . . . 9.1 Single-Particle or Extended Scheme . . . . . . . . . . . . . 9.2 Glass Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Phase Transitions Related to Molecular Deformation 9.4 Conformational Disordering of Alkyl Groups . . . . . . 9.4.1 Odd-Even Effect . . . . . . . . . . . . . . . . . . . . . 9.4.2 Systematics as Indispensable Tool . . . . . . . . 9.4.3 Entropy Reserver . . . . . . . . . . . . . . . . . . . . 9.4.4 Chains in Liquid Crystals . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Importance of Molecular Crystals . . . . . . . . . . . . . . . . . . . . . 10.1 Motional Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Significance and Difficulty . . . . . . . . . . . . . . . . 10.1.2 Entropic Detection . . . . . . . . . . . . . . . . . . . . . . 10.1.3 Dynamics in Quasi-one-dimensional Systems . . . 10.2 Molecular Crystals as Tunable Model System . . . . . . . . . 10.2.1 Basis of Tunability . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Unified Description of Structural Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Impurity Effects on Structural Phase Transitions . 10.3 Molecular Crystals as Stage for Novelties . . . . . . . . . . . 10.3.1 Coupling Between Molecular Dynamics and Electronic System . . . . . . . . . . . . . . . . . . . 10.3.2 Phenomena Involving Molecular Flexibility . . . . 10.3.3 Dipolar Ising System . . . . . . . . . . . . . . . . . . . . 10.3.4 Exotic Superstructure . . . . . . . . . . . . . . . . . . . . 10.3.5 Molecular Aggregation in a Gyroid Phase . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Molecules and Intermolecular Interactions

1.1 Hierarchy of Materials 1.1.1 Energetic Overview of Nature This book deals with the structure and properties of an ensemble of molecules. Before proceeding into details, let us see the typical magnitude of energies involved in various phenomena. It will be clear why such a description based on the identity of molecules is appropriate and what extent it is to. It is reasonable to start with nuclei of atoms. They are formed by nucleons (protons and neutrons) through the strong interaction. Although hadrons, including nucleons, are composed of quarks, the decomposition to quarks is impossible (known as the quark confinement). The minimal nuclear reaction is the formation of a deuteron from a proton and a neutron. This reaction accompanies the emission of a photon (γ-ray) with the energy of ca. 2.2 MeV (1 eV is ca. 96.5 kJ mol−1 ). According to the equivalence of mass and energy revealed by Einstein, E = mc2 (c is the speed of light in vacuum), the binding energy ΔE for an arbitrary nucleus with Z protons and N neutrons can be calculated from its mass defect ΔM as ΔE = ΔMc2   = Z Mp + N Mn − M c 2 ,

(1.1)

where Mp , Mn , and M are masses of proton, neutron, and the nucleus, respectively. Figure 1.1 shows the binding energy per nucleon (ΔE/(Z + N )). As seen in Fig. 1.1, the binding energy is typically several MeV and shows the maximum (≈ 8.7 MeV) for the nucleus of 56 Fe. Each neutral atom consists of a nucleus and electrons, the number of latter which accords to the charge compensation. The energy necessary to remove an electron from an atom in the ground state is the (first) ionization energy. The ionization energy is, within an excellent approximation, dependent not on the atom’s mass number but © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 K. Saito, Chemical Physics of Molecular Condensed Matter, Lecture Notes in Chemistry 104, https://doi.org/10.1007/978-981-15-9023-8_1

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1 Molecules and Intermolecular Interactions

[ΔE / (Z+N)] / MeV

9

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6 0

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Z Fig. 1.1 Binding energy per nucleon of most stable nuclei (having the longest half life) of elements as a function of atomic number

30 He Ne 20 ΔE1 /eV

Ar Kr Zn

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0

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20

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Z Fig. 1.2 First ionization energy of neutral atom as a function of atomic number

solely on the atomic number Z . The ionization energy is shown in Fig. 1.2. The figure indicates the energy scale of several eV, which is much (ca. 10−5 ) smaller than that of nuclear reactions. Atoms may bind through a chemical bond, resulting in the formation of a molecule. It has often been assumed that the energy gain upon the molecular formation is well approximated by a sum of those of formed chemical bonds, though some fundamental

1.1 Hierarchy of Materials Table 1.1 Energy scales in nature Phenomenon Nuclear reaction (binding energy of atomic nucleus per nucleon) Ionization energy of atom Energy of chemical bond Hydrogen bond

Intermolecular interaction

3

Energy scale

Environmental energy

10 MeV 10 eV > 3 eV ≤ 3 eV 10−1 eV 26 meV

Photon of visible light Thermal energy (kB T ) at 300 K

10−2 eV

difficulties have been recognized.1 The binding energy assigned to a bond is called the bond energy. Generally, bond energies are parallel to the order of chemical bonds. Indeed, the triple bond in a nitrogen molecule accompanies large bond energy of ca. 10 eV. Single bonds have about half of this. Thus, the energy scale of chemical bonding is smaller than the atomic ionization by about one order of magnitude. Ubiquitous interaction between molecules is of the dispersion character workable even between neutral molecules, as discussed in the next section. This interaction is approximately additive for interatomic contributions. The interaction energy itself may become significant when molecules are large (with a large number of atoms), accordingly. However, it seems adequate to see the interaction between two atoms as an element of this interaction. The order of magnitude is estimated as ca. 10−2 eV based on the lattice energies of solids of rare gas elements. On the other hand, a relatively strong interaction is known, such as the hydrogen bond. Its energy is ca. 10−1 eV. Thus, even considering this strong interaction, the energy of the intermolecular interaction is smaller than that of chemical bonds by one order of magnitude. However, note that, under the additivity of interactions, the molecular aggregation might compete with the chemical stability of substances if molecules are sufficiently large. Finally, environmental energy is to be considered. Although the light originally emitted at the surface of the sun (at ca. 6,000 K) contains notably components of ultraviolet light, a large portion of the component is absorbed by some components in the air (mainly by ozone). On the earth, thus, the visible light is dominant. Its energy lies between ca. 3 eV (violet) and 2 eV (red). The thermal energy is given by kB T . At 300 K, it is ca. 26 meV. Table 1.1 summarizes the energy scales mentioned.

example, the localized nature of chemical bonds is violated for delocalized π electrons in aromatic molecules. Even if the localization seems acceptable, the energy necessary to break a C–H bond of methane (CH4 ) is not a quarter of that to break all of its four bonds.

1 For

4

1 Molecules and Intermolecular Interactions

1.1.2 Molecular World It is evident from Table 1.1 that the energy involved in a nuclear reaction is much larger than those of other phenomena. The separation of the process from other processes is almost complete, accordingly. Nuclear synthesis (syntheses of elements below nickel) proceeds in stars. Indeed, the pressure and temperature at the solar core are, respectively, 2.5 · 1011 atm and 1.5 · 107 K, the thermal energy of the latter amounts to ca. 1.3 keV. Although this energy is still smaller than the binding energies of atomic nuclei, the extreme density caused by the high pressure facilitates the synthesis of helium nuclei from protons. Under the complete separation in the energy scale, any low-energy process is almost impossible to affect the nuclear process. This separation is the basis of the possibility of chemistry, a science of the diversity and “inhomogeneity” of the materials world. Note that the relevant energy to be compared with average translational energy, i.e., thermal energy, is not biding energies but the energy barrier to be overcome necessary for nuclear reactions. A similar situation is also encountered in molecular syntheses. The ignorance of nuclear reaction automatically offers a way of understanding materials based on atoms. This is the way traditionally developed in condensed matter physics, and certainly possible. However, for example, in the interpretation of temperature dependence of heat capacity of crystals, the analysis in this way assuming the number of atoms in a unit cell yields a high characteristic temperature (Debye temperature described in Sect. 5.5.1), which is practically useless because of too high value in comparison with melting temperatures. Although the separation in the energy scale is not complete in contrast with the above case, the energies of electronic processes such as ionization and formation of chemical bonds involve larger energy by, at least, about one order of magnitude than intermolecular interactions. Thus, it is reasonable and even better to consider this separation for understanding molecular materials. Indeed, returning to the example of heat capacity, we can readily explain even an apparent saturation of heat capacity to 6R if the number of degrees of freedom is adequately assumed (such as 6 for a rigid molecule). The last case exemplifies the benefit of the understanding of materials based on the molecular description. In some cases, this is done by replacing “atoms” with “molecules.” For example, in discussing electronic or magnetic properties of molecular systems (molecular conductor and magnet), their properties are interpreted based on a complicated (and often anisotropic) arrangement of sites. On the other hand, such a replacement cannot work for, for example, the issue of melting. The intrinsic difference between atoms and molecules lies in the necessity to take their anisotropy into account (except for cases of monatomic molecules, i.e., rare gasses). A partial melting, e.g., the melting of solely the positional degrees of freedom but the orientational ones, is only possible in cases of crystals consisting of molecules because atoms have no rotational degrees of freedom detectable from the outside. The possibility of molecular deformation is also an intrinsic difference. The main body of this

1.1 Hierarchy of Materials

5

monograph is devoted to showing how analysis and understanding of materials are possible based on molecules. Before concluding this introductory section, it is reminded that the separation in the energy scale is not perfect between intermolecular interaction and chemical phenomena. There exist some examples that reflect this incompleteness. There exist rare instances of phase transitions, in which molecular structures change with rearrangement of chemical bonds, include the melting of PCl4 ·PCl6 [1, 2] to molecular liquid and that in liquid sulfur (changing from S8 to linear polymer) [3].

1.2 Intermolecular Interaction 1.2.1 Charge Distribution in a Molecule Among the four fundamental interactions between particles in nature, i.e., strong and weak interactions, electrostatic interaction, and gravity, those workable between atoms and molecules are the two latter of them, which have no characteristic length scale. The interaction energy of both is inversely proportional to the distance between two interacting subjects. The gravity is always attractive, whereas the electricity can have not only attractive but also repulsive character depending on the signs of the electric charge. Because of the large difference in their magnitudes, however, the consideration only on the electric term is sufficient in molecular physics. Indeed, the gravitational and electrostatic interactions between two electrons (with the mass m ≈ 9.1 · 10−31 kg and the charge q ≈ −1.6 · 10−19 C) 1 Å apart are −5.5 · 10−61 J and 2.3 · 10−18 J, respectively. The electrostatic interaction between charges of molecules is dominant if they carry net charges, i.e., if they are ions. Even if molecules are neutral, electrostatic interaction governs the interaction between them. We can easily imagine that the antiparallel arrangement is more stable than the parallel arrangement for a pair of dipolar but neutral molecules because the head with the positive partial charge of each molecule is closer to the negatively charged tail of the other molecule. Thus, the consideration of charge distribution in a molecule, ρ(r), is necessary. Even for ions, a similar consideration is necessary to validate the naïve expectation that the interaction between them resembles that between point charges when the molecular size is non-negligible compared to their separation. The following identity is known as the multipole expansion for |R| > |r| in the three-dimensional space: ∞

 |r|n 1 = Pn (cos θ Rr ), |R − r| |R|n+1 n=0

(1.2)

where Pn (·) is n-th order Legendre polynomial, and θ Rr the angle between vectors R and r (cos θ Rr = R · r/|R||r|). Suppose that the center of gravity of a molecule with

6

1 Molecules and Intermolecular Interactions

the charge distribution ρ(r) be at the origin and that ρ(r) is negligible for |r| > rmol . Since the electrostatic potential is additive, that at R (outside the molecule) produced by the charge distribution ρ(r) in the molecule is a superposition of those formed by component charge elements. Thus, for |R| > rmol , V (R) =

 ∞ 1  1 |r|n Pn (cos θ Rr )ρ(r)dv, 4πε0 n=0 |R|n+1

(1.3)

where the integration is over the volume of the molecule. It is essential to notice that a successive term becomes smaller in the power of |R|. Thus, the first non-vanishing term is of primary importance. Let’s see some leading terms. Suppose first the term with n = 0 is non-vanishing with the total charge of q. Since P0 (·) = 1,  1 1 ρ(r)dv 4πε0 |R| 1 q . = 4πε0 |R|

V (R) ≈

(1.4)

This is precisely the same as the electrostatic potential produced by a point charge q. In other words, any charge distribution looks like a point charge if viewed from a point far away from the molecule. This guarantees the validity of the intuitive expectation mentioned before. When the net charge is null (q = 0), the leading term would be of n = 1. Since P1 (x) = x, the term is given by  1 1 |r| cos θ Rr ρ(r)dv, 4πε0 |R|2  R 1 · rρ(r)dv. = 4πε0 |R|3

V1 (R) =

(1.5)

In the second identity, the relation cos θ Rr = R · r/|R||r| is used. By defining2 the dipole moment p as  p=

rρ(r)dv ⎛ ⎞  r x ρ(r)dv = ⎝  r y ρ(r)dv ⎠ , r z ρ(r)dv

(1.6)

Equation 1.5 is rewritten as

2 In

this book, a resultant object consisting of components, such as a vector and matrix, after integrating components is simply expressed like Eq. 1.6.

1.2 Intermolecular Interaction

a)

7

b1)

c)

b2)

Fig. 1.3 Examples of dipolar (a), quadrupolar (b1 and b2) and octopolar (c) distributions of charge on a sphere. While b1 is axially symmetric similarly to the dipolar case (a), b2 has the twofold symmetry

V1 (R) =

1 R· p . 4πε0 |R|3

(1.7)

A dipole moment is a vectorial quantity characterizing the polarization of the molecular charge distribution (for primarily neutral molecules). The electric field E associated with this potential is obtained in accordance with standard static electromagnetism as a gradient for R, E 1 (R) = −∇ R V1 (R)  1 1 1 R · p∇ R + ∇R R · p =− 4πε0 |R|3 |R|3 1 1 [3(R · p)R − |R|2 p]. = 4πε0 |R|5

(1.8)

To characterize the charge distribution of symmetrical molecules such as nitrogen having the charge distribution essentially similar to b1 in Fig. 1.3, terms with n = 0 and n = 1 is insufficient because q = 0 and | p| = 0. The next term (n = 2) having the form  1 1 |r|2 P2 (cos θ Rr )ρ(r)dv V2 (R) = 4πε0 |R|3   2 1 1 2 3 cos θ Rr − 1 ρ(r)dv |r| = 4πε0 |R|3 2 1 1 t = RQR (1.9) 4πε0 |R|5

8

1 Molecules and Intermolecular Interactions

is called a quadrupolar term. Here, Q is a tensor of order 2, i.e., 3 × 3 matrix, characteristic of a molecule called quadrupole moment given by Qi j =

1 2



  3ri r j − |r|2 δi j ρ(r)dv

(1.10)

with i, j = x, y, z. Since Q is symmetric and traceless, its largest eigenvalue is often used to characterize the molecule. An example of the charge distribution relevant for this term is shown in Fig. 1.3. Higher-order terms are necessary for highly symmetric molecules such as the tetragonal ones like methane, and the octahedral ones like SF6 . The electrostatic potential is given by 1 1 Vn (R) = 4πε0 |R|n+1

 |r|n Pn (cos θ Rr )ρ(r)dv.

(1.11)

The associated electric field is calculated using ∇ R cos θ Rr =

 1 2 |R| r − (R · r)R 3 |R| |r|

(1.12)

as E n (R) = −∇ R Vn (R)    1 1 n ∇R P (cos θ )ρ(r)dv |r| =− n Rr 4πε0 |R|n+1  1 1 =− |r|n Pn (cos θ Rr )ρ(r)dv∇ R 4πε0 |R|n+1   1 + n+1 |r|n ρ(r)Pn (cos θ Rr )∇ R cos θ Rr dv |R|

=

n+1 Vn (R)R |R|2    1 2 1 |R| W n − (R·W n )R , − 4πε0 |R|n+4

(1.13)

where Pn (x) = Wn =

d Pn (x)  dx

r|r|n−1 ρ(r)Pn (cos θ Rr )dv.

(1.14) (1.15)

1.2 Intermolecular Interaction

9

The molecular characteristic that is tensor of the order n emerges in these terms. They are called a 2n -pole moment for arbitrary n. The first few are dipole (n = 1), quadrupole (n = 2), octopole (n = 3), hexadecapole (n = 4), etc.

1.2.2 Electrostatic Interaction In this section, we calculate electrostatic interactions while placing the charge distribution of another molecule in the electrostatic potential produced by a molecule (such as Eq. 1.3). It is noteworthy that only interactions involving up to dipole moments are necessary to consider in most real cases in comparison with the shape effects of molecules discussed later. When molecules are spherical, the interaction is simply of that between point charges 1 q0 q R , (1.16) E mm (R) = 4πε0 |R| where q0 and q R are their charges. For non-spherical molecules, the calculation can be performed as in the following example of the dipolar interaction. Suppose one molecule is at the origin and the other at R with the electron density ρ (R + r  ) around R. We expand |R + r  |−3 in r  assuming |R|  |r  | as 1 1 (r  ·R) ≈ − 3 . |R + r  |3 |R|3 |R|5

(1.17)

Thus, the interaction between two dipoles ( p0 at the origin and p R at R) is given by the integration over a space around R:     1 1 (r · R) (R + r  ) · p0 ρ (r  )dv  − 3 3 5 4πε0 |R| |R|      1 1 1        p ≈ ( p · R) ρ (r )dv + · r ρ (r )dv 0 4πε0 |R|3 0 |R|3    1 −3 5 ( p0 · R) R · Rρ (r  )dv  |R|   ( p0 · p R ) 3( p0 · R)( p R · R) 1 . (1.18) = − 4πε0 |R|3 |R|5

E dd (R) ≈

Note that the second line is obtained while neglecting the second-order term in r  , and the  last equality assumes the vanishing net charge for the molecule at R, i.e., q R = ρ (r  )dv  = 0 in the first term in the second line. This term is the interaction between a point charge and a dipole p0 . The last expression is exact for point dipoles irrespective of their species.

10

1 Molecules and Intermolecular Interactions

a)

b)

Fig. 1.4 Comparison of dipolar interaction (interaction between two dipoles) given by Eq. 1.18 ( p = | p| and R = |R|). Parallel arrangement has a lower energy than antiparallel one in tandem (a) (Vpara = −2 p 2 R −3 /4πε0 < 2 p 2 R −3 /4πε0 = Vanti ) while in (b) the relation is reversed (Vanti = − p 2 R −3 /4πε0 < p 2 R −3 /4πε0 = Vpara )

The interaction between a quadrupole and one of a point charge, dipole and quadrupole can be calculated similarly by truncating the series expansion of |R + r|−n up to the second-order in r. Note that the second-order term neglected in Eq. 1.18 is only a part of the interaction between a dipole and a quadrupole. Since 2n -pole moment has the form of [charge]×[length]n while the interaction between point charges has [charge]2 ×[length]−1 apart from the common prefactor (4πε0 )−1 , the interaction between a 2n -pole and a 2m -pole depends on their separation R as R −(n+m+1) . The interaction between two dipoles given by Eq. 1.18 indeed has the power of −(1 + 1 + 1) = −3. The interaction between them depends not only on the distance in between but also on their mutual orientations because 2n pole moments are tensorial quantities. Equation 1.18, for example, indicates that the interaction prefers parallel arrangement if two dipoles are in line (tandem) in Fig. 1.4a while it does antiparallel one if they are side-by-side as in Fig. 1.4b. It is interesting to see that not only the preferred arrangement but also the strength of the preference exhibits the anisotropy. There is a report that a seemingly isotropic (threedimensional) crystal exhibits the anisotropy (low-dimensionality) arising from the anisotropy of the dipolar interaction [4].

1.2.3 Polarizability and Dispersion Interaction In this section, we derive the dependence of the interaction between neutral molecules on their separation in two steps [5].

1.2.3.1

Dispersion of Polarizability

Consider a neutral molecule that is spherically symmetric and has a closed-shell electronic structure. The electronic wave function satisfies the following Schrödinger equation: ∂ (1.19) H0 Φ = i Φ, ∂t

1.2 Intermolecular Interaction

11

where H0 is a Hamiltonian operator of the molecule. Normalized stationary states are denoted by φn exp(−iωn t) (n = 0, 1, 2, . . .). Then, H0 φn = ωn φn .

(1.20)

This indicates that φn are eigen functions of the time-independent Schrödinger equation. Suppose the weak electric field is exerted on the molecules in the ground state (n = 0) along the x-axis. The time dependence of the field is assumed to be E x cos ωt. Then, the total Hamiltonian H is given by H = H0 − Px E x cos(ωt) = H0 + H ,

(1.21)

where Px is the operator corresponding to the x component of the molecular dipole moment. The wave function should fulfill the following time-dependent Schrödinger equation:   ∂ (1.22) H0 + H Φ = i Φ. ∂t Considering the weak filed, we assume the proportionality between the electric field and the dipole moment. Thus, 

Φ ∗ Px Φdv = α(ω)E x cos ωt,

(1.23)

where α(ω) is the frequency-dependent polarizability of the molecule. Assuming |H | |H0 |, we proceed with the time-dependent perturbation theory. We write the wave function as  φn exp(−iωn t)an (t). (1.24) Φ = φ0 exp(−iω0 t) + n

Here, |an (t)| 1 is assumed for n ≥ 1. Putting this expansion into Eq. 1.22 yields     ∂ H0 − i φn exp(−iωn t)an (t) φ0 exp(−iω0 t) + ∂t n    = −H φ0 exp(−iω0 t) + φn exp(−iωn t)an (t) .

(1.25)

n

Considering Eq. 1.20, we have − i

 n

φn exp(−iωn t)

d an (t) dt

(1.26)

12

1 Molecules and Intermolecular Interactions

for the left-hand side.3 Considering the smallness of |H | and |an (t)|, the second term in the right-hand side can be ignored. Then, we have − i



φn exp(−iωn t)

n

d an (t) = −H φ0 exp(−iω0 t) dt = Px E x cos(ωt)φ0 exp(−iω0 t)

(1.27)

By introducing ωn0 = ωn − ω0 , Eq. 1.27 is rewritten as − i



φn exp(−iωn0 t)

n

d an (t) = Px E x cos(ωt)φ0 . dt

(1.28)

Integration after multiplying φ∗n from the left yields − i exp(−iωn0 t)

d an (t) = E x cos ωt dt



φ∗n Px φ0 dv

(1.29)

because of the orthonormality of φn ’s 

φ∗m φn dv = δmn .

(1.30)

This is the equation an (t) should fulfill. By introducing  x = Pmn

φ∗m Px φn dv,

(1.31)

the equation is expressed as − i exp(−iωn0 t)

d x an (t) = E x Pn0 cos ωt dt

(1.32)

Considering the forced nature of the oscillation of the electric field, we can assume the periodic oscillation for an (t). Then, the solution is obtained by integrating Eq. 1.32 as  iωt e−iωt 1 x e Pn0 E x exp(iωn0 t) + . (1.33) an (t) = 2 ωn0 + ω ωn0 − ω The expectation value of the dipole moment is calculated to the first order in an (t) as

changed the derivative symbol to the normal one from the partial one because an (t) depends solely on t.

3 We

1.2 Intermolecular Interaction

13



Φ ∗ Px Φdv   x x ∗ ≈ exp(iωn0 t)an∗ (t)Pn0 + exp(−iω0n t)an (t)(P0n ) n

n

x 2 | ωn0 2  |Pn0 = E x cos ωt 2  n ωn0 − ω 2

(1.34)

x because of P00 = 0 derived from the spherical symmetry of the ground state of the spherical molecule with the closed-shell electronic structure. In the second equality, x ∗ x ) = P0n is assumed. A comparison with Eq. 1.23 yields the relation (Pn0

α(ω) =

x 2 2  |Pn0 | ωn0 . 2  n ωn0 − ω2

(1.35)

This formula indicates the frequency dependence of the polarizability. The dependence of a dielectric property (such as a polarizability) on the applied frequency is generally called its dispersion.

1.2.3.2

Dispersion Interaction

Consider two spherical molecules, each of which is the same as the one in the previous section. When their separation is large enough compared to molecular size, it is natural to treat two independent molecules as the unperturbed system. The interaction between them corresponds to the perturbation term. Since we discuss the system consisting of two neutral molecules, the interaction should arise from not the term involving net charges of the molecules but their higher-order moments. Let us see what is the lowest order term using two hydrogen atoms as an example. When two atoms are very distant, we may label protons and electrons. Using capitals for protons and lower case letters for electrons, the total Hamiltonian is written as   2  2 2  2 ∇I + ∇II2 − ∇1 + ∇22 2M 2m  1 1 q2 + Hp = − 4πε0 |r 1 − RI | |r 2 − RII |  1 1 q2 + − 4πε0 |r 1 − RII | |r 2 − RI |  1 1 q2 + . + 4πε0 |RI − RII | |r 1 − r 2 |

Hk = −

On the other hand, the Hamiltonian of atom 1 is given by

(1.36) (1.37)

14

1 Molecules and Intermolecular Interactions

HH = −

1 2 2 2 2 q2 ∇I − ∇1 − . 2M 2m 4πε0 |r 1 − RI |

(1.38)

Thus, Hk and the first line of H p are in the Hamiltonian of two independent atoms. Then, the perturbation term is HH =

q2 4πε0



1 1 1 1 . + − − |RI − RII | |r 1 − r 2 | |r 1 − RII | |r 2 − RI |

(1.39)

Using the multipole expansion, it can be confirmed that the lowest order term is of the form  3[(r 1 − RI ) · (RII − RI )][(r 2 − RII ) · (RII − RI )] q2 − 4πε0 |RII − RI |5  (r 1 − RI ) · (r 2 − RII ) (1.40) − |RII − RI |3 This is just the interaction between two dipoles p j = q(r j − R j ) ( j = 1, 2) with the distance |RII − RI |. Having seen the lowest order term in the perturbation being the dipolar interaction, we return to the general case. The operator of the dipole interaction is denoted by J . Then the total Hamiltonian is H = H0 + J

(1.41)

H0 = HH1 + HH2 1 1 J = [(P1 · P2 ) − 3(P1 · e)(P2 · e)] 4πε0 |R|3 1 1 t = P1 TP2 , 4πε0 |R|3 T = I − 3et e.

(1.42) (1.43)

(1.44)

where R is the vector between two molecular centers and e = R/|R|. Ignoring the necessity of antisymmetry for permutation of two electrons, the normalized eigen functions of the unperturbed Hamiltonian is given by ψmn = φ1m φ2n

(1.45)

H0 ψmn = (ωm + ωn )ψmn .

(1.46)

which satisfies Note that φ1m and φ2n in ψmn are around different centers. According to the standard treatment of the time-independent perturbation theory, the change in energy of the ground level is

1.2 Intermolecular Interaction

15



∗ ψ00 J ψ00 dv1 dv2

ΔE 1 =

(1.47)

in the lowest order. However, this term vanishes because, for example, 

∗ ψ00 P1x ψ00 dv1 dv2 =

 

=

φ∗10 P1x φ10 dv1



φ∗20 φ20 dv2

φ∗10 P1x φ10 dv1

=0

(1.48)

because of the spherical symmetry of the molecule. Therefore, the interaction comes from the second order term:   J00mn Jmn00 ωm0 + ωn0 m=0 n=0  = ψ ∗jk J ψlm dv1 dv2

ΔE 2 = −

(1.49)

J jklm

(1.50)

Since J jklm is proportional to |R|−3 , the interaction is attractive with the leading dependence of |R|−6 .4 To see the numerical factor, we write the interaction energy as 2 1 1 ΔE 2 = − μ 4πε0 |R|6   t00mn tmn00 μ= ωm0 + ωn0 m=0 n=0 

(1.51) (1.52)

t jklm = 4π 0 |R|3 J jklm  = ψ ∗jk t P1 TP2 ψlm dv1 dv2

(1.53)

jα

Since P0m (α = x, y, z and j = 1, 2) are quantities related to a spherical molecule, finally, we obtain   |P 1x |2 |P 2x |2 0m 0n μ=6 . (1.54) ω + ω m0 n0 m=0 n=0 Using the identity 1 2 = a+b π 4 The

series is in terms of |R|−1 .

 0

∞

ab (a 2

+

u 2 )(b2

+ u2)

du,

(1.55)

16

1 Molecules and Intermolecular Interactions

Equation 1.54 is rewritten as 3 μ= π



∞

α1 (iω)α2 (iω)dω,

(1.56)

0

where αi (ν) is the polarizability of atom i at a real frequency ν (given by Eq. 1.35) although the integration should be taken along the imaginary axis. This formula implies a close connection between the dispersion interaction and the molecular polarizability. For the interaction between unlike molecules, 1 and 2, some estimates have been proposed [6, 7]. Among them, the following form is often adopted: μ12 ≈

√

μ11 μ22 .

(1.57)

This approximation is a basis for combination rules often assumed in the atom–atom potential method described later.

1.2.4 Repulsion: Pauli’s Exclusion Principle The theory governing the microscopic world is the quantum mechanics, which has revealed a wavy nature of any moving body. The wavy nature becomes obvious and significant for microscopic particles. Thus, two microscopic particles are indistinguishable. A wave function expresses the state of such a system. Suppose that a state with two particles, a and b, in respective “coordinates,” α and β, is specified by a wave function ϕ(α, β). If two particles are interchanged, the expression should become ϕ(β, α). According to the indistinguishability of two particles, the two states are equivalent. This implies ϕ(α, β) = cϕ(β, α)

(1.58)

with a non-vanishing constant c. The repetition of the interchange of two particles once more yields (1.59) ϕ(α, β) = c2 ϕ(α, β), leading to c = ±1.

(1.60)

This simple argument shows that the indistinguishability gives an important classification of microscopic particles. Particles with c = 1 are called bosons while those with c = −1 are fermions. Electrons with a half-integer spin (s = ± 21 ) are fermions.

Fig. 1.5 Schematic representation of intermolecular interaction energy as a function of their distance

17

interaction energy / arbitrary unit

1.2 Intermolecular Interaction

0

Fermions have a notable property known as Pauli’s exclusion principle. Starting from the state where two particles have identical “coordinates,” the interchange of two particles yields ϕ(α, α) = −ϕ(α, α). (1.61) This forces ϕ(α, α) = 0. This result should be understood as an indication of the absence of such states. Namely, two electrons are not allowed to have the same coordinates. The exclusion principle imposes a critical effect on the intermolecular interaction. When two molecules in respective ground states approach, the overlap of electron distributions follows. The overlap causes the deformation of electron distributions. The deformation is achieved by the mixing of excited states’ wave functions to the ground state one, resulting in a significant increase in total energy. Thus, when the distance of two molecules is small enough to cause the overlap of electron distributions in the ground state, repulsive interaction becomes dominant. The increase in energy is steep in comparison with attractive interactions. The dependence on the distance (r ) is usually expressed as an exponential function (∝ exp(−r/r0 ) with a characteristic distance r0 ) or a power-law (∝ r −m ) with an index m ≥ 6). Total interaction between two molecules consists of both attractive and repulsive contributions and has a minimum, as shown in Fig. 1.5. It is noteworthy that the ground state wavefunction mostly determines the spatial range of the strong repulsive interaction except the details of energy increase, in contrast to other interactions. This property is the basis for practical van der Waals radii of atoms and ionic radii, both of which are defined as minimum distances found in crystals. The so-called excluded volume effects are also understood in terms of the repulsive interaction.

18

1 Molecules and Intermolecular Interactions

1.2.5 Practical Representation 1.2.5.1

First-Principle Computation

The fundamental physics that describes the behavior of molecules has long been established as the quantum mechanics, and numerous recipes to numerically solve the fundamental equations (wave equations) have been proposed. Besides, significant progress in computers was achieved in the past. Intermolecular interaction for a specific configuration for a molecular pair is an easy task for a modern computer. The most faithful way to follow quantum mechanics is to perform first-principle (quantum-chemical) calculations for every configuration upon necessity. Such computation may be possible but impractical in most cases. Alternatively, a prior construction of the intermolecular potential function, i.e., an approximation to the computed energy using some analytical functions, is more practical. In this case, computation is also necessary for a large number of configurations, i.e., combinations of molecular orientations and separations.

1.2.5.2

Atom–Atom Potential Method

Molecules are generally nonspherical. Any model of intermolecular interaction between them should reflect this anisotropy. Since the anisotropy of molecular shape is certainly based on atomic arrangements, it is naïve to assume that intermolecular interaction is expressed as a sum of interatomic interactions. This way of presenting intermolecular interaction is called the atom–atom potential method. It is, however, rather evident that the method has some problems for aromatic molecules that bear π-electrons spreading over many atoms. Indeed, a somewhat stronger interaction resulting in the stacking of planar aromatic molecules has been identified as the π–π interaction, though it is essentially the dispersion interaction. Note that atoms in a molecule are not conceptually straightforward and require theoretical development in quantum theory [8], in contrast to naïve intuition. In the atom–atom potential method, the intermolecular interaction between molecules 1 and 2 is expressed as V12 =



v(i, j)

(1.62)

(i, j)

where the summation is over atomic pairs, of which atom i is chosen from molecule 1 and atom j from molecule 2. The interatomic potential is usually assumed as a function of their distance ri j and atomic species. Taking asymptotic forms of attractive and repulsive parts of interaction into account, one of the most often assumed forms of the interatomic potential function is

1.2 Intermolecular Interaction

19



σ v(ri j ) = A ri j

12

 −B

σ ri j

6 ,

(1.63)

where σ is a characteristic distance of the potential. This is a special case of the interatomic potential known as the Lennard-Jones potential with the general form of  v(ri j ) = 4

σ ri j

p



σ − ri j

q  (1.64)

with p > q > 0. In the Lennard-Jones potential, σ has a meaning of the atomic radius, below which the interaction becomes repulsive (v(r ) ≥ 0 for r ≤ σ). The LennardJones potential with q and p is denoted as (q, p)–potential like (6,12)–potential. The other potential often used is  6  r  σ ij −b v(ri j ) = a exp − , σ ri j

(1.65)

which is called the Buckingham potential. It is noteworthy that the Lennard-Jones potential and the Buckingham potential are similar at a long distance but qualitatively different at unphysically short distances. The former diverges to positive infinity while the latter to negative infinity. Naïve minimization of crystal energy with the Buckingham type potentials may result in overlap of atoms, accordingly. Both empirical interatomic potential functions (Eqs. 1.63 and 1.65) contains three parameters to be determined through the fit with experimental data. Since the interactions are attractive at long distances, the minimization of energy will find a stable structure of crystals. Thus, crystal structures are usually primary input, yielding r0 , and ratio A/B or a/b. Other data such as lattice energy (estimated from enthalpies of sublimation), elastic properties, or frequencies of lattice vibration are necessary to fix the energy scale. Practically, in order to reduce the number of coefficients to be determined, the following combination rules are assumed for atomic species i and j, Ai j =



Aii A j j

(1.66)

Bii B j j

(1.67)

√ aii a j j  bi j = bii b j j .

(1.68)

Bi j =



or ai j =

(1.69)

for coefficients related to the energy scale following Eq. 1.57, and σi j =

1 (σii + σ j j ) 2

(1.70)

20

1 Molecules and Intermolecular Interactions

considering the meaning of σ as an atomic diameter. In some cases, it is necessary to take electrostatic interactions into account. This is undoubtedly the case for polar molecules. The electrostatic interactions are also divided into interatomic contributions by assigning partial charges to atoms. This way is adequate and more accessible than a sophisticated multipole expansion. The partial charges may be assigned based on either of the quantum-chemical calculation of an isolated molecule or kinds of fitting together with other coefficients. The former poses some difficulty in the transferability of potential functions. Anyway, potential functions with the contribution of (bare) charges, qi and q j , becomes  v(ri j ) = A or

σ ri j

12

 −B

σ ri j

6 +

1 qi q j 4π 0 ri j

 6  r  σ 1 qi q j ij −b + . v(ri j ) = a exp − σ ri j 4π 0 ri j

(1.71)

(1.72)

The attempt to determine interatomic potentials in this way was initiated by Kitaigorodstky [6], in the 1960s, assuming the Buckingham type of potentials for hydrocarbons and had continued and expanded to include contributions beyond the dispersion energy and repulsion coming from Pauli’s exclusion principle. Through its history of the atom–atom potential method, the method has been successfully described aggregation structure (primarily of crystals) and the dynamics of molecules inside such structures. The success implies that not only the energy (and structure) but also its derivatives (up to the second-order) are well approximated by the sum of interatomic potential functions, at least, around the equilibrium structure with the minimum energy. Interatomic potential functions with transferability have been proposed for elements composing organic molecules such as carbon, hydrogen, oxygen, nitrogen, sulfur, and halogens. Detailed description and discussion can be found in a monograph co-authored by the pioneer of this method [7]. The adoption of isotropic potential functions can express the anisotropy of interatomic interaction arising from the anisotropic molecular shape. However, it cannot represent the orientational dependence and “valence” saturation of some interactions between specific atomic pairs such as hydrogen bonds between hydrogen and electronegative (like oxygen) atoms. This fact is not severe for studying crystal properties but would be for searching stable structures (crystal structure or liquid structure) or molecular dynamics with significant fluctuation (in liquid). Note that the resultant potentials determined through a fit to experimental data are effective ones and dependent on temperatures at which the input data were taken. In turn, the effect of temperature may be discussed if this aspect is fully utilized. To this purpose, a fit of lattice vibration frequencies is necessary and promising because the number of input data on a single compound can be large enough. Once atom–atom potential functions have been established, it seems natural to assume their utility not only for intermolecular interaction. Indeed, the interatomic potential functions are also utilized in studying the conformations of flexible

1.2 Intermolecular Interaction

21

molecules. In such cases, however, it is reminded that other contributions may intrinsically exist in the related potential. For example, the torsional (twisting) potential of a biphenyl molecule (C6 H5 –C6 H5 ) is determined by not only the interactions between non-bonded atoms expressed by the atom—atom potential (mainly of the repulsion between ortho hydrogen atoms) but also the conjugation of π-electrons through the central C–C bond.

1.2.5.3

Computer Learning

Both ways described in the previous two sections are based on theories even though their levels of reliability may differ. Reflecting the significant improvement in computational power and the progress in information science (for artificial intelligence, i.e., AI), recently, an attempt to find interatomic potentials capable of reproducing input data (such as crystal structures at various temperatures for many compounds) has been reported [9]. The plots of resultant interatomic potentials against interatomic separation are different from those proposed in other ways, such as the LennardJones and Buckingham-type potentials. Even a maximum can be recognized. Their practical applicability for molecular materials is to be assessed in the future.

1.2.6 Other Interactions 1.2.6.1

Hydrogen Bond

Hydrogen bond (H-bond) is described as “a form of association between an electronegative atom and a hydrogen atom attached to a second, relatively electronegative atom” [10]. The “definition” implies that the hydrogen bond is strongly related to electrostatic interaction. This relationship is undoubtedly the case. However, the saturation in the number of the H-bond, suggests a partial character of valence bonds. Depending on the separation between the electronegative atoms, the potential energy curve takes a single-minimum or double-well form. Since the hydrogen nucleus is the smallest in mass, the so-called quantum effect is most significant in H-bonds [11, 12].5 Indeed, the substitution of hydrogen with deuteron results in a shift of location closer to the nearby electronegative atom. Notable changes in properties of phase transitions happen when the mechanism involves the dynamics or disordering of hydrogen atoms [14, 15]. There exist compounds, the deuteration of which induces a new phase transition [16, 17]. Conversely, the deuteration is useful in clarifying the mechanism of structural phase transitions. The strength (the energy gain) ranges from ca. a few to more than a hundred kJ mol−1 , which are in between those of intermolecular dispersion interaction and (nor5 The

situation is the same in methanes, in which the phase relation is significantly altered by deuteration [13], though the molecular symmetry also changes upon partial deuterations.

22

1 Molecules and Intermolecular Interactions

mal) chemical bonds. In this respect, the hydrogen bond often plays significant effects on the aggregation and properties of molecular systems. Anomalous properties of ice (such as relatively high melting temperature and the decrease in volume upon melting) are understood through considering the effect of the hydrogen bonds between molecules [18]. The formation and maintenance of DNA duplex via hydrogen bonds between conjugate pairs of bases is another striking example to demonstrate their importance. Many monographs are available on the hydrogen bond [19]. Readers are referred to them to learn more.

1.2.6.2

Entropic Interaction

Thermodynamics says that the equilibrium of any system is characterized by the minimum of the Gibbs energy, which is given by G = H − T S,

(1.73)

where G, H , T and S are Gibbs energy, enthalpy, thermodynamic temperature and entropy, respectively. The statement implies that the enthalpic term (the first term) and the entropic term (the second term) can be the source of effective interaction. This kind of interaction is often termed as entropic interaction. The entropic interaction is a weak interaction comparable to thermal energy (≈ kB T ). The entropic interaction is generally non-additive owing to its mechanism. It is often the case that a naïve two-body treatment fails. Although the entropic interaction is often discussed in multicomponent systems such as solution, the crystallization of hard spheres (known as Alder transition [20]) can be understood as a result of the entropic interaction in a neat system. By analogy, micro phase-separation in an ensemble of “amphiphilic” molecules may be understood in terms of the entropic interaction. One of the well-known examples of entropic interaction is the depletion interaction. In a fluidic system such as a solution containing two types of particles with a notable difference in size, an effective attraction works between large particles at a short distance. If their distance is smaller than the size of small particles, the small particles cannot enter between the large particles resulting in the asymmetry in the pressure that the small particles exert on the large particle. Since the asymmetry originates in the depletion of the small particles in between, the effect is called depletion effect or depletion interaction [21]. Another important interaction is the so-called Helfrich interaction [22], which is the effective interaction between not molecular entities but membranes. The Helfrich interaction is always repulsive and arises from the entropic penalty of membranes that independently fluctuate if they are close to each other. The interaction is a crucial component to govern the intermembrane distance in lamellar phases of lyotropic liquid crystals. It is noted that the apparent electrostatic interaction between charged membranes is, in reality, of entropic origin. The bare charges of the membranes are completely

1.2 Intermolecular Interaction

23

screened by counterions, which distribute asymmetrically in two sides of membranes, resulting in the unbalance of osmotic pressures. The effective interaction depends on the ionic distribution [23]. 1.2.6.3

Solvatophobic Effect

Solvatophobic effects, of which the representative is the hydrophobic effect, are often assumed as a source of some apparent interaction. Micro phase-separation, causing the formation of various structures in lyotropic liquid crystals, is a notable example. Although the effect is usually related to entropic effects, it can come from enthalpic (energetic) effects. The resultant difference in the affinity to the solvent works as the driving force to the equilibrium distribution of their orientation of chemical species (molecules or molecular ions). Roughly speaking, the effect appears as a daily experience that ions and polar molecules favor in polar solvents such as water and vice versa. In this hydrophobic effect, the breakage/formation of networks of hydrogen bonds has been believed to be crucially important. General degrees of effects of ions are organized as the so-called Hofmeister series [24].

1.3 What the Interaction Brings 1.3.1 Equation of State of Gas If the molecular size and intermolecular interaction are negligible, the equation of state of classical gas is given by that of the ideal gas (complete gas),

or

pV = n RT = N kB T

(1.74)

p =Z =1 ρkB T

(1.75)

with N being the number of molecules and ρ = N /V . Z is often called compressibility factor. Experiments have shown that Eq. 1.74 does not hold exactly though it is a good approximation at high temperatures. The equation of state of real gases is often expressed like

or

p = 1 + B(T )ρ + C(T )ρ2 + D(T )ρ3 + · · · ρkB T

(1.76)

p = 1 + B p (T ) p + C p (T ) p 2 + D p (T ) p 3 + · · · ρkB T

(1.77)

using temperature-dependent coefficients, B(T ), etc. These series are called virial expansions. Coefficients are called virial coefficients, accordingly. The name “expan-

24

1 Molecules and Intermolecular Interactions

sion” here is based on the fact that Eq. 1.74 holds in the limit of ρ → +0 and p → +0. Terms from the second in the right hand sides can thus be regarded as successive corrections arising from the non-ideality. The temperature where the lowest-order non-ideality vanishes, i.e., B(TB ) = B p (TB ) = 0, is known as the Boyle temperature, TB . A systematic series expansion of the compressibility factor for non-ideal gas consisting of spherical molecules has been derived assuming that the intermolecular interaction is of two-body interaction [25].6 First, the following function (the Mayer f function) reflecting the two-body interaction u i j between ith and jth molecules is defined:  ui j − 1. (1.78) f i j = exp − kB T The following function is further defined βk =

1 k!

 



f i j d r 2 · · · d r k;1 ,

(1.79)

k+1≥ j>i≥1

where the sum runs over every irreducible graph7 containing (k + 1) vertexes, connection between which corresponds to f i j . Note that this βk is obtained after integrating over (k + 1) molecules. Using this βk , the compressibility factor is given as ∞  k p =1− βk ρk . (1.80) ρkB T k + 1 k=1 Since the forms are exactly the same between Eqs. 1.76 and 1.80, the following correspondence is evident:  1 1 (1.81) f 12 d r 2 B(T ) = − β1 = − 2 2  2 1 C(T ) = − β2 = − (1.82) f 12 f 23 f 31 d r 2 d r 3 3 3 3 D(T ) = − β3 4 1 =− (1.83) d r 2d r 3d r 4 8 (3 f 12 f 23 f 34 f 41 + 6 f 12 f 23 f 34 f 41 f 13 + f 12 f 23 f 34 f 41 f 13 f 24 ) ···

6 Full

description of the cluster expansion is left for the literature [26]. term “irreducible graph” follows the convention in the formulation of statistical physics for many-body systems, which is beyond the level of this book. It is essential here that the graph corresponding to the two-body interaction is irreducible and survives accordingly.

7 The

1.3 What the Interaction Brings

25

The correspondence indicates that the virial expansion is a successive expansion in terms of contributions of molecular clusters. Even if the molecule is non-spherical. the intermolecular interaction can be averaged out spherically for its angular dependence upon discussing the properties of gas, since the intermolecular distance is large, For example, the largest contribution of the interaction can be estimated by    1 f 12 d r 2 , β1 = 2

(1.84)

where the · · ·  means the orientational average.

1.3.2 Liquefaction Along with the systematic expansion of the equation of state of non-ideal gas discussed in the previous section, there exists a famous equation. It was derived by applying corrections for the presence of both attraction and repulsion from the equation of the ideal gas, pV = N kB T , by van der Waals [27]. The equation is written as   2  N p+a (1.85) (V − N b) = N kB T, V where b stands for a molecular volume and a for a constant characterizing a two-body attraction.8 Depending on temperature T , the relation between pressure and volume changes, as shown in Fig. 1.6. At high temperatures, the dependence is monotonous and resembles the equation of state of the ideal gas. On the other hand, at low enough temperature, the dependence is transversally sigmoidal, giving three volumes at the same value of pressure. Two outer states are thermodynamically healthy, while the positive slope [(∂ p/∂V )T > 0] of the middle point means an unstable state. It is reasonable to assign the two outer states to the liquid (L) with a smaller volume and the gas (G) with a larger volume, ignoring the middle state. The pressure of the coexistence of two phases at a temperature is determined by the equality of their chemical potentials (molar Gibbs energy), μL = μG . By (∂G/∂ p)T = V , the condition is expressed as  G V dp = 0. (1.86) L

The integration is along the equation of states between coexisting liquid and gas specified by L and G, respectively. This condition is known as Maxwell’s rule. Figure 1.6 indicates that the difference in volume decreases with increasing temperature and that there exists a unique temperature, above which the coexistence of 8 In

terms of Sect. 1.3.1, a and b is related as B(T ) = (b − a/kB T ) and C(T ) = b2 .

26

1 Molecules and Intermolecular Interactions

Fig. 1.6 Pressure as a function of volume described by van der Waals equation of states in terms of reduced variables. The upper position of the curve corresponds to the higher temperature

2

p/p

c

vdW equation

1

0

0

1

2

3 V/V

4

5

c

gas and liquid cannot be realized. The temperature is known as a (liquid-gas) critical temperature. In reality, not only the temperature but also pressure and volume are fixed. Thus, the state is called a critical point. Figure 1.6 indicates that the following should hold at the critical points: (∂ p/∂V )T = 0 and (∂ 2 p/∂V 2 )T = 0. Quantities at the critical point are thus obtained as Vc = 3N b 1 a pc = 27 b2 8 a . RTc = 27 b

(1.87) (1.88) (1.89)

Then, by introducing the reduced variables, φR = V /Vc , πR = p/ pc , and θR = T /Tc , the van der Waals equation of state is simply written as 

3 πR + 2 φR



1 φR − 3

=

8 θR . 3

(1.90)

This simple form implies that the states of matter are “common” if expressed in terms of a distance from the critical point. Although the numerical values of coefficients (such as 38 ) are different from Eq. 1.90 reflecting the approximate nature of van der Waals’s treatment, it has been shown that a unified description holds well for some real systems. The correspondence between states in different materials implied by Eq. 1.90 is known as the principle of corresponding states [28], which is one of the best examples of a unified understanding of matter, though its validity is limited to simple molecular systems at most. The van der Waals equation of state does not say anything about crystallization but clearly indicates that the intermolecular attraction induces liquefaction.

1.3 What the Interaction Brings

27

1.3.3 Crystallization In contrast to the liquefaction by the attractive part of the intermolecular interaction, crystallization is not easy to understand. Suppose spherical molecules, between which the interaction works only for neighbors. Since the intermolecular interaction energy generally has a minimum, as shown in Fig. 1.5, molecular arrangements that guarantee minimum energy for all neighboring pairs correspond to the minimum energy of the whole system. This arrangement is the most stable at the absolute zero. The closest packing structures certainly offer a suitable situation for all particles. Note, however, that the so-called closest packing covers not only crystalline structures but also other disordered structures. Two simplest structures with the crystalline order consist of the closely packed layers, which is just the triangle lattice. The two crystalline structures are different only in relative stacking of the layers. The stacking in the face-centered cubic (FCC) structure is a repetition of three layers ABC (...ABCABC...)9 while that in the hexagonal closest packing (HCP) structure is of ...ABAB... The possibility of such descriptions of two crystalline structures means any aperiodic stacking can be a closest packing structure if neighboring layers are of different kinds. This simple example suggests that the origin of the periodicity is not merely the energetic effect but also from other sources. Even apart from the non-uniqueness of the aggregation structure, the crystallization process also put forward a problem. Consider a stable structure of clusters consisting of a small number of particles. With the increase of the number of particles (n) form 3 to 5, the structures of most stable clusters (with the minimum energy) changes from the equilateral triangle (n = 3), the regular tetrahedron (4) to the trigonal bipyramid (5). These are partial structures of the closest packing structures. For n = 6, regular octahedron has the minimum energy due to the formation of the equilateral triangles for the shortest bonds between neighboring particles. However, the situation changes for n = 7. Although the cluster shown in Fig. 1.7a has the same structure as in the FCC closest packing, it has a higher energy of −15.10 than the pentagonal bipyramid (b) with −15.94 for the Lennard-Jones (6,12)–potential (Eq. 1.63). Since the five-fold symmetry of the cluster (b) is incompatible with any translational periodicities, the growth of clusters cannot result in translationally symmetric structures (crystalline order). This also suggests that the origin of the crystalline orders is to be attributed to other sources. The indispensable source of the crystalline order was revealed as the entropic effect through computer simulations on classical particles by Alder and Wainwright [20]. They performed the simulations assuming only repulsive interactions (rigid spheres) while confining particles inside a box. They found that the equation of states consists of two branches, which correspond to fluid and crystal. The periodicity of the crystalline phase was clearly demonstrated by snapshots of trajectories of particles. Since the simulations are classical, the energy of the system is always the classical value ( 23 kB T per particle) of kinetic energy because of the prohibition of overlap of particles, only the source that can bring about the crystalline order is the so-called 9 The FCC structure can be represented as alternating stacks of square plane lattices as ...A’B’A’B’...

28

1 Molecules and Intermolecular Interactions

Fig. 1.7 Clusters of 7 spherical particles. Minimized energies for the Lennard-Jones (6,12)– potential are −15.10 for (a) and −15.94 for (b)

entropy term (−T S) in the free energy of the system. In this sense, the crystallization is driven by the entropy. This type of crystallization is called Alder transition. The simulation by Alder and Wainwright [20] resembles physical systems consisting of spherical molecules such as rare gasses. The attractive part of intermolecular interaction only plays as the role of the box to keep the volume. However, their results can be interpreted as indicating the vital role of molecular shape. If the intermolecular interaction depends on the mutual orientation of interacting molecules, the preference for the minimum energy may cause crystallization. Indeed, the crystal structure of the ice is the structure where hydrogen bonds are formed in a just enough way. In this sense, the discussion on aggregation structure based on intermolecular interaction representing molecular shape is a good starting point for crystal engineering of molecular crystals (Sect. 4.4.3).

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

D. Clark, H.M. Powell, A.F. Wells, J. Chem. Soc., 642–645 (1942) M.J. Taylor, L.A. Woodward, J. Chem. Soc., 4670–4672 (1963) B. Meyer, Chem. Rev. 76, 367–388 (1976) Y. Yamamura, H. Saitoh, M. Sumita, K. Saito, J. Phys.: Condens. Matter 19, 176219 (2007) F. London, Trans. Faraday Soc. 33, 8–26 (1937) A.I. Kitaigorodsky, Dokl. Akad. Nauk SSSR 137, 116–119 (1961) A.J. Pertsin, A.I. Kitaigorodsky, The Atom-Atom Potential Method (Springer, Berlin, 1987) R. Bader, Chem. Rev. 91, 893–928 (1991) D.W.M. Hofmann, L.N. Kuleshova, Chem. Phys. Lett. 699, 115–124 (2018) IUPAC. Compendium of Chemical Terminology, 2nd edn. (the “Gold Boo”), Compiled by A. D. McNaught and A. Wilkinson (Blackwell Scientific Publications, Oxford, 1997) E. Matsushita, T. Matsubara, Prog. Theor. Phys. 67, 1–19 (1982) K. Ando, Phys. Rev. B 72, 172104 (2005) T. Yamamoto, Y. Kataoka, J. Chem. Phys. 48, 3199–3216 (1968) P. Prelovšek, R. Blinc, J. Phys. C 15, L985–L990 (1982) E. Matsushita, T. Matsubara, Prog. Theor. Phys. 71, 235–245 (1984) K. Gesi, J. Phys. Soc. Jpn. 48, 886–889 (1980) Y. Moritomo, Y. Tokura, T. Mochida, A. Izuoka, T. Sugawara, J. Phys. Soc. Jpn. 64, 1892–1895 (1995)

References

29

18. D. Eisenberg, W. Kauzmann, The Structure and Properties of Water (Oxford University Press, New York, 2005) 19. Y. Maréchal, The Hydrogen Bond and the Water Molecule (Elsevier, Amsterdam, 2007) 20. J. Alder, T.E. Wainwright, J. Chem. Phys. 27, 1208–1209 (1957) 21. S. Asakura, F. Oosawa, J. Chem. Phys. 22, 1255–1256 (1954) 22. W. Helfrich, Z. Naturforsch. C 30, 841–842 (1975) 23. M. Hishida, Y. Nomura, R. Akiyama, Y. Yamamura, K. Saito, Phys. Rev. E 96, 040601(R) (2017) 24. F. Hofmeister, Arch. Exp. Pathol. Pharmacol. 24, 247–260 (1888) 25. J.E. Mayer, E. Montroll, J. Chem. Phys. 9, 2–16 (1941) 26. J.E. Mayer, M.G. Mayer, Statistical Mechanics (Wiley, Hoboken, 1940) 27. J. D. Van der Waals, Over de continuiteit van den gas- en vloeistoftoestand (On the Continuity of the Gaseous and Liquid States in English) (thesis submitted to Universiteit Leiden) (1873) 28. E.A. Guggenheim, J. Chem. Phys. 13, 253–261 (1945)

Chapter 2

Phase Transitions

2.1 Background 2.1.1 Thermodynamic Aspects The fusion of a crystal is, together with boiling/condensation, a representative of phenomena known as phase transitions. It is useful to summarize the terminology and properties of phase transitions in (macroscopic) thermodynamics. Note that the following descriptions assume thermodynamic equilibrium.1 A phase is not only a thermodynamic concept but also a reality. A phase must be uniform chemically and physically if viewed macroscopically, i.e., averaged over a volume that contains a sufficiently large number of molecules. All intensive variables (not only temperature, pressure, and density but also composition) are uniform in a single phase. A pure substance is assumed for a while. Generally, a substance has plural phases like gas, liquid, and solid, which are representative three phases of matter. Although these three phases are easily distinguished from each other under normal conditions, this is not generally true, as indicated by the gas-liquid critical point, where gas and liquid merge to a single state. Thermodynamics says that a phase is characterized by the Gibbs energy (G), a thermodynamic potential relevant to the conditions where temperature (T ) and pressure ( p) are independent variables. Now, we write the molar Gibbs energy of phase H as μH (T, p). Then, a surface in a (T, p, μ) space can be identified as phase H. When a substance has two phases, H and L (Fig. 2.1), two surfaces μH (T, p) and μL (T, p) generally cross with a crossing line. Since the Gibbs energy must be minimum at thermodynamic equilibrium, the stable phase is interchanged upon crossing the crossing line. Thus, the substance must change its form between phase H

1 We

here assume a usual meaning of “equilibrium.” It is seemingly stable and steady against time of our daily time scale. Some class of non-equilibrium and “relaxing” states is treated in Chap. 8.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 K. Saito, Chemical Physics of Molecular Condensed Matter, Lecture Notes in Chemistry 104, https://doi.org/10.1007/978-981-15-9023-8_2

31

32

2 Phase Transitions

µ

H L p T Fig. 2.1 Phase relation of two phases H (pink surface) and L (green surface) in (T, p, μ) space with Δtrs s > 0 (positive molar entropy of transition) and Δtrs v < 0 (negative molar volume of transition). In this case, H phase is a high-temperature phase and a high-pressure phase with respect to L phase because a phase having lower molar Gibbs energy, μ, is more stable at high temperatures and high pressures

and phase L upon the crossing. This change is a phase transition. On the crossing line, two phases have the same molar Gibbs energy. Both phases have equivalent stability. Namely, on the crossing line, two phases can coexist. The crossing line projected on the (T, p)-plane is identified as a coexistence line of the two phases, accordingly. The projected curve also serves as a phase boundary on the plane. Interestingly, when a substance has three phases, any combination of two of them always determines a coexistence line. Three determined coexistence lines then meet at a point known as the triple point, where three phases can coexist. The expression of the coexistence line is derived based on the fact that two phases have the same molar Gibbs energies on the line: μL (T, p) = μH (T, p).

(2.1)

Because of (∂μ/∂T ) p = −s (molar entropy) and (∂μ/∂ p)T = v (molar volume), we obtain, up to the first-order, − sL (T0 , p0 )δT + vL (T0 , p0 )δ p = −sH (T0 , p0 )δT + vH (T0 , p0 )δ p,

(2.2)

where (T0 , p0 ) is a state where two phases coexist. Thus, we have the so-called Clapeyron relation for the coexistence line, dT vH (T, p) − vL (T, p) = dp sH (T, p) − sL (T, p) Δtrs v(T, p) = . Δtrs s(T, p)

(2.3)

2.1 Background

33

Here, Δtrs s and Δtrs v are molar entropy and volume of transition, respectively. Although Eq. 2.3 is not affected by virtue of its form, the selection of their signs have a tradition: Δtrs s > 0 for the phase transition driven by a temperature change, i.e., phase H is a high-temperature phase, and Δtrs v < 0 for the phase transition driven by a pressure change, i.e., phase H is a high-pressure phase. Note that the correspondence of the signs and relative stabilities of two phases is related to the self-consistency of thermodynamics, which requests the absolute stability of thermodynamic systems.2 Phase transitions treated above accompany discontinuities in entropy and volume, first derivatives of the Gibbs energy because the crossing of surfaces accompanies them. Paying attention to the discontinuities, phase transitions of this type are called first-order phase transitions. It is also said that the phase transition is of firstorder. There exist phase transitions that do not accompany such discontinuities. Such phase transitions are classified, after Ehrenfest [1], according to the lowest order of derivative of thermodynamic potential that exhibits a discontinuity or a divergence. For example, a phase transition is of second-order if the Gibbs energy and its first derivatives, entropy s and volume v, are continuous but one of its second derivatives, isobaric heat capacity   ∂s , (2.4) cp = T ∂T p isothermal compressibility κT = −

1 v



∂v ∂p

 ,

(2.5)

T

and volume expansivity  ∂v ∂T p   1 ∂s =− , v ∂p T

β=

1 v



(2.6) (2.7)

is discontinuous or diverging at a transition point.3 A transition with the order higher than the second is theoretically possible, but the precise order is practically impossible to determine. For phase transitions with its order higher than the second, the understanding by assuming the crossing of two μ(T, p) surfaces is impossible. That is, in such cases, one phase does not extend beyond the phase boundary. This point is experimentally important because the observation of hysteresis such as supercooling in equilibrium can be the evidence of the first-order nature of the phase transition. 2 A positive Δ v trs

is possible for a temperature-driven transition (Δtrs s > 0) as commonly observed in the boiling of any liquids and fusion of most crystalline substances. The similar applies for pressure-driven cases (Δtrs v < 0 and Δtrs s < 0). 3 The last identity for the expansivity is due to the so-called Maxwell’s relations.

34

2 Phase Transitions

There exists another classification scheme of phase transitions. Landau [2] divided phase transitions into two kinds: first kind and second kind, the latter of which is a continuous transition between closely related phases. By expanding the difference in thermodynamic potentials between them in terms of the so-called order parameter, which is finite in one phase but null in the other phase, he derived some general conclusions for phase transitions from the thermodynamic point of view [2, 3]. All discontinuous (first-order after Ehrenfest) phase transition is classified as of the first kind. Note that some phase transitions of the first kind are discussed within Landau’s thermodynamic phenomenology, as will be discussed in Sect. 2.2. Note that Fig. 2.1 assumes that two phases are “reasonably” stable at any combinations of (T, p). The true equilibrium corresponds to the state that the compound is solely in the phase of a lower molar Gibbs energy. This phase is the (thermodynamically) stable phase. The other phase of a higher molar Gibbs energy is called a metastable phase. It is easy to identify which of two phases are stable and metastable if a phase transition (from one to another) occurs. This is, however, not always the case. A textbook example is the case of elemental carbon. Around the ambient condition, its stable phase is graphite, and all of diamond, any fullerenes, and even graphene are metastable. However, we never imagine the spontaneous transition of the diamond to graphite! This example demonstrates the fact that we cannot know the metastability only by simple observations. Figure 2.1 also implies another point, which is practically essential. Since both phases are seemingly stable around the phase boundary, overshooting such as superheating and supercooling can take place in this first-order transition. It is difficult to construct the logic that guarantees the loss of the stability of the phase on the phase boundary. We thus conclude that the possibility of the overshooting is intrinsic to any first-order transitions. In turn, the observation of overshooting is regarded as the most decisive way to assure the first-order nature of phase transitions, while assuming that the experiment correctly watches equilibrium states. In contrast, overshooting is impossible for higher-order transitions.

2.1.2 Entropy and Boltzmann’s Principle Entropy plays a central role in the thermodynamic study of phase transitions, as partly exemplified in Eq. 2.3. In classical thermodynamics, entropy is a state function that indicates the possibility of a change of thermodynamic state of a system (process) by mechanical means. Namely, the process is impossible if the entropy of the system is smaller after it than that before it, whereas it is possible if the opposite holds. This description is a form of the second law of thermodynamics. Boltzmann constructed the statistical mechanics to reproduce the classical thermodynamics relying solely on the assumption on entropy S, S = kB ln W,

(2.8)

2.1 Background

35

where kB is a universal constant currently known as the Boltzmann constant and W the number of microscopic states. Equation 2.8 is known as Boltzmann’s principle or Boltzmann’s relation. Note that its left-hand side is a macroscopic quantity while the right-hand side is calculated using a microscopic quantity. This simple equation bridges the macroscopic and microscopic worlds. Although the physical basis of Boltzmann’s principle (Eq. 2.8) remains under extensive study, there is no doubt about its validity (adequacy) for macroscopic systems. Then, it offers a novel way to determine the number of microscopic states. For example, suppose a system consisting of particles, each of which has two internal states. We distinguish two states by assigning σ = ±1. When the interaction among them is of two-particle type, the interaction may be expressed as −Ji j σi σ j . The total energy is given by  − Ji j σi σ j . (2.9) i

j=i

This model assigning two-state to each particle is known as the Ising model [4, 5], which will be roughly analyzed in Sect. 6.2.1. The model exhibits a phase transition under some conditions but does not under other conditions. Therefore, its thermodynamic properties show a wide variety. However, according to Boltzmann’s principle, the entropy of the system is N kB ln 2 at a sufficiently high temperature irrespective of the setting of {Ji j }. More precisely, S ≈ N kB ln 2 at T  |Ji j |max . It is astonishing. Since the entropy of transition is expressed in terms of the number of microscopic states of two phases, WH and WL , as Δtrs s = kB ln WH − kB ln WL WH = kB ln , WL

(2.10)

we can determine the change in the number of microscopic states occurring in the phase transition without referring microscopic details of the substance.4 If the transition is of purely first-order without any excess heat capacities on both (upper and lower) sides of the transition temperature, Δtrs s can be regarded as the excess entropy involved in the transition (Δex s). On the other hand, even in the case of continuous transitions such as one exhibited by the two-dimensional Ising model [6], we need to estimate Δex s for analyzing the transition mechanism. Since the classical thermodynamics applies to systems with any complexity, Boltzmann’s principle offers a novel strategy of microscopic study of complex systems, including molecular systems, if the entropy of real systems is experimentally available. Its experimental determination is performed, after Clausius, through dS = 4 Although

δq , T

(2.11)

Eq. 2.10 always holds, it depends on problems whether such an interpretation is scientifically fruitful or not. For example, the entropy of fusion is generally harder to rationalize than the entropy of spin systems like the Ising model.

36

2 Phase Transitions

where δq is infinitesimal heat absorbed by the system. By performing the integration along an idealized quasi-static path from state A to state B, the change in entropy is given by  B δq ΔS(A → B) = . (2.12) A T In practice, the integration is done using heat capacity C as  S(T2 ) − S(T1 ) =

T2 T1

C dT T

(2.13)

as far as the heat capacity is normally defined. On the other hand, at a first-order phase transition point, the entropy of a substance exhibits a jump by the entropy of transition. According to the first law of thermodynamics, we can write Δtrs s =

Δtrs h Ttrs

(2.14)

with Δtrs h being the molar enthalpy of transition, irrespective of the deviation of the experimental process from the quasistatic one. Finally, a comment is in order on why entropy is more convenient than the energy (or enthalpy) for analyses. Statistical mechanics enables calculations of thermodynamic quantities, not only entropy but also energy, based on molecular details. However, their calculations are only for thermal parts, which contribute to thermodynamic quantities upon temperature variation, but for the non-thermal part. For example, consider a harmonic oscillator with an angular frequency ω. As is well known, its energy levels are expressed as ε = ω(n + 21 ) with n = 0, 1, 2, . . . If we intend to analyze the energy, we need to take into account of the zero-point energy, the information of which is scarcely available from experimental thermodynamic quantities.

2.1.3 Interface and Phase Growth As described in Sect. 2.1.1, the Gibbs energy determines the most stable phase in the equilibrium. To realize the equilibrium state at varied conditions, however, a phase transition must occur. During the transition process, the interface between the two phases unavoidably emerges. The creation of the interface (surface) always consumes some work. The work necessary to create the surface of a unit area is the surface free energy. It is a simple parameter for isotropic liquids and is equivalent to the surface tension, γ. The Gibbs energy of a spherical droplet with a radius r is written as G=

4π 3 r g + 4πr 2 γ, 3

(2.15)

2.1 Background

37

where g is the Gibbs energy density (per unit volume). The relative magnitude of the second term vanishes as r −1 in large r . This conclusion is true as long as we assume the isotropic enlargement for any shape of the system.5 Thus, the surface energy is negligible in bulk substances. Since the lower Gibbs energy is more stable thermodynamically, a liquid droplet tends to minimize its surface area because of γ > 0. Its equilibrium shape is a sphere if we ignore the effect of gravity. On the other hand, the surface free energy depends on the created surface for crystals. We can thus imagine an equilibrium shape of crystalline grain, such as a cube for rock salt. In order for a phase transition to proceed, the effect of the interface is significant. Here, we see the existence of the minimal size of a new phase domain for proceeding the transition. Imagine a compound that undergoes a phase transition between two isotropic liquid phases, for simplicity. The difference in the Gibbs energy density (assigned to the new phase) is denoted by Δg(T ), and the interfacial tension between the two phases is by γ(T ). If a droplet of a new phase with a radius r exists, the difference in the Gibbs energy, denoted as ΔG(T ), relative to the uniform state of the mother phase is given by ΔG(T ) =

4 3 πr Δg(T ) + 4πr 2 γ(T ). 3

(2.16)

The condition for the droplet to grow is dΔG(T ) < 0. dr Thus, we have r > rc (T ) = −

2γ(T ) . Δg(T )

(2.17)

(2.18)

That is, the negative Δg(T ), which happens below the equilibrium transition point Ttrs , is insufficient to start the transition. Only if a droplet larger than the minimum size rc is formed as a result of the thermal fluctuation, the phase transition starts. The barrier hight necessary for thermal activation, Δ∗ G(T ), is calculated as ΔG(T ) at r = rc , 16π γ(T )3 Δ∗ G(T ) = . (2.19) 3 [Δg(T )]2 Since the departure from the transition temperature brings more negative Δg(T ), the critical size of the drop and the activation barrier hight to form the initial droplet decrease with the increase in |T − Ttrs |. Although the above analysis assumes two liquid phases, the presence of the minimum size of a new phase domain for growth is true for any first-order phase transitions. The formation of the domains to grow is termed as the nucleation. In the vicinity of Ttrs , the thermal nucleation is more 5 The

so-called “thermodynamic limit” implies the condition.

38

2 Phase Transitions

accessible at a higher side (Ttrs + ΔT ) than a lower side (Ttrs − ΔT ) because the necessary barrier hight (2.20) Δg(Ttrs ± ΔT ) = ΔT Δtrs s is mostly equal. This circumstance partly explains why the supercooling is more often than the superheating. It is noteworthy that both ΔG(T ) and γ(T ) depend not only on the mother phase but also on the new phase. If the mother phase has plural candidates of a resulting new phase, the minimal size of a critical domain differs depending on the resulting phase. There is no reason why the size is smaller for the most stable phase than others. Thus, the preference for a metastable phase can occur. The nucleation is a factor affecting kinetics and even the occurrence itself of phase transitions. However, there are cases that the nucleation does not play any role. If the transition is a consequence of the instability of a phase, i.e., the thermodynamic stability is lost at the transition point, the transition proceeds without any barrier to overcome. The so-called spinodal decomposition, a kind of phase separation, is a widely observed example. The nucleation is not the only factor to affect the time evolution of the phase transition (phase transition kinetics). If the new phase is anisotropic as a crystalline phase, the phase transition will proceed anisotropically, accordingly. For example, if a molecule is rodlike, the molecule coming from the isotropic liquid prefers to adhere laterally to the surface for the growth, because the contact area mostly controls the microscopic enthalpy gain. The resulting crystal would preferably grow into a disk. Thus, depending on the system, the growing domain will have a characteristic “dimension” (d = 1, 2, 3). In the case of the disk, d = 2 results. Similarly, highly anisotropic growth (d = 1) and isotropic growth (d = 3) lead to rodlike and spherical domains. It is well known [7–11] that the time (t) evolution of the transformed fraction (x) at a constant temperature is well described by   x = 1 − exp −Z t n ,

(2.21)

where n is a parameter known as the Avrami parameter, and Z is another parameter more specific to the system. The Avrami parameter involves the information of the mechanism of the proceeding of the transition through n = h + kd.

(2.22)

Here, h distinguishes the mechanism of the nucleation and takes unity (h = 1) for the homogeneous nucleation. On the other hand, it vanishes (h = 0) for the nonhomogeneous nucleation, where the nucleus exists at t = 0, and no further nucleation occurs. If the linear velocity of the growth remains constant, we have k = 1.6 √

t, yielding k = 21 , if the diffusion of particles (molecules) dominates, as in recrystallizations from dilute solution.

6 The growth rate will be proportional to

2.2 Landau’s Phenomenology

39

2.2 Landau’s Phenomenology The treatments described in this section has been proposed first by L. D. Landau, and widely known as Landau’s thermodynamic phenomenology of phase transitions. The phenomenology does not treat microscopic details of substances but gives fruitful insights [2, 3] as described below. In some respects, it puts a reference framework for discussing the nature and properties of any phase transitions.

2.2.1 Power Expansion of Thermodynamic Potential Suppose a thermodynamic system is under the condition characterized by the set of intensive variables τ . A set of structural variables of the system is denoted by {r}. The thermodynamic potential of the system is then written as F(τ , {r}). By introducing the set {r0 } for the equilibrium state, we can write F(τ , {r}) = F(τ , {r0 }) + ΔF(τ , {r}).

(2.23)

According to the variational principle of thermodynamics, ΔF(τ , {r0 }) = 0.

(2.24)

Namely, if a parameter φ is assumed to reflect the set {r} concisely, ΔF(τ , {r}) may be expanded in a power series of (φ − φ0 (τ )) around φ0 (τ ), the equilibrium value of φ under the condition, as ΔF(τ , φ) =



an (τ )(φ − φ0 (τ ))n

(2.25)

n≥2

with a2 (τ ) > 0. Note that φ is not restricted to a scalar but can be vector or tensor. Also, we may consider a parameter independent of φ, say σ, for additional degrees of freedom. Since Eq. 2.23 is assumed the thermodynamic potential of the system, it must be consistent with the full symmetry of the system. This requirement poses some restrictions on the form of the expansion of ΔF(τ , {r0 }) (Eq. 2.25). This restriction appears as the selection of terms retained in the expansion. For example, if the sign of a scalar φ is not a matter concerning the physical nature of the phase, all odd terms should vanish. It is emphasized that the form of the expansion crucially depends on the selection of φ.

40

2 Phase Transitions

2.2.2 Order Parameter In some phase transitions, we can identify φ that is null (φ0 = 0) in one but finite (φ0 = 0) in another phase involved in the phase transition. This quantity is called the order parameter relevant to the transition. For example, the amplitude of a periodic density modulation (density wave) may serve as an order parameter for the appearance of a layered structure like a smectic liquid crystal. Upon the formation of the layer, the translational symmetry by arbitrary length is broken. In this sense, the phase with null amplitude (φ0 = 0) is more symmetric than the one with finite φ0 (= 0). In terms of the group-theoretical notions, an order parameter belongs to an irreducible representation of the symmetry of the more symmetric phase, and the expansion (Eq. 2.23) is totally symmetric. If the irreducible representation is higher than one dimensional, the expansion of thermodynamic potential is forced to contain multiple, yet equivalent, order parameters. For investigations describing possible orders, it is an easy task to identify the order parameter. The main difficulty lies in writing appropriate expansion of thermodynamic potential compatible with the symmetry of a system considered. In contrast, the identification itself is an essential step for understanding a real phase transition based on experimental observations, some of which are crucially important and others not so.

2.2.3 Simplest Case: Continuous Transition We first consider the case where the relevant order parameter is a scalar and belongs to a one-dimensional representation. The physical equivalence between two states with ±φ is also assumed (though two states are distinct). The previous example of the formation of a one-dimensional density wave is an example of this case. Other examples include the uniaxial alignment of magnetic spins or dipoles. In this setting, a continuous phase transition between two phases with φ0 = 0 [symmetric (S) phase] and φ0 = 0 [symmetry-broken (B) phase] is deduced. We assume that the change in temperature induces the transition, and the S phase is stable at T > T0 while the B phase is so at T < T0 . As far as we consider states with small φ, we can truncate the expansion at a few first terms. Note that the order of the last term should be even, and its coefficient must be positive to guarantee the thermodynamic stability of the symmetry-broken phase. In the present case, the expansion is ΔF(φ, τ ) = a2 φ2 + a4 φ4 . with a4 > 0 around T0 .

(2.26)

2.2 Landau’s Phenomenology

41

Fig. 2.2 Thermodynamic potential (Eq. 2.26) as a function of the order parameter φ at various temperatures. T0 is the transition temperature of a second-order transition

T = T0

T > T0

T < T0

Since Eq. 2.23 is regarded as thermodynamic potential as a function of φ, its minimum should characterize the thermodynamic equilibrium. That is,

with

∂ΔF(τ , φ) ∂ F(τ , φ) = =0 ∂φ ∂φ

(2.27)

∂ 2 ΔF(τ , φ) < 0. ∂φ2

(2.28)

Thus, the transition at T0 is described by setting a2 = α(T − T0 )

(2.29)

with α > 0, because the state with φ = 0 is a (local) minimum of ΔF(φ, τ ) for T > T0 while it is a local maximum for T < T0 . ΔF as a function of φ at various temperatures (T ≶ T0 ) are shown in Fig. 2.2. While φ0 = 0 for T ≥ T0 , φ0 for T < T0 is obtained through 0= giving

 ∂ΔF(τ , φ) = 2φ α(T − T0 ) + 2a4 φ2 , ∂φ 

φ0 (τ ) = ± −

α(T − T0 ) 2a4

(T < T0 ).

(2.30)

(2.31)

The appearance of two solutions with a difference only in their sign is a consequence of the equivalence of two states with ±φ. Equation 2.31 gives φ0 (τ ) = 0 at T = T0 .

42

2 Phase Transitions

This phase transition is continuous in φ0 with a transition temperature of T0 . Thermodynamic quantities exhibit anomalies at T0 . By inserting Eq. 2.31 into Eq. 2.26, the thermodynamic potential in the equilibrium at T is obtained as Δf (T ) = −

α2 (T − T0 )2 . 4a4

(T < T0 ).

(2.32)

Hereafter, we use f and other small letters for the free energy minimized regarding the order parameter(s) and corresponding quantities. Thus, excess entropy Δs(T ) and heat capacity Δc(T ) as functions of temperature (T < T0 ) are Δs(T ) = −

α2 ∂Δf (T ) = (T − T0 ) ∂T 2a4

(2.33)

α2 ∂Δs = T. ∂T 2a4

(2.34)

Δc(T ) = T

Although thermodynamic potential (Δf (T )) and entropy (Δs(T )) are continuous, heat capacity (Δc(T )) exhibits a discontinuous step at T0 by α2 T0 /2a4 (downward on heating) because Δc = 0 for T > T0 . Thus, the phase transition described by Eq. 2.26 is a second-order phase transition. Note that a similar result is obtained while taking the temperature dependence of a4 into account as far as it remains positive. Landau’s phenomenology predicts larger heat capacity for the B phases irrespective of its relative location to the S phase. Namely, the heat capacity exhibits a stepped anomaly upward on heating if the B phase is a high-temperature phase. Indeed, there exist such examples [12–14]. If the order parameter is (spontaneous) polarization, the linear response to the external electric field is analyzed in the following way. A linear term expressing the effect of external field h is added to thermodynamic potential as ΔF(φ, τ , h) = a2 φ2 + a4 φ4 − φh.

(2.35)

Even in the symmetric “phase,” φ is no longer zero but finite. The equation determining its magnitude is ∂ΔF(φ, τ , h) = 2a2 φ + 4a4 φ3 − h = 0. ∂φ

(2.36)

For the S phase, the smallness of φ allows us to neglect the third-order term. 2a2 φ − h = 0. Thus, the (small) polarization of the S phase under the field is given by

(2.37)

2.2 Landau’s Phenomenology

43

φ0 (h) =

h (T > T0 ). 2a2

(2.38)

The (electric) susceptibility is defined as χ=

1 ∂φ , 0 ∂h

(2.39)

where 0 is the dielectric constant of vacuum. Taking Eq. 2.29 into account, we obtain χ(T ) =

1 1 . (T > T0 ) 20 α(T − T0 )

(2.40)

This divergence of the susceptibility is known as the Currie-Weiss law. For the symmetry-broken phase (B phase), assuming a small h, we put φ(h) = φ0 + δφ(h) with φ0 given by Eq. 2.31. Subtraction of Eq. 2.30 from Eq. 2.36 yields, in the lowest order of δφ(h), 2a2 δφ(h) + 12a4 φ20 δφ(h) − h = 0

(2.41)

Thus, h 2a2 + 12a4 φ20 h =− . 4a2

δφ(h) =

(2.42)

This gives χ(T ) = −

1 1 . (T < T0 ) 40 α(T − T0 )

(2.43)

The dielectric susceptibility diverges in the B phase too upon approaching the transition temperature T0 . Not only the divergence but also the two-times difference in the prefactor have been observed experimentally in dielectrics. At T = T0 , the order parameter grows as a function of the field h, 

h φ0 = 4a4

1/3 .

(2.44)

2.2.4 Discontinuous Transition If different signs of φ around φ0 = 0 correspond to physically different states, the expansion of thermodynamic potential contains odd-order terms. The minimal case

44

2 Phase Transitions

Fig. 2.3 Thermodynamic potential (Eq. 2.45) as a function of the order parameter φ at various temperature. Ttrs is the transition temperature of a first-order transition

T = Ttrs

T > Ttrs

T < Ttrs

is ΔF(φ, τ ) = a2 φ2 − a3 φ3 + a4 φ4

(2.45)

with a3 > 0 and a4 > 0. A negative sign in Eq. 2.45 is always possible by an arbitrary definition of the sign of φ. Figure 2.3 schematically shows the temperature dependence of Eq. 2.45 for a2 = α(T − T0 ). At high temperature, Eq. 2.45 has a single minimum at φ = 0, corresponding to the S phase. With decreasing temperature, a local minimum at φ = 0 appears below 9a32 + T0 (2.46) TU = 32αa4 though its value is still higher than that at φ = 0. The local minimum corresponds to a metastable B phase. Further decrease in temperature results in the equal value of the thermodynamic potential of the B phase to the S phase. This coincidence of the thermodynamic potential indicates the coexistence of two phases, a characteristic phenomenon of a first-order transition. The temperature is calculated as Ttrs =

a32 + T0 . 4αa4

(2.47)

At this temperature, equilibrium magnitudes φ0 of two phases are 0 and a3 /(2a4 ). Thus, the order parameter is discontinuous if the phase transition happens. Entropy also exhibits a discontinuity by

2.2 Landau’s Phenomenology

45

Δtrs s =

αa32 4a42

(2.48)

resulting in the presence of the latent heat Δtrs h = Ttrs Δtrs s. Further cooling causes the disappearance of the local minimum at φ = 0 at TL = T0 , below which thermodynamic stability of the S phase is lost. In summary, Eq. 2.45 predicts a discontinuous (first-order) phase transition. The existence of the lower limit of supercooling (TL ) and the upper limit of superheating (TU ) is also predicted. A first-order transition is not limited to the case of the presence of the third-order term in the expansion of thermodynamic potential. In the case where the expansion up to the fourth order (Eq. 2.26) is insufficient because of a4 < 0 even with a3 = 0, we need to treat (2.49) ΔF(φ, τ ) = a2 φ2 − a4 φ4 + a6 φ6 . A similar analysis to the case with a3 = 0 yields a first-order transition taking place at a2 (2.50) Ttrs = 4 + T0 4αa6 √ with a jump in the order parameter to a4 /2a6 from 0. The limits of the supercooling and superheating are TL = T0 and TU = a42 /(3αa6 ) + T0 , respectively. It seems natural to ask what happens when a4 = 0 because a continuous transition occurs for a4 > 0 in contrast to the present (a4 < 0) case.7 On approaching null from the negative side (a4 < 0), the transition temperature (Eq. 2.50) also approaches to T0 . Thus, the transition point with a4 = 0 is T0 . The transition point T0 with a4 = 0 (assuming a6 > 0), where a discontinuous (first-order) changes to a continuous (second-order) transition (or vice versa), is called a tricritical point. In this case, the order parameter grows as  α(T − T0 ) 1/4 φ0 (τ ) = − 3a6

(2.51)

below T0 . A tricritical behavior has experimentally been observed in some systems [15] with varying intensive parameters (e.g., pressure or composition in multicomponent systems) other than temperature.

2.2.5 Plural Equivalent Order Parameters It is often the case that the symmetry of the symmetric phase forces to take plural yet equivalent order parameters into account. Namely, the order parameters to be considered belongs to an irreducible representation (of the symmetry group) with 7 The

ignorance of the sixth-order term with a6 > 0 for a4 > 0 has no essential effect.

46

2 Phase Transitions

more than one dimension. For example, if the symmetric phase has a tetragonal unit cell (a = b = c), and expected orders are within the ab plane, a minimal expansion of thermodynamic potential up to the fourth-order has the form ΔF(τ , φa , φb ) = a2 (φa2 + φ2b ) + a4 (φa4 + φ4b ) + b4 φa2 φ2b

(2.52)

with a4 > 0 and b4 > −2a4 .8 Since the equilibrium state has minimum thermodynamic potential, we require ∂ΔF(τ , φa , φb ) = 2φa (a2 + 2a4 φa2 + b4 φ2b ) = 0 φa ∂ΔF(τ , φa , φb ) = 2φb (a2 + 2a4 φ2b + b4 φa2 ) = 0. φb

(2.53) (2.54)

The trivial solution, φa = φb = 0, corresponds to the symmetric phase for a2 > 0. To search for possible orders (|φa | + |φb | = 0), we set a2 < 0. Two types of solutions are possible: One is

a2 (2.55) |φa | = |φb | = − 2a4 + b4 with thermodynamic potential Δf 1 (τ ) = −

a22 . 2a4 + b4

(2.56)

The other has a form (φa , φb ) = (φ0 , 0) or (0, φ0 ) with

a2 φ0 = ± − 2a4 with Δf 2 (τ ) = −

a22 . 4a4

(2.57)

(2.58)

Depending on the relative magnitude of a4 and b4 , the phase appearing below T0 changes. Namely, the last type of order appears for b4 > 2a4 while the former for −2a4 < b4 < 2a4 . This undeterminacy indicates that the information about the symmetric phase (φa = φb = 0) is insufficient to predict the order to appear. Conversely, we can learn, from such an analysis, what is different for two materials that are essentially the same in their symmetric phase but different in symmetry-broken states.

8 The

last condition comes from the stability of the symmetry-broken (φa = φb = 0) phase.

2.2 Landau’s Phenomenology

47

2.2.6 Implication and Outcome Previous Sects. 2.2.3 and 2.2.4 revealed that the different nature of phase transition is related to the difference in the system symmetry. This result means that a detailed inspection over the system far from the transition point tells the nature of the transition. A remarkable example of this kind is a transition between isotropic liquid and nematic liquid crystal with uniaxial order [16]. Since the usual nematic liquid crystal is not polar, individual molecules can be regarded as rods symmetric concerning head and tail. This headless nature denies using the average of the orientation vector s because it can vanish even if all molecules align perfectly along an axis. Alternately, a proper order parameter for the nematic liquid crystal is given by φ = (3 cos θ2 − 1)/2 with θ being the angle from the uniaxial axis. This nematic order parameter indeed vanishes for the perfect disorder. It is interesting to see that the perfect alignment within a plane perpendicular to the nematic axis (θ = π/2) accompanies φ = − 21 . This fact indicates that the different signs of the nematic order parameter correspond to entirely different states. Thus, the expansion of the thermodynamic potential of the isotropic liquid in terms of the nematic order parameter unavoidably has a cubic (third-order) term. The analysis in Sect. 2.2.4 indicates that the transition should be discontinuous. Similar discussions are possible on the nature of phase transitions based on the symmetry of the disordered phase (with vanishing magnitude of the relevant order parameter). The restriction on the functional form of the thermodynamic potential of the disordered phase in terms of order parameters exerts some conditions for the possibility of continuous transition. Since representations of the space group, which characterizes the symmetry of crystals, depends on the wavevector of modulation wave, the order parameter accompanies the wavevector. For phase transitions between crystalline phases, conditions for continuous phase transition appear as selection rules for allowed wavevectors [2, 3, 17]. It is interesting to see that a general wavevector that drives a transition to an incommensurately modulated phase (incommensurate phase) is mostly compatible with a continuous transition while rational wavevectors leading to a simple multiplication of unit cell is often prohibited from driving a continuous one. It is often the case that plural order parameters, which are physically different from each other, are involved in a phase transition. In such cases, the argument within Landau’s phenomenology offers the restriction on the allowed form of the coupling of order parameters. Imagine a phase transition from the paraelectric phase to the ferroelectric phase with spontaneous polarization p. Since the ferroelectricity usually accompanies atomic (or molecular) displacement(s) in the crystal, the elastic deformation u is mostly induced upon the appearance of p. For such systems, the simplest form of thermodynamic potential contains a coupling term proportional to up 2 because the resultant deformation is independent of the sign of p. Another example is found with a detailed analysis in [18]. The coupling term of the last-mentioned form, up 2 , deserves another remark. To this point, we have implicitly assumed a single instability. Namely, we have

48

2 Phase Transitions

assumed that the order parameter not only characterizes the low-symmetry phase but also drives the phase transition. If another quantity, such as u in the previous paragraph, possesses the property of order parameters that are null in the highsymmetry phase but finite in the low-symmetry phase, this is a subsidiary order parameter. On the other hand, plural degrees of freedom, each of which has an order parameter, may possess potential instability to a respective symmetry-broken state. In this case, their virtual transition points (expected without coupling with others) would be different from each other. If there is no coupling, each degree of freedom undergoes an intrinsic phase transition. When the difference in the transition point is small with some coupling, however, transitions merge into a single phase transition [18, 19]. Furthermore, the merged transition may be of first-order even if each of two degrees of freedom has individually a continuous instability [19],9 i.e., the respective expansion being the form of Eq. 2.26. In short, the joint instability of many order parameters potentially brings about a first-order phase transition.

2.3 Critical Phenomena and Universality In the vicinity of a transition temperature of a continuous transition, many physical quantities exhibit singularity. Equations 2.40 and 2.43 are examples of susceptibilities in Landau’s phenomenology. Overall behaviors near a transition point of a continuous phase transition are generally termed as critical phenomena. In this context, the transition point is often denoted as a critical point. The critical phenomenon is the central issue in the physical study of phase transitions and discussed in many specialized monographs [20–22]. The description in this section is limited to issues having a possibility to encounter in the context of general studies of molecular systems. Some of the physical quantities exhibiting singularities such as heat capacity C and susceptibility χ are related to fluctuation. It is easy to verify10 1 (E − E )2

kB T 2 1 (σ − σ )2

χ∝ kB T

C=

(2.59) (2.60)

where E is (internal) energy and · means the average over the system and σ is (instantaneous) order parameter (i.e., φ = σ ). If these quantities diverge at the critical point, the fluctuation of the system also diverges, accordingly. It is known that the role of fluctuations in critical phenomena is crucial. Theories of critical phenomena neglecting the effect of fluctuations are classified as mean-field theory. 9 Although the reference [19] considers only one sign of the difference in the transition point, the other sign can also bring about a first-order transition. 10 While considering the possibility of difference in the definition of susceptibility, the formula is written as the proportionality.

2.3 Critical Phenomena and Universality

49

Landau’s phenomenology is its representative. The fluctuation strongly depends on the dimension of the system. In fact, for example, a phase transition is impossible for any one-dimensional system because significant fluctuation destroys any order except for the absolute zero even if the one-dimensional lattice is a priori given [2]. When the system under study is low-dimensional, e.g., highly-anisotropic magnetic or electronic systems, or surface phenomena, possible effects of fluctuation arising from the low-dimensionality should adequately be taken into account. It is usual to express singularities in physical properties near the critical point using the reduced temperature T − Tc . (2.61) t= Tc as χ ∝ |t|−γ −α

C ∝ |t| ξ ∝ |t|−ν

(2.62) (2.63) (2.64)

where γ, α, and ν are critical indices (or critical exponents) for susceptibility, heat capacity, and correlation length ξ, respectively. The correlation length is defined as the length that characterizes the decay of the spatial correlation expressed by the correlation function of the (local) order parameter. The correlation function as a function of distance r is defined as G(r ) = σi σi+r − σi σi+r ,

(2.65)

where σ j is a spin at site j (r j ) and · means the average over the ensemble. G(r ) decays as (2.66) G(r ) ∝ r −τ e−r/ξ at t = 0. That is, the decay is expressed by a product of a power-law (index, τ ) and an exponential law. This decay form defines the correlation length ξ. For t < 0, there is another critical index. Namely, φ ∝ |t|β

(2.67)

which defines a critical index for the order parameter. The critical indices that are meaningful also for t > 0 are distinguished by adding a prime (like α for one of heat capacity) though it has widely been believed that those for t > 0 and t < 0 are identical. At t = 0, other critical indices are defined through φ ∝ |h|1/δ G(r ) ∝ r

−d+2−η

(2.68) (2.69)

50

2 Phase Transitions

where d is the dimensionality of the system. It is emphasized that the functional dependences of the correlation function G(r ) on the distance (r ) are entirely different between Eqs. 2.66 and 2.69. The power-law dependence of Eq. 2.69 indicates the absence of a characteristic length at the critical point (t = 0). The spatial distribution of the local order parameter is fractal, accordingly. The divergence of the correlation length (Eq. 2.64) at the critical point implies that the physical significance of microscopic details of systems diminishes on approaching the critical point. Indeed, it is believed that critical behaviors are classified into a limited number of classes (universality classes), depending on essential features of systems such as spatial dimensionality, the number of freedom of the order parameter, and the range of interaction. Here, the degree of the freedom of the order parameter concerns with, for example, the difference between a discrete Ising spin [5] (with only two, up and down states) and a classical spin in the three-dimension (the spin head sweeps a spherical surface). Finally, the following relations are widely believed to hold irrespective of universality classes α + 2β + γ = 2

(2.70)

β(δ − 1) = γ (2 − η)ν = γ

(2.71) (2.72)

These are called scaling relations because they are intuitively rationalized by the argument while considering successive rescaling of the length scale of the system [23].

2.4 Formation Versus Collapse of Order There seems to be no need for discussing the validity of understanding that liquid crystals locate between gasses and crystals among aggregation states of matter. Indeed, the entropy of a liquid crystalline phase is between those of gas and crystal of the compound after monotonous temperature dependence. This fact suggests two ways to discuss liquid crystals, starting from either of the two limiting states. These ways correspond to the view of the formation or collapse of the liquid crystalline order. Alternatively, it can be said as the break or recovery of the relevant symmetry. The relation between the symmetry and the order brought by a phase transition is often controversial. It is essential to distinguish the symmetry and order. Imagine the liquid consisting of globular molecules. The liquid (in the limit of infinite volume) is symmetric against both any translation and rotation around any axis. The crystallization into a simple cubic lattice with spherically disordered molecular orientation breaks these symmetries: translational invariance is only satisfied for a sum of integer multiples of lattice vectors (na + mb + l c) and rotational invariance for specific angles around specific axes (e.g., π/2 around a or 2π/3 around (a + b + c)). When

2.4 Formation Versus Collapse of Order

51

the molecular orientation is ordered uniaxially along the a axis, the equivalence between the a axis and others is lost. That is, the symmetry lowers. The two views on an intermediate state are useful and necessary to understand a variety of molecular systems. They are, however, not equally possible. Generally, the discussion starting from the ordered state is easier to perform because relevant degrees of freedom are, even not specified, but restricted by the ordered state. The third law of thermodynamics is quite suitable for this view. Microscopic (spin) models of phase transitions are primarily within this view. Detailed investigation on the ordered states often leads to correct predictions about a phase transition to disordered state(s). A comprehensive discussion along this line will be given for the melting process of molecular crystals in Chap. 6. It is, however, emphasized that the question of why the specific crystal structure is the most stable is out of consideration from the beginning. The view starting from highly symmetric (disordered) states is much more difficult than the way mentioned in the previous paragraph. In contrast to the previous case, a wide variety of orders are potentially possible. In principle, a search for all possibilities and comparisons among them is necessary. Such a question to the most stable crystal structure for the ensemble of rigid spherical molecules is an example. Proper consideration of the effect of temperature is also necessary. For example, the face-centered cubic packing and the hexagonal closest packing are known as closest packings of spheres with the stacking of hexagonally close-packed layers of · · ·ABCABC· · · and · · ·ABAB· · ·. It is clear that they and random stacking of A, B, C are equally stable at the absolute zero as far as we consider interactions only between the contacting neighbors.

References 1. P. Ehrenfest, Proc. R. Neth. Acad. Arts Sci. 36, 153–157 (1933) 2. L.D. Landau, E.M. Lifshitz, Statistical Physics, 3rd edn. (Butterworth-Heinemann, Oxford, 1980) 3. J.-C. Tolédano, P. Tolédano, The Landau Theory of Phase Transitions (World Scientific, Singapore, 1987) 4. W. Lenz, Phys. Z. 21, 613–615 (1920) 5. E. Ising, Z. Phys. 31, 253–258 (1925) 6. L. Onsager, Phys. Rev. 65, 117–149 (1944) 7. A.N. Kolmogorov, Izv. Akad. Nauk SSSR, Ser. Mat. 3 355–359 (1937) 8. W.A. Johnson, R.F. Mehl, Trans. AIME 135, 416–459 (1939) 9. M. Avrami, J. Chem. Phys. 7, 1103–1112 (1939) 10. M. Avrami, J. Chem. Phys. 8, 212–224 (1940) 11. M. Avrami, J. Chem. Phys. 9, 177–183 (1941) 12. M. Tatsumi, T. Matsuo, H. Suga, S. Seki, J. Phy. Chem. Solids 39, 427–434 (1978) 13. H. Horner, C.M. Verma, Phys. Rev. Lett. 20, 845–846 (1968) 14. K. Saito, A. Sato, A. Bhattacharjee, M. Sorai, Solid State Commun. 120, 129–132 (2001) 15. J. Thoen, H. Marynissen, W. Van Dael, Phys. Rev. Lett. 52, 204–207 (1984) 16. P.G. de Genne, Phys. Lett. A 30, 454–455 (1969) 17. H.T. Stokes, D.M. Hatch, Isotropy Subgroups pf the 230 Crystallographic Space Groups (World Scientific, Singapore, 1988)

52

2 Phase Transitions

18. K. Saito, M. Hishida, Y. Yamamura, Soft Matter 11, 8493–8498 (2015) 19. R.B. Meyer, T.C. Lubensky, Phys. Rev. A 14, 2307–2320 (1976) 20. H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, New York, 1987) 21. J.J. Binney, N.J. Dowrick, A.J. Fisher, M.E.J. Newman, The Theory of Critical Phenomena (Oxford University Press, New York, 1992) 22. H. Nishimori, G. Ortiz, Elements of Phase Transitions and Critical Phenomena (Oxford University Press, New York, 2010) 23. L.P. Kadanoff, Physics 2, 263 (1966)

Chapter 3

Molecular Liquids

3.1 Liquid Structure Even when gasses consist of not spherical atoms but anisotropic molecules, they can mostly be treated as ideal ones that obey the ideal gas law, pV = n RT . It is, however, noted that their extensive properties certainly depend on not only masses of molecules but also their shapes.1 The situation is much more complicated in liquids due to their shape and interactions among them. Indeed, the presence of interparticle interaction and resulting local structure is the primary difference between gas and liquid.

3.1.1 Scattering from Molecular Liquid The intensity of the scattered radiation (such as neutron and electron beams, or X-ray) from a sample can generally be given as  2   I (q) ∝  ρ(r) exp(i q · r)d V 

(3.1)

V

with the density of scatterers, ρ(r). If the liquid consists of only one atomic species, the I (q) can be decomposed into the product of the squared atomic form factor f (q) and the so-called structure factor S(q) as I (q) = f (q)2 S(q).

(3.2)

1 For example, the heat capacity of a classical ideal gas (perfect gas) is 3R/2, 5R/2, and 3R

(R, the gas constant) for those of monatomic, linear (e.g., diatomic), and non-linear molecules, respectively, while ignoring internal vibrational degree(s) of freedom.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 K. Saito, Chemical Physics of Molecular Condensed Matter, Lecture Notes in Chemistry 104, https://doi.org/10.1007/978-981-15-9023-8_3

53

54

3 Molecular Liquids

This structure factor is related [1] to the radial distribution function, g(r ) = ρ(r )/ρave by g(r ) − 1 =

2 π



∞

q sin(qr )[S(q) − 1]dq.

(3.3)

(3.4)

0

Thus, the structure of liquids can be examined via scattering experiments. When the target liquid consists of many types of atoms, the situation is not so simple. For example, three radial distribution functions, gAA , gBB , and gAB ,2 are necessary to characterize the simplest case, “binary” liquid composed of A and B atoms. Although they can be determined by three independent scattering experiments utilizing beams having different scattering properties (e.g., X-ray, neutron, and electron) [2], such an investigation becomes impractical for systems consisting of more components. Alternately, it is often the case that the same analysis as neat liquids is performed for systems consisting of plural atomic species putting f (q) =



f i (q),

(3.5)

i

where summation runs over atoms. This procedure may be rationalized by somewhat similar q-dependences of f (q) except its magnitude (proportional to the atomic number at q = 0). However, analyses in this way result in the pursuit of the liquid structure in some atomic length scale. Although this is tractable and useful in simple molecular liquids [3], the experiments up to a high q range are unavoidably necessary because of the length scale l sensed by the X-ray is related to q by l = 2π/q. Besides, the information concerning the molecular arrangement becomes harder to deduce from the experimental results (intensity of the scattered X-ray) with increasing the complexity of the system. Therefore, a way of the “coarse-graining” is tempting if available. Suppose the scattering of X-rays by liquid. Since the electron density is the superposition of atomic electron densities, Eq. 3.1 can be written as 2      ρi (r − r i ) exp(i q · r)d V  . I (q) ∝ 

(3.6)

V

The sum can be further grouped into the contribution of “molecules” 

ρi (r − r i ) exp(i q · r) =

 j

2g

BA

ρi (r − r i ) exp(i q · r).

i

can be defined but carries the same information as gAB .

(3.7)

3.1 Liquid Structure

55

Here, the first sum runs over all molecules, whereas the second sum over atoms inside a molecule. Usually, a molecule consists of atoms of many types. The molecular sum is decomposed into partial sums over the same atomic species, l, which have the common electron density distribution around the nuclei, as 

ρi (r − r i ) exp(i q · r) =



i

l

ρl (r − r i ) exp(i q · r).

(3.8)

i

Integration over the whole volume yields f j (q) = exp(−i q · R j )



fl (q) exp[i q · (r li − R j )]

(3.9)

i

where R j specifies the location of the jth molecule. The molecular form factor is decomposed into partial form factors 

fl (q) exp[i q · (r li − R j )] =

i



sl (q, ω j ).

(3.10)

l

Note that s(q, ω j ) still depends on molecules at this stage because each molecule has different orientation ω j , which symbolically represents Euler angles of the jth molecule. If the orientation of neighboring molecules is independent of each other, the molecular form factor should become common for all molecules after averaging over the molecular orientation. The average of a function g(ω) over the orientation is given by  2π  2π  π 1 g(ω) = dψ dφ dθ sin θ · g(ω). (3.11) 8π 2 0 0 0 Thus, by defining | f av (q)|2 as  2     | f av (q)| =  sl (q, ω) ,   2

(3.12)

l

the net intensity given by Eq. 3.6 is written as I (q) ∝ N |S(q)|2 | f av (q)|2 ,

(3.13)

where N is the total number of molecules in the sample. The factor N , instead of the summation over molecules, reflects the fact that the molecular contribution is common for all molecules. The structure factor S(q) of the liquid is given by S(q) =

 j

exp(q · R j )

(3.14)

56

3 Molecular Liquids

Each partial form factor (l = C, H or O for typical organics) has the form sl (q, ω) = fl (q) pl (q, ω)

(3.15)

with the partial molecular structure factor pl (q, ω) =



exp[i q · (r i − R0 )]

(3.16)

i

with R0 being the center of molecule. Then, we have  2      sl (q, ω) = fl (q) f m (q) pl (q, ω) pm (q, ω)∗ .    m l

(3.17)

l

Since fl (q) does not depend on the molecular orientation ω, the orientational averaging is necessary for products of pl (q, ω). Using these quantities, the averaged scattering intensity from a single molecule is given by | f av (q)|2 =

 l

fl (q) f m (q) pl (q, ω) pm (q, ω)∗ .

(3.18)

m

The formulation given above is rigorous under the assumption, no correlation in orientation between neighboring molecules. The validity relies on the choice of the “molecules” as a constituting particle of the sample. Once we chose the molecule, the other structural information is included in the structure factor of liquid, S(q). Note that we only measure the net intensity of scattering. A naïve way to define the “molecule” is to assume that atoms within the orientational correlation length constitute a molecule. This strategy will only be the way for the molecular association to extend over a long distance, as in the case of chain and/or network. On the other hand, if the association mode is well defined and not-extending, molecular clusters with a well-defined structure may be regarded as a molecule. If dividing the experimental I (q) by | f av (q)|2 calculated under such an assumption gives S(q) resembling that for a simple liquid such as rare gas elements and hard spheres, the assumption seems plausible. In this context (or the way of proceeding analysis), the calculation of | f av (q)|2 based on available structural data is crucial. If we have the cartesian coordinates of atoms in a molecule (or a cluster), we redefine the origin of the coordinate by subtracting the coordinates of the center of electron density, (3.19) Ri = r i − R0 within a molecule with

3.1 Liquid Structure

57

 rρ(r)d V R0 = V ρ(r)d V V ci r i , ≈ i l ci

(3.20)

where ci is the number of electrons belonging to the ith atom. Adopting the approximation (Eq. 3.20) eliminates the use of atomic electron distributions in the real space. Rotation matrix M expressed using Euler angles (φ, ψ, θ) is given by ⎛

⎞ Cφ Cψ − Cθ Sφ Sψ −Cψ Sφ − Cφ Cθ Sψ Sψ Sθ M(φ, ψ, θ) = ⎝ Cψ Cθ Sφ + Cφ Sψ Cφ Cψ Cθ − Sφ Sψ −Cψ Sθ ⎠ , Sφ Sθ Cφ Sθ Cθ

(3.21)

where Cx and Sx stand for cos x and sin x, respectively. Thus, the partial molecular structure factor is given as pl (q, ω) =



exp[i q · M(φ, ψ, θ) · Ri ].

(3.22)

i

After putting this into Eq. 3.17, the subsequent averaging by Eq. 3.11 yields | f av (q)|2 . In practical calculation, by the orientational averaging arising from the isotropic nature of the liquid, we can freely choose the scattering vector q. By choosing q = t (0, 0, q), the integration over ψ can be avoided. The orientational average thus becomes  π  2π 1 pl (q, ω) pm (q, ω)∗ = dφ dθ sin θ · pl (q, ω) pm (q, ω)∗ . (3.23) 4π 0 0

3.1.2 Case Study Dicyclohexylmethanol [DCHM, (C6 H11 )2 CHOH] and tricyclohexylmethanol [TCHM, (C6 H11 )3 COH] are primary alcohols substituted with bulky cyclohexyl groups. The crystal structure reports say that a tetramer through cyclic H-bonds and an H-bonded dimer are structural units in crystals of DCHM [4] and TCHM [5, 6], respectively. Two compounds exhibit notable differences concerning H-bonds upon melting. While the H-bond inside a TCHM dimer starts to break below the melting temperature (94.25 ◦ C) while keeping the crystalline form [7, 8], H-bonds in the cyclic tetramer in DCHM survive almost wholly up to the melting temperature (64.36 ◦ C) [9]. In the liquid state above the normal melting temperatures, the H-bond is broken almost completely in TCHM [7, 8]. In contrast, a significant portion of molecules remains involved in H-bond in DCHM [9]. Since two compounds

58

3 Molecular Liquids

sum of fatom(q)

100

monomer (r = 4.0 Å)

|f(q)|

Fig. 3.1 | f av (q)| of monomer and tetramer of DCHM and sum of atomic form factors. Form factors of spheres having uniform electron density are shown by dotted lines. Adapted from Chem. Phys. Lett., 673, 74 (2017) [11]

(r = 1.0 Å)

50 DCHM tetramer (r = 6.5 Å) 0 0

1 -1 q/Å

2

exhibit contrasting behaviors despite similar molecular structures, they serve as good examples to test the coarse-graining.

3.1.2.1

Averaged Molecular Form Factor

The cartesian coordinates of a monomer and tetramer of DCHM were taken from the supplementary information of the literature [9], which reported those based on quantum chemical calculations. Atomic form factors were from another literature [10]. Calculated | f av (q)| are shown in Fig. 3.1, together with the sum of atomic form factors. Both of calculated | f av (q)| for monomer and tetramer are quite different from the simple sum. However, form factors of spheres having uniform electron density approximate well the calculated ones if appropriate radii are assumed. A comparison of | f av (q)| calculated for monomer and tetramer indicates that | f av (q)| does not always resemble that of a rigid sphere though the spherical averaging was certainly performed in the calculation over molecular orientation. Note that the averaging is applied after being squared, leading to non-negative | f av (q)|. The null value is unavoidably blurred. In this respect, the close coincidence having the first minimum at the same q with that of a uniform sphere (r = 6.5 Å) for the tetramer demonstrates its spherical shape viewed through the scattering process.

3.1.2.2

Molecular Radial Distribution Function

Figure 3.2 [11] shows the radial distribution functions calculated according to Eq. 3.4 while assuming a monomer as a “molecule.” Although the difference between the two compounds is notable, their first peaks are reasonably interpreted as an effective diameter of an individual molecule. This consistency indicates that the coarsegraining certainly works. It is emphasized that the experiment up to 2 Å−1 offers

3.1 Liquid Structure

g(r) - 1

0.5

0.0 DCHM

-0.5 0.5

g(r) - 1

Fig. 3.2 Radial distribution functions of molecules for DCHM (65 ◦ C–130 ◦ C) and TCHM (95 ◦ C–160 ◦ C). Red to blue color corresponds to high to low temperature at ca. 2 ◦ C interval. Reproduced with permission from Chem. Phys. Lett., 673, 74 (2017) [11]

59

0.0

TCHM -0.5

10

20

30

r/Å

1.02

rfp(T) / rfp(Tfus)

Fig. 3.3 Relative temperature dependence of the locations of the first peaks normalized by those at the temperature of fusion (Tfus ) in the radial distribution function of DCHM (65 ◦ C–130 ◦ C) and TCHM (95 ◦ C–160 ◦ C)

TCHM 1.01 DCHM 1.00 1.0

1.1

1.2

T / Tfus

information about the liquid structure at the molecular length scale (distribution of molecular centers). Although the difference between the highest and the lowest temperatures in this study is the same (65 ◦ C) for two compounds, the radial distribution function of DCHM exhibits a much stronger dependence on temperature than TCHM. The stronger dependence of DCHM is consistent with the previous study [9], according to which not only the number of H-bonds but also the relative populations of monomers and associates (up to the cyclic tetramer) change in this temperature region. The temperature dependence of the locations of peaks in the radial distribution function exhibits interesting behaviors. As shown in Fig. 3.3, that of the first peak −1 drfp /dT = 0.7 · 10−4 K−1 ) in comparison with that of DCHM is distinctly weak (rfp of TCHM (2.9 · 10−4 K−1 ). The magnitude of the latter can be interpreted as a result

60 100

8 6 4 2

n< r ( r)

Fig. 3.4 Number of molecules within a sphere of radius r . DCHM, solid curve (red, 130 ◦ C; blue, 65 ◦ C); TCHM, dashed curve (red, 160 ◦ C; blue, 95 ◦ C); liquid argon, black dotted curve calculated using the literature data [1]

3 Molecular Liquids

10

8 6 4 2

1 0.5

1.0

1.5

2.0

r / rfp

of thermal expansion considering volume expansivities of typical organic liquids (e.g., 1.3 · 10−3 K−1 for cyclohexane and benzene [12], and 1.0 − 1.2 · 10−3 K−1 for small primary alcohols [13]). Besides, other peaks of DCHM exhibit compatible temperature dependences (3.4 · 10−4 K−1 on average) with that. In this temperature range, a substantial portion of DCHM molecules associate through H-bond(s) [9]. Since the radial distribution function is a kind of weighted average of pair distributions (one molecule contributes twice), nearly fixed local geometry contributes substantially to the first peak. Thus, the weak temperature dependence of the rfp of DCHM can be interpreted as a reflection of the substantial population of the H-bond between neighboring molecules. The number of nearest molecules are, though only slightly, larger in DCHM than TCHM as seen in Fig. 3.4, which was calculated as 4π n 0. Unless it exists, a continuous transition takes place at u = 0. For the expansion to be meaningful, sums over wavevectors are taken from the common group of those. The leading second-order term requires each vector’s counterpart in the group. Because of the presence of the δ-function, the third-order term must satisfy the condition that three wavevectors form a triangle. The fourth-order term does not exert any condition. Groups fulfilling these requirements are limited to the following three cases: (i) six unit vectors from the same point (60 degrees in between) on a flat plane (edges of an equilateral triangle and its inverted image), (ii) edges of a regular octahedron (edges of a regular tetrahedron and its inverted image), (iii) edges of a regular icosahedron. The first group corresponds to the two-dimensional order and does not lead to threedimensional crystallization. The last group has fivefold symmetry, which is incompatible with any three-dimensional periodicity (but implicitly predicted the formation of icosahedral quasicrystals six years before its discovery in 1984 [6]). The second group is equivalent to the shortest vectors between particles in the face-centered cubic structure. Thus, the vectors can be specified as {1, 1, 0}, which covers (1, 1, 0) and equivalents [such as (−1, 0, 1), (0, −1, −1), etc.]. We see that these vectors certainly

68

4 Molecular Crystals

vanish if summed as (1, 1, 0) + (−1, 0, 1) + (0, −1, −1) = (0, 0, 0). Since we are dealing with the wavevectors, the lattice formed by the vectors is the so-called reciprocal lattice. The corresponding lattice in the direct space is the body-centered cubic (BCC) structure. In this treatment, possibilities of phase transitions and of emerging spatial order are discussed, in contrast to the standard use of the Landau theory, where a specific phase transition is described. The symmetric phase is the isotropic fluid common for all cases. For a meaningful comparison of emerging orders, it is necessary to normalize the free energy by the number of wavevectors in each group. Then, the phase transition with the highest transition temperature is concluded to occur. Through the analysis, the transition to the BCC structure was identified to occur at the highest temperature. Further analysis concerning temperature dependence revealed that the BCC structure is stable irrespective of temperature as far as the expansion (Eq. 4.3) is assumed. The above consideration leads to a striking (but rather strange) conclusion that globular molecules should crystallize into BCC structures. By incorporating slightly different wavevectors {1, 1, 1} and {2, 0, 0} that are within the experimental width of the wavevector of density fluctuation in isotropic liquid, the FCC structure can be stabilized [7]. The formation of wavevector triangles, though they are always mixed ones,√ is also essential in this case. Note that the difference in length (mismatch) is (2 − 3)/2 ≈ 0.13 in this case.

4.2 Diffraction and Modern Definition of Crystals In contrast to naïve understanding, the International Union of Crystallography (IUCr) adopts the definition of crystals as “any solid having an essentially discrete diffraction diagram” [1]. This definition is different from the traditional definition assuming translational symmetries. In this section, we see the relation between modern and traditional definitions. It is necessary because the modern definition is necessary for some molecular systems. We start with a naïve definition of crystals, i.e., a periodic arrangement of structural units. For simplicity, we consider a one-dimensional periodic array of point scatterers, of which the number and the period are N + 1 and a, respectively. The extension to three-dimensional cases is trivial. The density of scatterer is expressed as ρ(r ) =



δ(r − ja).

(4.4)

j=0,N

Here, we take the finite size of the array into account. When the scattering power of atoms is f j = f ◦ ,2 the complex strength of the scattered wave, A(q), is given by3 2 Generally,

the scattering power of scatterers, such as atoms, is a function of scattering vector q. = −1 while “ j” is an integer

3 Note that “i” in formulas in this section is the imaginary unit, i.e., i 2

index.

4.2 Diffraction and Modern Definition of Crystals



A(q) =

69

f j δ(r − ja) exp(iqr )

j=0,N

= f◦



exp(i jqa)

j=0,N

1 − exp(i N qa) 1 − exp(iqa)   N qa (N − 1)qa sin 2 = f ◦ exp −i . 2 sin qa 2

= f◦

(4.5)

The intensity of this wave, |A(q)|2 , becomes sharp as a function of q with increasing N around qa = nπ with n ∈ Z, because the width is less than the twice 2π/N a, which give the null amplitude. Since N of the real crystal is very large (typically, 106 − 107 ), the amplitude is effectively finite only when qa = nπ . The traditional crystals give a set of sharp diffractions. It is noteworthy that standard experimental techniques are capable of measuring only the intensity but not the phase of the scattered wave. The insensitivity to the phase means that the experiments yield identical results |A(q)|2 for ± f ◦ in Eq. 4.5. This fact has long been known as Babinet’s principle [8]. However, it is trifling for well-ordered crystals because we can choose plausible models based on the concentrated density of scatterers around atoms. When the object is highly disordered and anisotropic, however, Babinet’s principle exerts a severe difficulty in analyzing the distribution of scatterers [9]. As we will see in Chap. 6, many substances have structurally disordered solid phases. They are usually crystalline concerning the arrangement of molecular centers of mass. We can model such situations by considering a disordered array while writing f j = f ◦ + Δf j

(4.6)

with a random variable Δf j . We request the condition 

Δf j exp(ik j) = 0

(4.7)

j=0,N

for any k (including k = 0) to secure the randomness. In this case, the strength of diffracted wave is A(q) =

N  

 f ◦ + Δf j exp(i jqa)

j=0

=

N  j=0

f ◦ exp(i jqa) +

N  j=0

Δf j exp(i jqa)

70

4 Molecular Crystals



(N − 1)qa = f exp −i 2 ◦



sin N2qa , sin qa 2

(4.8)

where the last equality utilizes Eq. 4.7 while adopting k = qa. The last formula is precisely the same as the amplitude of the ideal array (Eq. 4.5). Thus, crystals possessing structural disorder again give a set of sharp diffractions. Next, we consider the case that positions of atoms are modulated by a displacement wave of which the period is incommensurate (irrational) to the original period: r j = ja + Δr0 sin jka.

(4.9)

√ The irrational modulation, such as k = 1/ 2, means that there are no atoms, of which the displacements from the ideal position, ja, are the same to each other in this atomic array. In this sense, the array has no translational symmetry. This type of crystalline phase is termed as an incommensurate phase because the modulated array gives a set of sharp diffractions as follows, as far as |qΔ|  1. The strength of the diffracted wave by the modulated array is calculated as A(q) = f ◦

N 

exp[iq( ja + Δr0 sin jka)]

j=0

= f◦

N 

exp(i jqa) exp(iqΔr0 sin jka)

j=0

≈ f◦

N 

exp(i jqa) (1 + iqΔr0 sin jka)

j=0

= f

◦

N 

exp(i jqa)

j=0

+f◦ = f◦

N qΔr0  exp(i jqa)[exp(i jka) − exp(−i jka)] 2 j=0

N 

exp(i jqa)

j=0

⎫ ⎧ N N ⎬ ⎨  qΔr 0 exp[i j (q + k)a] − exp[i j (q − k)a] (4.10) +f◦ ⎭ 2 ⎩ j=0 j=0 The sums in the last line have the same form as Eq. 4.5 with the substitution q → (q ± k). Thus, not only qa = nπ (with n ∈ Z) but also (q ± k)a = nπ is the condition of diffractions to appear. Namely, sharp diffractions appear at q = n(π/a) ± mk with m = −1, 0, 1. Thus, this modulated atomic array without any translational peri-

4.2 Diffraction and Modern Definition of Crystals

71

odicity belongs to the category of crystals defined by IUCr [1]. Diffractions with m = 0 are termed as satellite reflections in contrast to main reflections (m = 0). Although the sinusoidal modulation (Eq. 4.9) gives only the first-order satellite reflections, the details of structural modulation determine how higher-order satellite reflections can be detectable. The strength of higher-order components in the modulation governs those of satellite reflections. It is noteworthy that the position of the modulation wave relative to the underlying (regular) lattice does not alter the physical situation in the incommensurate phases as a whole if the crystal is sufficiently large (infinite in the ideal sense). This property brings about the possibility of the sliding of the modulation wave without any energy penalty.4 Indeed, this type of collective dynamics is characteristic of incommensurate phases [10–12]. Interestingly, the charge-carrying modulation wave (charge density wave) was proposed as a possible mechanism of superconductivity in the early days of its study [13]. Another important class of materials adopted as crystals by the IUCr definition is quasicrystals [6]. They possess fivefold or tenfold symmetry, which is incompatible with translational symmetry. Mathematically, quasicrystals are projections of sixdimensional regular crystals onto the three-dimensional space. Adequately designed block copolymers indeed exhibit the quasicrystalline order [14]. However, we give no further discussion of the quasicrystals in this book because no examples have been identified for matter composed of small molecules.

4.3 Molecular Shape and Crystal Structure 4.3.1 Within Landau Theory We pointed out that the molecular shape is crucial in crystallization. Here, attempted is a possible inclusion of the effects into the Landau theory of weak crystallization [15]. We assume that a molecule is a cylinder with a length larger than its diameter. Then, there are two characteristic lengths and two characteristic wavevectors in the reciprocal space, accordingly. Although we have as yet no rigorous proof, it is intuitively acceptable that the larger length scale (in the direct space) is much more important for establishing isotropic, i.e., cubic organization upon aggregation. Consequently, our expansion of free energy is in terms of this shorter wavevector. Assuming the cubic symmetry, we can describe groups of wavevectors that can form equilateral triangles. The possible combinations are {1, 1, 0}, {2, 1, 1},{2, 2, 0}, {3, 2, 1}, {4, 2, 2}, {4, 3, 1}, {5, 3, 2}, ... in the order of indexes of wavevectors. If we repeat the calculation by assuming equilateral triangles, the superior stability of the BCC structure results. However, this structure implicitly assumes, in reality, the 4 This

argument applies to smooth modulation waves, such as a sinusoidal one. If the structural modulation on each site is discrete, the energy is equal before and after its shift, but the activation energy is necessary.

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4 Molecular Crystals

spherical symmetry around the molecular center of gravity. The spherical symmetry means that the resulting crystal is a “plastic crystal” (see Chap. 6), where the molecules are isotropically disordered in their orientation. The formation of such plastic crystals is physically unrealistic for highly anisotropic molecules, as shown by simple theories [16, 17] and computer simulations [18, 19]. The lowest indexes are, however, clearly favored by nature because the higher the indexes, the longer the unit cell dimension that requires an effectively long-ranged interaction. As described in Sect. 4.1.2, the normalization of the free energy of the symmetric phase, i.e., the isotropic liquid, is mandatory for meaningful comparison. Expansion coefficients should remain unchanged irrespective of indexing methods because the identical wavevector is characterizing the most significant density fluctuation. The transition temperature is primarily determined by the number of the wavevectors we take into account. The number is 12 for {1, 1, 0}: (1, 1, 0), (1, −1, 0), (−1, 1, 0) , (−1, −1, 0) and their cyclic permutations. Similarly, they are 24, 12, and 48 for {2, 1, 1}, {2, 2, 0}, and {3, 2, 1}, respectively. The first two groups in the list of indexes are known to be crucially important for an exotic structure called the Gyroid phase, the basic structure of which is shown in Fig. 4.1a. For the Gyroid phase, which belongs to the space group I a3d, the two lowest surviving indexes of diffractions are {2, 1, 1} and {2, 2, 0} when irradiated by a suitable ray. Needless to say, {2, 1, 1} is more important because only {2, 2, 0} merely leads to the same structure as that in the BCC case, although the importance of {2, 2, 0} in enhancing the stability of the Gyroid phase was pointed out [20]. It is notable here that not only both {2, 1, 1} and {2, 2, 0} form equilateral triangles separately, but also the two wavevectors (2, 1, 1) and (2, 2, 0), and their equivalents can form isosceles triangles like (2, 1, 1), (−2, 1, 1) and (0, −2, −2). Two groups of wavevectors cooperatively enhance the stability of the Gyroid phase. The mismatch in length is the same as that in the FCC case. It is noted that the present discussion is free of microscopic details of systems, and, consequently, is equally applicable even to other soft matter than liquid crystals consisting of small anisotropic molecules. The Gyroid phase is characterized by a triply periodic minimal surface (TPMS) called gyroid [21]. The gyroid surface forms a family with other TPMS, called P and D surfaces. These TPMSs are related to each other by a mathematical transformation called the Bonnet transformation, which keeps the local geometry at all points on the surfaces [21]. This property implies that the structures are degenerate in energy if the energy is a function of local properties on surfaces. In reality, however, the physical appearance of structures characterized by the P and D surfaces has been infrequent in contrast to the Gyroid phase. Similar analyses of mesophases characterized by P or D surfaces indicate that such stabilization is only operative in the Gyroid phase [15]. A structure closely related to higher indexes (Fig. 4.1b ) is also known [15]. The most important index is {3, 2, 1} in this case. The wavevector having the nearest length is {4, 0, 0}. This wavevector can form isosceles triangles √ as (3, 2, 1), (−3, 2, −1), and (0, −4, 0). The mismatch is calculated as (4 − 14)/4 ≈ 0.06, the smallest in all combinations considered. Because of this good matching, the wavelength dependence of the expansion coefficients may be ignored. Under some

4.3 Molecular Shape and Crystal Structure

73

Fig. 4.1 Crystal structures predicted by the Landau theory of weak crystallizatiion. a Gyroid phase driven by {2, 1, 1} density waves, b cubic phase formed mainly by {3, 2, 1} density waves. Shown are regions with higher (or lower) density than transparent region(s)

conditions, the transition temperature becomes higher than that of the phase characterized by the single density fluctuation {2, 1, 1} relevant for the Gyroid phase. There are physical systems showing intense diffractions specifically for this combination of wavevectors [22–24] in low-molecular-mass thermotropic liquid crystals. These systems also have the Gyroid phase. The results of structure analyses on these indicate that the spatial regions depicted by gray in Fig. 4.1 are filled by centers of rodlike molecules [25–27].

4.3.2 Close Packing The crystallization of real systems is brought about by attractive interactions. The interaction most ubiquitous is of the dispersion interaction and effectively approximated rather well by the sum of interatomic interaction, as described in Sect. 1.2. Since the attractive interatomic interactions are certainly stronger at a shorter distance unless the repulsive one becomes dominant at a very short distance, it is natural to expect that molecules tend to crystallize for denser packing. Indeed, packing densities of molecular crystals estimated by assuming van der Waals radii of atoms usually lie around √ 70%. Although this is slightly smaller than that of the closest packing of spheres (π 2/6 ≈ 0.74),5 this should rather be regarded as close to that while taking into account the fact that irregular molecular shape would be unsuitable for achiev√ packings with complete mechanical stability as low as its packing density of π/4 2 (≈ 0.55) are known for hard spheres [28].

5 Crystalline

74

4 Molecular Crystals

ing good packing. In this respect, close packing is the most important guideline for understanding the structure of molecular crystals [29, 30]. Kitaigorodskii [29] performed analyses of possible space groups suitable for the close packing (sixfold coordination within a layer) of a physically reasonable range of molecular shape. He classified close-packed layers into closest-packed, limitingly close-packed, and permissible ones. Here, the terminology for layers is: Closestpacked layers are layers that can be made no denser at a given molecular volume. Limitingly close-packed layers (for a given symmetry) are closest-packed ones in which a molecule occupies a special position, i.e., one in which the molecule retains its inherent symmetry. Permissible layers are those that, while close-packed, are neither closest-packed nor limitingly close-packed. Although his analyses are not rigorous, the result is useful as a general background of understanding the structures of molecular crystals. A full list can be found in his old book published originally in 1955. Kitaigorodskii showed that molecular symmetry crucially affects the range of available space groups. According to his analysis [29], the closest packing is allowed only for P1, P21 , P21 /c, Pca, Pna, and P21 21 21 , if the molecule to crystallize has no special symmetry (C1 ). Note that the molecular symmetry under discussion is one in the crystal but not of an isolated molecule. Thus, all space groups are available to achiral molecules. If the molecule is a chiral enantiomer, on the other hand, only non-centrosymmetric space groups (P21 and P21 21 21 ) are allowed. The counterpart is necessary for a chiral molecule to complete the crystal packing for centrosymmetric space groups (P1, P21 /c, Pca, and Pna). Namely, only the racemic mixture can crystallize with the closest packing. It is interesting to see that two (P1, P21 /c) of centrosymmetric space groups are also favorable for centrosymmetric molecules. Reflecting this fact, these occupy significant parts in reported crystal structures recorded in the Cambridge Crystallographic Data Centre [31], as seen in Fig. 4.2. Not only them, all space groups in Fig. 4.2 were identified as favorable ones by Kitaigorodskii. This fact indicates that his analyses caught some of the essential features of structures of molecular crystals.

4.3.3 Polymorphism When a molecule is merely a sphere as an argon atom, the only variable to characterize their relative geometry is the distance. In contrast, necessary for anisotropic molecules are additional six variables, i.e., three Euler angles for each. Thus, a possible “space” for a variety of molecular packing is vast for molecular crystals. Some of these possible crystal structures really occur often. When plural crystalline phases can exist under the same condition in equilibrium (typically at room temperature under the ambient pressure), each phase with a different crystal structure from other(s) is called polymorph. The phenomenon of the presence of plural polymorphs is termed as polymorphism. There is no limit on the number of possible polymorphs.

4.3 Molecular Shape and Crystal Structure Fig. 4.2 Relative frequencies of space groups of crystal structures recorded in the Cambridge Structural Database 2016 [31]. The pie of “others” is cumulative for those with a relative frequency smaller than 1%

75

others Cc Pn21 (< 1.0%) Pnma Pbca

P21/c

P21 P212121 C2/c

– P1

Four polymorphs have been identified in an organic conductor, (ET)2 I3 [32–35],6 and three exhibit superconductivity. Note that the polymorphs share the chemical composition (see the footnote of this page, for example). The pure (non-solvated) and solvated crystalline phases are not polymorphs with each other, accordingly. Only a single polymorph is the thermodynamically stable phase, and the other(s) is metastable. In this respect, the polymorphism involves the practical impossibility of a phase transition to proceed between crystalline phases. The term “practical” covers the case that thermal agitation is insufficient to activate the transformation in a reasonable time (of observation).7 That is, the polymorph is essentially characteristic of crystals, in which molecular mobility is limited. However, polymorphs may transform into another one during prolonged observation or upon the variation of external conditions, such as temperature or pressure. In this case, relative thermodynamic stability can be deduced from the transition behavior. However, it is often the case that no transition can be induced between polymorphs by no means. Even in such cases, the transition into a common phase reflects relative stability. The melting into the isotropic liquid is a typical example. The phase with the highest melting point (temperature) is the most stable thermodynamically near the melting point. In this situation, the melting of the metastable phase may result in the crystallization of the stable phase and the successive melting on further heating. This type of melting behavior is often termed as “double melting.” When the transitions to the common phase are well characterized, we may locate the equilibrium transition point between two polymorphs [40], even if the transition never occurs in experiments. On the other hand, some compounds, such as charge transfer salts mentioned as organic 6 The structure of an ET molecule is shown in Fig. 8.4. Other charge-transfer salts such (ET)·I 3

[36] and solvated ones such as (ET)·I3 (TCE)0.33 (TCE: trichloroethane) [37] are known, but these are not polymorphs of (ET)2 I3 . 7 If the transition takes several tens hours [38, 39], one may miss it, leading to a practical identification of polymorphism.

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4 Molecular Crystals

conductors, exhibit polymorphs, among which the thermodynamic relation cannot be assessed experimentally. Different polymorphs appear depending on the process of its formation. The variety of the way of polymorph formation includes deposition of vapor (including the sublimation), crystallization from solution (the so-called recrystallization), crystallization from its melt, and transition from another crystalline phase. In the case of crystallization from the solution, the solvent often affects the resulting polymorph. However, other factors, such as temperature or thermal history of the system, can be effective. We discussed one underlying aspect with somewhat broad applicability in Sect. 2.1.3. In this context, it seems valuable to mention the so-called cold crystallization. It is a crystallization much below the equilibrium transition temperature upon heating. Its realization means that the compound could be supercooled deeply without crystallization. Usually, the cold crystallization occurs as a subsequent phenomenon to the softening8 of some liquidus glasses (discussed in Chap. 8). The crystal structure primarily governs the physical properties of crystalline materials. For example, p-NPNN (an abbreviation of p-nitrophenyl nytronyl nitroxide), the first compound for purely organic ferromagnetism, exhibits not only the ferromagnetic polymorph but also an antiferromagnetic one [39, 41]. (ET)2 I3 is also another example. The appearance itself of the superconductivity depends on the polymorph besides the variety of transition temperature to a superconducting state [32–35]. The polymorphism is thus essential in both fundamental study and practical application of molecular crystals. Unfortunately, there is no universal rule to control polymorphism. The mechanism of the formation of polymorphs is too specific to the target polymorph. Finding the condition through many trial and error is practically the only strategy as long as the bulk material is considered. On the contrary, the identification of a useful polymorph can give a target local-structure for nanotechnology, such as molecular manipulation.

4.4 Cohesive Energy 4.4.1 Experimental Information The state with minimum energy is certainly chosen at the absolute zero. The energy gain by cohesion, i.e., cohesive energy, is a fundamental quantity to characterize the crystal, accordingly. For molecular crystals, estimates of cohesive energy can be obtained as the heat of sublimation. The heat of sublimation can be estimated by direct calorimetry if the rate of sublimation is relatively high. It is noteworthy that the heat of sublimation does not coincide with the enthalpy of sublimation because of the effects of pressure. Appropriate corrections are necessary. When the direct calorimetry is challenging to 8 The

glass transition on heating.

4.4 Cohesive Energy

77

apply, estimation through vapor pressure is often utilized. The temperature dependence of the saturated vapor pressure, which is the pressure of vapor coexisting with a condensed phase, is thermodynamically given as sv − ss dpv = dT vv − vs Δh ≈ , T vv

(4.11)

where thermodynamic quantities written by lower case letters are those per mole. Ignoring the temperature dependence of the difference in enthalpy between the vapor and crystal (Δh) while assuming the ideal gas behavior for the vapor, the integration gives Δh + const. (4.12) ln pv (T ) = − RT The formula is known as the Antoine equation. Thus, the information on the vapor pressure gives the enthalpy of sublimation. Since the molecular shape is not kept upon the change in its environments, the experimental estimates of cohesive energy, in principle, always contains the energy assignable to the change in the molecular geometry. For example, benzene molecules sit on centers of inversion in crystal, resulting in the loss of its ideal sixfold symmetry. However, the symmetry recovers in a usual gaseous state resembling the isolated state. When the compound is ionic, the sublimation is practically impossible. For such ionic crystals, the cohesive energy is estimated through the so-called Born-Harber cycle [42, 43], which is based upon the first law of thermodynamics. For example, consider the cohesive energy of the rocksalt, NaCl. The difference in energy between crystalline NaCl and (mixed) gas of Na+ and Cl− can be decomposed as: 1 Decomposition to elements : NaCl (c) → Na (c) + Cl2 (g) 2 Sublimation : Na (c) → Na (g) 1 Atomization : Cl2 (g) → Cl (g) 2 Ionization : Na (g) → Na+ (g) + e− Electron attachment : Cl (g) + e− → Cl− (g), where (c) and (g) distinguish states of substance, crystalline or gaseous. The important aspects here are that each reaction is practically achievable for measuring energies. Notably, the energies related to two last reactions are known as ionization energy (or potential) and electron affinity. For both ionic and nonionic cases, experimental estimates of cohesive energy are smaller than the ideal ones by the zero-point energy of lattice vibration, which is treated in Chap. 5. The so-called harmonic approximation becomes appropriate in most cases. Since the zero-point energy of a harmonic oscillator with the angular

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4 Molecular Crystals

frequency, ω, equals to 21 ω, the estimation of vibrational zero-point energy results in obtaining the average angular frequency

ω =

ωg(ω)dω,

(4.13)

where g(ω) is the vibrational density of states. For simple ionic crystals, this can be done through analyzing moments of g(ω):

Mn =

ωn g(ω)dω,

(4.14)

some (but n = 1) of which are available from experimental heat capacity at low temperatures [44]. Similar attempts were also reported for crystals of relatively simple molecules [45, 46]. In general, the information about the vibrational density of states is necessary. Although scattering techniques can, in principle, be utilized, more difficulties are encountered for systems consisting of plural atomic species than the case of the structural study discussed in Sect. 3.1.1. Indeed, this issue involves not only the difference in scattering power of atoms but also ways of involvement in vibrational modes, i.e., eigenvectors of lattice vibration. Computational means would now be accessible rather easily.

4.4.2 Lattice Sum Unless the cohesive energy is calculated based on the quantum mechanics, often called first principle or ab initio calculation, estimating cohesive energy is through a summation of interatomic potentials, v(|r|), which is a function of the interatomic distance |r|. The calculations involve terms of the form  i= j

v(|r i − r j |) =

1  v(|ha + kb + l c + r n − r m |), 2 m n=m h,k,l

(4.15)

where indexes i and j are for atoms in the system, indexes m and n for atoms in a unit cell spanned by lattice vectors a, b and c. Sums of the form 

v(|ha + kb + l c + r n − r m |)

(4.16)

h,k,l

are generally called the lattice sum. Although we encounter no difficulty if the convergence is reasonably quick, functions of distance r with negative powers usually need special care because of slow convergence. Representative is the electrostatic (Coulomb) potential for ions. If the crystal is one-dimensional with a as the interionic distance, the sum, i.e., electrostatic energy is

4.4 Cohesive Energy

79

  1 1 1 1 − + − + + ... a 2a 3a 4a ∞ e2  1 = 2· (−1)n 4π 0 j=1 ja

e2 E = 2· 4π 0

=−

(4.17)

e2 ln 2 2π 0 a

On the other hand, if all ions bare the same charge q, the sum diverges to positive infinity. This is also true for two and three-dimensional lattice. In contrast, if the power n is larger than 3 for r −n , the sum would converge finite because of

∞  1 1 ≈ 4πr 2 n dr n r r a 1 4π = n − 3 a n−3

(4.18)

for isotropic cubic lattices. This property is the basis of the usefulness of the LennardJones potentials given by Eq. 1.63. Even for the dispersion interaction with n = 6, which is ubiquitous for neutral molecules, the lattice sum does not quickly converge for practical calculation. A basic strategy widely used to overcome this difficulty is a use of an auxiliary function Φ(r ), originally proposed by Ewald [47], though another method proposed by Bertaut [48] is known (but was shown to be placed within the same framework [49]). The sum is divided into two parts as 

v(r j ) =



Φ(r j )v(r j ) +



[1 − Φ(r j )]v(r j ).

(4.19)

The auxiliary function Φ(r ) is chosen so as to accelerate the first sum, thus leading to the ease in estimating the first sum numerically. The second sum is now less convergent than the original summation. This means, however, that its Fourier transform has a dominant contribution in short wavelengths. The Fourier transform results in a discrete series due to the periodicity of the lattice. Thus, by applying Fourier transform to the second sum, the summation is performed in the reciprocal space based on Parseval’s theorem. Ewald [47] applied the method with Φ(r ) = erfc(r )

∞ 2 exp(−t 2 )dt = √ π r

(4.20)

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4 Molecular Crystals

to calculate the electrostatic energy (Madelung energy) of simple ionic compounds. The Madelung energy of simple ionic compounds Az+ Xz− depends on a cubic lattice constant a like z 2 e2 E = −α , (4.21) a where α is the Madelung constant that solely depends on the lattice type (such as NaCl, CsCl, or fluorite).9 The use of an auxiliary function to calculate lattice sums is not limited to electrostatic interaction but extended to other interactions [49–53]. For the lattice sums for a negative power function r −n , the use is suggested as an auxiliary function [49] of   Γ n/2, K 2 πr 2 Φ(r ) = Γ (n/2)

(4.22)

where Γ (m) = Γ (m, 0) and Γ (m, x) is the upper incomplete gamma function:

∞

Γ (m, x) =

t m−1 e−t dt.

(4.23)

x 1 The adoption of this function was reported to reduce the computational time to 10 . Note that the summation radius of 1 nm in the direct space is necessary even if this method is employed. Since the Ewald method and its analogs are based on the periodicity of the system under consideration, their use is not limited to crystalline systems but benefits other periodic systems. Indeed, they are widely used in molecular dynamics simulations while assuming the so-called periodic boundary condition to reduce the effects of boundaries. The application on the potential energy is acceptable. However, some care should be paid for its derivatives [54]. Consider, for example, that the force acting on a Na+ ion in an infinitely large rocksalt crystal upon an infinitesimal displacement from the ideal initial position, say the origin. To calculate the force, the increment of energy accompanied by this displacement is necessary to evaluate. If the Ewald method (or its analogs) is adopted, the assumed periodicity results in the simultaneous movement of all equivalent ions. This is of a completely different situation from that imagined at the beginning. Fortunately, however, the result is correct for the usual situation because the original Na+ ion is on the inversion center in the lattice formed by the equivalent ions. By virtue of the inversion symmetry, the force exerted by the equivalent ions completely vanishes. On the other hand, this method miscalculates the force constants (second derivatives by coordinate).

9 Different

definitions may be used for the Madelung constant. For example, it can be based on not the lattice constant but the distance between adjacent ions.

4.4 Cohesive Energy

81

4.4.3 Crystal Engineering The success of the analyses by Kitaigorodskii [29, 30] described in Sect. 4.3.2 suggests that the effect of thermal agitation is not significant in many cases on structures of molecular crystals, because crystallographic experiments are performed most often at room temperature. Thus, we may expect that the minimization of intermolecular interaction energy is a promising strategy to understand and/or even to design crystal structure. Besides, the crystal structure is effectively the most fundamental basis for understanding any physical properties of crystals. For example, the electronic properties of crystalline material are basically and well understood through the so-called band theory. The same is true for molecular conductors [55, 56]. Furthermore, the effects of thermal agitation are possibly taken into account based on crystal structures. In this respect, designing crystal structure is a key step in materials science not only for fundamental aspects but also for applications. The term, crystal engineering [57], is thus used while emphasizing this aspect. For the search for the most stable crystal structure, the lattice energy is treated as a function of structural parameters. There is no systematic way to cover all possibilities of arbitrary crystal structures. Usually, the number of atoms (molecules) in a unit cell and the space group is a priori assumed. For molecular crystals, the molecular geometries are also assumed when the identity of molecules is utilized in such cases as the atom-atom potential method. At this point, there are two choices about the molecular geometry, idealized one vs. one referring experimental data of known or related compounds. Namely, concerning benzene, for example, which of the sixfold or the inversion symmetry is more suitable for computation? Although both have been used in literature, it is necessary to recognize where the assumed atomic positions come from. The author thinks that the idealized geometry is better than the experimental one. Finally, we must minimize the “energy” while taking into account the potential energy for a flexible degree of motion, if the degree is significantly flexible like the rotation around a single bond. Once the lattice energy is expressed in terms of structural parameters within some constraints (such as molecular geometry and space group), numerical minimization can be performed in a standard way. For this purpose, some care is necessary for the atom-atom potential of the Buckingham type (Eq. 1.65). Since this potential function diverges to −∞ at r → +0, naïve and careless minimization of the lattice energy may reach unphysical results (overlapping of atoms). Practically, designing crystal packing becomes more manageable if we can identify a small number of strong intermolecular interactions. In this respect, the hydrogen bonding, which is stronger than dispersion interactions, directional and saturable, is often utilized. We must keep in mind the following issues. First, there is no way to cover all possible crystal structures. The ability of nature (a real molecular system) to search a stable structure(s) is generally superior to a sophisticated computer, even if it is the most advanced. Second, one needs to assess the “stability” of found structures unless

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the target is the most stable structure. The stability against a possible transformation to other structures under the thermal environment is a key to realize metastable phases (polymorphs, see Sect. 4.3.3).

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32. T. Mori, A. Kobayashi, Y. Sasaki, H. Kobayashi, G. Saito, H. Inokuchi, Chem. Lett., 957–960 (1984) 33. K. Bender, I. Hennig, D. Schweizer, K. Dietz, H. Endres, H.J. Keller, Mol. Cryst. Liq. Cryst. 108, 359–371 (1984) 34. M. Kurmoo, D.R. Talham, K.L. Pritchard, P. Day, A.M. Stringer, J.A.K. Howard, Synth. Metals 27, A177–A182 (1988) 35. A. Kobayashi, R. Kato, H. Kobayashi, S. Moriyama, Y. Nishio, K. Kajita, W. Sasaki, Chem. Lett., 459–462 (1987) 36. R.P. Shibaeba, R.M. Lobkovskaya, E.B. Yagubskii, E.E. Laukhina, Sov. Phys. Crystallogr. 32, 530–532 (1987) 37. R.P. Shibaeba, R.M. Lobkovskaya, V.F. Kaminskii, S.V. Lindeman, E.B. Yagubskii, Sov. Phys. Crystallogr. 31, 546–549 (1986) 38. Y. Tozuka, Y. Yamamura, K. Saito, M. Sorai, J. Phys. Chem. 112, 2355–2282 (2000) 39. Y. Nakazawa, M. Tamura, N. Shirakawa, D. Shiomi, M. Takahashi, M. Kinoshita, M. Ishikawa, Phys. Rev. B 46, 8906–8914 (1992) 40. Y. Yamamura, Y. Suzuki, M. Sumita, K. Saito, J. Phys. Chem. B 116, 3938–3943 (2012) 41. M. Tamura, Y. Nakazawa, D. Shiomi, K. Nozawa, Y. Hosokoshi, M. Ishikawa, M. Takahashi, M. Kinoshita, Chem. Phys. Lett. 186, 401–404 (1991) 42. M. Born, Verhand. Deutsch. Phys. Gesell. 21, 679–685 (1919) 43. F. Haber, Verhand. Deutsch. Phys. Gesell. 21, 750–768 (1919) 44. T.H.K. Barron, W.T. Berg, J.A. Morrison, Proc. Roy. Soc. A 242, 478–492 (1957) 45. T. Atake, H. Chihara, J. Chem. Thermodyn. 3, 51–60 (1971) 46. T. Atake, H. Chihara, Bull. Chem. Soc. Jpn. 47, 2126–2136 (1974) 47. P.P. Ewald, Ann. Physik 64, 253–287 (1921) 48. E.F. Bertaut, J. Phys. Rad. 13, 499–505 (1952) 49. D.E. Williams, Acta Crystallogr. Sect. A 27, 452–455 (1971) 50. B.R.A. Nijboer, F.W. de Wette, Physica 23, 309–321 (1957) 51. B.R.A. Nijboer, F.W. de Wette, Physica 24, 422–431 (1958) 52. A.S. Oja, M.M. Salomaa, J. Phys. C 17, L187–L193 (1984) 53. J.W. Weenk, H.A. Harwig, J. Phys. Chem. Solids 36, 783–789 (1975) 54. H. Nakayama, K. Saito, M. Kishita, Z. Naturforsch. a 45a, 375–379 (1990) 55. T. Ishiguro, K. Yamaji, G. Saito, Organic Superconductors (Springer, Heidelberg, 1998) 56. T. Mori, Electronic Properties of Organic Conductors (Springer, Heidelberg, 2016) 57. G.R. Desiraju, Crystal Engineering: The Design of Organic Solids (Elsevier, Amsterdam, 1989)

Chapter 5

Lattice Dynamics of Molecular Crystals

5.1 Scope of This Chapter Lattice vibration is the description of atomic and molecular dynamics within the lowest-order approximation, i.e., harmonic approximation. Despite its limitation originating from the approximated nature, it is fundamental for understanding the properties of crystalline materials at temperatures lower than characteristic temperatures (to be introduced later). Its study has a long history accordingly. The monograph by Born and Huang [1] describes the results in the early stage. Although some methods were proposed and even utilized in specific problems for incorporating the anharmonic effects neglected in the treatments [2–5], they are not described in this book because, nowadays, molecular simulations become handy (but can be blind with physical mechanisms behind phenomena). On the other hand, it may be noteworthy that a kind of elementary excitations, solitons, is known to exist in intrinsically anharmonic systems [6, 7] and was also suggested for molecular systems [8, 9]. These are, however, beyond the scope of this chapter.

5.2 Atomic Case We start with a simple case that treats atomic arrays (in one-dimension) and crystals (in three-dimension) to let this chapter be self-contained. The extension to molecular crystal is given in the next section. Readers familiar to the atomic case (described in any textbooks of solid-state physics) may skip this section.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 K. Saito, Chemical Physics of Molecular Condensed Matter, Lecture Notes in Chemistry 104, https://doi.org/10.1007/978-981-15-9023-8_5

85

86

5 Lattice Dynamics of Molecular Crystals

5.2.1 Atomic Array 5.2.1.1

Simple array

Suppose an array of N identical atoms with mass m. Atoms are allowed to move only along the x-axis. By labeling atoms with l = 0, 1, . . . , (N − 1) while choosing the most left atom to be the origin of labeling, the instantaneous (time-dependent) coordinate of the lth atom is denoted as xl (t). Adjacent atoms are connected by a (harmonic) spring, of which a force constant and a natural length are k and a, respectively. The equation of motion of the lth atom (1 ≤ l ≤ N − 2) is written as m

d 2 xl (t) = −k[xl (t) − xl−1 (t) − a] − k[xl (t) − xl+1 (t) − a] dt 2

(5.1)

Considering that the most stable arrangement of atoms is of the array with an equal distance a between neighboring atoms unless an external force is exerted, we introduce rl (t) = xl (t) − la. Then, the above equation is rewritten as m

d 2 rl (t) = −k[rl (t) − rl−1 (t)] − k[rl (t) − rl+1 (t)]. dt 2

(5.2)

To proceed further, we adopt the so-called cyclic boundary condition while assuming that N is so large that the effects of two ends are not matters. In most cases, the first assumption is true in usual cases because a macroscopic system consists of a vast√number of particles (of an order of 1018 or more), resulting in an estimate 3 N  1018 = 106 . In reality, we impose a condition r0 (t) = r N (t). This setting is equivalent to transform the array into a ring by connecting two ends. Assuming the boundary condition, the equations of motion (Eq. 5.2 for all l) have plane wave solutions of a form, rl (t) = r 0 exp[i(−ωt + laq)],

(5.3)

where r 0 (= 0), ω, and q are the amplitude, angular frequency, and wavenumber, respectively. The q is related to the wavelength λ of the vibration as λ = 2π/|q|.

(5.4)

By putting these solutions into Eq. 5.2, we have mω 2 rl (t) = k(1 − e−iqa + 1 − eiqa )rl (t) = 2k(1 − cos qa)rl (t). Thus, the angular frequency ω as a function of q must fulfill the relation,

(5.5)

5.2 Atomic Case

87



2k (1 − cos qa) m  qa k =2 sin . m 2

ω(q) =

(5.6)

The amplitude r 0 is, in principle, also a function of q, but has no meaning in this simplest case. Owing to the cyclic boundary condition, N qa must be an integer multiple of 2π, i.e., N qa = 2nπ (n ∈ Z). Besides, spatial arrangements of displacements are the same because of corresponding to q and q + 2π a     2π r exp i −ωt + la q + = r 0 exp[i(−ωt + laq)]. a 0

(5.7)

Thus, we can set n max − n min = N − 1. Considering that the total number of motional degrees of freedom equals N , we restrict the range of n as −N /2 < n ≤ N /2. The range of q is π π (5.8) − 0 are called optical branches. One of the conditions for direct interaction, which is allowed in the lowest-order perturbation treatment, is the coincidence of frequencies of the vibrational mode and the electromagnetic wave while assuming the fulfillment of symmetry requirements. Since the wavelength of the interacting electromagnetic wave is much longer than interatomic distances,2 the net polarization carried by a vibrational mode with finite q vanishes. Thus, only the optical mode with q = 0 is responsible for optical properties in the usual situation. Finally, the number of optical branches increases with the increase of the independent atoms in a unit periodicity. Next, we consider a variant of the alternate array as another example. Suppose the equilibrium locations of atoms with mass M are Rl◦ = (l + δ)a (0 < δ < 21 ), example, the vibrational frequency of a hydrogen molecule, 4,401 cm−1 , is the highest frequency of atomic vibrations. Its wavelength is ca. 2 µm, which is larger than the interatomic distance by a factor of 104 .

2 For

90

5 Lattice Dynamics of Molecular Crystals

and then put M = m (all atoms being of the same type). This array has symmetry operations that relate atoms at la and (l + δ)a. For example, we can imagine a mirror perpendicular to the array axis at x = δ/2. Thus, groups of atoms at la and (l + δ)a are crystallographically equivalent though they are translationally inequivalent to each other (not related by any translations of integer multiples of the unit vector a). Since two interatomic distances, δa and (1 − δ)a, exist, the corresponding force constants would differ. They are denoted as k1 and k2 . In this case, using the Rl (t) = X l (t) − Rl◦ , the dynamical matrix D(q) to be diagonalized is given by  D(q) =

m −1 0 0 m −1



−(k1 + k2 e−iqa ) k1 + k2 iqa k1 + k2 −(k1 + k2 e )

 .

(5.20)

The eigenvalues are 2 ω± (q) =

   1 k1 + k2 ± k12 + k22 + 2k1 k2 cos qa . m

(5.21)

This shows that the number of branches is equal to that of translationally independent atoms in a unit periodicity.

5.2.1.3

Longitudinal and Transverse Modes

The consideration given above covers most issues for purely one-dimensional problems. It is, however, valuable to consider that atoms can move parallel and perpendicular to the array direction. Apart from physical realities, suppose the following equations of motion for a simple array of atoms (mass m):

d 2 rl (t)





= −k [rl (t) − rl−1 (t)] − k [rl (t) − rl+1 (t)], dt 2 d 2 rl⊥ (t) ⊥ ⊥ = −k⊥ [rl⊥ (t) − rl−1 (t)] − k⊥ [rl⊥ (t) − rl+1 (t)], m dt 2 m

(5.22) (5.23)

for the motion parallel ( ) and perpendicular (⊥) to the array, respectively. Note that two independent equations exist for the latter because of the presence of two directions perpendicular to the array. It is reasonable to assume k > k ⊥ , considering the situation of ideal springs (k ⊥ = 0). Since Eqs. 5.22 and 5.23 are entirely decoupled, we can solve them independently, yielding 

qa k

sin , m 2  k⊥ qa ω⊥ (q) = 2 sin . m 2 ω (q) = 2

(5.24) (5.25)

5.2 Atomic Case

91

The displacements in the former and the latter are parallel and perpendicular to not only the array direction but also the direction of wave propagation expressed by the wavevector q. Emphasizing the latter facts, they are called longitudinal and transverse modes. The sound velocity is larger for the longitudinal modes reflecting the assumption k > k⊥ .

5.2.2 Atomic Crystals It is in order to proceed to cases of three-dimensional crystals. Let us start with the simplest case for three-dimensional atomic crystals consisting of N identical atoms. Suppose a crystal of atoms (mass m) on a simple cubic lattice. For three-dimensional crystals, models assuming ideal springs as interatomic bonds are rather unphysical. For example, a simple cubic lattice of point atoms connected by ideal springs is unstable to any shear. We must consider the issue of what are force constants to be assumed. Suppose V (R) be the potential energy of a crystal. The R is a 3N dimensional vector containing all information of atomic coordinates. At the static equilibrium, the coordinates of all atoms r l◦ satisfy 

∂V ∂rα,l

 = 0,

(5.26)

eq

for all l with α = x, y, or z with the subscript “eq” for the values at equilibrium. Thus, we can expand the potential in terms of small displacements r l from the equilibrium position and may truncate the expansion at the second-order around r l◦ as V ≈ V0 + = V0 +

1 2 l l 1 2 l l

α,β=x,y,z





∂2 V ∂rα,l ∂rβ,l

 rα,l rβ,l eq

φαβ (l, l )rα,l rβ,l

(5.27)

α,β=x,y,z

where φαβ (l, l ) are generalized force constants between the atomic pair (l, l ). Note that φαβ (l, l ) = φβα (l , l), leading to the double appearance of the same terms in Eq. 5.27 except for l = l . Lagrangian L = T − V with T being the kinetic energy of the system serves a fundamental role in the description of a mechanical system in terms of generalized coordinate Q j and its time derivatives Q˙ j [10]. The equation of motion is given by P˙ j = ∂ L/∂ Q j with the generalized momentum P j = ∂ L/∂ Q˙ j . Since the kinetic energy in the present problem is given by T =

1

d r l

2 m

dt , 2 l

(5.28)

92

5 Lattice Dynamics of Molecular Crystals

the equation of motion reads m

d 2 rα.l (t) = − φαβ (l, l )rβ,l (t). dt 2

l

(5.29)

β=x,y,z

Now, we assume the following form for a solution r l (t) = r 0 exp[i(−ωt + r l◦ q)],

(5.30)

where q is a wavevector. After putting this solution into the equation of motion and dividing the common factors, we have the dynamical matrix ⎞ ⎛ F F F 1 ⎝ xx xy xz ⎠ Fyx Fyy Fyz D(q) = m F F F zx zy zz

(5.31)

with Fαβ =



φαβ (l, l ) exp(−i d ll q),

(5.32)

l

d ll = r l◦ − r l◦ .

(5.33)

Note that d ll is always a lattice vector. A similar consideration to the one-dimensional case indicates that we can choose −

π π < qα ≤ (α = x, y, z) a a

(5.34)

as a region of q = (qx , q y , qz ). This region is the first Brillouin zone of a simple cubic crystal. To treat general (three-dimensional) atomic crystals, we need to generalize the above treatment in two respects. One is a generalization to cover crystals having non-orthogonal lattice vectors, a, b, and c with v 1 v 2 = 0 (v 1 , v 2 = a, b or c) for at least one pair. For such situations, there are no problems with Eqs. 5.29–5.33. We only need to consider an independent region of q. To this end, we define the following vectors, called reciprocal lattice vectors, 2π b×c vcell 2π b∗ = c×a vcell 2π c∗ = a×b vcell

a∗ =

(5.35) (5.36) (5.37)

5.2 Atomic Case

93

where vcell = a(b × c)

(5.38)

is the volume of a unit cell. Note that asterisks are used to indicate not complex conjugate but reciprocal lattice vectors, according to the tradition. It is easy to verify the following identities, 2π = a∗ a = b∗ b = c∗ c 0 = a∗ b = a∗ c = b∗ c = b∗ a = c∗ a = c∗ b

(5.39) (5.40)

If q and q  differ by a sum of integer multiples of reciprocal lattice vectors, the relation (5.41) exp(i qall ) = exp(i q  all ) holds because all is a lattice vector (Eq. 5.33). The most straightforward choice of the region, therefore, is a parallelepiped corresponding to a unit cell of reciprocal space. However, it is usual to adopt as the region the first Brillouin zone, which is the region consisting of points, of which the closest reciprocal lattice point is the origin q = 0. The other point to be generalized is the possibility of plural atoms in a unit cell. Representatives are simple salts such as rock salt (NaCl). Suppose n atoms exist in a cell. An additional index, j, distinguishing plural atoms, is necessary. Then, the coordinate of an atom is specified like r jl , and the force constant becomes φαβ ( j, j ; l, l ) =



∂2 V ∂rα,l ∂rβ,l

 .

(5.42)

eq

Using these force constants, the equation of motion is given by mj

d 2 r jα.l (t) =− φαβ ( j, j ; l, l )rβ,l (t). 2 dt

j l

(5.43)

β=x,y,z

The dynamical matrix is a (3n × 3n)-matrix D(q) of the form D(q) = M−1 F(q)

(5.44)

where M−1 and F(q) are composed of (n × n) small matrices of the size (3 × 3) as ⎞ m1−1 . . . 0 ⎟ ⎜ = ⎝ ... m−1 ... ⎠ j 0 . . . mn−1 ⎛

M−1

(5.45)

94

5 Lattice Dynamics of Molecular Crystals

⎞ m −1 0 0 j ⎝ 0 m −1 0 ⎠ m−1 j j = 0 0 m −1 j ⎛ ⎞ f11 . . . f1n ⎜ ⎟ F(q) = ⎝ ... f j j ... ⎠ ⎛

⎛

f j j

(5.46)

(5.47)

fn1 . . . fnn

⎞ Fj x j x Fj x j y Fj x j z = ⎝ Fj yj x Fj yj y Fj yj z ⎠ . Fjzj x Fjzj y Fjzj z

(5.48)

Components of the force matrix are given by F jα j β =



φαβ ( j, j ; l, l ) exp(−i d ll q).

(5.49)

l

5.3 Crystals of Rigid Molecules 5.3.1 Formulation Molecular crystals are surely crystals having plural atoms in a unit cell. The treatment given in the previous section can naïvely be applied. However, there are reasons to take the hierarchy of molecular systems into account [11]: It makes the understanding clear and coherent, and, more importantly, necessary computational cost is drastically reduced by virtue of the reduction in the dimension of the dynamical matrix, which is (6n mol × 6n mol ) if a molecule is regarded as a fundamental rigid particle (n mol , the number of molecules in a unit cell).3 Starting with such treatment, assuming rigid bodies, we may introduce relevant degrees of freedom into the formulation [12]. In classical mechanics of atoms, their masses are sole quantities characterizing them. Other quantities are necessary for rigid bodies [10]. First, we define the center of mass (center of gravity) of the body by R0 = m=

1 m 

 Rρm (R)d V, ρm (R)d V,

(5.50) (5.51)

where ρm (R) is a mass density, and the integration is over the body’s volume. The other necessary quantity is the moment of inertia i, which is a (3 × 3)-matrix with components 3 The

number becomes (5n mol × 5n mol ) if molecules are linear in shape.

5.3 Crystals of Rigid Molecules

95

 i αβ =



ρ(r)

(rγ2 δαβ − rα rβ )d V,

(5.52)

γ=x,y,z

where r = R − R0 . Using these quantities, the kinetic energy of a rigid body is given by

1 d r 2 1 T = m



+ t ωiω. (5.53) 2 dt 2 Here, ω is an angular momentum around the center of mass and is related to angular coordinates Θ by dΘ . (5.54) ω= dt If the potential energy is written as U (Θ), thus, the equation of angular motion around the center of mass reads i

d 2Θ = −∇Θ U (Θ). dt 2

(5.55)

For the molecular crystal, a unit cell of which contains n molecules consisting of p atoms, the potential energy can be expanded in terms of displacements from equilibrium structure, r p = R − R0 p θ p = Θ − Θ0 p

(5.56) (5.57)

where r p is the coordinate of the center of mass of the pth molecule with p that symbolically indicates all indexes necessary for distinguishing molecules. The expansion of potential energy (lattice energy) up to the second-order is V ≈ V0 +

1 2 p 

α,β=x,y,z

1



p

+ + +

2

p



p



p

  φRT αβ ( p, p )θα, p r β, p

p α,β=x,y,z

1 2

  φTR αβ ( p, p )r α, p θβ, p

p α,β=x,y,z

1 2

  φTT αβ ( p, p )r α, p r β, p

p





α,β=x,y,z

Here, the force constants are defined as

  φRR αβ ( p, p )θα, p θβ, p .

(5.58)

96

5 Lattice Dynamics of Molecular Crystals

φab αβ ( p,





p)=

∂2 V ∂xα, p ∂xβ, p

 ,

(5.59)

eq

where superscripts a and b are “T” or “R”, xαa is the α component of r with a = T or θ with a = R. Thus, the equations of motion are   d 2 rα, p  TR    φTT (5.60) =− αβ ( p, p )r β, p + φαβ ( p, p )θβ, p , 2 dt p β=x,y,z ⎛ RT ⎞   φxβ ( p, p )rβ, p + φRR xβ ( p, p )θβ, p 2 ⎜ d θp ⎟ RT RR   ip 2 = − ⎝ φ yβ ( p, p )rβ, p + φ yβ ( p, p )θβ, p ⎠ . (5.61) dt  β=x,y,z RR   p   φRT zβ ( p, p )r β, p + φzβ ( p, p )θβ, p

mp

Now, we assume the following forms for solutions r l (t) = r 0j exp[i(−ωt + r l◦ q)],

(5.62)

r l◦ q)],

(5.63)

θ l (t) =

θ 0j

exp[i(−ωt +

where j is the index distinguishing molecules in a unit cell, and r ◦l is a vector specifying the unit cell, to which the lth molecule belongs. Putting the above solutions into the equations of motion yields the following conditions for solutions, ω 2 Mu(t) = Fu(t),

(5.64)

where  M= ⎛

mT 0 0 mR

m1 . . . ⎜ .. m = ⎝ . mj 0 ... T

 ,

(5.65) ⎞

0 .. ⎟ , . ⎠ mn

⎞ mj 0 0 mj = ⎝ 0 m j 0 ⎠ , 0 0 mj ⎞ ⎛ i1 . . . 0 ⎟ ⎜ mR = ⎝ ... i j ... ⎠ ,

(5.66)

⎛

0 . . . in

(5.67)

(5.68)

5.3 Crystals of Rigid Molecules

97

⎛

⎞ r 1 (t) ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ r n (t) ⎟ ⎜ ⎟, u(t) = ⎜ ⎟ ⎜ θ 1 (t) ⎟ ⎜ . ⎟ ⎝ .. ⎠ θ n (t)  TT TR  F F F= , FRT FRR ⎞ ⎛ ab ab F11 . . . F1n ⎜ . . ⎟ Fab = ⎝ .. Fab .. ⎠ ,

Fab j j =

jj ab ab Fn1 . . . Fnn ⎛ ab F j x j x F jabx j y ⎜ ab ab ⎝ Fjyj x Fjyj y F jabz j x F jabz j y

F jabx j z

(5.69)

(5.70)

(5.71) ⎞

⎟ F jaby j z ⎠

(5.72)

F jabz j z

and ab F jα j β =

dl =



 φab αβ ( p, p ) exp(−i d l q),

p ◦ r0 −

r ◦l ,

(5.73) (5.74)

where r 0 is the position of an arbitrarily chosen reference cell, to which the pth molecule belongs. Thus, the dynamical matrix to be diagonalized to obtain the dispersion relations is (5.75) D(q) = M−1 F.

5.3.2 Properties of Problems In this section, some issues useful for computations are analyzed and discussed. We begin with the moment of inertia. Since an atomic mass is highly centered on its nucleus because of a light mass of an electron, we obtain  Rjm j , R0 =  mj

(5.76)

where m j is the mass of the jth atom in a molecule and sums run over atoms in the molecule. The moment of inertia i of a molecule is explicitly given by

98

5 Lattice Dynamics of Molecular Crystals

⎛

  ⎞ m j (y 2j + z 2j ) − m j x j y j − m j x j z j    m j (z 2j + x 2j ) − m j y j z j ⎠ . i = ⎝ −  m j yj x j   m j (y 2j + z 2j ) − m jz j x j − m jz j yj

(5.77)

The definitions of force constants (Eq. 5.59) indicate any of those has the symmeba   try, φab αβ ( p, p ) = φαβ ( p , p). Thus, considering Eqs. 5.73 and 5.74, the force matrix F is a Hermitian matrix. Now, we consider Eq. 5.64. Since the matrix M is a real symmetric matrix, it can be diagonalized using an orthogonal matrix B4 : t

 BMB i j = λi2 δi j .

(5.78)

Here, we write eigenvalues of M as λi2 (λi > 0) because these are either a mass of a molecule or principal values of the molecular moment of inertia tensor i. We denote λ for a diagonal matrix having diagonal components λi . By putting v = λt Bu, Eq. 5.64 becomes ω 2 v = λ−1 t BFBλ−1 v.

(5.79)

= D (q)v This relation means that D (q) has the same eigenvalues as the original D(q). The i j-component of D (q) is calculated as Di j =

l1

l2

l3

λi−1 δil1 Bl2 l1 Fl2 l3 Bl3 l4 λl−1 δl4 j 4

l4

1 = Bl2 i Bl3 j Fl2 l3 λi λ j l l 2

3

3

2

1 = Bl3 j Bl2 i (Fl3 l2 )∗ λi λ j l l = (D ji )∗

(5.80)

Therefore, D (q) is Hermitian, leading to real eigenvalues. Since a Hermite matrix is easier to handle than a general complex matrix D(q), ω 2 is usually obtained in this route. Original eigenvectors u j0 ( j = 1, . . . , 3n) are recovered by u = λ−1 Bv.

(5.81)

For the practical application of lattice dynamical theory to real systems, it is usually assumed that a sum of two-body interactions expresses the lattice energy. We consider some issues further under this assumption. Suppose that the total potential energy can be expressed as a sum of two-body interactions, v(r 1 , r 2 ) = v(r 2 , r 1 ). The potential energy is written as 4t BB

= Bt B = I, unit matrix.

5.3 Crystals of Rigid Molecules

99

1 v(r p , r p ). 2 p 

V =

(5.82)

p

Then, the force constants for p = p are expressed as  φab αβ ( p,



p)=

∂ 2 v(r p , r p ) a ∂x b ∂xα, p β, p

 ,

(5.83)

eq

where xαa is the α component of r (a = T) or θ (a = R). The range of the intermolecular interaction determines how many force constants are necessary for the lattice-dynamical calculation. On the other hand, for p = p , the self-force constant is given by   ∂ 2 v(r p , r p ) 1 φab , (5.84) αβ ( p, p) = 2 2  ∂xα, p p

eq

which consists of many terms. However, it is not only unnecessary but also harmful to evaluate them according to Eq. 5.84 because of the so-called sum rules of force constants. The rules are necessary in order to guarantee physical consistency for calculation results. To get the rules, we return to the equations of motion, Eqs. 5.60 and 5.61. Uniform translation of the crystal does not exert any force and torque on any molecules. Then, put r p = t (x0 , 0, 0) and θ p = 0 for all p , for example. The equations of motion result in φTT (5.85) 0 = −x0 αx ( p, p) p

0 = −x0

p

⎛

⎞  φRT x x ( p, p )  ⎠ ⎝ φRT . yx ( p, p ) RT φzx ( p, p )

(5.86)

Since x0 is arbitrary and the same is true for r p = (0, y0 , 0) and r p = (0, 0, z 0 ), we have  φTT (5.87) αβ ( p, p ) = 0, p

p

 φRT αβ ( p, p ) = 0.

(5.88)



Thus the self-force constants with superscripts “TT” and “RT” are given by φaT αβ ( p, p) = −



p = p

 φaT αβ ( p, p ).

(5.89)

100

5 Lattice Dynamics of Molecular Crystals

TR   Further, the symmetry between φRT αβ ( p, p ) and φβα ( p , p) yields RT φTR αβ ( p, p) = φβα ( p, p).

(5.90)

For the “RR” sector, we need another consideration. Let R0 p = t (X 0 p , Y0 p , Z 0 p ) be the location of the center of mass of the pth molecule in equilibrium. We consider the rotation of the crystal around the axis passing through R0 p parallel to the x-axis by θ0 and assume θ0 being so small that sin θ0 ≈ θ0 holds. Then, θ p = t (θ0 , 0, 0) for all p including p = p . On the other hand, displacements becomes ⎛

0

⎞

r p = ⎝ −θ0 (Z 0 p − Z 0 p ) ⎠ θ0 (Y0 p − Y0 p ).

(5.91)

Putting these into Eq. 5.61, we have ⎛

⎞  TR  TR    φRR x x ( p, p ) − (Z 0 p − Z 0 p )φx y ( p, p ) + (Y0 p − Y0 p )φx z ( p, p )  TR  TR  ⎠ ⎝ φRR   0= yx ( p, p ) − (Z 0 p − Z 0 p )φ yy ( p, p ) + (Y0 p − Y0 p )φ yz ( p, p )  TR  TR    φRR p zx ( p, p ) − (Z 0 p − Z 0 p )φzy ( p, p ) + (Y0 p − Y0 p )φzz ( p, p ) (5.92) ( p, p) has already been known, this equation deterbecause θ0 is arbitrary. Since φTR αβ RR mines φαx ( p, p). For example,

φRR x x ( p, p) = −



 φRR x x ( p, p )

p  = p

+

p

−

 (Z 0 p − Z 0 p )φTR x y ( p, p )





 (Y0 p − Y0 p )φTR x z ( p, p ).

(5.93)

p RR Similar relations necessary to determine φRR αy ( p, p) and φαz ( p, p) are derived by considering small rotations around axes parallel to y- and z-axes.

5.4 Crystals of Deformable Molecules Any motional degrees of freedom may be additionally introduced into a Lagrangian considered in the previous sections as long as they have no effects on translational and rotational degrees of freedom. Practically, this requirement is equivalent to that the new degrees of freedom keep the position and orientation of molecules. Intramolecular vibrations just fit the requirement if the so-called normal coordinates ξ for them are used. Although a limited number of internal degrees of freedom are often nec-

5.4 Crystals of Deformable Molecules

101

essary for reality [12], it is preferable to formulate the problem using the normal coordinates for intramolecular vibrations [13]. As well known, there are (3n − 6) or (3n − 5) intramolecular vibrational modes for a molecule of non-linear or linear shape consisting of n atoms, respectively. Thus the Lagrangian of deformable molecules can be written as L =T −V





1 dξ p 2 1

d r p 2 1 t

+

m p

ω pi pω p + = −V +

dt , 2 dt 2 2 p

(5.94)

where the inner sum runs over intramolecular modes. Note that the potential energy V in the above Lagrangian includes not only the lattice energy but also the intramolecular potential energy. The force matrix has the following form, ⎛

⎞ FTT FTR FTI F = ⎝ FRT FRR FRI ⎠ , FIT FIR FII

(5.95)

where the superscript “I” is for intramolecular vibrations. Components of the force matrix are ab  F jα φab (5.96) j β = αβ ( p, p ) exp(−i d l q), p

with d l defined in Eq. 5.74. Here, α and β are used to distinguish intramolecular modes if the superscript “a” and “b” are “I”. The force constants are defined as φab αβ ( p,





p)=

∂2 V ∂xα, p ∂xβ, p

 ,

(5.97)

eq

similarly to Eq. 5.59, while understanding x = ξ for a = I. The consideration on the sum rule similarly to the case of rigid molecules yields φaT αβ ( p, p) = −



 φaT αβ ( p, p ),

(5.98)

p  = p aT φTa αβ ( p, p) = φβα ( p, p),

(5.99)

with a = T, R or I. The sum rules for the RR sector remain the same (see the end of the previous section). Special consideration is necessary for the “II” sector. Suppose a simple crystal, a unit cell of which contains two diatomic molecules (n mol = 2). Since a single molecule has only one intramolecular vibration (stretching), the “II” sector of the mass and force matrices are (2 × 2)-matrices. That of the mass matrix is

102

5 Lattice Dynamics of Molecular Crystals

 MII =

μ0 0μ

 ,

(5.100)

where μ is the reduced mass of two atoms constituting the molecule. It is important to remember that the intramolecular vibrations are involved in FII . Indeed, FII is decomposed into two parts, FII = FII(0) + FII(1)   II II   F11 F12 φin 0 + . = II II F21 F22 0 φin with FiII(1) = j



φiIIj ( p, p ) exp(−i d l q).

(5.101)

(5.102)



p = p

Here, φin is a force constant for the stretching vibration of a molecule. The presence of φin is one of the reasons for the absence of sum rules for the “II” sector. It is easy to extend this treatment for general molecules having more internal vibrations. The inclusion of many modes is unnecessary in most cases because of generally weak interaction between lattice and molecular vibrations. However, there exist cases where a limited number of intramolecular modes having frequencies similar to those of lattice vibrations are crucially important for understanding properties of molecular crystals, as will be exemplified in the next section. Note that FII is not a diagonal matrix. Intramolecular vibrations of n mol translationally inequivalent molecules are not equivalent to one another but split into n mol even for a common wavevector q. The splitting also occurs at q = 0, which is responsible for most optical spectroscopy as discussed in the case of alternate array of atoms. This splitting is an example of a phenomenon called the factor group splitting, also known as the Davydov splitting.

5.5 Related Issues and Examples 5.5.1 Heat Capacity and Debye Temperature The number of states as a function of energy is generally called as the density of states. A density of states of lattice vibrations is a phonon density of states. The phonon density of states is fundamental for the thermodynamic properties of non-metallic crystals because lattice vibrations in the harmonic treatment described in this chapter are mutually independent of each other. That is, we can calculate thermodynamic functions by adding those of harmonic oscillators. For example, the heat capacity of a crystal is expressed using a phonon density of state, g(ω) as

5.5 Related Issues and Examples

103



 C(T ) =

g(ω)CE

 T dω, ΘE

(5.103)

where ΘE = ω/kB is a characteristic temperature, and CE [T /Θ E ] a universal function expressing the heat capacity of a harmonic oscillator having ΘE . They are called the Einstein temperature, and the Einstein heat capacity, respectively, after the Einstein model of heat capacity of crystals, which is the first quantum model capable of explaining its small magnitude at low temperatures and the saturation to the classical value at high temperatures. The Debye model is an established model of the heat capacity of solids. It correctly describes the temperature dependence of heat capacity at low temperatures (proportional to a cube of temperature, ∝ T 3 ). The Debye temperature, ΘD is the model’s only parameter, which differs from material to material. In reality, the Debye model is a model taking only sound waves into account. As seen in Sect. 5.2.1.2, the sound velocity is given by the slope of acoustic branches near q = 0, where their dispersion relations are linearly dependent on the wavevector, i.e., ω ∝ |q|, in any direction. Thus, the surface area of equal frequency in the q space, A(ω), increases proportionally to |q|2 for three-dimensional solids, irrespective of the degree of anisotropy. Since A(ω)dq is proportional to the number of vibrational modes with ω, the number of vibrational modes, i.e., a phonon density of states g(ω), increases proportionally to ω 2 : g(ω) = cN ω 2 (c, a constant related to the sound velocity). On the other hand, the number of motional degrees of freedom is finite because of the atomic nature of any solids. If the solid consists of N atoms, the total number of degrees of freedom should be three times the number of atoms (N ). Thus, we have 

∞

3N =

g(ω)dω.

(5.104)

0

To fulfill this requirement, Debye assumed the presence of the highest frequency, ωD , which is determined by 

ωD

3N =

cN ω 2 dω.

(5.105)

0

Namely, ωD = (9/c)1/3 . The relation ΘD = ωD /kB gives the Debye temperature, a characteristic temperature that reflects both the (averaged) sound velocity and the highest frequency of the lattice vibration. Note that the “averaging” sound velocity is taken over not only three acoustic branches but also the direction of the wave vector. A simple formula to express this averaged sound velocity is unavailable for general molecular crystals having anisotropy, in contrast to the case of simple isotropic (cubic) crystals. When the solid under consideration is a crystal, a unit cell of which contains only a single atom, the meaning of the Debye temperature is safely duplicate, as described in the preceding paragraph. When a unit cell of an atomic crystal contains plural atoms, such as cesium chloride (CsCl), two characteristic temperatures

104

5 Lattice Dynamics of Molecular Crystals

should be different because there are acoustic and optical branches in the dispersion relation. The sound velocity solely depends on the former, while the optical branch is undoubtedly involved in the highest frequency of the lattice vibration. With the increase in the number of atoms in a unit cell (n atom ), the number of the optical branches increases while those of acoustic branches remain the same at 3 per unit cell (or 3/n atom per atom). Although the highest frequency of the lattice vibration can be significantly larger than that of the acoustic branch, it can serve as measures for both purposes if taking the number of atoms correctly into account. When the solid is a crystal consisting of rigid molecules, which have translational and rotational degrees of freedom, the lattice vibration includes optical branches of rotational degrees of freedom. There exist 6 (not 3) degrees of freedom per molecule, accordingly. Only the three translational degrees of freedom can potentially contribute to the acoustic branches, which carry 3 degrees of freedom per unit cell. On the other hand, all 6 degrees of freedom contributes to the lattice heat capacity. The total degrees of freedom increase accordingly if we count internal vibrational modes. In analyses of the temperature dependence of heat capacities of solids, an equivalent Debye temperature determined according to  Cexp (T ) = CD

T ΘD (T )

 (5.106)

is often used. Here, the left-hand side is the experimental heat capacity, and CD [T /Θ D ] is a universal function for the Debye heat capacity as a function of the reduced temperature T /ΘD . This analysis is useful because ΘD is constant if the temperature dependence of heat capacity strictly follows the Debye model. Although the real material does not follow precisely the model, the weak deviation is reflected in the temperature dependence of ΘD . An example is shown in Fig. 5.1, where a weak anomaly in heat capacity is visible in the plot of equivalent Debye temperatures assuming 9 degrees of freedom per molecule.

50

160

-1

ΘD / K

40 150

Cp / J K mol

30

-1

Fig. 5.1 Experimental heat capacities of crystalline biphenyl (C6 H5 –C6 H5 ) and corresponding equivalent Debye temperatures deduced assuming 9 degrees of freedom per molecule. A weak anomaly due a structural phase transition is enhanced in the latter. (plotted using the data in Bull. Chem. Soc. Jpn., 61, 679 (1988) [14])

140 20

20 30

40 T/K

50

60

5.5 Related Issues and Examples

105

To make such analyses on non-simple solids such as molecular solids, the assumption of the number of degrees of freedom is necessary. If three is assumed, the limiting value to the absolute zero, lim T →0 ΘD (T ), reflects the (averaged) sound velocity, which can be compared with simple solids. On the other hand, the assumption of 6 (3 translational and 3 rotational degrees of freedom) yields the measure of the highest frequency of the lattice vibration. If the interest is solely on the demonstration of anomalous behavior, the number of degrees of freedom may be chosen arbitrarily, as in the example shown above.

5.5.2 Debye–Waller Factor Standard diffraction crystallography practically utilizes the so-called Debye–Waller factor to describe the intensity of diffracted radiation (X-ray, neutron, or electron) relative to the ideal intensity expected for the complete crystal. The factor has originally been introduced to describe the effect of thermal motion [15, 16]. Nowadays, however, the factor is experimentally obtained as the parameter that reflects the random distribution of an atom around the ideal (i.e., averaged) position. When all atoms independently exhibit isotropic displacement around the averaged position, the Debye–Waller factor has the following form:   1 exp − |q|2 σ 2 . 2

(5.107)

Here, q is the scattering vector, and σ 2 is the mean squared displacement. Although the distribution of atomic displacement may be anisotropic in reality, the factor reflects the mean squared displacement. In practice, U = σ 2 and B = 8π 2 U are reported as the temperature parameter or mean squared displacement parameter. In the quantum mechanical treatment of a harmonic oscillator with a mass m and the potential energy function 21 kx 2 , the expectation value of the squared displacement xn2 associated with the nth energy-eigenstate is given by xn2 =

  1 ω n+ (n = 0, 1, 2 . . .), k 2

(5.108)

√ where ω = k/m is the angular frequency. The above expression is equivalent to the energy of the nth state E n divided by k. That is, xn2 = E n /k. This equality indicates that the equipartition of the total energy to the kinetic and potential energies holds for each energy-eigenstate. Thus, the same equality holds for the thermal average: x 2  =

1 E k

106

5 Lattice Dynamics of Molecular Crystals

ω = k



 −1  1 ω + exp −1 . 2 kB T

(5.109)

The above formula indicates that the squared displacement saturates at a finite magnitude at T = 0 due to the so-called zero-point vibration and is proportional to temperature at high temperatures as x 2  ≈ kB T /k. Since the frequency range of the external vibration is generally low in molecular crystals, the proportionality holds in many cases (typically at T  100 K). Thermal vibration of molecules (and consequently riding atoms) is expressed by a superposition of normal modes as long as we assume the displacement is small (harmonic approximation). The weight of each normal mode depends on the initial condition and temperature. However, the superposition of normal modes practically yields the same temperature dependence of x 2  ∝ T because the zero-temperature magnitude of the squared displacement is small for modes with high angular velocity as ω/k = 1/mω. Thus, if we ignore the effect of the correlation of atomic motions, we can usea single-particle description for the averaged information. A rough estimate yields x 2  ≈ 0.1 Å around room temperature. Note that a temperature parameter in the Debye–Waller factor is, irrespective of its definition, proportional to the mean squared displacement. We can thus use the proportionality to check whether an atom (or a group of atoms) is well localized at a single position or disordered over plural positions [17–19]. If the atom is disordered over two equivalent positions apart by d and the squared displacement is σ 2 at each position, the net squared displacement becomes x 2  = σ 2 + ≈

d2 4

d2 kB T + . k 4

(5.110)

The plot of an apparent temperature parameter against temperature exhibits a finite positive intercept at T = 0 unless d = 0, i.e., the atom is well localized.

5.5.3 Lattice Instability and Structural Phase Transition The effects of intramolecular vibrations can be significant on lattice vibrations and related properties when their frequencies are within a range of the lattice vibrations. This situation often happens for twisting vibrations of some atomic groups around a single bond because the bond itself is axially symmetric. Crystalline biphenyl (C6 H5 –C6 H5 ) is a typical example to demonstrate the importance of the coupling between the external and internal motional degrees of freedom. The relevant degrees of freedom is only the mutual twist of two phenyl groups around the connecting C–C single bond in the molecule. Besides, the moment of inertia rel-

5.5 Related Issues and Examples 100

ν / cm–1

Fig. 5.2 Dispersion relations of branches formed by translational and twisting degrees of freedom in the direction of b∗ of the room temperature phase of crystalline biphenyl with (solid line) and without (pure twist, broken line; translational, dot-dash line) their coupling. Symmetric and antisymmetric modes are shown separately for clarity. Reproduced with permission from Phys. Status Solidi b, 118, 129 (1983) [5]

107

50

antisymmetric

0 0

b* –– 2

symmetric

0

evant is much larger than that of the methyl group by more than 30 times, resulting in its low characteristic energy in the isolated state. Although the first few lattice dynamical calculations assumed rigid molecules [20, 21], the importance of the coupling between the lattice vibration and the internal twist has been identified [5, 22–24]. Meanwhile, a structural phase transition associated with a change in the internal twist angle was discovered around 40 K (Fig. 5.1) for the crystal: The molecule is planar above the transition temperature while it is twisted below the transition temperature [25]. The transition is of the so-called displacive type accompanying a soft mode. The soft mode is a vibrational mode, the frequency of which strongly depends on temperature and vanishes at the transition temperature. The vanishing of the vibrational frequency means that the eigenvector (the combination of displacements in the crystal lattice) is statically realized below the transition temperature. Unless the mode belongs to the totally symmetric species of the symmetry of the higher symmetry phase, the appearance of the static displacement causes the lowering of the symmetry. If the wavevector is at a general point in the Brillouin zone, the structural modulation appearing below the transition temperature is incommensurate to the original (unmodulated) lattice. The fact that the modulation is incommensurate to the mother lattice implies the absence of any mechanism to lock the modulation wavevector k∗IC at a special wavevector. Thus, the k∗IC is generally dependent on temperature. The phase transition in crystalline biphenyl is just a displacive transition to an incommensurate phase with modulation wavevectors k∗ ≈ 21 b∗ . Figure 5.2 shows the dispersion relations of phonons that potentially couple to molecular twisting degrees of freedom calculated for the room temperature phase. The room temperature phase belongs to the space group P21 /a with two molecules on inversion centers in a unit cell. We can easily see that translational and rotational displacements of a molecule on an inversion center are decoupled owing to their different symmetries, e.g., the former changes the direction of the displacement while the latter keeps it upon the spatial inversion. Since the twisting displacement has the same symmetry as the translational displacements, the translational degrees

108

5 Lattice Dynamics of Molecular Crystals

of freedom can couple with the molecular twist. Thus, Fig. 5.2 contains 8 = (3 + 1) × 2 branches formed by 3 translational and 1 twisting degrees of freedom per molecule. The results without the coupling are drawn by a broken line for the twisting branches and by dot-dash lines for translational branches. It is evident that, even without the coupling, the twisting branches exhibit a notable dispersion against the wavevector. The absence of the coupling allows the crossing of branches. When the coupling is taken into account, their crossing disappears, and a dip in the dispersion relation appears near b∗ /2. Remember that the phonon frequencies are square-roots of eigenvalues of a kind of dynamical matrix. Negative eigenvalues are equivalent to the instability of an assumed crystal structure. Thus, a dip in a phonon dispersion relation is, in general, a symptom of some potential lattice instability. The appearance of the dip in Fig. 5.2 indicates that the presence of the soft twisting degree of freedom and its coupling produces the lattice instability. The importance of the coupling in this lattice instability was demonstrated through varying its strength artificially. The authors of [24] indicated that the crystal structure loses stability for some conditions. Later, the other group [26] indicated that the negative intramolecular contribution to the force field for the twisting is essential. Crystalline bis(4-chlorophenyl)sulfone [BCPS, (ClC6 H4 )2 SO2 ] is another example to exhibit a structural phase transition in which intramolecular motional degrees play an important role. The crystal at room temperature belongs to the space group C2/c with molecules on the two-fold axis. The crystal undergoes a displacive phase transition around 220 K to an incommensurate phase having the modulation wavevector k∗ ≈ 0.8c∗ . The lattice-dynamical calculation incorporating molecular deformations [27] was used to clarify the instability of the crystal structure. An idealized molecular model is assumed: all bond angles are 2π/3, and benzene rings are regular hexagons. Only two twisting degrees of freedom are incorporated if applicable. To separate the twisting degrees of freedom from the translation and rotation of a whole molecule, they are included in the calculation as symmetric and antisymmetric combinations concerning the twofold symmetry around the axis that coincides with the bisectors of ∠OSO and ∠CSC. Intermolecular interaction is expressed by sums of atom-atom potentials discussed in Sect. 1.2.5.2. Before the lattice-dynamical calculations, the lattice energy is minimized while keeping the symmetry of the crystal structure experimentally known, in order to ensure the structure is minimum in the lattice energy. Figure 5.3 shows the phonon dispersion relations calculated under three different force constants for the twist of chlorophenyl groups. A dip in the dispersion relation in the lowest-lying mode can be recognized at q ≈ 0.8c∗ in the result assuming rigid molecules. This dip can be regarded as a symptom of the potential instability for the modulation with this wavevector. On the other hand, the crystal is unstable as indicated by imaginary frequencies for some wave vectors if the intramolecular restoring force is assumed null. Indeed, with assuming the intramolecular twisting frequency being 70 cm−1 , the lattice vibrations have positive frequencies everywhere in the q space, indicating the stability of the crystal structure. Thus, the intramolecular twisting degrees of freedom does not cause instability but enhances it in contrast to the case of crystalline biphenyl.

5.5 Related Issues and Examples

109

a

b

c

ν / cm

–1

100

50

0

0

c*

0

c*

0

c*

Fig. 5.3 Phonon dispersion relations in the direction of c∗ of the room temperature phase of crystalline bis(4-chlorophenyl)sulfone assuming a rigid molecule model, b flexible molecule with a force constant corresponding to 70 cm−1 for the twisting vibration of a chlorophenyl group around the single bond to the central sulfur atom, and c flexible molecule with free twisting degrees of freedom. Imaginary frequencies are drawn below zero. Reproduced with permission from Solid State Commun., 81, 241 (1992) [27]

It is noteworthy that the calculations indicating the potential instability of crystal structures are performed for high-temperature phases in both examples. Hightemperature phases have a higher symmetry than the low-temperature phase. Besides, utilized intermolecular interactions are of a simple atom-atom potential model. An unknown phase transition occurring at a low temperature is possibly predicted based on only a known crystal structure [26].

5.5.4 Anharmonicity and Thermal Expansion The volume thermal expansivity α is defined as   1 ∂V α= V ∂T p   ∂ ln V = . ∂T p

(5.111) (5.112)

In the case of the ideal gas, the enhancement of the thermal motion of particles results in the expansivity of 1 (5.113) α= . T

110

5 Lattice Dynamics of Molecular Crystals

The relation is the well-known Charles’ law. A counterintuitive situation, however, occurs in the case of crystalline solids within the harmonic approximation. To see this, we treat isotropic solids within the classical treatment.5 Suppose the harmonic oscillator with the potential energy function, 21 kx 2 . The thermal average of displacement is " ! ∞ kx 2 −∞ x exp − 2kB T d x " ! . (5.114) x = ∞ kx 2 −∞ exp − 2kB T d x The denominator is the normalization constant. Since the integrand in the numerator is an odd function of x, the thermal average is identically null. This result indicates that the thermal agitation cannot bring about thermal displacement (expansion) if the potential energy is purely harmonic (quadratic). We need the deviation from the harmonicity, i.e., anharmonicity, to discuss thermal expansion. We assume the following form of the potential energy function in the vicinity of x = 0, 1 2 1 3 kx − f x . (5.115) 2 3 We approximate the Boltzmann factor by truncating the series at the lowest order in the anharmonicity ( f ) as  exp −

− 13 f x 3 kB T

1 kx 2 2



    f x3 kx 2 exp = exp − 2kB T 3kB T     kx 2 f 3 x exp − . ≈ 1+ 3kB T 2kB T

(5.116)

Since the non-vanishing result comes only from the even integrand, we need, as in the case of Eq. 5.114, to evaluate simply √   2π kx 2 d x = 5/2 (kB T )3/2 f. x exp − 3kB T −∞ 2kB T 3k 

f

∞

4

(5.117)

for the numerator and 

    √ kx 2 kB T 1/2 d x = 2π exp − . 2kB T k −∞ ∞

(5.118)

for the denominator. Finally, we reach x ≈ 5 The

f kB T. 3k 2

(5.119)

anisotropy brings about a complexity because of the tensor form of the expansivity, whereas the quantum treatment does not alter the fundamental properties of the issue.

5.5 Related Issues and Examples

111

We see that the anharmonicity brings about a thermal variation in the atomic position, which is linear both in temperature and the anharmonicity parameter within the lowest-order approximation. Since any vibrational modes in real crystals are more or less anharmonic, thermal “expansion” always occurs. It is noteworthy that the direction of the variation depends on the sign of the anharmonicity parameter f . It seems useful to describe the general background of the thermal expansion briefly [28]. First, using Maxwell’s relations, we have  ∂ ln V α= ∂T p     ∂p ∂ ln V =− ∂ p T ∂T V   ∂p = κT ∂T V   ∂S = κT , ∂V T 

(5.120)

where κT is the isothermal compressibility defined as 1 κT = − V



∂V ∂p

 .

(5.121)

T

If we can decompose the system into non-interacting subsystems that share the system volume, the last expression of the expansivity (Eq. 5.120) indicates that we can decompose the thermal expansivity into those of subsystems as α=



αi .

(5.122)

subsystems

because the entropy is an extensive quantity and the isothermal compressibility is common for subsystems. The lattice vibration contributes as a subsystem(s)6 in any real cases. The analysis of the thermal expansivity often utilizes a thermodynamic function, called the Grüneisen function. The Grüneisen function is defined as   ∂ ln T (5.123) γ(T, V ) = − ∂ ln V S   ∂S 1 = , Cv ∂ ln V T where Cv is the isochoric heat capacity (one at constant volume). The second equality uses one of Maxwell’s identities. 6 Each

vibrational mode can, in principle, play as a role of individual subsystem because of their mutual independence.

112

5 Lattice Dynamics of Molecular Crystals

Using the Grüneisen function, we can express the thermal expansivity as α=

κT C v γ(T, V ), V

(5.124)

which results from  dS =

∂S ∂T



 dT + V

= Cv d ln T +

∂S ∂V



αV d ln V. κT

dV T

(5.125)

Since the isochoric heat capacity and isothermal compressibility are non-negative based on the requirement of the stability of thermodynamic systems, the sign of thermal expansivity generally coincides with that of the Grüneisen function. The second line of Eq. 5.123 enables its definition of each subsystem. Then, we can express the Grüneisen function of the system using those of subsystems as  γ(T, V ) =

i

γi (T, V )Cv,i (T, V )  . i C v,i (T, V )

(5.126)

This expression indicates that the total Grüneisen function is the Cv -weighted average of those of subsystems. Although the temperature dependence of Grüneisen function is generally weak, the total Grüneisen function depends on temperature because the isochoric heat capacity depends on temperature, as seen in the previous section. Many solid substances exhibit a positive thermal expansivity. If a substance shrinks upon heating, its thermal expansivity and Grüneisen function are negative. The negative Grüneisen function requires a subsystem(s) that has a negative contribution at the temperature after Eq. 5.126. This property enables the identification of the subsystem responsible for the negative expansivity if we have a way to know Grüneisen functions of subsystems. When a single characteristic energy ε specifies the internal energy of a system, the Grüneisen function is given by γ(T, V ) = −

d ln ε . d ln V

(5.127)

The condition exactly holds for harmonic oscillators, each of which serves as a subsystem, because ωi of each vibrational mode characterizes its energy levels. Mode Grüneisen parameters defined as γi = −

d ln ωi . d ln V

are accessible through experiments under pressure.

(5.128)

5.5 Related Issues and Examples

113

5.5.5 Large Amplitude Motion A molecule of trans-stilbene (C6 H5 –CH=CH–C6 H5 ) has two “soft” intra-molecular vibrational modes: the twist of phenyl groups around respective single bonds to the central CH=CH moiety. Spectroscopic studies [29–31] have revealed that the characteristic energy of these twisting degrees of freedom is within a range expected for external lattice vibrations of molecular crystals. trans-Stilbene crystallizes into a monoclinic lattice with a unit cell containing two molecules on inversion centers [32– 34]. One molecule is orientationally disordered while the other has an abnormally short C=C bond without seeming disorder. Based on the principles and practical application of structural determination through scattering experiments (irrespective of X-ray or neutron beam), a large amplitude of a molecular motion was suggested as a possible cause for the short bond in the central moiety [34]. The motion suggested is a combination of the overall libration of a molecule and the intramolecular twisting of phenyl rings while keeping the orientation of rings (Fig. 5.4). A lattice-dynamical calculation was performed to see whether such a composite motion happens in the crystalline lattice [35]. For lattice-dynamical calculations, an idealized molecular model is assumed. Only two twisting degrees of freedom are incorporated if applicable. To separate the twisting degrees of freedom from the translation and rotation of a whole molecule, they are included in the calculation as gerade and ungerade combinations concerning the inversion symmetry at the molecular center (of mass). Intermolecular interaction is expressed by sums of atom-atom potentials discussed in Sect. 1.2.5.2. Minimization of the lattice energy while keeping the symmetry of the crystal structure experimentally known yields a reasonable convergence when the intramolecular potential for the twists is assumed to be completely flat. The intramolecular force constants for two combinations of the twisting degrees of freedom are set to null, accordingly. Even assuming so, the twisting degrees of freedom exhibit the frequency corresponding to ca. 140 cm−1 at q = 0 (all equivalent molecules oscillate in phase).

Fig. 5.4 Suggested motion of a trans-stilbene molecule in crystal as a possible cause of a seeming shrinkage of the central C=C bond. Hydrogen atoms are omitted for clarity. Reproduced with permission from Bull. Chem. Soc. Jpn., 69, 909 (1996) [35]

114

5 Lattice Dynamics of Molecular Crystals

Fig. 5.5 Phonon dispersion relations of crystalline trans-stilbene assuming rigid and flexible molecules. Adapted with permission from Bull. Chem. Soc. Jpn., 69, 909 (1996) [35]

Lattice dynamical calculations are performed while assuming rigid and flexible molecules. All possible sum rules for force constants are considered in calculations. Figure 5.5 shows the resultant dispersion relations and phonon density of states. A comparison of the two results indicates that the highest frequency of the lattice vibrations (the high-frequency end of the density of states) are shifted by ca. 30 cm−1 . Besides, the density of states around 50 cm−1 significantly increases. The increase means that the twisting degrees of freedom of phenyl groups significantly contribute to the lattice vibration in two frequency regions. Close inspection over eigenvalues (motional patterns) at q = 0 indicates that the suggested motion as a cause for the short C=C bond is included in the lower frequency region in an almost pure form. Since the representative frequency, 50 cm−1 , corresponds to ca. 35 K in temperature, such a mode is well excited at, say, 200 K or above. The identification of the composite motion as the cause of the short C=C bond length seems acceptable, accordingly. Anharmonic effects may also contribute to the phenomenon. The phonon density of states obtained through integration of the dispersion relations over the whole q range is shown in Fig. 5.6. In the lowest frequency region, two models give essentially the same results. This coincidence results from that the finite frequency of internal modes having even vanishing internal force constants.

5.5 Related Issues and Examples

115

Fig. 5.6 Phonon density of states (g(ν)) for crystalline trans-stilbene assuming rigid (dotted line) and flexible (solid line) molecules. That calculated while incorporating only the twisting degrees of freedom of four phenyl groups (possesed by two molecules in a unit cell) are also shown by broken line. Reproduced with permission from Bull. Chem. Soc. Jpn., 69, 909 (1996) [35]

5.5.6 Degrees of Freedom for Membrane Dynamics The understanding of lattice-dynamical treatment gives valuable insights into the dynamics of membranes consisting of molecules, such as lipid bilayers. Discussions on membranes often consider the dynamics of membranes within the continuum approximation, quite similar to the Debye model of heat capacity. Although the continuum approximation is handy and attractive, we need to take into account appropriately the underlying molecular (atomic) nature of the system. The Debye model assumes only acoustic branches within the approximation and, instead, introduces the cut-off frequency to adjust the total number of motional degrees of freedom. We assume an isolated bilayer with the flat geometry. Even if the membrane is fluidic within the membrane, the molecular displacement normal to the membrane feels the restoring force as long as the membrane is stable. We consider only displacements of this kind, hereafter. Since the membrane is two-dimensional, the wavevector (the direction of propagation) is also two-dimensional restricted within the membrane layer. All modes we discuss are transverse accordingly. The wavelength has the upper limit 2l, where l is a typical (averaged) distance between neighboring molecules. Because of the bilayer nature of the membrane, “lattice” vibrations split into two branches, acoustic and optical ones. When the length of the wavevector is small, i.e., long wavelength, the “vibration” belonging to the acoustic branch is a part of the deformation of the membrane as a whole, which brings the repulsive interaction between membranes, known as the Helfrich interaction [36]. It is not trivial what range of the wavevector is relevant to this interaction. Each fluctuation mode plausibly obeys the equipartition law. However, fluctuations with the short-wavelength possibly may contribute little to the repulsion, though Helfrich assumed equal contributions of all modes. Since the numbers of the vibrational modes and molecules coincide, the equipartition law does not hold for molecules if a significant part of modes scarcely contributes to the repulsion. The equipartition law can be assumed only when the mode-specific phenomena or techniques are subject to discussion [37].

116

5 Lattice Dynamics of Molecular Crystals

The optical branch of the vibration normal to the membrane is responsible for the fluctuation of membrane thickness [38]. The mode with the vanishing wavevector corresponds to uniform breathing of the volume and is out of consideration under the assumption of the volume conservation. The physically reasonable length of wavevector is necessary, accordingly, to describe the thickness fluctuation. It is not trivial what range of the wavevector is relevant to the thickness fluctuation depending on assumed models. The equipartition law seems not to hold for molecules, accordingly. When the membrane is not a molecular bilayer but a single layer, the dispersion relation has only the acoustic branch. The discussion on the deformation as a whole remains the same as that of bilayers, while one about the thickness fluctuation becomes significantly different. Its absence can be a claim when one believes the anti-phase displacements of molecules being essential. On the other hand, the dynamics with a sufficiently short wavelength of the molecular scale may serve as a microscopic origin of the thickness fluctuation sensed by dull techniques without atomic resolution. In both cases, the equipartition law does not hold in these cases. Discussions on membranes often proceed on the area basis. In this respect, it is noteworthy that the average area of molecular occupancy should correctly count the number of molecules involved. No special attention is necessary for the singlelayered membrane. In the case of bilayers, the numbers of modes involved in acoustic and optical branches are exactly half of the total number of molecules. The energy carried by each branch is the same as one in the single-layer case, in which only the acoustic branch exists.

References 1. M. Born, K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, Oxford, 1954) 2. D.J. Hooton, Phil. Mag. 3, 49–54 (1958) 3. S. Takeno, Prog. Theor. Phys. 45, 137–173 (1970) 4. A. Nakanishi, T. Matsubara, J. Phys. Soc. Jpn. 39, 1415–1416 (1975) 5. N.M. Plakida, A.V. Belushkin, I. Natkaniec, T. Wasiutynski, Phys. Status Solidi b 118, 129–133 (1983) 6. N.J. Zabusky, M.D. Kruskal, Phys. Rev. Lett. 15, 240–243 (1965) 7. M. Toda, J. Phys. Soc. Jpn. 22, 431–436 (1967) 8. S. Takeno, Prog. Theor. Phys. 71, 395–398 (1984) 9. F. Fillaux, C.J. Carlile, Phys. Rev. B 42, 5990–6006 (1990) 10. L.D. Landau, E.M. Lifshitz, Mechanics, 3rd edn. (Butterworth-Heinemann, Oxford, 1982) 11. S. Califano, V. Schettino, N. Neto, Lattice Dynamics of Molecular Crystals. Lecture Notes in Chemistry, vol. 26 (Springer, New York, 1981) 12. D. Kirin, J. Chem. Phys. 100, 9123–9128 (1994) 13. G.S. Pawley, S.J. Cyvin, J. Chem. Phys. 52, 4073–4077 (1970) 14. K. Saito, T. Atake, H. Chihara, Bull. Chem. Soc. Jpn. 61, 679–688 (1988) 15. P. Debye, Ann. Phys. 348, 49–92 (1913) 16. I. Waller, Z. Phys. A 17, 398–408 (1923) 17. M.H. Lemee, L. Toupet, Y. Delugeard, J.C. Messager, H. Cailleau, Acta Cryst. B 43, 466–470 (1987)

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18. H. Saitoh, K. Saito, Y. Yamamura, H. Matsuyama, K. Kikuchi, M. Iyoda, I. Ikemoto, Bull. Chem. Soc. Jpn. 66, 2847–2853 (1993) 19. Y. Yamamura, E. Saito, H. Satoh, N. Hoshino, K. Saito, Chem. Lett. 41, 119–121 (2012) 20. T. Luty, Mol. Cryst. Liq. Cryst. 17, 327–354 (1972) 21. I. Natkaniec, M. Nartowski, A. Kulczycki, J. Mayer, M. Sundnik-Hrynkiewicz, J. Mol. Struct. 46, 503–508 (1978) 22. E. Burgos, H. Bonadeo, E. D’Alessio, J. Chem. Phys. 65, 2460–2466 (1976) 23. T. Wasiutynski, I. Natkaniec, A.V. Belshkin, J. Phys. (Paris), Colloq. C6, 599–601 (1981) 24. H. Takeuchi, S. Suzuki, A.J. Dianoux, G. Allen, Chem. Phys. 55, 153–162 (1981) 25. H. Cailleau, Incommensurate Phases in Dielectrics. II Materials, eds. by R. Blinc, A.P. Levanyuk (North-Holland, Oxford, Amsterdam, 1986) 26. K. Saito, M. Asahina, Y. Yamamura, I. Ikemoto, J. Phys.: Condensed Matt. 7, 8919–8926 (1995) 27. K. Saito, K. Kikuchi, I. Ikemoto, Solid State Commun. 81, 241–243 (1992) 28. T.H.K. Barron, J.G. Collins, G.K. White, Adv. Phys. 29, 609–730 (1980) 29. Z. Meíc, H. Güsten, Spectrochim. Acta A 34, 101–111 (1978) 30. A. Bree, M. Edelson, Chem. Phys. 51, 77–88 (1980) 31. T. Suzuki, N. Mikami, M. Ito, J. Phys. Chem. 90, 6431–6440 (1986) 32. C.J. Finder, M.G. Newton, N.L. Allinger, Acta Cryst. B 30, 411–415 (1974) 33. J.A. Bouwstra, A. Schouten, J. Kroon, Acta Cryst. C 40, 428–431 (1984) 34. K. Ogawa, T. Sano, S. Yoshino, Y. Takeuchi, K. Toriumi, J. Am. Chem. Soc. 114, 1041–1051 (1992) 35. K. Saito, I. Ikemoto, Bull. Chem. Soc. Jpn. 69, 909–913 (1996) 36. W. Helfrich, Z. Naturforsch C 30, 841–842 (1975) 37. J.F. Faucon, P. Mitov, P. Méléard, I. Bivas, P.J. Bothorel, J. Phys. (Paris) 50, 2389–2414 (1989) 38. M. Nagao, E.G. Kelley, R. Ashkar, R. Bradbury, P.D. Butler, J. Phys. Chem. Lett. 8, 4679–4684 (2017)

Chapter 6

Melting of Molecular Crystals

6.1 Melting in Reality Before proceeding to analysis and understanding, it is necessary to see the melting behavior in the real world. Many molecular crystals of small molecules melt into isotropic liquid in a single step along with the vertical and central arrow in Fig. 6.1. Although crystals of both rare gas elements and benzene belong to this group, there exists a significant difference in between. While the molecule of any rare gas, the atom in reality, only has the translational degrees of freedom, the molecule of benzene has the rotational degrees of freedom additionally. In this sense, two types of structural order are lost during the melting of benzene crystal simultaneously. This understanding naturally leads to the expectation that two types of melting may occur separately and successively. Then, depending on the order which takes place first, two types of partially molten phases are expected during the process of an ordered crystal to the isotropic liquid phase. Such a phase appearing in the course of melting is generally called a mesophase. Representatives of mesophases are known as plastic crystal and liquid crystal. The plastic crystal is a crystalline phase in the sense that the molecular centers of gravity locate periodically. However, molecular orientations around them should be utterly disordered because of the melting of the orientational degrees of freedom: the degree of the completeness of disordering is assessed through entropy, as discussed later. In reality, many substances exhibit various crystalline phases with a moderate disorder in molecular orientations. Such phases are regarded as not a plastic crystalline phase but simply a disordered phase. On the other hand, the liquid crystal expected here should be an anisotropic fluid because it appears as a result of the melting of the translational degrees of freedom. This type of liquid crystalline phase is called the nematic phase (see Sect. 7.1.2). In contrast to the case of plastic crystals, the term, liquid crystalline phase, covers not only the nematic phase with complete disorder concerning the translational degrees of freedom but also many other phases with the partial disorder. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 K. Saito, Chemical Physics of Molecular Condensed Matter, Lecture Notes in Chemistry 104, https://doi.org/10.1007/978-981-15-9023-8_6

119

120

6 Melting of Molecular Crystals

Isotropic Liquid positional melting

orientational melting

Plastic Crystal

melting

Liquid Crystal (Nematic)

positional melting

orientational melting

Ordered Crystal Fig. 6.1 Three representative routes of melting of molecular crystals from the ordered crystal to the isotropic liquid (schematic). Melting takes place successively via mesophases (plastic or nematic crystal phases) if the anisotropy in the molecular shape is significant. Dark circular regions around cylindrical molecules in the plastic crystal represent the excluded region by the molecular reorientation.

Let us see the melting behavior quantitatively in terms of entropy. Table 6.1 summarizes the thermodynamic properties of some selected molecular crystals together with those of rare gas elements.1 It is noteworthy that the cohesive dispersion force primarily forms all of them. Although the collected data corresponds to those at a respective triple point, resulting in the difference in pressure, we can regard them as those under constant pressure ignoring the pressure dependence because this is small enough. The first point to be noticed is that despite a wide variety of their temperatures of fusion, the entropies of fusion of crystals of rare gas elements are mostly the same at ca. 14 J K−1 mol−1 (≈1.7NA kB ). Note that the molecule of rare gasses has only the translational degrees of freedom. Remembering Boltzmann’s principle (Eq. 2.8), this constancy indicates that the translational melting under constant pressure at a respective triple point accompanies a similar change (in terms of a factor) in the microscopic number of states.

1 Helium

is omitted because of the absence of the triple point among gas, liquid, and crystal.

6.1 Melting in Reality

121

Table 6.1 Examples of thermodynamic properties of fusion and solid–solid phase transitions in selected molecular crystals. Tfus in this table refers to the triple point for consistency with Δtrs S Substance

Ttrs K

Δtrs S J K −1 mol−1

Tfus K

Δfus S

J K −1 mol−1



ΔS

J K −1 mol−1

references

Ne

24.57

13.4

13.4

[1]

Ar

83.81

14.1

14.1

[2]

Kr

115.78

14.2

14.2

[3]

Xe

161.3

14.3

14.3

[4]

N2

35.61

6.4

63.14

11.4

17.8

[5]

O2

43.76

17.1

54.36

8.2

29.0

[6]

23.84

3.6

F2

45.55

16.0

53.54

9.5

25.5

[7]

CO

60.85

10.0

67.95

12.2

22.2

[8]

Ethane

89.81

25.5

90.34

6.5

32.0

[9]

Cl2

172.12

37.2

37.2

[10]

Br2

265.90

37.2

39.8

[11]

CCl4

250.3

10.2

30.7

[12]

SiCl4

205.5

38.0

38.0

[13]

GeCl4

221.7

38.9

38.9

[14]

SnCl4

240.2

38.1

38.1

[15]

Benzene

278.7

35.3

35.3

[16]

Naphthalene

353.38

53.9

53.8

[17]

Biphenyl transAzobenzene

225.4

40.4

20.5

0.40

342.09

54.3

54.7

[18, 19]

341.9

66.2

66.2

[20]

Similar constancy can be recognized for substances of which molecules are tetrahedral. That is, ca. 38 J K−1 mol−1 for SiCl4 , GeCl4 , and SnCl4 . Those of chlorine and bromine are almost the same in magnitude, and benzene also has a similar entropy of fusion. Since no phase transitions are known between crystalline phases for these compounds, their crystals experience the fusion in a single step with simultaneous losses of both positional and orientational orders. It is thus reasonable to regard the excess in the entropy of fusion beyond that of rare gas elements as the contribution of orientational melting. The following facts support such interpretation: the summation of entropies of fusion and phase transition(s) between crystalline phases recovers a rough constancy if the tails in the temperature dependence of heat capacity are taken into account. Thus, we can conclude that the orientational contribution is larger than the positional contribution because of 38 − 14 > 14. Substances of which molecules are anisotropic, i.e., naphthalene, biphenyl, and trans-azobenzene, have a larger entropy of fusion. This fact implies that the entropic contribution of

122

6 Melting of Molecular Crystals

the orientational melting depends on molecular anisotropy. The internal molecular flexibility might contribute further in the case of trans-azobenzene. Since the crystals of rare gas elements have only the positional order to be lost upon their melting, the accompanied change in entropy of fusion is expected to be minimum. However, this is not the case. In Table 6.1, crystals of some substances melt with a smaller entropy of transition than rare gas crystals. We can interpret the smaller entropy of transition as a symptom of severe orientational disorder in the crystal because we expect a similar entropy of fusion while assuming the complete disorder in both translational and orientational degrees of freedom in the isotropic liquid. Timmermans [25], thus, defined “plastic crystals” as crystals that melt with a smaller entropy of fusion than those of crystals of rare gas elements. According to this definition, crystals at a respective triple point among gas, liquid, and solid are plastic crystal phases for nitrogen, oxygen, fluorine, ethane, and CCl4 in the table. As described above, the melting behavior of molecular crystals is qualitatively rationalized. However, there exists a difficulty in interpreting its magnitude in terms of microscopic view. For example, recent molecular dynamical simulation of the Lennard-Jones particles mimicking argon [26] indicates that the entropy of fusion under constant volume is ca. 0.5N kB in contrast to the experimental one under constant pressure 1.7N kB shown in Table 6.1. However, the heat capacity under constant volume cV exhibits excessive contribution in the premelting region below the temperature of fusion due to the activation of diffusion. The entropy increment involved in this excess heat capacity is ca. 0.5N kB . The difference between entropies of transition under constant pressure and at constant volume is thus calculated as ca. 0.7N kB , which is larger than the entropy increment of the ideal gas N kB ln(Vliq /Vcrystal ) (≈0.1N kB ). The larger increment indicates that the entropy of liquid has a stronger dependence on volume than gasses. Although the melting occurs as a result of an accidental coincidence of the molar Gibbs energies of a liquid and a crystal (Sect. 2.1.1),2 physical properties often exhibit anomalies in the close vicinity below the melting temperature, even in an ideal measurement. These are called premelting behavior. The apparent premelting anomaly consists of two contributions. One comes from the effect of impurities, which cause the so-called melting point depression. By this effect, the sample starts to melt below the true melting point. After suitably correcting this effect, however, the anomalous increase in heat capacity survives. This excess is due to the formation of some defects in the crystalline lattice.3 By assuming the equilibrium between normal molecules (on the lattice) Mnorm and defects D, Mnorm  D,

2 Here,

the word “accidental” indicates that two phases are stable in both sides of the coexistence border and that the locating the transition point requires their comparison, in contrast to an ideal second-order transition with a diverging susceptibility at the critical point. 3 Observations for bulk samples implies that the tail is mainly not a surface effect but a bulk effect.

6.1 Melting in Reality

123

the fraction of the defect xD is related to the chemical potential of formation, Δf μ = Δf h − T Δf s as   Δf μ xD = exp − kB T     Δf h Δf s exp − . (6.1) = exp kB kB T This relation can be confirmed in some ways. One being the most intuitive is the difference between the volume expansivities determined macroscopically (dilatometry) and microscopically (e.g., by X-ray diffraction). While the former is affected by the formation of defects such as lattice vacancies or interstitial molecules, the latter reflects only the expansions of the averaged lattice spacing. The former is always larger than the latter, and the difference is due to the formation of defects. Assuming the formation of vacancies, which is expected to appear more readily than interstitials, the difference is directly proportional to the fraction of the defects. The fraction of the defect amounts to a few percents at the melting temperature. Another way to sense the formation of defects is the temperature dependence of the heat capacity. Experimental (or apparent) heat capacity usually exhibits a notable tail on the low-temperature side. This tail comes from not only the effects of impurities (because real compounds cannot be perfectly pure) but also defects formed in the crystalline lattice. The excess heat capacity Cexcess is obtained as  Cexcess =

∂ x D Δf h ∂T

 .

(6.2)

p

Assuming constancy of both Δf h and Δf s based on a narrow width of the temperature range of the interest (ca. a few 10 K), the excess heat capacity Cexcess becomes Cexcess =

    Δf s (Δf h)2 Δf h exp exp − . kB T 2 kB kB T

(6.3)

Thus, the plot of T 2 Cexcess against the inverse temperature T −1 gives both Δf h and Δf s (and xD ). In the case of crystals of rare gas elements, the resultant Δf h is comparable to the lattice energy [2, 3]. The formation of lattice vacancies rationalizes this coincidence. On the other hand, a similar analysis yields smaller Δf h for crystals of a simple polyatomic molecule [21]. The smallness is attributed to the formation of orientational defects, which can be more accessible with a small energetic penalty to appear than the formation of a vacancy for molecular crystals

124

6 Melting of Molecular Crystals

6.2 Plastic Crystals and Liquid Crystals 6.2.1 Mean-Field Theory of Ising Model 6.2.1.1

Ising Model

Although the Ising model was originally a concise model of magnetism [22, 23], it has served as one of the essential models in the study of phase transitions and critical phenomena. Noteworthy is that a two-dimensional version of it assuming short-ranged interaction is the first model that has been proven analytically to exhibit a continuous phase transition [24]. In this section, we analyze the model in the lowest level approximation to prepare for the understanding of some phase transitions based on microscopic models. Readers familiar with such treatments may skip this section. Suppose an ensemble of a vast number of “spins” (σ = ±1) on sites of a regular lattice. Here, the regular lattice means only the following: All sites are equivalent and have z neighboring sites. We always assume the simple cyclic boundary condition to eliminate the effect of the periphery of the system.4 Assuming that the interaction between spins work only when two spins are on neighboring sites, the energy of the ensemble is written as 1   σi σ j , (6.4) E =− J 2 i j∈{NN} i

where the second sum runs over z sites surrounding the i th site (the nearest neighbors). Here, we assume J > 0, which favors the parallel alignment of neighboring spins. The most stable state with the minimum energy is the ferroic state where all spins have the same state either of σ = ±1. Thus, the average of σ over all spins becomes m = σ  nU − nD = nU + nD = ±1,

(6.5)

where · is an average over all spins calculated using the numbers of Up (σ = 1, ↑) and Down (σ = −1, ↓) spins, n U and n D . Obviously, N = n U + n D . On the other hand, at sufficiently high temperatures, the entropy of the ensemble should tend to N kB ln 2 according to Boltzmann’s principle (Eq. 2.8). In this state, m=0

(6.6)

is expected. 4 The

book.

detailed description is necessary for some cases with J < 0, which is not considered in this

6.2 Plastic Crystals and Liquid Crystals

125

It has been established that the properties of the ensemble strongly depends on the dimensionality of the lattice. If the lattice is one-dimensional, no phase transition occurs at finite temperature. On the other hand, as described previously, the model can be rigorously analyzed for some cases on two-dimensional lattices with z = 3 (honeycomb lattice), z = 4 (square [24] and Kagomé lattices) and z = 6 (triangular lattice). All of these show a continuous phase transition between the ferro (m = 0) and para (m = 0) phases at finite temperature. In the three-dimensional cases, a similar phase transition is proven to occur [27], but exact expressions of thermodynamic functions have not been derived.

6.2.1.2

Bragg–Williams Approximation

According to statistical mechanics, thermodynamic properties of a system having an energy expression of Eq. 6.4 are obtained from the partition function Z (T ) given by Z (T ) =



⎞   J σi σ j ⎠ , exp ⎝ 2kB T i j∈{NN} ⎛

(6.7)

i

where the first summation runs over all possible states of the system. Although many efforts have been devoted to tackling the problem, they have been successful only in limited cases in one or two dimensions [24]. Some approximation is necessary to proceed with, accordingly. One of the most primitive approximations is after Bragg and Williams [28]. By virtue of its simplicity (with many issues to be corrected), however, it is intuitively applicable to a wide range of problems. The analysis strategy of the Ising model consists of two steps, constructing an approximate expression of the Helmholtz energy F(m) as a function of m, which serves as the order parameter, and determining m as one that minimizes F(m). The number of states having the same m is given by a binomial coefficient  N W = nU N! . = n U !n D ! 

(6.8)

Thus, the entropy is given by N! n U !n D ! ≈ kB [N (ln N − 1) − n U (ln n U − 1) − n D (ln n D − 1)]

n nU nD nD  U ln + ln = −N kB N N N N 1 = N kB [2 ln 2 − (1 + m) ln(1 + m) − (1 − m) ln(1 − m)] , 2

S = kB ln

(6.9)

126

6 Melting of Molecular Crystals

where Stirling’s formula, ln N ! ≈ N (ln N − 1) for N 1, is adopted while considering identities like m + 1 = 2n U /N . The energy of the ensemble may vary from state to state even if m is specified. An average energy of a pair of neighboring spins is evaluated by merely considering the probabilities of finding four arrangements, ↑↑, ↓↓, ↑↓, and ↓↑, as

n 2

n  n  D U D −J + 2J N N N N   nU − nD 2 = −J N

ε ≈ −J

n 2 U

= −J m 2 .

(6.10)

Since there are z N /2 pairs in the ensemble, an estimate of the total energy is E =−

zJ N 2 m . 2

(6.11)

Thus, the Helmholtz energy per spin becomes F(m) 1 = [E(m) − T S(m)] N N zJ = − m2 2 1 − kB T [2 ln 2 − (1 + m) ln(1 + m) − (1 − m) ln(1 − m)] (6.12) 2 This expression of the Helmholtz energy corresponds to assumptions that only terms having the common energy is dominant in the partition function (Eq. 6.7) and the common energy is given by the “average” energy calculated above. This type of approximation is known as saddle point approximation or stationary state approximation. Having written the expression of an approximate Helmholtz energy as a function of m, it is in order to find the equilibrium m, which should minimize F(m). Namely, the followings are requested: d F(m) =0 dm m=m eq d 2 F(m) >0 dm 2 m=m eq The former is reduced to

(6.13) (6.14)



 zJ m = tanh m . kB T

(6.15)

127

1.0

1.0

0.5

0.5

0.0 0.1

2

3

4

6 7 8 9

5

1

2

S/NkBln2

m

6.2 Plastic Crystals and Liquid Crystals

0.0

T / TMF Fig. 6.2 Temperature dependence of the order parameter m (solid red curve, left axis) and entropy per spin (solid black curve, right axis) of the Ising model calculated by a simple mean-field (MF), i.e., the Bragg–Williams or molecular-field, treatment. A dotted black curve also shows the latter of the exact solution for the two-dimensional rectangular lattice (the transition temperature is Tc ≈ 0.567TMF with z = 4) [24] for the sake of comparison

To see the dependence of the approximate Helmholtz energy (Eq. 6.12) on m around 0, which would be essential if a phase transition occurs between the ferro (m = 0) and para (m = 0) states, we expand it up to the fourth order in m as F(m) 1 1 ≈ −kB T ln 2 + (kB T − z J )m 2 + kB T m 4 + . . . N 2 12

(6.16)

The coefficient of the second-order term changes its sign at TMF =

zJ . kB

(6.17)

This change suggests the instability of the para state below this temperature. Numerical solution of Eq. 6.15 yields the temperature dependence of m shown in Fig. 6.2. The ensemble exhibits the ferro state below TMF but the para state above it. The transition is predicted as of the second-order with the continuous vanishing of m but accompanying an abrupt jump in heat capacity as in a continuous transition treated in a Landau’s phenomenology in Sect. 2.2. It is noted that the total entropy involved in this disordering process is N kB ln 2 by Boltzmann’s principle (Eq. 2.8), while no jump appears at all. This is an example that the entropy of transition (corresponding to the latent heat) accompanied by a first-order phase transition is insufficient to discuss the nature of phase transitions, as pointed out in the preceding Sect. 2.1.2.

6.2.1.3

Molecular Field Approximation

It is known to exist another route to reach the same result as that of the Bragg– Williams approximation. We rewrite the energy of the ensemble as

128

6 Melting of Molecular Crystals

⎛ ⎞  1 ⎝ σi E =− J σj⎠ 2 i j∈{NN} i

1 ≈− z J σi σ i 2 i 1 =− εi , 2 i

(6.18)

where σ i is the average of σ surrounding the i th spin. Now, the problem is equivalent to that of “independent” spins in a molecular field given by z J σ i . Considering the equivalence of all sites on the lattice, we request σi  = σ i . Namely,

 σ exp − zkJBσT  σ

 σ  =  z J σ  exp − σ σ =±1 kB T 

σ =±1

= tanh

zJ σ  kB T

(6.19)

This is precisely the same equation as Eq. 6.15 because of σ  = m. Along this line, the denominator in the first line of the above equation should be regarded as a partition function q of a spin. However, to obtain the Helmholtz energy per spin (F/N ), we must correct the double-counting in the interaction energy as F zJ 2 = −kB T ln q + m , N 2

(6.20)

because the interaction between a pair of the i th and j th spins is counted twice: once as the energy of the i th in the molecular field formed by its neighbors including the j th spin and again as a part of the molecular field acting on the j th spin. Although the Bragg–Williams approximation may intuitively be more straightforward for problems dealing with spins with discrete degrees of freedom, understanding along with the molecular-field spirit enables the enumeration of entropy for continuous spins, leading to broader applicability as will be seen in Sect. 7.2.2.

6.2.1.4

Deficiency of Mean-Field Treatments

The Bragg–Williams and molecular-field approximations, together with Landau’s phenomenology of phase transitions, are classified into a category called meanfield approximation. Approximations treated in this section reduce the problem of a system consisting of a vast number of particles to that of a single particle in an averaged field (mean-field). Such a way of treatment and understanding is intuitively convenient and practically tractable. However, this merit has a trade-off relation with reality. The essential factor neglected in any mean-field treatment is generally

6.2 Plastic Crystals and Liquid Crystals

129

termed as fluctuations. The treatment erroneously predicts the occurrence of a phase transition, even for a one-dimensional case (z = 2). In the case of two-dimension, the temperature dependence of entropy is notably inconsistent, even qualitatively, as seen in Fig. 6.2, whereas that of the order parameter m is qualitatively correct in the sense that m is finite below but null above the transition temperature. The inconsistency is due to neglecting the local correlation arising from the interaction workable even above the transition temperature. Such a deficiency should be kept in mind.

6.2.2 Simple Theory of Melting of Atomic Crystal 6.2.2.1

Free Volume Model and Communal Entropy

The most remarkable difference in macroscopic properties between a liquid and a crystal is their mechanical properties. Namely, the former exhibits fluidity while the latter does not. This contrast is due to a difference in microscopic states of particles inside: they may visit a whole volume in the former, whereas they are confined in a small volume in the latter. The volume assumed to be available to a particle is often termed as free volume. The difference between a liquid and a solid is its vast difference in terms of the free volume. We now assume the same functional form for a partition function q(V ) of a single particle (atom). Considering it of a free particle (mass m) in a box (volume v), we adopt     u 2π mkB T 3/2 , (6.21) v exp − q(v, u) = h2 kB T where u is a potential energy (per particle) that is assumed to be uniform inside the box and independent of temperature. Using this q(v, u), we express the partition functions of a liquid and a solid, assuming that a particle in a solid is confined to a small volume V /N (N being the number of particles), as q(V, u liq ) N N! N  V , u sol =q . N

Z liq =

(6.22)

Z sol

(6.23)

It is reasonable to assume u sol < u liq . The energy and entropy of a liquid are larger by N Δu = N (u liq − u sol ) and N kB than those of a solid, irrespective of temperature. This difference in entropy, coming from the confinement in a solid, is called communal entropy. The difference in the Helmholtz energies of a liquid and a solid is (6.24) ΔF = N Δu − N kB T.

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6 Melting of Molecular Crystals

Fig. 6.3 Two interpenetrating lattices (α and β) assumed in the model of melting by Lennard-Jones & Devonshire with the number of the neighboring sites on the same lattice as a site belongs to being z = 6 and that on the other lattice being z = 8. This figure corresponds to a crystal phase because of the apparent preference of atoms to occupy the α lattice

lattice α

lattice β

It changes its sign at T ∗ = Δu/kB , above and below which the liquid and solid are more stable than the other, respectively. Thus, a phase transition can be imagined between the two states. The total change in entropy is N kB . The above consideration is not a model of melting phase transitions but a simple comparison of two states based on crude approximations. Besides, it should be emphasized that consideration assumes a constant volume. However, it is interesting to note that the expected difference in entropy is surprisingly close to that reported for the melting under the constant volume if that accompanied by atomic diffusion is taken into account [26]. Another point to be noticed is that the consideration highlights the atomic confinement (small free volume). On the other hand, the consideration is expected to apply equally to the softening of glasses, which is different from the melting. It is thus necessary to incorporate the spatial order and its rupture on melting.

6.2.2.2

Theory by Lennard-Jones and Devonshire

Two essential factors, i.e., the confinement and the structural order in crystals, were incorporated by using a lattice model into a theoretical model of melting by LennardJones and Devonshire [29]. They assumed two interpenetrating lattices with N sites for a system consisting of N atoms, as shown in Fig. 6.3. Since the total number of sites is 2N , atoms can, in principle, visit any sites in the system. If atoms equally occupy the sites of two lattices, the system is identified as the liquid.5 On the other hand, if one lattice is preferred, structural order emerges. Suppose that two identical lattices (α and β) interpenetrate each other. Each lattice has N sites with z neighbors for each. As a result of the interpenetration, each site 5 Ordered

occupation on two lattices can be imagined but does not occur in their treatment.

6.2 Plastic Crystals and Liquid Crystals

131

on one lattice also has neighbors on the other lattice. Let this number be z. Since all sites are equivalent, the partition function of a single particle without any atoms on its neighbors (both on α and β) is assumed to depend only on the total volume V (and temperature). The partition function is written as q(V ). Considering only the interaction between neighboring atoms on different lattices, we can express the microscopic information of each state by specifying the number of such neighboring pairs denoted as n αβ . The partition function of the system is then expressed as Z=



  n αβ w , q(V ) exp − kB T N

(6.25)

where w is the energy penalty for the disordering. The summation runs over all possible states. Since an exact calculation is impossible for the partition function, we introduce a new variable ξ , which is the occupancy of sites on the α lattice. This ξ is unity for the perfect order (perfect crystal) and 21 for the maximum disorder (liquid).6 Using ξ , we estimate the “average” of n αβ as n αβ ≈ z N ξ(1 − ξ ).

(6.26)

in the spirit of the Bragg–Williams approximation.7 Similarly, the multiplicity of this term is estimated as 2

N! (6.27) (ξ N )![(1 − ξ )N ]! by considering the configurations on both lattices. Then, an approximate partition function is given by

Z ≈ q(V ) N

N! (ξ N )![(1 − ξ )N ]!

2

  z N ξ(1 − ξ )w . exp − kB T

(6.28)

Thus, the Helmholtz energy F(ξ ) as a function of ξ is obtained as F(ξ ) = −N kB T ln q(V ) +2N kB T [ξ ln ξ + (1 − ξ ) ln(1 − ξ )] + z N ξ(1 − ξ )w

(6.29)

The minimization with respect to ξ yields the following equation  zw(2ξ − 1) = 2ξ − 1. tanh 4kB T 

(6.30)

of 21 from ξ and σ appearing later yields order parameter(s), which fit the ordinary definition of order parameters; null in the disordered state and finite in the ordered state. 7 Since the counterpart of the central spin is on the other lattice, the division by a factor 2 is unnecessary. 6 Subtraction

132

6 Melting of Molecular Crystals

Since this is the self-consistency equation having virtually the same form as Eq. 6.15, this theory predicts a continuous transition between the crystal and liquid at TLJD = zw/4kB under the constant volume. The entropy involved in this transition is 2N kB ln 2 ≈ 1.4N kB . Melting is usually observed under not the constant volume but the constant pressure. The discussion about the melting under the latter condition requires some consideration of the volume dependence. There exist two terms that depend on the volume. One is the contribution of a single atom involved in q(V ) whereas the other is in w. For the latter, Lennard-Jones and Devonshire [29] assumed that the repulsive part of the interatomic potential is dominant. Considering the repulsive part of the Lennard-Jones potential as a function of interatomic distance r given by φ(r ) = 4ε



 r0 12 r0 6 , − r r

they assumed w = w0

v 4 0

v

.

(6.31)

(6.32)

On the other hand, they assumed the free volume model (Eq. 6.21) with v = V /N for q(V ) for the former. Here, it is essential to be aware that the potential energy u sensed by each atom (considered in Sect. 6.2.2.1) does not affect any for the melting under constant volume but becomes effective for the melting under the constant pressure. However, its introduction requires another characteristic energy in the model, leading to more complexity. Thus, a possible contribution from its volume dependence is neglected in the following, though this contribution may affect the location of equilibrium pressure between two phases. Under this condition, the pressure of the system is calculated as   zwξ(1 − ξ ) N kB T + p= V 4

(6.33)

with ξ that minimizes the Helmholtz energy. The last term is the pressure arising from the “disorder” and does not vanish even in the complete disorder (ξ = 21 ). Figure 6.4 shows isotherms on the p − V plane. The isotherms at high temperatures are not monotonic. This behavior indicates that, under such pressures, two phases having different densities, i.e., a crystal and a liquid, coexist, as seen in the melting in daily life. Indeed, they were successful in explaining the melting properties of argon by suitably choosing some parameters to reproduce the melting temperature [29].

6.2 Plastic Crystals and Liquid Crystals Fig. 6.4 p − V isotherms for the model of melting by Lennard-Jones and Devonshire for selected temperatures, which is expressed in terms of T0 = zw0 /kB

133 5

pv0 / kB

4

2.0T0

3

1.5T0

2

1.0T0

1 0.5T0 0 0.4

0.6

0.8

1.0

v / v0

6.2.3 Translational and Orientational Melting Since molecules generally have not only the translational (positional) but also the orientational degrees of freedom, any theory of melting of molecular crystals should take them into account. Pople and Karasz [30] first considered this issue. They introduced additional quantities into the model by Lennard-Jones and Devonshire described in the previous section. They were primarily interested in possibilities of orientational melting while keeping translational order (crystalline state), and even denied discussions on a possibility of liquid crystals. However, the model is viewed from a broader perspective here. Suppose on each lattice (α or β) two molecular orientations (say, U and D). An energy penalty w is assumed for a pair of neighboring molecules having “wrong” relative orientations (e.g., UD) on the same lattice. By introducing the numbers of such misoriented pairs, n αα and n ββ , on lattices, the partition function of the system can be expressed as Z=



 n αβ w + (n αα + n ββ )w

, q(V ) exp − kB T 

N

(6.34)

where the summation runs over all possible states. To proceed further, we express the fraction of molecules with U orientation over both lattices by a parameter σ . Then, the following estimates result8 : n αβ ≈ z N ξ(1 − ξ )

n αα ≈ z N ξ σ (1 − σ ) n ββ ≈ z N (1 − ξ )2 σ (1 − σ ).

8 For

zN.

2

(6.35) (6.36) (6.37)

n αα and n ββ , the factor 2 coming from UD and DU pairs is cancelled by the double count by

134

6 Melting of Molecular Crystals 0.7 0.5

IL kBT/zw

Fig. 6.5 Phase diagram of Pople-Karasz model agains ν = z w /zw under constant volume. OC, orientationally ordered crystal (ξ = 21 , σ = 21 ); DC, orientationally disordered crystal (ξ = 21 , σ = 21 ); OL, orientationally ordered liquid (ξ = 21 , σ = 21 ); IL, isotropic liquid (orientationally disordered liquid, ξ = 21 , σ = 21 )

OL

0.3 0.2

DC OC 0.1 0.2

0.3

0.5

0.7

1

2

3

Accordingly, the estimate of energy is E ≈ N [zwξ(1 − ξ ) + z w ξ 2 σ (1 − σ ) + z w (1 − ξ )2 σ (1 − σ ) = N [zwξ(1 − ξ ) + z w (1 − 2ξ + 2ξ 2 )σ (1 − σ )].

(6.38)

The corresponding estimate of entropy based on the combinatorial number of states is given by S = −N kB {2[ξ ln ξ + (1 − ξ ) ln(1 − ξ )] + σ ln σ + (1 − σ ) ln(1 − σ )}. (6.39) In the model by Pople and Karasz, the following ratio ν=

z w

zw

(6.40)

reflects the relative importance of positional and orientational energy penalties. Namely, ν is a parameter conceptually related to the anisotropy of the molecular shape. The phase diagram, while fixing w and w (corresponding to constant volume), is shown in Fig. 6.5. In a region with small ν, on heating, the orientationally ordered crystal (ξ = 21 , σ = 21 ) first exhibits the orientational melting to orientationally disordered crystal (ξ = 21 , σ = 21 ), and then to isotropic liquid (ξ = 21 , σ = 21 ). On the other hand, the positional melting takes place first into the orientationally ordered liquid (ξ = 21 , σ = 21 ) with a large ν. Transition temperatures between the partially disordered states and the isotropic (orientationally disordered) liquid are obtained by locating the temperature where the sign of the second-order term of the expansion of the Helmholtz energy changes. The expansion around ξ = 21 and σ = 21 yields zw TDC = (2 − ν) (6.41) 8kB and

6.2 Plastic Crystals and Liquid Crystals

135

TOL =

zw ν. 4kB

(6.42)

In an intermediate region of ν, the melting occurs to the isotropic liquid in a single step. The locating this boundary requires numerical minimization of the Helmholtz energy concerning ξ and σ , and numerical comparison of the Helmholtz energies of two phases because the transition is of the first-order. Practically, the following simultaneous equations are to be solved   zw z w

ξ = − σ (1 − σ ) (2ξ − 1) ln 1−ξ 2kB T kB T σ z w

ln = (1 − 2ξ + 2ξ 2 )(2σ − 1) 1−σ kB T

(6.43) (6.44)

and, the corresponding Helmholtz energy is compared with that with ξ = 21 and σ = 21 . Although transitions predicted assuming constant volume are mostly of secondorder, they might become of first-order under constant pressure conditions, as shown in the previous section for simple melting. Indeed, assuming the same v −4 dependence of energy penalties (w and w ) as that assumed by Lennard-Jones and Devonshire [29]. Pople and Karasz [30, 31] indicated that phase transitions became of the first-order in most cases. Some issues can be pointed out to be examined in the model by Pople and Karasz. As we see in Sect. 6.1, plastic crystals appear in systems of globular molecules. In such cases, the number of possible molecular orientations is much larger than the two assumed in the model. Amzel and Becka [32] later made an extension concerning the available orientations. Another variation came from the side of ordered liquids (liquid crystals). The same dependences of energy penalties on volume imply their ratio, ν, remains the same upon the volume variation. Chandrasekhar et al. [33, 34] claimed that the volume dependence of the orientational penalty, w , is weaker. Assuming w ∝ v −3 , they reported an improved phase diagram under constant pressure. Even taking these modifications into account, the parameter ν cannot practically be related to the anisotropy of molecular shape, except ν = 0 for the completely globular case. In this respect, models are to be treated as not quantitative but qualitative ones to understand the general trend of melting behavior depending on molecular anisotropy. A variety of partial melting can be imagined for either orientational and positional order. Plural crystalline phases of methane (CH4 ) [35–37] and varieties of liquid crystals discussed in Sect. 7.1.2 are such examples. Realistic models based on molecular details were constructed for systems consisting of simple molecules such as diatomic molecules [38] and methanes [36, 39]. Due to their small moments of inertia, proper consideration of possible quantum effects are sometimes necessary for good descriptions. Especially in cases of methanes, quantum effects bring a drastic difference in physical properties among

136

6 Melting of Molecular Crystals

systems consisting of molecules having a different isotopic composition of hydrogens, CH4−n Dn (n = 0 − 4) [35]. Although much can be learned from such models, we do not treat them here because they deeply involve details specific to substances.

6.3 Molecular Deformation The melting process described in the previous sections from the perfectly ordered crystal to isotropic liquid is the one during which molecules acquire structural entropy. If we consider the properties of molecules with the shape, another way of acquiring structural entropy can be imagined. Such acquiring can happen in reality. Molecular conformation plays an important role here. A representative series of molecules, the properties of which are deeply related to molecular flexibility, is n-alkanes, H–(CH2 )n –H (n = 2, 3, . . .). Compounds with sufficiently large n are known as poly(ethylene), a representative of linear polymers. Properties of linear polymers are understood based on their flexibility coming from conformational degrees of freedom [40]. The C–C bonds in n-alkanes are the socalled single bonds and allow the internal rotation. To be concise, we first focus on a molecule of ethane (CH3 –CH3 ). The molecular energy has a threefold symmetry for the mutual rotation of methyl groups due to the tetrahedral bonding of the carbon atoms. Dihedral angles between planes defined by Ha Ca Cb and Ca Cb Hb (a and b distinguish carbon atoms) in the molecular ground state are either of π or ± π3 (while neglecting the width coming from quantum effects). In the case of n-alkanes with n ≥ 4, similarly, the dihedral angles are essentially either of π or ± π3 for two planes defined by Ca Cb Cc and Cb Cc Cd . The conformations of Ca Cb Cc Cd with a dihedral angle of π and ± π3 are called trans and gauche, respectively.9 The trans form is more stable in energy than the gauche form in n-alkanes (and n-alkyl groups). Figure 6.6 shows cumulative entropies of transition starting from an ordered crystal to isotropic liquid of n-alkanes (n = 2 − 29 with n being the number of carbon atoms in a molecule). As is evident, the dependence on n is linear with an averaged slope (in the sense of least-squares) of 10.1 J K−1 mol−1 though they split into two depending on the parity of n (even or odd) if inspecting in detail [49, 50]. The dependence on the parity is known as the odd-even effect, which in thermodynamic properties generally comes from the difference in aggregation structure due to the different orientation of terminal –CH2 –CH3 bonds between molecules with odd and even n, as explained in Sect. 9.4.1. The magnitude of the slope is comparable with and slightly larger than R ln 3 (≈ 9.1 J K−1 mol−1 ), which is the expected magnitude for the threefold disorder (over a trans and two gauche conformations) around a C–C single bond. Namely, the n-dependence of the cumulative entropy indicates that the threefold disorder realizes certainly in the isotropic liquid. Acquiring entropy by the internal structural degrees of freedom in the course of melting also occurs in compounds having alkyl groups. Figure 6.7 illustrates the assignment of acquired entropies to individual degrees of freedom [51]. The assign9 Italic

style is usually used for trans and gauche.

6.3 Molecular Deformation

137

200

fusS

-1

S / J K mol

-1

300

+

trsS

100 trsS

0

30

20 10 n in H-(CH2)n-H

0

Fig. 6.6 Entropy of solid-solid transition (Δtrs S, cross) and sum with entropy of fusion (Δfus S, circle) as a function of n, the number of carbon atoms in a molecule of odd (blue) and even (red) members of n-alkane. Drawn using data from literatures [9, 41–48]

3 J K-1 mol-1

orientational (27) positional (14) alkyl-chain (15)

crystal

55 J K-1 mol-1

56 J K-1 mol-1

ΣΔ trsS

nematic order (3)

head-to-tail (Rln2) orientation about long-axis (20) positional (14) alkyl-chain (15)

crystal

crystal

pentylbiphenyl

orientational (17) positional (14)

SmE 36 J K-1 mol-1

IL nematic

IL

31 J K-1 mol-1

IL (9)

5CB

head-to-tail (Rln2) front or rear faces (Rln2) alkyl-chain (15) polar group (9)

5TCB

Fig. 6.7 Contributions of motional/conformational degrees of freedom to the sum of entropies of transition in pentyl biphenyl, 5CB (4’-pentyl-4-cyano-1,1’-biphenyl) and 5TCB (4’-pentyl-4isothiocyanato-1,1’-biphenyl). IL, isotropic liquid; nematic, nematic liquid crystal; SmE, a smectic (layered) liquid crystal having three-dimensional long-range order. The unit of numbers in parentheses is J K−1 mol−1 . (Reprinted with permission from J. Phys. Chem. B, 114, 4870 (2010) [51]. Copyright 2010 American Chemical Society

ment is based on the experimental magnitudes of respective contributions discussed in the previous and present sections. It is noteworthy that the entropy capacity of the alkyl group is significant compared to both entropies necessary for translational (ca. 14 J K−1 mol−1 ) and orientational (ca. 25 J K−1 mol−1 ) melting discussed in Sect. 6.1. Indeed, it is widely accepted that the low melting point of ionic liquids, which are salts having melting points lower than room temperature, at least partially, originates in the large entropy acquired in alkyl groups in the liquid state [52], as mentioned in Sect. 3.3.

138

6 Melting of Molecular Crystals

Fig. 6.8 Disordered two conformations (a) and an “averaged” structure (b) of a p-terphenyl molecule (schematic). Hydrogen atoms are omitted for clarity

By identifying the energetic penalty between neighboring molecules on the same sublattice, w , like that for the wrong combination(s) of their conformations, a parallel discussion can be conducted to that in Sect. 6.2.3. Namely, in many cases, both positional and conformational orders are lost in a single step from crystals to isotropic liquid. Phase transitions mainly related to conformational order/disorder have been identified in limited cases. A series of compounds known as p-polyphenyl is representative. Their molecules have a linear structure consisting of benzene rings represented as H–(C6 H4 )n –H (n = 2, 3, . . .). They have a twisting degree(s) of freedom around each C–C single bond between benzene rings. The competition of the delocalization of π -electrons and steric repulsion between hydrogen atoms on neighboring rings yields the stable conformation of alternate twist. Indeed, molecules of compounds with n = 3 − 5 are alternately twisted in crystals at low temperatures [53–55].10 Each molecule takes one of two twisted conformations (red or green), as exemplified in Fig. 6.8a for the case of p-terphenyl (n = 3) depending on its location. Upon heating, the crystals undergo phase transitions [56–58], above which molecular conformation is disordered, resulting in seemingly planar conformations determined by X-ray crystallography [54, 59–62] (Fig. 6.8b). Further heating results in the fusion of crystals as usual [63]. It is interesting to see that the entropy increment involved in the twist transitions remains almost the same as N kB ln 2, irrespective of n [56–58], even if the number of possible conformations increases as 2n−1 per molecule leading to the entropy increment of N (n − 1)kB ln 2. The constancy of the entropy increment indicates that the relative orientation of benzene rings is correlated. For example, in the case of p-terphenyl, only two conformations represented as (+)–(–)–(+) and (–)–(+)– (–) are allowed but (+)–(0)–(–) and (–)–(0)–(+) are not. Namely, the two terminalrings’ reorientations are synchronized in the disordered phase (at room temperature). Such a motional correlation, either intramolecular or intermolecular, is not limited to (n = 2) is omitted from the present discussion because its twist phase transition is different character from those in others as briefly discussed in Sect. 5.5.3.

10 Biphenyl

6.3 Molecular Deformation

139

molecular crystals [64–68]. We will discuss other examples and the physical meaning of the issue in Sect. 10.1. Conformationally disordered states are usually impossible to discern from the orientationally disordered state in terms of crystallography using scattering techniques. Indeed, the energetic situation is also expected to resemble that in a supposed disordered state. Thus, it is hard to imagine a phase sequence in which an orientational melting occurs at a higher temperature than an intramolecular conformational disordering transition, though such a sequence is theoretically possible. On the other hand, an orientational melting obscures the interaction between different conformations of neighboring molecules. Thus, the disordering of internal conformation indeed happens but proceeds without a phase transition. CFCl2 –CFCl2 shows a symptom of such a situation as a glass transition (the subject of Chap. 8) concerning the thermal change of conformational equilibrium [69]. In this compound, the difficulty in obtaining conformational order is suggested as a possible origin of difficulty in reaching an orientationally ordered crystal obeying the third-law of thermodynamics. Ironically, a liquid may exhibit a phase transition related to the change in molecular conformations. Some achiral liquid crystalline compounds exhibit optically active isotropic liquids [70–73]. They are achiral in a sense as a dynamical average of two chiral conformations, which are easily interchangeable into each other as a result of thermal motion. Since their molecules can adopt one of the chiral conformations in response to their chiral environment (adaptive chirality), they can form chiral phase(s) in a limited situation. At high temperatures, needless to say, they become an achiral liquid.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

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18. K. Saito, T. Atake, H. Chihara, Bull. Chem. Soc. Jpn. 61, 679–688 (1988) 19. R.D. Chirico, S.E. Knipmeyer, A. Nguyen, W.V. Steele, J. Chem. Thermodyn. 21, 1307–1331 (1989) 20. J.C. van Miltenburg, J.A. Bouwstra, J. Chem. Thermodyn. 16, 61–65 (1984) 21. T. Atake, H. Chihara, Bull. Chem. Soc. Jpn. 47, 2126–2136 (1974) 22. W. Lenz, Phys. Z. 21, 613–615 (1920) 23. E. Ising, Z. Phys. 31, 253–258 (1925) 24. L. Onsager, Phys. Rev. 65, 117–149 (1944) 25. J. Timmermans, J. Phys. Chem. Solids 18, 1–8 (1961) 26. Y. Kataoka, read at the Spring meeting of the Chemical Society of Japan (2017) 27. R.B. Griffiths, Phys. Rev. 136, A437–A439 (1964) 28. W.L. Bragg, E.J. Williams, Proc. Roy. Soc. London 145A, 699–730 (1934) 29. J.E. Lennard-Jones, A.F. Devonshire, Proc. Roy. Soc. London 169A, 317–338 (1939) 30. J.A. Pople, F.E. Karasz, J. Phys. Chem. Solids 18, 28–39 (1961) 31. F.E. Karasz, J.A. Pople, J. Phys. Chem. Solids 20, 294–306 (1961) 32. L.M. Amzel, L.N. Becka, J. Phys. Chem. Solids 30, 521–538 (1969); ibid, 30, 2495 (1969) (errata) 33. S. Chandrasekhar, R. Shashidhar, N. Tara, Mol. Cryst. Liq. Cryst. 10, 337–358 (1970) 34. S. Chandrasekhar, R. Shashidhar, N. Tara, Mol. Cryst. Liq. Cryst. 12, 245–250 (1971) 35. K. Clusius, L. Popp, Z. Phys. Chem. B46, 63–81 (1940) 36. H.M. James, T.A. Keenan, J. Chem. Phys. 31, 12–41 (1959) 37. A.I. Prokhatilov, A.P. Isakina, Acta Crystallogr. Sec. B 36, 1576–1580 (1980) 38. C.A. English, J.A. Venables, Proc. Roy. Soc. London A 340, 57–80 (1974) 39. T. Yamamoto, Y. Kataoka, Phys. Rev. Lett. 20, 1–3 (1968) 40. P.-G. de Gennes, Scaling Concepts in Polymer Physics (Cornel University Press, Ithaca, 1979) 41. J.D. Kemp, C.J. Eagan, J. Am. Chem. Soc. 60, 1521–1525 (1938) 42. J.G. Aston, G.H. Messerly, J. Am. Chem. Soc. 62, 1917–1923 (1940) 43. J.F. Messerly, G.B. Guthrie, S.S. Todd, H.L. Finke, J. Chem. End. Data 12, 338–346 (1967) 44. D.R. Douslin, H.M. Huffman, J. Am. Chem. Soc. 68, 1704–1708 (1946) 45. J.C. van Miltenburg, G.J.K. van den Berg, M.J. van Bommel, J. Chem. Thermodyn. 19, 1129– 1137 (1987) 46. H.L. Finke, M.E. Gross, G. Waddington, H.M. Huffman, J. Am. Chem. Soc. 76, 333–341 (1954) 47. A.A. Schaerer, C.J. Busso, A.E. Smith, L.B. Skinner, J. Am. Chem. Soc. 77, 2017–2019 (1955) 48. R.J.L. Andon, J.F. Martin, J. Chem. Thermodyn. 8, 1159–1166 (1976) 49. M. Sorai, K. Tsuji, H. Suga, S. Seki, Mol. Cryst. Liq. Cryst. 59, 33–58 (1980) 50. M. Sorai, K. Saito, Chem. Rec. 3, 29–39 (2003) 51. K. Horiuchi, Y. Yamamura, R. Pełka, M. Sumita, S. Yasuzuka, M. Massalska-Arod´z, K. Saito, J. Phys. Chem. B 114, 4870–4875 (2010) 52. Y. Shimizu, Y. Ohte, Y. Yamamura, K. Saito, Chem. Phys. Lett. 470, 295–299 (2009) 53. J.L. Baudour, Y. Delugeard, H. Cailleau, Acta Crystallogr. Sec. B 32, 150–154 (1976) 54. J.L. Baudour, L. Toupet, Y. Délugeard, S. Ghémid, Acta Crystallogr. Sec. C 42, 1211–1217 (1986) 55. J.L. Baudour, Y. Delugeard, P. Rivet, Acta Crystallogr. Sec. B 34, 625–628 (1978) 56. K. Saito, T. Atake, H. Chihara, J. Chem. Thermodyn. 17, 539–548 (1985) 57. K. Saito, T. Atake, H. Chihara, Bull. Chem. Soc. Jpn. 61, 2327–2336 (1986) 58. K. Saito, Y. Yamamura, M. Sorai, Bull. Chem. Soc. Jpn. 73, 2713–2718 (2000); Acta Crystallogr. Sec. C 42, 1211–1217 (1976) 59. H.M. Rietveld, E.N. Maslen, C.J.B. Clews, Acta Crystallogr. Sec. B 26, 693–706 (1970) 60. J.L. Baudour, H. Cailleau, Acta Crystallogr. Sec. B 33, 1773–1780 (1976) 61. Y. Delugeard, J. Desuche, J.L. Baudour, Acta Crystallogr. Sec. B 32, 702–705 (1976) 62. K.N. Baker, A.V. Fratini, T. Resch, H.C. Knachel, W.W. Adams, E.P. Socci, B.L. Farmer, Polymer 34, 1571–1587 (1993) 63. G.W. Smith, Mol. Cryst. Liq. Cryst. 49, 207–209 (1979)

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64. Y. Yamamura, H. Saitoh, M. Sumita, K. Saito, J. Phys.: Condens. Matter 19, 176219 (2007) 65. Y. Yamamura, H. Shimoi, M. Sumita, S. Yasuzuka, K. Adachi, A. Fuyuhiro, S. Kawata, K. Saito, J. Phys. Chem. A 112, 4465–4469 (2008) 66. Y. Miyazaki, Q. Wang, A. Sato, K. Saito, M. Yamamoto, H. Kitagawa, T. Mitani, M. Sorai, J. Phys. Chem. B 106, 197–202 (2002) 67. S. Ikeuchi, Y. Yamamura, Y. Yoshida, M. Mitsumi, K. Toriumi, K. Saito, Bull. Chem. Soc. Jpn. 83, 261–266 (2010) 68. S. Ikeuchi, K. Saito, Curr. Inorg. Chem. 4, 74–84 (2014) 69. K. Kishimoto, H. Suga, S. Seki, Bull. Chem. Soc. Jpn. 51, 1691–1696 (1978) 70. C. Tschierske, G. Ungar, ChemPhysChem 17, 9–26 (2016) 71. C. Dressel, F. Liu, M. Prehm, X. Zeng, G. Ungar, C. Tschierske, Angew. Chem. Int. Ed. 53, 13115–13120 (2014) 72. C. Dressel, T. Reppe, M. Perhm, M. Brautzsch, C. Tschierske, Nat. Chem. 6, 971–977 (2014) 73. S. Kutsumizu, S. Miisako, Y. Miwa, M. Kitagawa, Y. Yamamura, K. Saito, Phys. Chem. Chem. Phys. 18, 17341–17344 (2016)

Chapter 7

Liquid Crystals

7.1 Classification 7.1.1 Thermotropic Versus Lyotropic Liquid crystal, in a broad sense, is liquid having structures. Usually, the structure induces the anisotropy, which is the basis for its numerous applications, such as an LC display. Liquid crystals are often classified into two groups of thermotropic and lyotropic. The underlying mechanism bringing them about has been regarded as much different: Excluded volume effect and microphase separation. The thermotropic liquid crystal appears upon temperature variation. Liquid crystals of this type can be a neat compound. Their formation comes from strong shape anisotropy of molecules, one of the characteristics of systems consisting of not independent atoms but molecules. The formation of thermotropic liquid crystals is understood within a unified view of the melting of molecular crystals, as described in Chap. 6. On the other hand, the lyotropic liquid crystal appears in multicomponent systems like a concentrated aqueous solution of soap. Internal structures result from microphase separation. There is no doubt that each mechanism solely works well in some liquid crystals. Both kinds of liquid crystals exhibit various phases of different structures. This chapter mainly concerns with the thermotropic liquid crystals (except the last section). However, it is noteworthy that the distinction between thermotropic and lyotropic liquid crystals is not so clear in real systems, as will be discussed later in Sect. 9.4.4.2.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 K. Saito, Chemical Physics of Molecular Condensed Matter, Lecture Notes in Chemistry 104, https://doi.org/10.1007/978-981-15-9023-8_7

143

144

7 Liquid Crystals

7.1.2 Various Thermotropics This section is not intended to cover all liquid-crystalline phases identified in real materials but to give an idea of liquid crystal structures necessary to understand the importance of molecular natures described in the preceding sections. However, it would be fruitful for more profound studies to learn which molecular properties we retain and ignore to describe respective phases adequately. The readers who need the information to identify specific phases through experiments should consult specialized books [1].

7.1.2.1

Nematic Phase

The liquid crystal introduced in Chap. 6 exhibits fluidity due to the dynamically random location of molecules and the anisotropy in its physical property. This phase is regarded as the one that is brought about by the positional melting while keeping the orientational order of molecules. This phase has the highest symmetry among various liquid crystals and is called a nematic phase. The nematic (N) phase is uniaxial due to the uniaxial alignment of anisotropic molecules. A unit vector called the director, usually written as n (Fig. 7.1), specifies the uniaxial direction. The uniaxial nature of the nematic phase implies that the molecular anisotropy around the uniaxial axis is completely averaged out by thermal motion. That is, the molecules in nematic phases can be assumed axially symmetric. Besides, no reports of the polar nematic phase in the director exist, even if constituting molecules are polar (having the head and tail). Thus, n and −n are physically equivalent. The equivalence implies that the molecules in nematic phases have no distinction between its head and tail. It is, therefore, suitable to imagine a highly anisotropic spheroid, irrespective of prolate or oblate, as an abstract molecule in nematic phases. A nematic order parameter expresses the degree of anisotropy. Because of the lack of the distinction of head and tail described above, a simple average of, say, a molecular director defined as a vector from the head to the tail, has no meaning. Instead, the nematic order parameter s is defined as s = P2 (cos θ), where P2 (x) =

1 (3x 2 − 1) 2

(7.1)

(7.2)

is the second-order Legendre polynomial, θ the angle between the molecular axis and the director n, and  f  is the ensemble average of f (over the system). If all molecules orient their axes along n (θ = 0 for all molecules), s = 1 is obtained. On the other hand, for the random (uniform) distribution of molecular orientations, an integration over polar coordinates yields s = 0 because of

7.1 Classification

145

Lateral View

Axial View

Isotropic Liquid n

Nematic LC

Smectic LCs

Smectic A

Disordered Crystal (Smectic LC?)

Smectic B

Crystal E (Smectic E)

Ordered Crystal

Fig. 7.1 Relation of orthogonal liquid crystalline phases of calamitic molecules starting from an ordered crystal. Arrows indicate possible phase transitions

146

7 Liquid Crystals





2π

π/2

dφ 0

0

1 (3 cos2 θ − 1) sin θdθ = 0. 2

(7.3)

It is clear that deviations from the perfect order continuously diminish s. Thus, the s defined by Eq. 7.1 indeed serves as an order parameter indicating whether the nematic order exists. It is, however, interesting to see the case with θ = π/2, for example. This case corresponds to the one where all molecules are perpendicular to n. The nematic order parameter becomes s = − 21 , clearly indicating that the positive and negative s correspond to physically different states. This asymmetry in the property of s brings about significant consequence on the thermodynamic property of the phase transition between the nematic phase and isotropic liquid. If the transition is described in Landau’s thermodynamic phenomenology (Sect. 2.2), there always exists the third-order term in s in the expansion of free energy. According to the general discussion, therefore, the transition should be of first-order (Sect. 2.2.4). Since the director concisely reflects the molecular arrangement, its direction prefers its continuity as a function of the position. Thus, we can assign the director at its point as a vector field to each point. This field is the director field. The uniform field is the most stable, and the deviation from it costs some energy penalty. We can thus define some elastic constants for the director field deformation [2, 3], despite the fluidity of nematic liquid crystals. The director field and accompanied elastic constants are the basis for the elastic theory of liquid crystals [4]. The theory enables concise yet effective treatments of characteristic phenomena to liquid crystals. Those include the formation of topological defects and the dynamics of the director field crucial for applications. However, no further details are given here, because the theory stands on a higher hierarchy as a continuum theory than the molecular description of systems, which is the subject of this book.

7.1.2.2

Smectic Phases

Another liquid crystalline phase appears in the phase diagram reported based on the computer simulation of ensembles of spherocylinders. This type of liquid crystalline phase is called smectic after the smectite, a clay mineral having a layered structure. Different smectic phases have different internal structures. In any phase, molecules are nearly normal to layers. The appearance of a layered structure can be sensed by some scattering experiments (X-ray, neutron beam, electron beam, or visible light in limited case). Since such a scattering process physically performs the Fourier transform of the density of scatterer (see Chap. 3), the fundamental scattering (often indicated as the 001 reflection with the z-axis along the layer normal) may serve as an order parameter characterizing the smectic order. It is, however, to be remembered that the scattering intensity is “weighted,” depending on the scattering beam used. The smectic phase that has the highest symmetry is called a smectic A (SmA) phase. In this phase, the nematic director defined locally within a small region of a layer is normal to the layer. There is no order within the layer except the nematic order. That is, the phase is circularly symmetric around the layer normal. We can imagine

7.1 Classification

147

a spheroid as an abstract molecule, as in the case of nematic phases. The distinction from nematic phases is the existence of layers. If the degree of the layer formation is high enough, each layer can be regarded as a two-dimensional fluid. However, it is often the case that only the fundamental reflection is discernible experimentally. In such cases, the naïve description of a smectic A phase as stacked layers of twodimensional fluid is not adequate. Smectic phases having not the circular symmetry, but the six-fold symmetry around the layer normal are known to exist. Such phases are termed smectic B (SmB) phases. SmB phases are further divided into hexatic B (HexB) and crystal B (CrB) phase. In the HexB phase, molecules are arranged roughly on a triangular lattice. Since the spatial order in two-dimension is fragile, the arrangement cannot extend over a long-range. There is no structural correlation between neighboring layers, accordingly. In contrast, the spatial order in the CrB phase extends not only within the layer but also to adjacent layers. For the reason that the spatial order extends in three dimensions, the CrB phase is a kind of a three-dimensional crystal after the definition by the International Union of Crystallography (IUCr) [5]. In comparison with the SmA phase, the appearance of reflections indicating the triangular lattice within the layer indicates the formation of these phases. Although reflections with mixed indexes between hk0 (from the order within each layer) and 00l (from the stack of layers) are expected for the CrB phase, such observation is rare. In practice, HexB and CrB phases are discriminated through the assessment of the linewidth for hk0 (narrower for CrB phase than for HexB phase). Further lowering of the symmetry from the CrB phase results in another smectic phase called the smectic E (SmE) or crystal E (CrE) phase. In this phase, the effective molecular symmetry becomes of a brick (rectangular parallelepiped). Molecules without distinctions between the head and tail and between the front and back form herringbone arrays within respective layers, which stack on each other along the zaxis. The illustration certainly looks like a crystal of the orthorhombic system. This resemblance is the reason why this phase is also called the crystal E phase. The CrE phase seems an anisotropic counterpart of plastic crystals for globular molecules while taking the assumed dynamical disorder into account. On the other hand, the highly anisotropic shape of molecules suffices to pose a question of whether such a disorder is possible. This issue is impossible to resolve by only considering the averaged molecular shape. A possible resolution will be discussed in Sect. 9.4.4.2. Although the orientation of the nematic director in the smectic phases mentioned above coincides with the layer normal, which is the unique and rational direction for the layered structure, there is generally no reason why they should coincide with each other. Smectic C phase is a representative of such tilted smectic phases. Note that there is no possibility for a uniformly tilted nematic phase because the uniform tilt merely rotates the director. The tilt of the director from the layer normal induces additional degrees of freedom. For example, the phase having the alternate tilt in adjacent layers within a common plane perpendicular to the layer is called SmCA phase in contrast to the simple SmC phase having uniform tilt within a common plane. A wide variety can be imagined for the arrangement of the tilting in a layer by layer. Even a successive twist of the tilt direction is possible.

148

7.1.2.3

7 Liquid Crystals

Columnar Phases

The (uniaxial) nematic phase is common in ensembles of highly anisotropic molecules, whereas layered smectic phases are characteristic of ensembles of elongated molecules. The counterpart of the latter in ensembles of flat molecules are columnar phases, though columnar phases made of rod-shaped mesogens are also known to exist in reality. Molecules stack on each other to form separate columns, which aggregate in two dimensions forming a square or triangular (hexagonal) lattice. Many columnar phases are discriminated based on aggregation modes of columns or inside the columns (e.g., titled or normal). For details, the readers are directed to specialized books of liquid crystal science [1].

7.1.2.4

Cubic Phases

Some liquid crystalline compounds exhibit mesophases having a cubic symmetry. These are generally called cubic phases. Figure 4.1a shows a basic structure of the Gyroid phase, which appears most often. The cubic symmetry accompanies a threedimensional periodicity, leading to an understanding of such structures as crystals. The number of molecules within a unit cell is of the order of 103 . They share space groups with cubic phases in lyotropic liquid crystals and block copolymers, both of which contain plural “components.” In the case of cubic phases consisting of small molecules, requisites for their formation are under extensive study. A concise description of cubic phases is unavailable, accordingly. However, the molecular packing recently revealed (see Fig. 10.8) [6] suggests the molecular shape is essential for them, as will be discussed in Sect. 10.3.5.

7.1.2.5

Chiral Phases

If a body and its mirror image cannot be exactly superimposed (like your right and left hands), the body is termed “chiral.” When the body is a molecule, two types of molecules (right- and left-handed) are equivalent in an isolated state, including chemical reactivity with an achiral (= “not chiral”) molecule. In contrast, they are different in interaction with chiral molecules. Such a difference remains even in macroscopic properties of a macroscopic system unless the system consists of equal amounts of the two counterparts. Adding a small number of chiral molecules to the ensemble of achiral molecules also produce a chiral macroscopic system. Such systems exhibit chiral phases not only for a crystal and isotropic liquid but also for liquid crystals. Chiral liquid crystalline phases are indicated by adding an asterisk to the abbreviation of phase (such as N* or SmC*). The effect of the molecular chirality appears as a local tilt of neighboring molecules. Note that the local tilt cannot extend over the three-dimensional space. Some “defects” are necessary. In the chiral nematic (N*) phase, often called also as

7.1 Classification

149

Fig. 7.2 A structural model of a chiral cubic phase consisting of achiral rodlike molecules. While eight defects are assumed on each sphere, jungle gyms are filled without any defects by a twisted arrangement of molecules similarly to those on the sphere. Reproduced from Phys. Chem. Chem. Phys., 18, 3280 (2016). [7] with permission from the PCCP Owner Societies

a cholesteric (Ch) phase after its first identification in a cholesterol derivative, the chirality produces a spatial rotation of the director having (ideally) a single singular line (the rotation axis). The direction of the axis and the pitch of the rotation emerge as new characteristic properties. When the tilting power is strong enough, the defect lines exhibit some spatial order regularly. The resulting three-dimensional periodic lattice comprises a vast unit cell containing ca. 106 molecules. Since the lattice constant has a length scale comparable to visible light, such phases are called “blue phases.” Three different blue phases have been identified. In chiral smectic phases, the local tilt is incompatible with smectic A structure but compatible with that of the smectic C phase. Thus, the SmA* phase, SmA phase consisting of chiral molecules, is indistinguishable from a normal SmA phase of achiral molecules. On the other hand, SmC* having a local tilt in the SmC structure can have a net polarization arising from the electric dipole pointing perpendicular to the molecular figure (long) axis. Recently, chiral liquid crystalline phases consisting of seemingly achiral molecules have been reported [8–10]. Figure 7.2 shows a structural model for a highly complicated phase known as a chiral cubic phase. Although some structural models can be constructed as exemplified, mechanisms to establish and stabilize such an organization have not been clarified yet.

7.2 Effects of Molecular Anisotropy 7.2.1 Liquid Crystal of Hard Particles: Onsager Theory As mentioned in Chap. 6, the appearance of mesophases, including liquid crystals, has been well established through molecular simulation [11, 12]. However, it is not easy to identify the primary mechanism of the formation of liquid crystals. Here, a seminal treatment by Onsager [13], which shows the formation of a nematic liquid crystal solely by the anisotropy of molecular shape, i.e., the repulsive interaction arising from the excluded volume effect, is briefly described. Since the calculation is

150

7 Liquid Crystals

complicated and specific to the model (beyond the scope of this book, accordingly), we here only trace the logic. The treatment is based on the lowest-order perturbative (virial) expansion (Eq. 1.76) of thermodynamics of the classical gas (fluid) discussed in Sect. 1.3.1. At the lowest order, the free energy density of the fluid consisting of identical particles is given by   F = c ln c − 1 + f (ω) ln[4π f (ω)]dω kB T V    1 β1 f (ω1 ) f (ω2 )dω1 dω2 , − c 2 with

1 β1 = V

      u 12 exp − − 1 dr1 dr2 kB T

(7.4)

(7.5)

where c is the number density of particles, f (ω) the (single-particle) distribution function of the molecular orientation (specified by ω = (θ, ϕ)) normalized to unity, and u i j the interparticle interaction between the ith and jth particles, and ri position of the ith particle. The integration in the definition of β1 (Eq. 7.5) is taken over the whole volume while keeping orientations of particles 1 and 2. When only the interaction between rigid bodies is assumed, i.e., u = ∞ (the integrand = −1 in Eq. 7.5) if two particles overlap and u = 0 (integrand = 0) unless so, it is easy to verify that −β1 equals to the excluded volume assignable to a single πr 3 for rigid spheres with the radius r . For highly particle. For example, −β1 = 32 3 anisotropic cylinders (with length l and diameter d with l/d  1), Onsager showed −β1 (θ12 ) ≈ dl 2 sin θ12 with θ12 being the angle between the long axes of two particles 1 and 2. A half of its orientational average, bex = π4 dl 2 , serves as an effective volume of a single particle. Comparing this quantity with the volume of a cylinder π4 d 2 l, we see that the effective volume is larger by a factor l/d. Using bex , thus, we can write − β1 =

8 bex sin θ12 . π

(7.6)

Note that the enhancing factor for volume mentioned above (l/d) is valid for l/d  1. Its minimum value was shown to be larger than ca. 5 irrespective of l/d. According to thermodynamics, free energy should be minimum in the equilibrium of any system. Here, it is noteworthy that there is no contribution in “energy” in the present model: kinetic energy is always equal to 23 kB T per particle because of the classical nature of the model while the potential energy is always zero since the overlap of particles is prohibited. The free energy density divided by kB T (Eq. 7.4) is, in fact, the entropy density of the system. The entropy, therefore, determines the equilibrium state in this treatment. It is, however, emphasized that thermal motion is necessary to “obtain” the equilibrium state though the phase transition boundary is independent of temperature.

7.2 Effects of Molecular Anisotropy

151

The orientational distribution function f (ω) should minimize the free energy 1 , corresponding to density given by Eq. 7.4, The uniform distribution f (ω) = 4π the isotropic fluid, satisfies the requirement, irrespective of the density c. On the other hand, a variational determination of f (ω) in a general form is a formidable task in general. Onsager showed that the integration in Eq. 7.4 could be performed analytically if the following functional form is assumed: f (ω) =

α cosh(α cos θ) 4π sinh α

(7.7)

with α being a parameter characterizing the anisotropic distribution of particles’ orientations. The null value of α corresponds to the uniform distribution, and a more positive α represents a more anisotropic distribution. After the analytical integration in Eq. 7.4, free energy is obtained as a function of α. Standard procedure for finding the minimum (d F(α)/dα = 0 and d 2 F(α)/dα2 > 0) yields a solution α as a function of bex c, as shown in Fig. 7.3. Finally, the comparison of free energies between anisotropic solution (with α > 0) and isotropic fluid (α = 0) reveals that the former is more stable in bex c  4.5. Although this solution while assuming Eq. 7.7 as a trial function is not necessarily the best one, its presence guarantees that the isotropic fluid is less stable than the unidirectionally aligned state, at least in the lowest order perturbative expansion. That is, only the repulsive interaction reflecting the anisotropy of molecular shape produces the nematic order. After the original paper by Onsager [13], many papers have treated a similar issue, not only for cylinders but also for disks, in more detailed and/or expanded manner(s) [14–16]. All these have shown that the isotropic fluid is less stable than the nematic state in bc  4 with a slightly different definition of b, the effective excluded-volume of a particle. Recognizing that the minimum of the ratio between the effective and true volumes of a particle is 4 (for the isotropic sphere), we see that there is a minimum anisotropy to stabilize the nematic state.

Fig. 7.3 Concentration dependence of α characterizing anisotropic distribution of the particle’s orientation in the Onsager theory [13]. Dotted curve is in a metastable region with respect to the isotropic fluid (F(α) > F(0))

30

20

10

0 3.5

4.0

4.5 bexc

5.0

5.5

152

7 Liquid Crystals

It is noteworthy that the volume of the considered system is kept from outside because the treatment assumes the classical gas consisting of anisotropic particles. The result can, therefore, be compared directly with gasses at extremely high density, and colloidal dispersion, the volume of which is kept by that of the solvent. Attractive interaction between particles is necessary for a system having its inherent volume.

7.2.2 Maier–Saupe Theory The Onsager theory described in the previous section surely works for some class of liquid crystals, especially of lyotropic ones. However, the theory is abstract for a framework of molecular understanding of liquid crystals. A model for treating, at least, “smooth” intermolecular interaction is necessary. The most concise yet successful one is known as the Maier–Saupe theory [17]. The Maier–Saupe theory considers only an angle-dependent part of intermolecular interaction between rodlike (often termed “calamitic”) molecules. The assumed interaction is v(θi, j ) =

V (θi, j ) = −P2 (cos θi, j ) V0

(7.8)

Here, θ is the angle between the figure axes of interacting molecules, i and j, and P2 (x) the second-order Legendre polynomial (Eq. 7.2). The adoption of the secondorder formula reflects the absence of the distinction between the head and tail of molecules. This interaction has minimum energy −V0 for θi, j = 0 while a maximum V0 /2 for θi, j = π/2. Note that a state specified by θ is not unique in general except for the minimum energy. Molecular orientations are specified by the spherical coordinate (θi , φi ). By virtue of the addition theorem for spherical harmonics Ylm l 4π ∗ Pl (cos θi, j ) = Ylm (θi , φi )Ylm (θ j , φ j ), 2l + 1 m=−l

(7.9)

the interaction is decomposed into three terms (corresponding to m = 0, ±1, ±2) as P2 (cos θi, j ) = P2 (cos θi )P2 (cos θ j ) 3 3 − cos Δφ sin 2θi sin 2θ j + cos 2Δφ sin2 θi sin2 θ j (7.10) 4 4 with Δφ = φi − φ j . As far as the discussion is limited to axially symmetric systems consisting of axially symmetric molecules, it is reasonable to assume their cancelation after summation over respective surrounding molecules. Thus, we need to handle a model having only θ for each molecule:

7.2 Effects of Molecular Anisotropy Fig. 7.4 Potential function v(θ) [= V (θ)/V0 ] as a function of r . The solid and dotted curves alternate by a step of Δr = 0.1. The dot-dash curve represents 9 v(θ) with r = rc (= 79 ). Reproduced from J. Phys. Soc. Jpn., 86, 084602 (2017) [19]

153

v( )

2

r=1

1

0

r=0

-1 0.0

0.3

0.2

0.1

0.4

0.5

/π

v(θi, j ) = −P2 (cos θi )P2 (cos θ j )

(7.11)

Figure 7.4 shows the dependence of the interaction energy on θ (the angle between the long axes of interacting molecules). Since the intermolecular interaction (Eq. 7.11) is a bilinear product of functions of a variable assigned to a single particle as in the case of spin models on lattices described in Chap. 6, the analysis can proceed through a similar way. The internal energy of the ensemble consisting of N molecules is given by E=



v(θi, j )

{i, j}

=−



⎛ ⎝ P2 (cos θi )

i



⎞ P2 (cos θ j )⎠

(7.12)

j

where the sum in the first line runs over the interacting pairs and the sum over j in the second line over molecules interacting with molecule i. Suppose the interaction is short-ranged and affecting z molecules. Since all molecules are equivalent to each other, the internal energy can be rewritten as E≈− ≈−

z P2 (cos θi )P2 (cos θ) 2 i

(7.13)

zN 2 s 2

where s = P2 (cos θ) is the average. This s serves as the order parameter for the nematic order in reality (see Eq. 7.1) and should fulfill the self-consistency

154

7 Liquid Crystals

 π/2 1 s= P2 (cos θ) exp [βzs P2 (cos θ)] sin θdθ q(β) 0  1 1 = P2 (cos θ) exp [βzs P2 (cos θ)] d(cos θ). q(β) 0  1 1 = P2 (x) exp [βzs P2 (x)] d x q(β) 0 with

 q(β) =

1

exp [βzs P2 (x)] d x,

(7.14)

(7.15)

0

where β = 1/k B T is an inverse temperature. The number of states appearing in the model is always infinite due to the classical nature of the model except for the perfect order θi = 0 for all i, in contrast to lattice problems treated in Chap. 6 assuming discrete states for each molecule. Therefore, the entropy cannot be evaluated in this problem by counting the possible number of states based on the Boltzmann principle. Alternatively, Eq. 7.15 is regarded as the partition function (state sum) for a single molecule in a mean-field −zs P2 (cos θ). It is, however, essential to know that the internal energy E/N derived naïvely from this partition function differs from that derived above (Eq. 7.13). This difference is due to the double count of the interaction, as is usually the case in any mean-field treatments. The corrected free energy is given by z F = −β −1 ln q(β) + s 2 . N 2

(7.16)

We can not solve the self-consistency equation analytically, but find solutions numerically. The Maier–Saupe theory predicts a first-order transition between the disordered state (isotropic liquid) and the nematic liquid crystal, in accordance with the symmetry consideration (Sect. 7.1.2.1). The comparison of free energies of the nematic phase and disordered state (with s = 0) is necessary. The free energy of the disordered state is F = −kB T ln 1 = 0 by Eq. 7.15. The Maier–Saupe theory predicts a first-order transition at Ttrs = 0.2201zV0 /kB . Figure 7.5 shows the temperature dependence of the nematic order parameter s. Since the intermolecular interaction prefers the parallel arrangement, the perfect nematic order appears at the absolute zero. The Maier–Saupe theory explains the formation of only the nematic phase. To treat other phases, at least, the introduction of a length scale is necessary. Indeed, a simple introduction of the molecular length explains the formation of layered liquid crystal, SmA phase, as shown by McMillan [18].

7.3 Effects of Molecular Shape 1.0

s

Fig. 7.5 Temperature dependence of the nematic order parameter s = P2 (cos θ) of the Maier–Saupe model [17] within the original mean field treatment

155

0.5

0.0 0.0

0.1

0.2

T / zV0

7.3 Effects of Molecular Shape Even if we assume the axial symmetry for particles without the distinction between its head and tail, we can expect the preference for the twisted alignment for a broad class of rodlike mesogens. Indeed, the axial symmetry of a non-polar molecule allows the presence of the quadrupolar interaction, which prefers the vertical alignment for two neighboring molecules. Depending on its strength relative to other components of interaction, it may produce a maximum at the parallel alignment in the orientation-dependent part of intermolecular interaction. The preference for twisted arrangements between neighboring molecules is investigated [19] through computer simulations of an extended version of the Maier–Saupe model of nematic liquid crystals [17]. A phase diagram of a model capable of expressing such preferences has been constructed for a lattice model with a nearest-neighbor interaction. Molecules are assumed to sit on the simple cubic lattice. The preference for the twisted alignment without a preferred twist sense for neighboring spins is expressed by the following interaction between the nearest neighbors (0 ≤ r ≤ 1): v(θ) =

  7 V (θ) = − (1 − r )P2 (cos θ) − r P4 (cos θ) . V0 3

(7.17)

1 (35x 4 − 30x 2 + 3) 8

(7.18)

Here, P4 (x) =

is the 4th-order Legendre polynomial. Figure 7.4 shows the change in the shape of interaction potential v(θ)√with r . The θmin that minimizes v(θ) grows from 0 in 9 to arctan(2/ 3) < π2 at r = 1. We can correlate this change with that r ≤ rc = 79 in molecular shape, as illustrated in Fig. 7.6. A finite θmin means that the interaction is not ferroic. Since two senses of twist (left and right) are energetically equivalent, a kind of alternate order may be expected on the simple cubic lattice. The hump of the potential at θ = 0 grows as a function of r for r > rc .

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7 Liquid Crystals

r

rc

Fig. 7.6 Image of the geometrical change (from spindle to anti-spindle via cylinder) for a non-polar 9 molecule having the axial symmetry. rc (= 79 for the potential expressed by Eq. 7.17) is a threshold for twist. Reproduced from J. Phys. Soc. Jpn., 86, 084602 (2017) [19]

1.5

T / V0

1.0

disordered 0.5

uniaxial (nematic) alternate SB1

0.0 0.0

SB2 0.5

1.0

r Fig. 7.7 Phase diagram of extended Maier–Saupe model on the simple cubic lattice as a function of r in Eq. 7.17. SB phases possess both uniaxial and alternate orders. Solid line, first-order phase transition driven by the nematic instability; dotted line, second-order phase transition with the alternate twist order. The dot-dashed line indicates the height of the energy hump at θNN = 0. Reproduced from J. Phys. Soc. Jpn., 86, 084602 (2017) [19]

Monte Carlo simulations revealed the presence of two types of instabilities (tendencies to respective orders) depending on the degree of preference if only the nearest neighbor interaction is taken into account. A weak next-nearest-neighbor interaction induces other instabilities with different spatial orders depending on its sign. Figure 7.7 is a phase diagram without next-nearest-neighbor interaction. The tendency to the uniaxial (nematic) order brings about a first-order phase transition while a continuous transition results from that to the alternate order, which appears for resolving the frustration in local twists in the arrangement of molecules. The two instabilities behave almost independently. The phase sequence on cooling is thus either the disordered → the uniaxial → the uniaxial and alternate (SB), or the disordered → the alternate → SB.

7.3 Effects of Molecular Shape

157

To go beyond the nearest-neighbor interaction, the following interaction was assumed for next-nearest neighbors: VNNN (θ) = −vNNN P2 (cos θ) V0

(7.19)

The next-nearest-neighbor interaction, irrespective of its sign, induces another phase transition from the SB phase to a phase showing a local chiral order while retaining the macroscopic achirality. The achiral nature of the new phase implies that the presence of only a nearest-neighbor interaction is insufficient to induce a chiral order even if locally. The local chiral order induced by the next-nearest-neighbor interaction forms a weak spatial order if the next-nearest-neighbor interaction does not favor the nematic order (vNNN < 0). The local chiral order does not produce the net chirality but contains two types of linear domains (chains) of opposing local chiralities with equal amounts. The simulations indicated that no spatial order concerning the local chirality occurs without the next-nearest-neighbor interaction. In contrast, spatially linear segregation of the local chirality emerges with it. These findings seem to contribute to understanding cubic or chiral phases, such as the Gyroid phase introduced in Sect. 4.3.1

7.4 Molecular Shape and Aggregation in Lyotropics Liquid crystals formed upon the variation of composition in a system are classified as lyotropic in general [20]. Their formation is driven by the amphiphilicity, which means that molecules favor both oil and solvents such as water. In reality, a part of a molecule favors a contact with, say, water while other parts contact with oil. Although complicated molecules are reported recently to produce exotic aggregation structures, representative amphiphilic molecules are elongated in one direction, as seen for soaps and phospholipids. In this section, therefore, the discussion is limited to such molecules consisting of two parts, hydrophobic hydrocarbon chain(s) and a hydrophilic head group. Assuming the axial symmetry for molecules, we imagine that the molecular shape is a cylinder or a circular truncated cone (including a complete cone as a limiting case). A parameter, called the packing parameter, has been used to characterize the shape of an amphiphilic molecule. Assume that v and l are the volume and length of hydrophilic hydrocarbon chain of the molecule. The sectional area of the hydrophilic head group projected on the plane perpendicular to the molecular axis (see Fig. 7.8) is denoted as a . The packing parameter is defined by the ratio v/al. It is easy to verify that v/al = 1 for a cylinder, v/al = 13 for a complete cone, and 1 < v/al < 1 for a circular truncated cone. The two numerical magnitudes, 13 and 3 1, are often termed critical packing parameters because they are closely related to aggregation structures, as discussed below. However, it is noteworthy that the molecular geometrical quantities, v, a, l, are not predetermined ones for individual

158

7 Liquid Crystals

l

a

Fig. 7.8 In the definition of the packing parameter, v/al, an amphiphilic molecule with volume v is approximated by a cone like object with the area of the basal plane a and the hight l

molecules, but a set of equilibrium properties determined as a result of competitions among many factors. Let us start the consideration with v/al = 13 . Since the molecules are amphiphilic, they aggregate each other. For v/l  1, a spherical aggregate with a radius l can be formed by molecules, the number of which is calculated as 43 πl 3 /( 13 al) = 4πl 2 /a. When they are in the water, the resultant aggregates are favorable in the sense that hydrophilic head groups completely cover the surface and that there is no contact with water for the hydrophobic part. This type of aggregates is well known as spherical micelles. Micelles can crystallize like Fig. 7.9 if they are monodispersed. For v/al = 1, the aggregation of molecules results in the formation of a stack of monolayer sheets. Because of the amphiphilicity of the molecules, each sheet has the sidedness. The sidedness of neighboring layers should be alternate in a stack. If a solvent is present, the solvent will penetrate one of two types of interlayer spaces, depending on its property. Namely, if the solvent is water, it will penetrate between layers of hydrophilic head groups. The resultant state is a lamellar phase consisting of molecular bilayers, resembling the smectic phase of thermotropic liquid crystals. When the axial symmetry of molecules is abandoned, another critical magnitude of packing parameter, v/al = 21 , is identified. It is easy to verify that the volume V and the surface area A of a cylindrical rod (infinitely long) and the radius R satisfy the relation V /A = π R 2 /(2π R) = 21 R. Namely, molecules having v/al = 21 can form in just enough way a cylindrical rod with a surface filled by their head groups and the inside by the hydrophobic parts. The resultant rod can be regarded as a rod-shaped micelle. Arrangement of rod micelles can produce ordered states such as a representative hexagonal phase, in which the rod micelles align parallel with a triangular arrangement on the plane perpendicular to rods. Having identified critical packing parameters, we proceed to intermediate magnitudes. Imagine an increase in the size of the head group starting from v/al = 13 while keeping v constant. A slight increase of a may be adjusted by a change in l because of the flexibility of the hydrophobic part, leading to the retainment of the stability of the spherical micelles. A decrease in a can be achieved in two different ways. A slight decrease of a would need an oily component to be held in a central part of the spherical aggregate of molecules having a shape of a truncated cone. This type of aggregates is a crucial ingredient for the physical mechanism of detergents. The other

7.4 Molecular Shape and Aggregation in Lyotropics

159

Fig. 7.9 Ordered states formed by micelles. Body centered cubic “crystal” made of spherical micelles (left) and hexagonal phase made of rod-shaped micelles (right)

way is to deform the shape of a micelle. Since the infinitely long cylinder requires v/al = 21 as discussed above, the formation of a micelle with its shape of an elongated (prolate) spheroid results in 13 < v/al < 21 in average. The consideration of a state “on average” is reasonable because the aggregation of molecules discussed here is always dynamical. Molecules change their position (and their shape) as a result of thermal motion. Depending on the shape anisotropy of a micelle, their ensemble may form ordered states. If the anisotropy is large enough, according to the discussion in Sect. 7.2.1, they will form a nematic liquid crystalline state called the micelle nematic phase. It is crucial to keep in mind that such a “particle” can change its size, number, and shape in contrast to cases of preformed colloidal particles and individual molecules. A further increase in the packing parameter from v/al = 21 , on average, causes formations of junctions between rod-shaped micelles. At a junction, the ratio of the surface area to the inner volume (sjunction /vjunction ) is undoubtedly smaller than a cylinder. The spatial location of junctions may be regular. Bicontinuous cubic phases are examples of such states. These states can be described starting from the side of the lamellar phase (v/al = 1). If a tubing made of a bilayer bridges two neighboring bilayers, the averaged packing parameter decreases because of the finite thickness of the bilayer. States with random connections of non-parallel bilayers in such a way are seemingly isotropic and called a sponge phase. On the other hand, when the connections are regular in space, such states exhibit periodicity. If two sides of the bilayer are equivalent everywhere, the central surface of such bilayers must mathematically be a minimal surface [21], which has a minimal area with a fixed frame. Indeed, some bicontinuous cubic phases possess respective minimal surface called triply periodic minimal surface (TPMS) or infinitely periodic minimal surface (IPMS) [21]. The two descriptions of a bicontinuous cubic phase using jointed rods (jungle gyms) and a minimal surface are complementary to each other if two jungle gyms are embedded in two spaces divided by a minimal surface. The situation in the ¯ can be seen in Fig. 10.8 case of the so-called Gyroid phase (with space group Ia3d) if particles are ignored.

160

7 Liquid Crystals

Cases with small head groups formally correspond to v/al > 1. This situation can be discussed by the interchanging roles of two parts in a molecule. For example, water is confined in a central part of a rod-shaped micelle in the inverse hexagonal phase and water channel inside rods of jungle gyms in the inverse bicontinuous cubic phase. General understanding briefly described above is the same in polymers [22]. However, there exists a significant difference. In the case of polymers, the length scale of various organizations is much larger than that of a monomer. A quantity primarily controlling a way of organization is the relative composition of components. The shape of monomers hardly plays any role.

References 1. J.W. Goodby, P.J. Collings, T. Kato, C. Tschierske, H.F. Gleeson, P. Raynes (eds.), Handbook of Liquid Crystals (Wiley, Weinheim, 2014) 2. C.W. Oseen, Trans. Faraday Soc. 29, 883–899 (1933) 3. F.C. Frank, Disc. Faraday Soc. 25, 19–28 (1958) 4. P.G. de Gennes, J. Prost, The Physics of Liquid Crystals (Oxford University Press, Oxford, 1994) 5. International Union of Crystallography, Acta Crystallogr. A 48, 922–946 (1992) 6. Y. Nakazawa, Y. Yamamura, S. Kutsumizu, K. Saito, J. Phys. Soc. Jpn. 81, 094601 (2012) 7. K. Saito, Y. Yamamura, Y. Miwa, S. Kutsumizu, Phys. Chem. Chem. Phys. 18, 3280–3284 (2016) 8. C. Tschierske, G. Ungar, ChemPhysChem 17, 9–26 (2016) 9. C. Dressel, F. Liu, M. Prehm, X. Zeng, G. Ungar, C. Tschierske, Angew. Chem. Int. Ed. 53, 13115–13120 (2014) 10. C. Dressel, T. Reppe, M. Perhm, M. Brautzsch, C. Tschierske, Nat. Chem. 6, 971–977 (2014) 11. P. Bolhius, D. Frenkel, J. Chem. Phys. 106, 666–687 (1997) 12. J.A.C. Veerman, D. Frenkel, Phys. Rev. A 43, 4334–4343 (1991) 13. L. Onsager, Ann. N. Y. Acad. Sci. 51, 627–659 (1949) 14. A. Ishihara, J. Chem. Phys. 19, 1142–1147 (1951) 15. K. Lakatos, J. Stat. Phys. 2, 121–136 (1970) 16. R. Zwanzig, J. Chem. Phys. 39, 1714–1721 (1963) 17. W. Maier, A. Saupe, Z. Naturforsch. A 13, 564–566 (1958) 18. W.L. McMillan, Phys. Rev. A 4, 1238–1246 (1971) 19. K. Saito, M. Hishida, Y. Yamamura, J. Phys. Soc. Jpn. 86, 084602 (2017) 20. A.M.F. Neto, S.R.A. Salinas, The Physics of Lyotropic Liquid Crystals (Oxford University Press, London, 2005) 21. S. Hyde, S. Andersson, K. Larsson, Z. Blum, T. Landh, S. Lidin, B.W. Ninham, The Language of Shape (Elsevier, Amsterdam, 1997) 22. I.W. Hamley, The Physics of Block Copolymers (Oxford University Press, Oxford, 1998)

Chapter 8

Molecular Glasses

8.1 Glass as Frozen-in State 8.1.1 Simple View on Glass Transitions 8.1.1.1

Relaxation and Dispersion

When a macroscopic system is brought slightly out of its equilibrium by changing an external parameter (such as electric field), the system usually exhibits a continuous change into a new equilibrium under the condition. This change is called relaxation. A typical example is the relaxation of the electric polarization of dielectric material after the magnitude of applied electric field E is suddenly changed, e.g., E 0 → 0 at t = 0. Assume that the stationary polarization is p0 before the sudden change (t < 0), and p0 = χ0 E 0 , in which χ0 is a (static) electric susceptibility. Since the polarization relaxation is driven by the deviation in polarization from the equilibrium one, it is reasonable to write d p = −τ −1 p, (8.1) dt where τ is a time constant known as a relaxation time. This equation is easily integrated as   t . (8.2) p(t) = p0 exp − τ The relaxation, in this case, obeys an exponential decay with a time constant τ . Note that the relaxation of a system does not always exhibit an exponential decay. To get an idea of the relation between the microscopic dynamics of particles (molecules) and the relaxation of the system, let us consider the particles having

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 K. Saito, Chemical Physics of Molecular Condensed Matter, Lecture Notes in Chemistry 104, https://doi.org/10.1007/978-981-15-9023-8_8

161

162

8 Molecular Glasses

two states (+ and −). Their numbers in two states are denoted as n + and n − with n = n + + n − being constant. We write the rate equations describing time evolutions of n + and n − as d n ± = −k∓ n ± + k± n ∓ dt = −(k− + k+ )n ± + k± n,

(8.3)

using microscopic (but averaged) rate constants k− and k+ responsible for transition + → − and − → +, respectively. For

we have

Δn = n − − n + ,

(8.4)

d Δn = −(k− + k+ )Δn + (k− − k+ )n. dt

(8.5)

In the stationary state with dΔn/dt = 0, we find Δn =

(8.6)

k− − k+ n. k− + k+

(8.7)

Thus, Eq. 8.3 is rewritten as d dt

 Δn −

   k− − k+ k− − k+ n = −(k− + k+ ) Δn − n , k− + k+ k− + k+

(8.8)

yielding a solution as Δn(t) = C exp[−(k− + k+ )t] +

k− − k+ n k− + k+

(8.9)

with an integral constant C to be determined by the initial condition. Equation 8.9 indicates that this system consisting of n particles relaxes to the equilibrium with the relaxation time, (8.10) τ = (k− + k+ )−1 , which is expressed in terms of microscopic rate constants. We next assume that the asymmetry in rate constants originates in the effect of the applied electric field E. Assuming weak field, we write k− − k+ = χ0 E. k− + k+

(8.11)

8.1 Glass as Frozen-in State

163

Using an averaged polarization per particle, p = Δn/n, the rate equation (Eq. 8.3) becomes 1 dp = − ( p − χ0 E) . (8.12) dt τ When the electric field is oscillating like E(t) = E 0 e−iωt ,

(8.13)

the averaged polarization also oscillates in the same frequency ω as p(t) = p0 e−iωt

(8.14)

in the steady state. Putting these into Eq. 8.12 leads to 1 − iω p(t) = − [ p(t) − χ0 E(t)] , τ

(8.15)

χ0 E(t) = (1 − iωτ ) p(t).

(8.16)

yielding

Thus, the constant characterizing the polarization response to the electric field is given by 1 χ0 1 − iωτ   1 ωτ = χ0 +i 1 + (ωτ )2 1 + (ωτ )2   = χ (ω) + iχ (ω).

χ(ω) =

(8.17)

This formula defines the (complex) electric susceptibility χ(ω). Note that the susceptibility is a function of the frequency of the applied field. Such frequency dependence of susceptibilities is termed as dispersion. Real and imaginary parts of Eq. 8.17 satisfy the following relations;  ∞   χ (ω )  1 dω χ (ω) = P −ω π ω −∞  ∞   χ (ω ) − χ∞  1 χ (ω) = − P dω , π ω − ω −∞ 

where P



(8.18) (8.19)

dω  takes the Cauchy principal value, and χ∞ = lim χ(ω), ω→∞

(8.20)

164

8 Molecular Glasses

Fig. 8.1 Frequency dependences of real (χ ) and imaginary (χ ) parts of complex electric susceptibility calculated by a simple Debye model

1.0

0.5

0.0 10

-4

10

-2

10

0

10

2

4

10

which expresses the instantaneous response to the applied field. The above couple of equations is known as the Kramers-Kronig relation. The relation originates in the causality and applies to any linear susceptibilities of a system in response to weak perturbations. Equation 8.17, known as the so-called Debye relaxation, exhibits very clearly the frequency dependence of complex electric susceptibility and its meaning. We assume for a while, a constant τ , which is only the time constant specific to the system. Figure 8.1 shows the frequency dependences of real and imaginary parts of the χ. The observer sees a large amplitude of polarization oscillation with ω  τ −1 in-phase with E(t) while a small amplitude with ω  τ −1 . This dependence is interpreted in the following way; the oscillation of the applied electric field is slow enough for dipoles to attain equilibrium distribution with ω  τ −1 , but it is too fast to follow with ω  τ −1 . In other words, the equilibrium state is sensed with ω  τ −1 , but the frozen-in state is observed with ω  τ −1 .

8.1.1.2

Molecular Dynamics and Arrhenius Law

Microscopic molecular motions are generally stochastic. They are well described as an activation process characterized by respective activation barrier  j for each motional mode j. Overall internal dynamics of the system are dominated by the slowest mode, on which we focus attention here. The rate of overall dynamics of the system, τ −1 , depends on the reachability to the top of the energy barrier (a ). Assuming the Boltzmann distribution, the number of particles having the energy a is proportional to n ± exp[−a /(kB T )]. Considering the stochastic nature of the dynamics, we assume the escape frequency, or the rate, k, of passing the barrier top, is proportional to the number of particles. Thus, we may write   a 1 . exp − k= 2τ∞ kB T

(8.21)

8.1 Glass as Frozen-in State

165

−1 This is the so-called Arrhenius law. The τ∞ is often called an attempt frequency. If two states differ in their energy by Δ = − − + > 0, the rate constant depends on the direction of the change of states because the apparent hight of the energy barrier differs from one side to another side. In the equilibrium, the following should hold



Δ n + : n − = 1 : exp − kB T This demands

 .

(8.22)

  k− Δ . = exp − k+ kB T

(8.23)

This is an example of a requirement of detailed balance. The attempt frequency usually has an order of those of molecular vibrations. Equation 8.21 implies that the rate of the slowest mode decreases very rapidly with decreasing temperature. Since thermal equilibrium is attained through visiting representative microscopic states by the system, the system cannot attain a thermal equilibrium within a reasonable time if k becomes very small. Namely, on going to low temperatures, one should consider the possibility of the state being out of thermal equilibrium for any systems. Note that τ = k −1 serves as a relaxation time for thermalization because it determines the overall speed of the relaxation to the equilibrium state. Even if the Arrhenius formula (Eq. 8.21) is assumed for the temperature dependence of the relaxation time τ , it rapidly grows with cooling, as shown in Fig. 8.2. The real part of the electric susceptibility shows the “freezing” around the temperature where ωτ ≈ 1 happens.

8.1.1.3

Naïve Definition of Glass Transitions

Depending on a characteristic and intrinsic frequency (i.e., time scale) of measurements, we observe the crossover between equilibrium and non-equilibrium states.

10 1.0

10 10

0.5

10 10

0.0 0

100

200

T/K

300

10 400

3

0

-3

-6 -9

-12

/s

Fig. 8.2 Temperature dependences of real (χ ) and imaginary (χ ) parts of complex electric susceptibility at ω/(2π) = 106 Hz calculated by a simple Debye model (τ∞ = 10−13 s) with the Arrhenius formula for temperature dependence of τ (a /kB = 3000 K.)

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8 Molecular Glasses

However, there exists its limit on the low-frequency side, the existence of which is practically unavoidable. Namely, any measurements in science on materials are to be done by a human. Her/his daily time scale practically limits the time to wait for thermal equilibrium. If the τ becomes sufficiently more prolonged than the daily time scale (typically, 100 s – 1 h), the system looks macroscopically stable in time, resembling a system in thermal equilibrium, but undergoing, in reality, a prolonged approach to equilibrium. Thus, we cannot observe, at sufficiently low temperatures, the equilibrium state, which is the subject of the third law of thermodynamics. If the change from the equilibrium state to a non-equilibrium state on cooling accompanies detectable anomalies in properties of the system, the change is often called a glass transition. The glass transition has been regarded as one of the most challenging issues in condensed matter science with a long history [1–3]. The above description of glass transitions based on the comparison of the relaxation time τ of the system and the daily time scale implies that any systems undergo a glass transition. This expectation is because τ goes to infinity on approaching the absolute zero according to the Arrhenius law (Eq. 8.21). This naïve outcome is, however, not the case. Essential is a distinction between relevance and irrelevance of the slowest mode for the system’s property. For example, the reorientation of a benzene molecule around its sixfold axis is a classic example of molecular motion in the crystal [4]. Its rate is well described by the Arrhenius equation, resulting in excessive length than the daily time scale around 50 K (τ ≈ 103 s). However, the reorientation is well described as hopping between six equivalent states. Even if the reorientation ceases, the averaged configuration inside the crystal remains the same. Noticeable anomalies are not expected in the properties of the crystal, accordingly. Indeed, there has not been known the presence of glass transitions for crystalline benzene [5]. On the other hand, if the slowest mode involves the transition between nonequivalent configurations of the system, the change between equilibrium and non-equilibrium states would be a glass transition.

8.1.1.4

Thermodynamic Symptoms of Glass Transitions

According to the above definition of glass transitions, they differ from phase transitions in the usual sense of thermodynamics. The glassy state is out of equilibrium and may differ from state to state, depending on how the relevant degrees of freedom became frozen-in. The occurrence of a glass transition on cooling depends on the time scale used in observation, or, more practically, on a cooling rate. For a glass transition on heating, i.e., for a transition from a non-equilibrium state to the equilibrium state, not only the time-scale but also the given state of the glass affects behaviors. Since, at least, a motional mode is practically prohibited below the glass transition, degree(s) of freedom involved in the mode cannot change their states below the glass transition. This immobility means that the thermodynamic functions related to the first law of thermodynamics (enthalpy in usual experiments) no longer change their magnitudes even with cooling. Thus, the temperature dependence of enthalpy becomes weaker, as shown in Fig. 8.3. Namely, the contribution of the degrees of freedom to

8.1 Glass as Frozen-in State

167 liquid

rapid cooling slow cooling

H

Tg

T

0

liquid-quenched glass

supercooled liquid

-1

(dT/dt) / (mK min )

Tg

crystallization

2

TCHM

-2 200

250

300

T/K

Fig. 8.3 Enthalpy relaxation around a glass transition temperature Tg . Upper: The enthalpy curve (with a fixed microscopic structure) of a glass formed by rapid cooling exists higher than that of a glass formed by slow cooling. On heating, the glass formed by a rapid cooling slightly relaxes to the equilibrium with an exothermic effect at T < Tg and with an endothermic effect at Tg < T < Teq . Here, Teq is a temperature where a relaxation time becomes sufficiently shorter than the time scale of the heating. The glass formed by a slow cooling exhibit notable effect only at Tg < T < Teq . Lower: Experimental heat evolution observed for TCHM (Tg = 265 K) in the calorimetric experiment reported in [6]

heat capacity vanishes below the glass transition. In other words, the glass transition appears as a stepped decrease in heat capacity on cooling. A schematic example will be shown in Fig. 8.6 for the case of a liquid quenched glass. The occurrence of a glass transition is often detected by calorimetric experiments. Upon increasing the “frequency” of observation (measurement), the step in heat capacity moves towards a higher temperature, as shown through the heat capacity spectroscopy [7]. The temperature of a glass transition (often indicated as Tg ) is defined not thermodynamically but practically. The time scale, either of 102 s or 103 s, is often used to locate it. Around Tg , the τ of the system is comparable to the daily time scale. This situation means that the approach of the system to the corresponding equilibrium occurs at a noticeable rate there. This behavior is typically known as enthalpy relaxation schematically shown in Fig. 8.3. In practice on heating the glassy sys-

168

8 Molecular Glasses

tem, the exothermic effect is observed for T  Tg whereas the endothermic one for T  Tg . The observation of a set of a stepped anomaly in heat capacity and enthalpy relaxation is regarded as concrete evidence of the detection of a glass transition.

8.1.2 Structural Chemistry of Glass Transitions Even if we restrict ourselves to motional degrees of freedom, there exist many kinds of them in molecular systems, as discussed in previous chapters. Each of them has an intrinsic time scale with different temperature dependence. On cooling, therefore, the order of their time scales may vary. Dynamics slows down with the decrease in temperature. When a relaxation time of the slowest mode surmounts the daily time scale (ca. 102 − 103 s), a freezing-in indeed happens. If this involves inequivalent states, a kind of glass transition takes place. Because of the presence of many degrees of freedom, plural kinds of glass transition have been identified. When isotropic molecular liquid, in which both positional and orientational degrees of freedom are isotropically disordered, is subject to rapid cooling (quench), a liquid-quenched glass (LQG) or glassy liquid is formed. Extensive studies have been devoted to this type of glass transition in real systems. However, theoretical and computational studies have targeted on an abstract and simpler LQG consisting of spherical particles without rotational degrees of freedom (neglecting even kinetic energy of rotational degrees of freedom). Most of such studies nowadays seem to assume the existence of an ideal thermodynamic phase transition (Sect. 8.3) as a counterpart of the kinetic glass transition defined in Sect. 8.1.1.3. Not only isotropic liquids but also other systems possessing partial long-ranged order(s) can exhibit glass transitions. A glassy state derived from liquid crystals is a glassy liquid crystal [8], whereas that derived from orientationally disordered crystal is a glassy crystal [9]. For glassy crystals, the degree of the frozen-in disorder can be qualified through the determination of the so-called residual entropy. The most famous example of residual entropy of a crystal and its resolution can be found for the ice [10–13]. Note that the frozen-in disorder does not guarantee the presence of detectable glass transition even if the residual entropy (discussed in the next section) is undoubtedly resolved [14, 15] in contrast to the successful case, such as the ice [16]. As in the case of the melting process of molecular crystals discussed in Chap. 6, internal motional degrees of freedom can be involved in glass transitions. Clear examples were found in organic conductors containing a half structural unit of a famous donor molecule abbreviated as BEDT-TTF or ET, of which the molecular structure is shown in Fig. 8.4. In agreement with the chemical common sense, the six-membered rings are not flat. In some crystals of their charge-transfer complexes with counter anion(s), the ring exhibits structural disorder. The freezing-in of this structural disorder causes glass transitions [17–20]. The presence of this glass transition resolved the mysterious history-dependence of conducting properties of an organic superconductor, κ-(ET)2 Cu[CN(CN)2 ]Br [18, 19], which exhibited the highest transition temperature Tc = 10.8 K at that time.

8.1 Glass as Frozen-in State Fig. 8.4 Molecular structure of an organic donor, BEDT-TTF (a), and two possible conformation of six-membered rings viewed along the molecular figure axis (b). Reproduced with permission from Solid State Commun., 111, 471 (1999) [18]

169 a) S

C H 2C

C

S

S

S

S

S

C

H 2C

C C

S

C S H

CH2 S

H

b)

CH2

C

H H

H S

S

C

C

H

H

C S H

Coupling between intramolecular dynamics and other motional degrees of freedom is often rather weak. This decoupling can lead to the possibility that the glass transition arising from intramolecular dynamics occurs in a glassy state of other degrees of freedom frozen-in. Indeed, such an occurrence of multiple glass transitions in the same compound is known. The freezing of the conformations between trans and gauche forms in CFCl2 -CFCl2 [21], and of the direction of hydrogen bond in the glassy liquid of tricyclohexylmethanol [6] are examples. Note that the glass transitions discussed in this section are examples of the simple freezing of residual structural disorder of any kind. This fact means that the disorder can be either relevant or irrelevant for the symmetry of the system. Within the examples discussed above, relevant disorders are frozen in LQG and glassy liquid crystals. The glass transition in the ice is also the freezing of the relevant disorder, as evidenced by the successful induction of a proton-ordering transition [22]. On the other hand, intramolecular dynamics and resulting disorder does not bring about a new symmetry. Namely, the degrees of freedom are irrelevant in the context of ordering. In crystalline C60 [23], on cooling an order-disorder transition to the ordered phase with some dynamical disorder occurs at 262.1 K, and then a glass transition due to the freezing of this residual disorder in the ordered phase does around 85 K.

8.2 Properties of Glasses 8.2.1 Fragility Glass and glass transition have first been recognized for liquid quenched glasses. It is thus reasonable that the temperature dependence of the viscosity, which is roughly proportional to the mechanical relaxation time and related to the rate of polarization relaxation, was widely studied. The temperature dependence of relaxation times [τ (T )] on approaching glass transition of a wide variety of materials is often compared in a modified Arrhenius plot known as the Angell plot, where the

170

0

log10( / s)

Fig. 8.5 Schematic Angell plot showing the relation between the relaxation time τ and the temperature normalized by the glass transition temperature T /Tg

8 Molecular Glasses

strong

-5

-10

fragile

0.0

0.5 Tg / T

1.0

abscissa (inverse temperature) is normalized by Tg−1 . Figure 8.5 shows a schematic example. In this form, the plot is always within the range of 0 ≤ Tg /T ≤ 1 and log τ∞ ≤ log τ (T ) ≤ log τ (Tg ), where τ (Tg ) is the chosen relaxation time to define a glass transition for the plot (can be arbitrarily chosen as discussed before). Although τ∞ may differ from material to material, that is physically limited by a molecular attempt frequency, and thus it is of the order of 10−13 − 10−16 s. Angell [1] showed that the dependence of τ (T /Tg ) is not universal but dependent on materials, and defined the fragility, m, a non-dimensional parameter characterizing the dependence, by  ∂ log τ  . (8.24) m= ∂(T /T )  g

T =Tg

The minimum fragility (m ≈ 16) happens if the relaxation time correctly obeys the Arrhenius law while a larger fragility such as 100 or more is also known. Glass formers with a relatively small and large fragility are often called as strong and fragile in this research field.

8.2.2 Step in Heat Capacity at Tg The stepped decrease in heat capacity (upon cooling) is a characteristic symptom of a glass transition, as shown in Fig. 8.6, though there exist examples of phase transitions accompanying such an anomaly [24–26]. Thus, the properties of glass transitions of materials have often been compared in terms of their Tg /Tfus and ΔC p /C p,crystal (Tg ). These forms have been accepted as a result of past struggles because some normalization is necessary for comparison. However, they are not rational enough to accept from a logical point of view. First, it is doubtful as they are normalized by the quantities of the other state (crystal) than the glass. Second, the preceding consideration put the other issue. The heat capacity of a harmonic

8.2 Properties of Glasses

171

oscillator depends on temperature due to a quantum effect. Since some colloidal suspension, which is free from quantum effects, also undergoes a glass transition, the effects should be accounted for before comparison. Third, in ΔC p /C p,crystal (Tg ), no distinction is made between degrees of freedom relevant and irrelevant to the glass transition. No universality in the relative contribution of many degrees of freedom to heat capacity can be expected, accordingly The first and second issues can be treated in rather reasonable ways. Due to the difference in energy scale by one order of magnitude, intramolecular vibrational frequencies scarcely change depending on aggregation state, leading to the possibility of reasonable corrections. Concerning anharmonicities, the so-called (C p − Cv ) correction can be applied. These corrections applied to some organic LQG formers [6] indicated that the heat capacity of LQG is similar to that of crystals in magnitude (≈ 6kB per molecule for translational and librational vibrations) at Tg and that there remain significant differences in heat capacity of liquids. The resultant difference in heat capacity should be attributed to the difference in degrees of freedom relevant to the glass transitions (the third issue) or the number of particles in aggregation (unit entities) most reasonable for the glass transition. Indeed, the step in heat capacity was huge for the compound, of which the equilibrium of the hydrogen bond formation is frozen at Tg together with motional (translational and rotational) degrees of freedom in the glass transition [6].

8.2.3 Residual Entropy Since the heat capacity coming from the freezing degrees of freedom ceases below the glass transition temperature, seeming entropy of the glass at the absolute zero does not follow the third-law of thermodynamics, i.e., lim T →+0 S(T ) > 0. This magnitude is known as the residual entropy. In reality, the residual entropy Sres is calculated by Sres = Sabs (T ) − Scal (T )  = Sabs (T ) − lim T0 →+0

(8.25) T T0

Cexp (T ) dT T

where Sabs (T ) and Scal (T ) are absolute and calorimetric entropies, respectively, and Cexp is the experimental heat capacity. The absolute entropy is termed the statistical entropy when calculated based on the statistical mechanics for simple molecules [12, 14, 15]. For more complex cases, the absolute entropy can be estimated from the calorimetric entropy evaluated for the equilibrium phase sequence of the same compound [27, 28]. Figure 8.6 helps to understanding. The residual entropy quantitatively characterizes the degree of disorder frozen-in at the glass transition. For some simple cases [12, 14], the residual entropy is quantitatively interpreted by a microscopic disorder through Boltzmann’s principle, S = kB ln W . Note that Sres < 0 is an indication of the failure in obtaining Sabs (T ). Such a case may happen when the

8 Molecular Glasses

S

Cp

172

T Fig. 8.6 Temperature (T ) dependences of thermodynamic quantities related to a glass transition. Upper: Heat capacities (C p = T (∂ S/∂T ) p ) of crystal, liquid, and glass. In the equilibrium sequence, the crystal melts at Tfus with the latent heat (vertical line) to the liquid. Upon cooling the liquid, it becomes “supercooled” if it fails to crystallize below Tfus . Further cooling brings about a glass transition around Tg , around which the heat capacity exhibits a stepped decrease to the magnitude comparable with the crystal. Since the time scale of the observation is comparable with that of the enthalpy relaxation around Tg , the step in apparent heat capacity is rounded. Lower: Entropies (S) of crystal, liquid, and glass. The jump at Tfus is the entropy of fusion, Δfus S. The difference between the glass and crystal entropy is the (temperature-dependent) configurational entropy. The residual entropy is its extrapolation to the absolute zero if one can safely assume the third law of thermodynamics. Because of the larger heat capacity of the supercooled liquid (above Tg ) than the crystal, a naïve extrapolation of the liquid entropy predicts the crossing with the crystal one at the Kauzmann temperature TK

experiment is done down to very low temperatures, where the fine-levels of nuclei are experimentally accessible. A related quantity to the residual entropy 

Tfus Cglass (T ) dT Sconf (T ) = Sliq (Tfus ) − T T  Tfus Ccrystal (T ) − Cglass (T ) Δfus H dT + = T Tfus T

(8.26)

8.2 Properties of Glasses

173

is often discussed as a measure of the temperature-dependent degree of the disorder. This quantity is called the configurational entropy in the research field of glasses [29]. Here, Cglass indicates both heat capacities of the glass below Tg and the supercooled liquid. By definition, Sres = Sconf (0) assuming the third law of thermodynamics for the crystal.

8.2.4 Effects of Structural Inhomogeneity Irrespective of a kind of the frozen-in degree(s) of freedom, glasses intrinsically possess, to some extent, structural disorder, which may be difficult to characterize quantitatively beyond the residual entropy. The structural disorder inevitably leads to the inhomogeneity in its structure. If the degree of the disorder is weak, and only a limited kind of disorder can exist, it can be regarded as a point defect. However, such a situation is difficult to imagine in real glasses. On the contrary, we should consider situations where an extent affected by each “defect” overlaps with that of others. In this case, effects on each molecule vary from molecule to molecule, resulting in the nearly continuous distribution of the state of surroundings. Let consider such situations. Although liquid-quenched glasses are plausibly more isotropic than crystalline solids, for which the Debye T 3 –law of heat capacity had been established, the violation of the law has been reported for LQG. Heat capacities of nonmetallic1 glasses at low temperatures roughly obey the law, C = αT 3 + βT [30–32]. Since the cubic term of the heat capacity of solids originates in the dispersion relation of the sound (acoustic) waves (see Chap. 5), the heat capacity of glasses ought to have the cubic term. The primary issue is thus to understand the origin of the linear term. Anderson [33] and Phillips [34] independently explained the presence of the linear term by assuming a continuous distribution of level splitting of tunneling states. The non-vanishing density of states at the null excitation energy, n(0), leads the linear term with β ∝ n(0). Further, their models that are mostly the same to each other explain well the temperature dependence of the acoustic attenuation in disordered solids. Later, the density of states excess to the Debye one has been well established below ca. 1 meV. The peak in the density of states (or equivalently in C p T −3 ) is often called the boson peak. Excitations responsible for the peak in the density of states are believed to be involved in anomalous physical properties of glasses. Some models predict the presence of the boson peak while assuming the inhomogeneous distribution of structural disorders [35, 36]. It is noteworthy that all of the studies on the anomalous properties of glasses described in the previous paragraph implicitly assume the universality of the anomalies. However, there is an example that would imply the dependence of the anomalous property on the degree and/or type of frozen-in disorder [37]. The heat capacity of 1 Conduction

electrons well approximated by the Fermi gas also contribute linearly on the temperature to heat capacity, irrespective of crystalline or amorphous states.

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8 Molecular Glasses

the glassy plastic crystal of ethanol suggests the presence of the boson peak [38] whereas that of the glass of the disordered phase of p-chloronitrobenzene, the residual entropy of which implies the freezing of the head-to-tail disorder of molecules [28], does not but with additional heat capacity beyond the Debye one [37].

8.3 Possibility of Ideal Glass Transitions Since the heat capacity of liquids (irrespective of stable or supercooled) is larger than that of the crystal, a naïve extrapolation of their entropies toward the low-temperature side leads to their crossing at some temperature, often called the Kauzmann temperature, TK . Namely, (8.27) Sliquid (T ) < Scrystal (T ) for T < TK . Figure 8.6 illustrates the situation. Considering Boltzmann’s principle (Eq. 2.8), however, it is expected that the liquid with (at least) positional disorder should have a larger entropy than the crystal that has positional order. Equation 8.27 certainly contradicts this expectation [39]. This contradictory situation is known as the Kauzmann paradox. The easiest way to avoid this catastrophic situation is to assume a phase transition above TK . Indeed, Tg is always higher than but close to TK . Another support for the presence of “ideal” glass transition comes from the temperature dependence of the relaxation time. The temperature dependence of the relaxation time of fragile glass formers is empirically well expressed by the so-called WLF formula (after Williams, Landel, and Ferry [40]),  τ (T ) = τ∞ exp

 WLF , kB (T − T ∗ )

(8.28)

where WLF is the apparent activation energy, and T ∗ is a constant. Purely Arrhenius behavior of the strong glass former recovers by setting T ∗ = 0. Since the relaxation time diverges at T ∗ , Eq. 8.28 has been regarded as a possible symptom of a hidden thermodynamic transition. In this view, the non-Arrhenius behavior can be regarded as a symptom of the growth of the structural or dynamical correlation length of the system on approaching the critical point. The volume characterized by this correlation length is often termed as a cooperatively rearranging region, assumed in theory [29] that naturally explains the temperature dependence of the relaxation time expressed by Eq. 8.28. Stronger temperature dependence of the relaxation time than the Arrhenius behavior can be rationalized in another way if the change in liquid structure upon the temperature variation is acknowledged [41]. It is reasonable to imagine that the activation energy responsible for molecular dynamics grows upon cooling due to the growth of local structure (inside the liquid) even if only a single molecular dynamics is assumed. Indeed, structural relaxations in a liquid structure characteristic to a fixed temperature have been reported to obey the Arrhenius law [42, 43]. In this

8.3 Possibility of Ideal Glass Transitions

175

view, experimental dependences of relaxation time on temperature represented by the Angell plot (Fig. 8.5) reflect the variation of the activation energy sensed by a rearranging molecule in varied surroundings. Note that the kB [d(ln τ )/d(1/T )] does not reflect the activation energy at T , although the Arrhenius law is assumed in this view. It is a critical issue to identify the difference between the equilibrium liquid and the glass of the same material in order to discuss the possibility of ideal glass transitions as not a crossover but a phase transition. Since there exists some disorder in the system, inhomogeneity itself, irrespective of static or dynamic, cannot be an intrinsic indicator of the “glassiness.” In this respect, it is interesting to mention a recent report [44], which identified a structural difference between modeled liquid and glass by applying an emerging methodology based on applied mathematics. There are some theoretical attempts to rationalize the presence of ideal glass transition(s) [3, 45]. Even so, every glass transition ever reported is practically due to the prolonged relaxation time of systems, leading to the fact that no one has ever observed the ideal glass transition. It is also emphasized that theoretical attempts have treated idealized systems consisting of spherical particles without rotational degrees of freedom. Little has been clarified about molecular nature, such as rotational degrees of freedom with anisotropy and the possibility of molecular deformation, all of which potentially contribute to the occurrence and properties of glass transitions. Indeed, it is inevitable that internal motional degrees of freedom (configurational change) are involved deeply in real glass transitions in polymers.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

C.A. Angell, Science 267, 1924–1935 (1995) P.G. Debenedetti, F.H. Stillinger, Nature 410, 259–267 (2001) L. Bertier, G. Biroli, Rev. Mod. Phys. 83, 587–645 (2011) E.R. Andrew, R.G. Eades, Proc. Roy. Soc. London 218A, 537–552 (1953) H. Kawaji, Thesis for the Master’s Degree of Science (in Japanese), Osaka University, Graduate School of Science (1986) Y. Yamamura, Y. Suzuki, M. Sumita, K. Saito, J. Phys. Chem. B 116, 3938–3943 (2012) N.O. Birge, Phys. Rev. B 34, 1631–1642 (1986) M. Sorai, S. Seki, Bull. Chem. Soc. Jpn. 44, 2887–2887 (1971) K. Adachi, H. Suga, S. Seki, Bull. Chem. Soc. Jpn. 41, 1073–1087 (1968) W.F. Giauque, M.F. Ashley, Phys. Rev. 43, 81–82 (1933) W.F. Giauque, J.W. Stout, J. Am. Chem. Soc. 58, 1144–1150 (1936) L. Pauling, J. Am. Chem. Soc. 57, 2680–2684 (1935) J.F. Nagle, J. Math. Phys. 7, 1484–1491 (1966) T. Atake, H. Chihara, Bull. Chem. Soc. Jpn. 47, 2126–2136 (1974) Y. Tozuka, H. Akutsu, Y. Yamamura, K. Saito, M. Sorai, Bull. Chem. Soc. Jpn. 73, 2279–2282 (2000) O. Haida, T. Matsuo, H. Suga, S. Seki, J. Chem. Thermodyn. 6, 815–825 (1977) H. Akutsu, K. Saito, Y. Yamamura, K. Kikuchi, H. Nishikawa, I. Ikemoto, M. Sorai, J. Phys. Soc. Jpn. 68, 1968–1974 (1999) K. Saito, H. Akutsu, M. Sorai, Solid State Commun. 111, 471–475 (1999) H. Akutsu, K. Saito, M. Sorai, Phys. Rev. B 61, 4346–4352 (2000)

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20. K. Saito, H. Akutsu, K. Kikuchi, H. Nishikawa, I. Ikemoto, M. Sorai, J. Phys. Soc. Jpn. 70, 1635–1641 (2001) 21. K. Kishimoto, H. Suga, S. Seki, Bull. Chem. Soc. Jpn. 51, 1691–1696 (1978) 22. Y. Tajima, T. Matsuo, H. Suga, Nature 299, 310–312 (1982) 23. Y. Miyazaki, M. Sorai, R. Lin, A. Dworkin, H. Szwarc, J. Godard, Chem. Phys. Lett. 305, 293–297 (1999) 24. M. Tatsumi, T. Matsuo, H. Suga, S. Seki, J. Phy. Chem. Solids 39, 427–434 (1978) 25. H. Horner, C.M. Verma, Phys. Rev. Lett. 20, 845–846 (1968) 26. K. Saito, A. Sato, A. Bhattacharjee, M. Sorai, Solid State Commun. 120, 129–132 (2001) 27. S. Ishimaru, K. Saito, S. Ikeuchi, M. Massalska-Arod´z, W. Witko, J. Phys. Chem. B 109, 10020–10024 (2005) 28. Y. Tozuka, Y. Yamamura, K. Saito, M. Sorai, J. Phys. Chem. 112, 2355–2282 (2000) 29. G. Adam, G.H. Gibbs, J. Chem. Phys. 43, 139–146 (1965) 30. A.A. Antoniou, J.A. Morrison, J. Appl. Phys. 36, 1873–1876 (1965) 31. C.L. Choy, R.G. Hunt, G.L. Salinger, J. Chem. Phys. 52, 3629–3633 (1971) 32. R.C. Zeller, R.O. Pohl, Phys. Rev. B 4, 2029–2041 (1971) 33. P.W. Anderson, B.I. Halperin, C.M. Varma, Philos. Mag. 25, 1–9 (1972) 34. W.A. Phillips, J. Low Temp. Phys. 7, 351–360 (1972) 35. L. Gil. M.A. Ramos, A. Bringer, U. Buchenau, Phys. Rev. Lett. 70, 182–185 (1993) 36. A. Lindqvist, O. Yamamuro, I. Tsukushi, T. Matsuo, J. Chem. Phys. 107, 5103–5107 (1997) 37. K. Saito, H. Kobayashi, Y. Miyazaki, M. Sorai, Solid State Commun. 118, 611–614 (2001) 38. C. Talón, M.A. Ramon, S. Vieira, C.J. Cuello, F.J. Belmejo, A. Criado, M.L. Senent, S.M. Bennington, H.E. Fischer, H. Schober, Phys. Rev. B 58, 745–614 (1998) 39. W. Kauzmann, Chem. Rev. 43, 219–256 (1948) 40. M.L. Williams, R.F. Landel, J.D. Ferry, J. Am. Chem. Soc. 77, 3701–3707 (1955) 41. M. Oguni, J. Non-Cryst. Solids 210, 171–177 (1997) 42. O.V. Mazurin, Yu.K. Startsev, S.V. Storjar, J. Non-Cryst. Solids 52, 105–114 (1982) 43. H. Fujimori, M. Oguni, Solid State Commun. 94, 157–162 (1995) 44. Y. Hiraoka, T. Nakamura, A. Hirata, E.G. Escolar, K. Matsue, Y. Nishiura, Proc. Nat. Acad. Sci. 113, 7035–7040 (2016) 45. T. Maimbourg, J. Kurchan, F. Zamponi, Phys. Rev. Lett. 116, 015902 (2016)

Chapter 9

Molecular Flexibility and Material Properties

9.1 Single-Particle or Extended Scheme Among molecular deformation, the easiest is the twisting around a single bond, which mostly has the axial symmetry. The reorientation of methyl groups has long been a subject of extensive studies as typical dynamics capable in condensed states [1]. Most experimental results have successfully been analyzed within a single-particle scheme. Namely, reasonable descriptions are possible of the dynamics as a motion of the entity that senses an averaged potential, a kind of mean-field (molecular-field). The intramolecular potential dominates the potential while surrounding molecules also contribute. It is, however, interesting to note that some literature claims the necessity of an extended scheme [2]. When a twisting body is large and anisotropic like a phenyl ring, a characteristic frequency becomes smaller than 100 cm−1 , a typical upper bound of the external lattice vibrations in molecular crystals. The twisting degree strongly couples with lattice degrees of freedom, as discussed in Sect. 6.3. Since the lattice vibration intrinsically exhibits the so-called dispersion, an extended scheme is necessary to correctly describe the molecular deformation.

9.2 Glass Transitions The intramolecular motional degrees can be involved in phase transitions or glass transitions. The example introduced in Sect. 8.1.2 concerns the conformational freezing in organic conductors. As other examples of a glass transition arising from the freezing-in of an intramolecular motion, we can list those in crystals of trans-azobenzene (C6 H5 –N=N–C6 H5 ), trans-stilbene (C6 H5 –CH=CH–C6 H5 ), and related charge-transfer compounds [3–5]. The orientational disorder is reported for © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 K. Saito, Chemical Physics of Molecular Condensed Matter, Lecture Notes in Chemistry 104, https://doi.org/10.1007/978-981-15-9023-8_9

177

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9 Molecular Flexibility and Material Properties

them at room temperature. Some studies [6, 7] suggest that this orientational disorder is brought about not by an overall reorientation of a flat molecule but by a crankshaft motion (or, alternately, pedal motion), which keeps the orientations of two benzene rings mostly while the central moiety is reversed. Since this dynamics resembles an elementary dynamics in linear polymers, the glass transition can be regarded as a model of freezing of such local motion in (locally) disordered polymers. The glass transition temperature is the lowest for crystalline trans-azo-benzene among them. For trans-stilbene and its charge-transfer salts, those are well correlated with a volume available to each trans-stilbene molecule in crystals. These facts reflect that such molecular motion is involved in glass transitions. In the case of the charge-transfer salt of trans-stilbene and tetracyanoquinodimethane (often abbreviated as TCNQ), the two states (orientations) of each trans-stilbene molecule are crystallographically equivalent above the glass transition temperature. This equivalence implies that the glass transition occurs in a highsymmetry, i.e., disordered phase. On the other hand, two orientations are inequivalent in crystalline trans-stilbene, indicating the glass transition due to the freezing-in of residual disorder in a low-symmetry, i.e., symmetry-broken ordered phase. These exemplify that a glass transition is possible in either a disordered or ordered phase.

9.3 Phase Transitions Related to Molecular Deformation Although the examples identified so far are limited, the acquiring entropy by molecular deformation (internal structural degrees of freedom) can bring about a phase transition in crystalline states, as discussed in Sect. 6.3, where the described were the twist transition in crystalline p-polyphenyls. This section briefly describes other examples. The crystal of bis(4-chlorophenyl)sulfone (BCPS) mentioned in Sect. 5.5.3 undergoes a structural phase transition of the displacive type caused by the softening of the lowest branch of lattice vibration, in which the twisting degrees of freedom of phenyl groups are involved (Fig. 5.3). The situation is somewhat similar to the case of crystalline biphenyl, the first member of p-polyphenyls. Crystals of molecules with similar structures to BCPS have also been studied within this context [8–11]. Crystalline trichloroacetamide (CCl3 CONH2 , abbreviated as TCAA) is a wellknown organic ferroelectric [12, 13]. In the crystal, sheets formed by N–H/O hydrogen-bonds (H-bonds) are stacked [14]. Each layer consists of dimers. The dimer is cyclic and also formed by two H-bonds between two molecules (A and B), which have two opposing orientations and are not related by any symmetry operations. Due to the non-equivalence of molecules A and B, the dimer bears a small electric dipole (ca. 10% of that of an isolated molecule), the orientational order of which brings about the ferroelectricity. An incommensurate phase appears in a small temperature interval of ca. 2 K between the two successive transitions. Through the successive phase transitions on cooling, the size of the unit cell remains essentially the same. However, the space group loses the inversion symmetry: P21 /c of the paraelectric

9.3 Phase Transitions Related to Molecular Deformation

179

(PE) phase above the higher transition temperature versus P21 of the ferroelectric (FE) phase. The orientational dynamics of trichloromethyl and amino groups and the partial orientational disorder of the trichloromethyl group of molecule B have been reported. The report of the disorder and dynamics suggests that the phase transitions are basically of order-disorder type. The excess entropy acquired in the course of phase transitions confirmed this expectation [14]. It is noteworthy that the magnitude of the excess entropy distinguishes whether the motional correlation is significant or not in this disordering process (as will be discussed in Sect. 10.1). The experimental result indicated that not a dimer but a molecule is the disordering entity: the PE phase is a “mixture” of four types of dimers A:B, (A:B)inv 1 (this is equivalent to Ainv :Binv ), A:Binv , and Ainv :B. Crystalline TCAA serves as an example of the disordering process involving molecular deformation and a novel strategy to design molecular ferroelectrics [14]. Although the majority of molecules are certainly polar (asymmetric), only a limited number of ferroelectrics have been discovered and developed in neat molecular substances. Even if polar molecules crystallize in a polar structure that has a spontaneous polarization (under ultimate electric field, for example), the polarization reversal, which is the most important property for ferroelectrics, is hard because of severe steric hindrance due to molecular anisotropy. The polarization reversal is essential for achieving a large dielectric constant. In crystalline TCAA, the intramolecular reorientation of rotors, the trichloromethyl groups (CCl3 –), promotes the reversal, as schematically shown in Fig. 9.1. Methyl groups and those substituted with three identical atoms often reorient themselves in crystals because of their globular shape. Despite their nearly globular shape, the reorientation can establish the inversion symmetry, resulting in the disappearance of a finite dipole moment of a dimer. The ease in reorientation efficiently contributes to the polarization reversal in crystalline TCAA. A variety of intramolecular deformations such as reorientation, twist, or crankshaft motion observed in trans-azobenzene (Sect. 9.2) potentially work as an underlying mechanism of the polarization reversal because they can break/establish the inversion symmetry of a unit cluster of interacting molecules. Since a dimer is the easiest to be formed and widely observed in crystals, we reach the following strategy for the development of molecular ferroelectrics: Design a crystal consisting of dimers of polar molecules that must have, at least, a partial structure easy to be deformed. Although the saturated polarization realized by this strategy is smaller than the fully ferroelectric arrangement of the molecular dipole moments, the strategy automatically implements the mechanism of the polarization reversal in the crystal. It is noteworthy that the strategy applies solely to molecular crystals because the intramolecular deformation is intrinsic to molecular systems.

1 The

subscript “inv” means the inverted counterpart of the original one.

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9 Molecular Flexibility and Material Properties

Fig. 9.1 Mechanism of polarization reversal through reorientation of rotating groups. In the case of crystalline TCAA, the reorientation of trichloromethyl groups (tetrahedrons) promotes the reversal of the dipole moment (red arrow) of a cyclic dimer (oblate spheroid). Adapted from CrystEngCom, 13, 2693 (2011) [14] with permission from The Royal Society of Chemistry

9.4 Conformational Disordering of Alkyl Groups A linear alkyl group, represented as Cn H2n+1 –, is monovalent and capable of substituting a hydrogen atom, accordingly. The group is often used as a substituent to enhance the solubility in organic solvents for the ease in handling, to adjust the spatial arrangement of functional groups, or to control chemical/physical properties of compounds such as stability or melting point. Such potential usages result in a vast number of alkyl-substituted compounds. The alkyl group is flexible and capable of exhibiting the disordering, as seen in Sect. 6.3. Thus, the alkyl group deserves special consideration due to its importance in the material world.

9.4.1 Odd-Even Effect Since a natural parameter characterizing the linear alkyl group is self-evident, i.e., n, the number of carbon atoms in the group, properties of series of compounds with alkyl chains are often summarized as a graph against n. Figure 9.2 is an example. The figure shows the boiling and melting temperatures under ambient pressure as functions of n. While the boiling temperature is a function mostly smooth, the melting temperature exhibits an alternation, or “zig-zag.” Since the zig-zag disappears if we see the dependence separately for odd and even members, which are discriminated by the parity of n, the zig-zag dependence of properties on n is called an odd-even effect. The behavior of the melting temperature for a series of compounds with alkyl chains is a kind of odd-even effect. Although contrasting behaviors of the melting and boiling temperatures imply that the odd-even effect does not appear in all properties, the effect appears in many properties of alkyl-substituted compounds. We can attribute their cause to the crystal structure for the effects in which crystals are involved. Imagine naïvely crystals of

9.4 Conformational Disordering of Alkyl Groups

181

500

T/K

400 300

Tboil Tfus

200 100 0

4

6

8

10

n in CnH2n+2 Fig. 9.2 Boiling (Tboil ) and melting (Tfus ) temperatures of n-alkane (Cn H2n+2 ) as functions of n under ambient pressure

fancy “alkanes” in Fig. 9.3. In the case of “butane,” the crystal possesses the periodicity c along the long axis of the molecule. The periodicity matches the length of a single molecule. Note that this structure is not symmetric concerning the direction c and its inversion −c. On the other hand, in the case of “propane,” a similar packing (Fig. 9.3b) is possible but not useful for the close packing, though the symmetry concerning c and −c stands. Instead, the structure consisting of doubled layers (Fig. 9.3c) is more effective for the packing. Thus, the parity of n affects the desired packing. This effect should realize not only in crystals but also in any aggregation. Namely, the odd and even members of alkyl compounds plausibly prefer different packing structures. Since the aggregation structure is a basis for all properties of condensed matter, the odd-even effect is a natural and straight consequence of the difference in aggregation. For example, in the case of melting temperature shown in Fig. 9.2, it is the temperature where the crossing occurs between the Gibbs energies of the crystal and liquid. Since the cohesive energy, the origin of the enthalpy, and the lattice vibrations, which retain thermal energy, strongly depend on the crystal structure, the melting temperature would separately depend on n for odd and even cases even if no parity dependence exists in the liquid phase. For liquid crystals, there exists literature that not only the main component but also a minor component added as a solute is responsible for odd-even effects [19, 20]. Indeed, the phase boundary between the nematic and SmA phases is discriminated in liquid crystalline mixtures. Such behaviors have found rational reasoning within the framework proposed for neat systems [21], which considers a different preference for aggregation according to details of molecular chain structure.

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9 Molecular Flexibility and Material Properties

a)

b)

c)

c

c

c

Fig. 9.3 Supposed crystal packings of fancy “alkanes.” Layered packings of “butane” (a) and “propane” (b and c). The periodicity normal to the molecular layer is denoted as c in each case. Single-layered “butane” (a) and double-layered “propane” (c) share the local relation between successive layers though a and b consist of the simple stacking of monolayers

9.4.2 Systematics as Indispensable Tool Figure 9.2 is a typical way to summarize the physical properties of a series of compounds. Further, the plot indicates the existence of a new phenomenon, the odd-even effect in this case. If we pursue the systematics based on extensive quantities, the output becomes not only qualitative but also quantitative [20, 22]. Indeed, the ndependence of the layer spacing of layered smectic crystals undoubtedly reflects the (averaged) length normal to layers. Namely, the slope of this dependence should be the contribution of a methylene group (–CH2 –). The comparison of the slope and the expected contribution yields the averaged angle of the alkyl group from the layer normal [20, 23, 24]. We explain its implication further in Sect. 9.4.4.4. The other example is the analysis of the entropy of transition in a series of compounds with varying lengths of alkyl chains [22]. The melting of n-alkanes accompanies a linear increase in the cumulative entropy of fusion with increasing the length of a molecule, as shown in Fig. 6.6. The slope, the increment per methylene group, indicates the entropy difference per methylene between the ordered crystal and isotropic liquid. The reasoning leading to this conclusion should apply to any phase transitions as long as the two phases involved in the phase transition remain the “same.” Namely, a series of compounds with different linear alkyl groups exhibits the same phase transition irrespective of the complexity of the system under consideration. We call this analysis as “ΔS analysis.” The ΔS analysis is unique for the time scale covered in the study of chain dynamics. Spectroscopies such as NMR are widely used to study molecular dynamics. Each spectroscopic method has a characteristic time scale to sense the molecular dynamics successfully. The characteristic time scales of spectroscopies are generally equal to the inverse of the frequency used. For NMR, which is the most powerful and widely used method, that is 10−8 –10−10 s. On the other hand, as the ΔS analysis bases on

9.4 Conformational Disordering of Alkyl Groups Fig. 9.4 Molecular structures of cubic mesogens, ANBC(n) and BABH(n)

183 O

R

O OH O2N ANBC(n) (R = CnH2n+1) O

R

O HN

NH O

R

O BABH(n) (R = CnH2n+1)

thermodynamics, the time scale is extremely long, just as sensed by our daily life. The characteristic time is about 103 s. Very slow dynamics, therefore, are favorably studied. Besides, this time scale of thermodynamics corresponds to the lowest limit of molecular dynamics that can be detected. Fast dynamics (e.g., that detected by NMR) are also covered by this thermodynamic method, though no information is available concerning the time scale of the dynamics. Here is an example of the application of the ΔS analysis. ANBC(n) and BABH(n) are famous mesogens exhibiting mesophases of cubic symmetry in a broad range of the chain-length, n [25–29]. Figure 9.4 shows their molecular structures. Figure 9.5 shows the chain-length dependence of the entropy of transition between the SmC phase and cubic phases [30]. In each series, the dependence is seemingly linear, implying the slope successfully reflects the difference of the chain entropy per methylene group. In ANBC(n), there is a break in the dependence because two different cubic phases exist depending on chain-length, though this fact is ignored here for simplicity. For BABH(n), it is essential to remember that the cubic phase locates at the low-temperature side of the SmC phase. The negative slope means that the plot starts from a negative value with a positive slope if plotted for the phase transitions from the SmC phase to the cubic phase, similar to the case of ANBC(n). The absolute magnitude of the slope in BABH(n) is twice of that in ANBC(n) and, therefore, quite reasonable if we consider the molecular structure of BABH(n), which has chains at both molecular ends while ANBC(n) has only one. This consistency strongly suggests that the aggregation modes in each cubic phase are quite similar despite the inverted phase sequence against temperature. It is noteworthy that the molecular details are unnecessary to the ΔS analysis. Indeed, the molecular arrangement in cubic phases of thermotropics has long been an attractive but challenging issue because of the large number of molecules in a unit cell (typically of 103 ) and severe thermal disorder. The estimated entropy difference of the chain between the cubic and SmC phases [ca. 0.5 J K−1 (mol of CH2 )−1 ] is equivalent to an increase of only 6% in the number of microscopic states, according to

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9 Molecular Flexibility and Material Properties

ΔtrsS / J K-1 mol-1

6

4 BABH ANBC SmC cub*

– Ia3d SmC

2

– SmC Ia3d 0

0

10

20

30

nC

Fig. 9.5 ΔS analysis for the phase transition between SmC and cubic (I a3d and cub*) phases of ANBC(n) and BABH(n). The break in ANBC(n) reflects the change in the space group of cubic phases [27] (see Fig. 9.9a)

Boltzmann’s principle. At that moment, the analysis of the chain-length dependence of entropy of transition offered the most microscopic information based on the most macroscopic quantity. Now, after struggling with critical problems [15, 17, 31], the primary core arrangement has experimentally been established for the cubic phase with the I a3d symmetry [16], as in Fig. 10.8. Moreover, the packing of alkyl chains is also under discussion [32]. The aggregation structure that had thus revealed is consistent with the results of ΔS analysis: The entropic competition between the positive and negative contributions of chain and core in the SmC – cubic phase transition. The above example shows that the systematics is highly useful in elucidating properties and phenomenon exhibited by compounds having alkyl chains. The systematics is especially helpful for molecular systems because they are generally complicated.

9.4.3 Entropy Reserver The cubic phases discussed in the previous section possess the same space group I a3d despite the inverted phase sequence against temperature: The cubic phase locates at the low-temperature side of the SmC phase in BABH(n) whereas it is at the high-temperature side in ANBC(n). This fact is undoubtedly nontrivial in the sense that one phase has its intrinsic order, distinct from others, and the order usually accompanies a natural sequence of entropy. For example, the disordered liquid always locates at the high-temperature side of the ordered crystal. Also, the magnet loses its magnetic power above a Curie temperature. The result of the previous ΔS analyses resolves the curious behavior.

9.4 Conformational Disordering of Alkyl Groups

185

Fig. 9.6 Alkyl chain as entropy reserver. Different capacity of the reserver (due to different alkyl length) explains the inverted phase sequence in ANBC(n) and BABH(n). Reproduced from [33]

Since the chain contribution is positive for the SmC – cubic phase transition from the slopes of Fig. 9.5, there must be a negative contribution to yield the negative net entropy of phase transition for BABH(n). The other part of the molecule from the alkyl chain is its core, which remains one per molecule. That is, the core is more ordered in the cubic phase than the SmC phase. Thus, it becomes clear that there is an entropic competition between the alkyl chain and molecular core in this SmC – cubic phase transitions. The inverted phase sequence between ANBC(n) and BABH(n) can be understood, as illustrated in Fig. 9.6 [30, 33]. The entropy contribution of the core is negative in SmC – cubic transition and assumed the same in ANBC(n) and BABH(n) in the figure, whereas the contribution of the chain is positive and depends on its length. ANBC(n) having long chains shows the SmC – cubic phase transition on heating because the sign of the net entropy change is positive. On the other hand, the entropy gain by the chain is insufficient to overcome the negative core contribution in BABH(n), resulting in the cubic – SmC phase transition on heating. Here, the alkyl chain serves as the entropy reservoir. In other words, the entropy reserved in one degree of freedom moves to another degree upon the phase transition. This logic leads to the expectation that the cubic phase appears at the high-temperature side of the SmC phase by elongating the chain in BABH(n). This expectation was confirmed under pressure [34–39]. In the above analysis, the contribution of cores is assumed common for them because the aggregation structures are supposed the same while taking into account of the formation of dimers of ANBC(n) (except for the isotropic liquid) [40]. It is noteworthy that we cannot deduce the difference in the core contribution in two phases from the intercept at n = 0 in the ΔS analysis because the effect of the core plausibly extends over methylene groups near the core. The inverted phase sequence also occurs within the phase sequence of BABH(n) [18]. This inversion is related to the reentrant behavior of the I a3d phase. While the cubic I a3d phase is at the low-temperature side of the chiral cubic phase in BABH(13), the phase sequence is inverted in BABH(15) and BABH(16). This inversion is explained similarly to that between the cubic and SmC phases [or, equivalently between ANBC(n) and BABH(n)], based on the chain-length dependence of the entropy of I a3d—chiral cubic transition [18].

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9 Molecular Flexibility and Material Properties

The entropy transfer phenomenon has also been recognized in a series of inorganic complexes (MMX complexes) [41–44], which have been studied mainly from interests on their variety in electronic states arising from its one-dimensional nature. The entropy transfer between the alkyl chain and other moieties in ligands upon phase transition demonstrates the universal importance of the role of alkyl chains as the entropy reserver. The entropy “capacity” of alkyl chains amounts to ca. 10 J K−1 (mol of CH2 )−1 as revealed in the case of fusion of n-alkanes (Fig. 6.6). Only two methylene groups suffice to surmount the entropy increment expected for the positional melting. In terms of the free energy, its contribution (T ΔS) at room temperature amounts to a quarter of the hydrogen bond, which is the strongest interaction between molecules (Sect. 1.2.6.1). Finally, a comment is in order concerning the thermodynamic implication of the entropy transfer. The transfer seemingly accompanies the decrease in entropy. At a glance, the decrease contradicts the fundamental property of entropy: The entropy is a monotonic non-decreasing function of temperature.2 The entropy transfer and the associated decrease in entropy are, however, compatible with thermodynamics because the considered system is inseparable into “component” systems.

9.4.4 Chains in Liquid Crystals Real thermotropic mesogens most often have long and flexible alkyl group(s) at its end(s). Even for such mesogens, the excluded volume effect is, not explicitly but implicitly, assumed as a driving force to form the smectic and nematic phases [45, 46], because the assembly of hard rods undoubtedly exhibits these liquid crystalline phases besides the isotropic liquid [47, 48]. A theory by McMillan [49] assumes that the chain elongates the molecular length. There is no doubt concerning this point. In the context of the melting of molecular crystals described in Chap. 6, however, some questions arise for the alkyl chains: When and how does the chain melt during the process? Is there any active role played by the alkyl chains in liquid crystals? Although some roles played by alkyl chains were theoretically suggested in the 1980s [50–57], little efforts have been paid experimentally to this subject.

9.4.4.1

Chain Melting and Entropic Stabilization

It is trivially unnecessary to seek for the particular role(s) in the liquid crystal formation for the alkyl chains if they remain rigid and ordered in the liquid state because the molecules behave as (semi)rigid rods. However, the chain-length dependence of the cumulative entropies of transitions up to the isotropic liquid (Fig. 6.6) indip = T (∂ S/∂ T ) p ≥ C v > 0. The last inequality is a condition of the thermodynamic stability of the system.

2C

9.4 Conformational Disordering of Alkyl Groups

187

cates linear dependences separately depending on the parity of the length due to the odd-even effect. The slope favorably compares with R ln 3, which corresponds to the entropy increment due to the threefold disorder around a C–C bond. Namely, the slope indicates that the molecular conformation is (almost) fully disordered in the isotropic liquid of linear alkanes. This finding naturally leads us to the expectation that the alkyl chain attached to the mesogenic core is (almost) fully molten in the isotropic liquid exhibited by mesogens. To reach a conclusion with broad applicability about the melting of alkyl chains, we need to analyze the most ordered liquid crystalline phase. Among orthogonal mesophases exhibited by calamitic mesogens, the closest to an ordered crystal (OC) is the smectic E (SmE, also known as crystal E [CrE]) phase (Sect. 7.1.2.2). The SmE phase has a layered structure but has no two-dimensional fluidity. Although the rotation of the molecule around the molecular long axis perpendicular to the layer is restricted, the molecules possess distinctions of neither head and tail nor front and rear. Representative SmE mesogenic series includes nTCB, the molecular structure of which is in the inset of Fig. 9.7. Figure 9.7 shows the cumulative entropies of transitions of nTCB from the OC up to the isotropic liquid, which serves as the reference state in this graph [58, 59]. Two characteristics are evident. One is the constancy of the entropy change from the SmE phase to the isotropic liquid (shaded regions), even if the phase sequence includes the smectic A (SmA) phase. This constancy demonstrates that the SmE phase is mostly the same as the isotropic liquid in the entropic state of the chains. The other characteristic feature is the roughly linear dependence of the entropy change on the chain-length with the odd-even effect indicated by a dotted line. Its slope is ca. 10 J K−1 (mol of CH2 )−1 , which is equal IL SmA SmE

-1

−Σ Δ trs S / J K mol

-1

0

-50

-100 NCS

R

OC

-150 2

4

6 8 n of alkyl chain

10

12

Fig. 9.7 ΔS analysis using cumulative enthalpies transitions of nTCB (n being the number of carbon atoms in alkyl group R = Cn H2n+1 , as shown in the inset) for phase transitions from the ordered crystal (OC) to the isotropic liquid (IL) via smectic E (SmE) and A (SmA) phases. Reprinted with permission from J. Phys. Chem. B, 116, 9255 (2012) [59]. Copyright 2012 American Chemical Society

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9 Molecular Flexibility and Material Properties

to that of alkanes in Fig. 6.6. This coincidence indicates that the chain of nTCB is, starting from the completely ordered state in OC, certainly molten, or disordered in the same degree as that in the isotropic liquid of alkanes. Thus, the constant entropic state of chains in the SmE phase shows that it is already molten in the SmE phase. The molten state of chains in the liquid crystalline state revealed by thermodynamic analyses has support from a spectroscopic technique [59]. Although a remarkable change happens upon the transition from the OC to the SmE phase, little change occurs upon the phase transitions not only from the SmA phase to IL but also from the SmE phase to the SmA phase. Similar behaviors in the ΔS analysis and vibrational spectroscopy appeared in another mesogenic series exhibiting the SmE, nPA [60, 61]. In summary, alkyl chains attached to a rigid core at its end(s) gain the conformational disorder comparable to the isotropic liquid already in the SmE phase, which is one of the liquid crystalline mesophases with the highest order. Since it is hard to imagine that highly disordered chains recover their order in more disordered mesophases, we conclude that the alkyl chains are mostly molten in any liquid crystalline phases. It is interesting to note here that, through thermodynamic analyses, Sorai et al. [62] had interpreted the successive phase transitions in solid states of a discotic mesogenic series as the successive chain melting process. IR [62] and NMR [63] measurements supported their interpretation. They concluded that the chain melting prepared liquid crystalline states. Its active mechanism was, however, left unrevealed. Now, we can understand the mechanism. Under the ambient condition, the thermodynamic stability of a given phase is governed by the Gibbs energy (G) that consists of enthalpy (H ) and entropy (S) terms as G = H − T S. Molecular motions (disorder) have a direct effect on the entropy of a substance and modulates the Gibbs energy. Unless we take the chain entropy into account, the entropy gain of the SmE phase, which comes from the loss of head/tail and front/rear discriminations, is merely 2R ln 2 ≈ 11 J K−1 mol−1 . The thermodynamic analyses described in the previous paragraph reveal that the chains are molten even in the most rigid liquid crystalline phase (SmE phase). The large entropy reserved in the chain enhances the thermodynamic stability of liquid crystalline phases through the entropy term in the Gibbs energy. This mechanism is summarized in Fig. 9.8. This is just predicted theoretically by Dowell [50–54]. The statement “Chain melting prepares LC states” by Sorai [62] can also be understood in the same context. Although the excluded volume effect drives the formation of some liquid crystalline phases such as N, SmA, and HexB phases (as exemplified by molecular simulations [47, 48]), this fact does not guarantee its role in real mesogens. The analysis described here indicates that, contrary to the common belief, the presence of the alkyl chain(s) attached to the rigid core moiety in most real mesogens is not merely accidental but vital. The rareness of the mesogenic behavior of small molecules without alkyl chain(s) like p-sexiphenyl [64] should instead be considered as reasonable though liquid crystalline states driven by the excluded volume effect are well known in colloidal systems [65–67].

9.4 Conformational Disordering of Alkyl Groups

IL without chain LC without chain

G = H – T •S

Fig. 9.8 Mechanism of stabilization of liquid crystalline mesophases for mesogens having alkyl group(s) at its end(s)

189

crystal LC with chain

T•Schain

IL with chain

T 9.4.4.2

Chains as Intramolecular Solvent

The analyses described in Sect. 9.4.4.1 indicate that, in liquid crystals, the alkyl chains attached to the rigid core of mesogenic molecules may behave differently from the core. The inspection over the variation in their “composition” yields another fruitful insight. Figure 9.9 compares phase diagrams of the neat (pure) ANBC(n) against the chain length and of ANBC(16)–n-tetradecane binary system [68]. The axis for the binary system is against n ∗ , which is the number of paraffinic (or aliphatic) carbon atoms per core in the system and alternately regarded as an effective chain length of the mesogen. The cubic phase of the binary system covers both of the I a3d and cub* phases in the neat series. The resemblance between them is evident if we omit some structural details of cubic phases. Similar behaviors were obtained for other mixtures of ANBC(n) (n = 14, 16, 18) and long alkanes (n-tetradecane and n-hexadecane) [68, 69]. It is noteworthy that the cubic phase appears n ∗  15 in the mixture despite the absence of cubic phases in neat ANBC(14). Kutsumizu et al. [69] also confirmed the change in symmetry of the cubic phases between n ∗ ∼ 17 and 18 in the ANBC(14)–n-hexadecane system. Since long alkanes are real solvents, it is reasonable to conclude that the intramolecular chain serves as the solvent (selfsolvent) in a neat substance. These resemblances in the phase behavior lead to the concept, “Quasi-Binary (QB) picture of Thermotropics” [70, 71]. Another example that contributed to the establishment of the QB picture lies in a simpler, even most famous, mesogenic series, nCB (R–(C6 H4 )2 –CN, R = alkyl group) [71]. nCB is one of the most representative series of rodlike mesogens exhibiting much simpler liquid crystalline phases than those in ANBC(n). Short-chain nCBs (n ≤ 7) exhibit only the N phase in between the ordered crystal and the isotropic liquid while nCBs with a long chain (n ≥ 10) exhibit only the SmA phase. Both SmA and N phases appear in 8CB and 9CB in this order upon heating. Phase diagrams have been reported for binary systems of 7CB and n-heptane [71] and n-octane [72]. Although the structural characterization in the latter is insufficient, the phase dia-

190

9 Molecular Flexibility and Material Properties 500

500 SmA

– Ia3d

450

cub*

SmC

T/K

IL

IL – Ia3d

SmA 450

Cubic SmC

400

400

a) 350

b) Crystal

Crystal 350 10

15

20 n

25

16

18 n*

20

Fig. 9.9 Phase diagrams of a neat ANBC(n) against the chain length n [29] and b ANBC(16)–ntetradecane mixture against the effective chain length n ∗ [68]. The cub* phase is chiral but its space group is unknown. The symmetry of the Cubic phase in the mixture (b) is not determined, resulting in the coverage for both the I a3d and cub* phases of the neat series. Adapted from [33]

grams are mostly the same if drawn against n ∗ . In both phase diagrams, the addition of alkane induces, around n ∗ ∼ 7.5, a new phase, which was undoubtedly the SmA phase for the n-heptane system [71]. These behaviors show that the chains serve as a solvent in the liquid crystalline phases. The layer spacing (repeat distance) of the mixture linearly depends on n ∗ and lie on the extrapolated dependence of neat nCBs (n = 8, 9). The smooth variation of the layer spacing means that the QB picture holds not only for the general phase behavior but also for the structural aspect (molecular aggregation). Since intermolecular interaction and segregation are evident in the crystalline state [73, 74], the above understanding of the SmA phase results in the following view. The melting process of nCB in the phase sequence, ordered crystal–SmA phase–N phase–isotropic liquid, is regarded as a shift of major interaction from lyotropic (amphiphilic) to thermotropic one. Namely, the melting of 8CB and 9CB to the isotropic liquid via SmA and N liquid crystals is a process involving the thermotropic–lyotropic crossover. The QB picture even offers the reason why the averaged chain length has been utilized as an essential parameter to characterize the liquid crystalline systems. Indeed, in the study of the critical behavior of nCB mixtures, the averaged chain length, n ∗ , has been used to control their critical behavior [75]. Depending on n ∗ , the SmA – N transition changes from the second-order to the first-order. The point where the change happens is a triclitical point (Sect. 2.2.4). Although the range of n ∗ is narrow in the research of in this context, n ∗ in a much broader range gives a unified phase diagram, which interestingly includes the reentrant N phase that has finally been discovered in this famous series after a long history of the research [19, 20].

9.4 Conformational Disordering of Alkyl Groups

9.4.4.3

191

Structural Implications for Layered Smectics

The almost fully molten state of alkyl chains leads to a question of whether the traditional view of layered structures of smectics is valid or not. The traditional view assumes the formation of layers by whole molecules. More precisely, the centers of molecules form planes. To make more explicit the question, suppose the following mesogenic molecule: It consists of two parts, the rigid aromatic core and alkyl chain, the lengths of which are equal to each other. Molecules are mostly normal to the smectic layers, i.e., the (local) director is normal to each layer. Since smectics ever reported are non-polar, each layer is formed by a mixture of equal amounts of molecules with opposite orientations (Fig. 9.10a). Because of the equal lengths of two parts of the molecule, the layer is, on average, uniform in “composition.” The smectic structure as a whole is also uniform except for the presence of stacking planes. It is even troublesome to identify its layered structure, which is the characteristic of smectic phases. The uniformly mixed layer of aromatic cores and chains is unfavorable for the electronic/hole conduction within the commonsense of organic conductors[76, 77]. The so-called π –electrons (holes) on aromatic parts are responsible for the conduction and that the electronic interaction between aromatic parts, i.e., the overlap of wave functions of π –electrons (π –orbitals) of neighboring molecules, is essential. The situation becomes more severe if the length of the alkyl chain is longer than that of the core part of a mesogen having a core–chain structure. In this case, no overlap of π –orbitals is likely. This consideration contradicts the better conductivity of smectic phases than nematic phases [78, 79] and the excellent carrier mobility exhibited by a SmE phase [80]. The uniformly mixed layer is doubtful again in the view of the mechanism of structure formation. As described in the previous Sect. 9.4.4.1, the entropy reserved in molten alkyl chains significantly contributes to the thermodynamic stabilization of liquid crystalline phases [61]. The kinetic entropy can play a significant role [51, 81, 82]. In the mixed layer, the dynamics of alkyl chains is more or less suppressed by

a)

l

b)

chain

c)

chain

core

core

chain

chain

core

core

chain

chain

Fig. 9.10 Models of smectic structure. a Traditional model assuming layers of whole molecules. b Nano-segregated model consisting of an alternate stack of core layers and chain layers. c Modified nano-segregated model consisting of an alternate stack of core layers and chain layers, which further consist of two half layers attached to the adjacent core layers

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9 Molecular Flexibility and Material Properties

the presence of surrounding rigid cores, resulting in a reduction in the chain entropy. At least, the tendency to increased entropy would disfavor the mixed layer. As described in the preceding section, the SmA phase in nCB continuously swells with alkanes [71]. The increment in the layer spacing is consistent with that calculated while assuming that the neat liquid alkane preserves its density in the swelled state. Thus, it is highly plausible that the molten chains, together with “real” solvent, liquid alkane, form separate layers from those of aromatic cores. This structural model implies that the molten chains and cores separately form layers even in the neat system, as shown in Fig. 9.10b. The similarity of the quasi binary systems of thermotropic liquid crystals with real lyotropic systems extends over not only the phase behavior but also the formed structure. Mostly the same local aggregation structure (as expressed in terms of the ratio between the molecular length and the average repeat distance between molecules) in the N phase of 5CB and 7CB [83, 84] seems consistent with the “lyotropic” formation of the SmA phase in neat nCB and nCB-alkane systems. Note that the view of “lyotropic” nature for the formation of smectic phases is historically not entirely new [85–88]. Indeed, Guillon et al.[87, 88] reported some experimental works showing that the core and flexible chains segregated though weakly. The decisive evidence of the nano-segregated structure of smectic phases comes from the investigation of the SmE phase [89, 90]. Although the orientation and conformation of molecules are highly disordered, the SmE phase has the threedimensional periodicity, which means that well-defined lattice constants characterize its averaged structure [91, 92]. If the SmE phase of a mesogen with the core-chain structure swells with alkane, the swelling itself serves as supporting evidence. The phase diagram determined for the binary system of 9TCB (the inset in Fig. 9.7) and n-nonane indicates the continuous swelling of the SmE phase structure [89]. Further, its three-dimensional periodicity offers stronger evidence. The lattice constant along the stacking direction (along the molecular long axis) increases continuously upon swelling, while those within the layer scarcely depend on the extent of swelling. No effect on the structural order within the layer indicates the invasion of alkane molecules between cores is negligible, and all alkane molecules reside within the chain layers. The three-dimensional periodicity of the SmE phase also enables an attempt to analyze its aggregation structure in neat nTCBs [90], even though the full (three-dimensional) analysis of SmE structures through the X-ray diffraction is impossible because of severe structural disorder. The so-called Fourier syntheses3 of series of diffraction peaks assignable to the stacked layers (up to fifth order) of aligned samples indicate the stepped electron density along the stacking axis (layer normal). The length of the high electron density region remains constant and equal to the length of the core part for a broad range of the alkyl chain length (2 ≤ n ≤ 10). These results indicate that the structure of the analyzed SmE phase of the nTCB series is described as not the stack of uniform molecular layers but as the alternate stack of the core and chain layers (Fig. 9.10b). 3 Remember

that the square root of the intensity of diffraction is a Fourier transform of the density of scatterer as explained in Sect. 4.2.

9.4 Conformational Disordering of Alkyl Groups

193

The nano-segregated structure thus revealed has a layered structure characteristic to smectics, and seems reasonable concerning its formation mechanism. The structure is also consistent with the excellent carrier mobility reported for SmE compounds [80] because the electron transfer is possible even in the dynamically disordered state [4, 5]. The fact that the SmE phase is the closest to the ordered crystal again plays essential importance here. The above description has not relied on any specific properties of compounds under discussion beyond the possession of an alkyl chain attached to the core moiety. The supposed mechanism to stabilize such a segregated structure should apply equally to a wide variety of real mesogens. In this respect, the nano-segregated structure should be a fundamental structure of any smectic phases. Although the nano-segregation may compete with two-dimensional fluidity, which is deeply involved in the formation mechanism of the smectic phase of rigid rodlike molecules [47, 48], its entropic gain would be smaller than the chain entropy in real mesogens with an alkyl chain(s). It is noteworthy that the molecular aggregation in smectic phases may be significantly different between asymmetric (chain–core) mesogens and seemingly symmetric (chain–core–chain) mesogens, in which the two-dimensional fluidity can play a substantial role in its stabilization.

9.4.4.4

Alkyl Chains in Smectics: Revisit

The layer spacing (d in Fig. 9.10) is significantly larger than the length of nCB molecules with the fully extended alkyl chain, in contrast to the “ideal” SmA phase naïvely expected for a system of hard rodlike objects [47, 48]. Thus, the structure of the SmA phase in nCB was the object of extensive studies. The assumption of interdigitated bilayers [83, 84] finally resolved this discrepancy. The resultant model of the smectic phase is denoted as the SmAd phase. This understanding mainly concerns the cores. The information concerning the aggregation structures of the alkyl chains, on average, lies in the systematic dependence of the layer spacing on the chain length, as noticed in Sect. 9.4.2. The discussion here concerns only orthogonal phases, properties of which are isotropic around the layer normal [93]. Since the polarizability of the core part dominates, the cores are widely believed to be normal to the smectic layers. When the alkyl chain is in the so-called all-trans conformation shown in Fig. 9.11, the length of the chain increases with the rate of a sin θ2 per methylene.4 Here, a is the C–C bond √ length (ca. 1.5 Å), and θ is the bond angle, which is ideally equal to√2 cos−1 1/3 ≈ 109.5◦ . Using these typical values, we obtain an estimate as a 2/3 ≈ 1.2 Å(CH2 )−1 . However, we know that the alkyl chains in liquid crystalline phases, including the SmE phase, are mostly molten. The assumption of the all-trans conformation is inapplicable accordingly. Since the estimation needs the magnitude appropriately averaged, the logical deduction is not only complicated but also impractical. 4 Hereafter,

“per methylene” is indicated by (CH2 )−1 in numerical expressions.

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9 Molecular Flexibility and Material Properties

Fig. 9.11 Idealized geometry of the carbon backbone of an alkyl chain with the all-trans conformation

Lipid bilayers dispersed in aqueous media undergo a phase transition between a chain-ordered phase (called gel phase) and a chain-disordered phase (liquid crystal phase). Although the increment of entropy of transition per methylene [94] is smaller than those of n-alkanes and the usual liquid crystals, the conformational disorder of the alkyl chains in the liquid crystalline phase has been assumed mostly complete. The length of the alkyl chain has been a subject in this research field. The experimental data of the bilayer thickness [95] indicates the increment of 1.9 ∼ 2.1 Å per methylene in a molecule slightly depending on temperature. The negative temperature dependence possibly reflects a gradual chain melting in the bilayer of the liquid crystal phase. Anyhow, this increment is consistent with the old theoretical estimate [96, 97]. Therefore, the same should also apply in the usual liquid crystalline mesophases under consideration. Since a methylene group in a lipid molecule contributes twice because of the bilayer structure, the expected increment is, therefore, 0.95 ∼ 1.05 Å(CH2 )−1 if the chain is, on average, normal to the smectic layers. Figure 9.12 shows the alkyl-chain length dependence of layer spacing of various smectic phases for some rodlike mesogens [24]. The selection of compounds bases on the availability of reliable systematic data (e.g., the data in a single paper or papers by the same researcher) and the simple shape of molecules. Interestingly, we recognize two slopes, 1.9 Å and 1.4 Å per methylene in a molecule. Since both slopes are larger than the expected slope for a single molten chain, we must conclude that the chain layer in these smectic phases consists of multiple layers of chains. Considering the presence of core layers at both sides, we can readily imagine the participation of alkyl chains from both sides, as shown in Fig. 9.10c. The structure looks to recover the feature compatible with the two-dimensional fluidity, at a glance. However, it is not the case because a molecule under consideration possesses a single alkyl chain. If rather strong interaction leads to a formation of a temporal antiparallel pair (“dimer”) of molecules, the dimer may quickly diffuse within the layer. The larger slope in Fig. 9.12 is comparable to that of the thickness of lipid bilayers [95], in which the alkyl chains are normal to the bilayer. The molecules of compounds showing the larger slope have, thus, the fully molten alkyl chain normal to the smectic layer [20, 24]. Since the cores have been believed normal to the layer, the molecules have straight rod form. We can construct an example of molecular geometry with a disordered chain without a severe energetic penalty [20]. On the other hand, the smaller slope, 1.4 Å per methylene per in a molecule, is significantly smaller than the larger one. This smallness implies that chains incline from the normal to the smectic layer [23]. The inclination is estimated as cos−1 (1.4/1.9) ≈ 43◦ based on

9.4 Conformational Disordering of Alkyl Groups

40

d /Å

Fig. 9.12 Alkyl-chain length dependence of layer spacing of various smectic phases. The mark and color distinguish the phase and compound, respectively. Reproduced from Phys. Chem. Chem. Phys., 19, 25518 (2017) [24] with permission from the PCCP Owner Societies

195

1.9 Å

30

SmAd

1.4 Å

20

5

10 Chain length

SmB SmE

15

the magnitudes of two slopes. Assuming the core normal to the layer, we need to imagine a bent form of molecules in this case. Interestingly, a bent molecular form with a disordered alkyl chain has been identified for a mesogenic series in Fig. 9.12 through a crystallographic experiment of the crystalline phase [23]. The presence of a real example of molecules with the bent form implies that the inclination magnitude originates in the conformational degrees of freedom of alkyl chains. We recognize no relation among the averaged molecular form (rod or bent), the phase-type (SmAd , SmB, and SmE) and the compound. For example, the same form corresponds to plural phase types, and a mesogenic series exhibits a change in the molecular form. Such observations indicate that the distinction of molecular form is an independent characterization of liquid crystalline phases from existing classifications based on the macroscopic and/or microscopic symmetries of the phase [93]. The last statement, however, does not mean the absence of microscopic structural difference between those of molecules with rod and bent forms. It is easy to imagine that the positional fluctuation of a molecule in the direction normal to the layer is easier with the rod form than with the bent form. The segregation of core and alkyl layers would be more significant in the latter. This expectation is the case. The diffraction peak intensity (of X-ray or neutron beam) from stacked layers is significant only at the lowest order in the former. In contrast, it is readily observable up to higher orders in the latter [24]. A search for other differences in properties has just started. The analysis of the chain-length dependence of the layer spacing should apply to more complicated phases that the deformed “layers” characterize [15] if we can identify an appropriate characteristic length. For the Gyroid phase (Fig. 10.8), the proper choice is the body diagonal of the cubic unit lattice [32]. The analysis shows that the chain layers can interestingly be not bilayers but monolayers despite two chains at both ends of the core.

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9 Molecular Flexibility and Material Properties

There remains plenty of issues to be investigated further. One concerns the role of the effective molecular form on molecular dynamics such as diffusion and reorientation. The dependence of anisotropy of self-diffusion [98] may be clarified by taking the molecular form into account. The dynamics in neighboring phases are interesting from the viewpoint of the relation between molecular form and aggregation states. Another issue is the applicability of the discussed idea in non-orthogonal smectic phases, such as the SmC phase, in which the cores cooperatively incline in one direction. The investigation of SmC phases would be valuable to clarify such points as what in reality incline in non-orthogonal phases, whether the core is normal to the layer in orthogonal phases, or what causes the averaged inclination of the disordered alkyl chain.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

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Chapter 10

Importance of Molecular Crystals

10.1 Motional Correlation 10.1.1 Significance and Difficulty It is usual to consider and describe molecular dynamics in the condensed phase as the dynamics of a single particle (molecule or moiety) except for the description of lattice vibrations. Since the interactions among particles, including repulsion, are the source of all properties of ensembles, the description based on the single-particle is an imitation. For the thermodynamic equilibrium, the well established is that all quantities can mostly be represented in terms of static correlation functions [1, 2]. Further, the fluctuation-dissipation theorem gives the representation of frequencydependent susceptibilities in terms of two-time correlation functions. The knowledge of the correlation among them should, therefore, be indispensable for a deeper understanding of the dynamics of constituent particles of ensembles. Here, we can expect a unique advantage of molecular systems because we can choose systems where only a limited number of degree(s) of freedom suffice to consider. An example of an intramolecular correlation was briefly described in Sect. 6.3. Before proceeding into details, we notice the difficulty of studying the motional correlation. First, it is mostly beyond the equilibrium properties, which the statistical mechanics gives correct answers on, as in the case of the single-particle dynamics. Second, the above point implies that the apparent effect is challenging to identify in many experimental results. Each experiment must prepare a theoretical model(s) to detect effects [3]. The preparation unavoidably accompanies arbitrariness. It is also noteworthy that each experimental technique relies on theoretical models, even in analyzing results interpretable within the single-particle description. Third, preparations of theoretical models and experimental setup are demanding. Unless a good target is in mind, researchers will not start the preparation. In this respect, the © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 K. Saito, Chemical Physics of Molecular Condensed Matter, Lecture Notes in Chemistry 104, https://doi.org/10.1007/978-981-15-9023-8_10

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identification of systems where the effect of the correlation is significant is currently of great significance. It is essential to understand that routine analysis of crystal structure by diffraction, which take only Bragg reflections arising from the long-range periodicity into account, is insensitive to the correlation at a short distance. A detailed analysis of diffuse scattering at high angles is necessary because of the short-ranged and intrinsically incoherent nature of the motional correlation [3, 4]. The utilization of the XAFS (X-ray absorption fine structure) technique, which unveils local structures, may be useful [5].

10.1.2 Entropic Detection Concerning the second difficulty in the last section, the enumeration of entropy has a unique character. Suppose two Ising spins, each of which has two states (↑ and ↓). All possible states are (↑↑), (↑↓), (↓↑), and (↓↓). If the interaction strongly favors the same state for interacting spins, the available states are limited to (↑↑) and (↓↓). On the contrary, the interaction strongly hating the same results in (↑↓) and (↓↑). These situations share the property that the specification of the state of one spin suffices the specification of the other. If the interaction extends over n spins, the number of available states for N spins is not 2 N but 2 N /n .1 Since the entropy is proportional to the (natural) logarithm of the number of states, the entropy directly reflects the presence of strong interaction, which is equivalent to the perfect correlation. Indeed, the example in Sect. 6.3 is based on the magnitude of entropy of transition. The drawback of the simple adoption of entropic detection is that it gives no details of the rate of dynamics. In this respect, the application of the heat capacity spectroscopy [6] seems promising. The main difficulty exists in its operation frequency, which is generally low in comparison with the rate of either correlated or uncorrelated molecular dynamics. An example of intramolecular motional correlation is the correlated disordering of ligands in inorganic complexes. Quasi-one-dimensional complexes known as MMX have a repeat unit shown in Fig. 10.1. Two metal atoms (platinum or nickel) are bridged by four ligands (RCS− 2 , R = alkyl group), forming an MM unit. A halogen (X) bridges two MM units, resulting in an infinite chain of MMX units. The distance between two sulfur atoms of a bridging CS2 moiety is longer than the M–M distance. Thus, the two sulfur atoms are not on but slant from the plane defined by the MMX chain and the carbon atom. Many MMX complexes exhibit a phase transition between the slant-ordered state and the slant-disordered state. Although the entropy increments involved in such transitions are well described by the entire disorder in some complexes [7–9], those of two complexes are significantly smaller than that of the fully disordered state [10, 11]. In particular, that of Pt2 (CH3 CS2 )4 I is comparable 1 We only consider the ferroic case, which favors the same states for interacting spins, to avoid issues

related to the so-called frustration.

10.1 Motional Correlation

201

left

right

Fig. 10.1 Correlated ligand dynamics deduced from the magnitude of the entropy of transition between the slant-ordered state and the slant-disordered state of MMX complex [12]. Only two states are possible for a unit complex

with N kB ln 2. This magnitude indicates that only two states are possible for a unit complex, which has four disordering moieties. Namely, all four CS2 moieties change their slants simultaneously. We can imagine the situation, as in Fig. 10.1 based on possible steric effects. No information is currently available concerning the rate of the change between the two states. Note that routine analyses of crystal structure by diffraction usually do not distinguish the correlated disordering over two states and the uncorrelated disordering over 16 (= 24 ) states [12]. The above example yields only experimental evidence of the almost perfect correlation in the temperature range under study but not the temperature evolution. The next case offers an entire story of the temperature dependence. Tricyclohexylmethanol [TCHM, (C6 H11 )3 COH] is a primary alcohol with bulky cyclohexyl groups as substituents. Its bulkiness forces the hydrogen bond (H-bond) restricted in a dimer at best [13, 14] without extending as in usual cases (network or chain). In the crystal at room temperature, the electric dipole moments mainly resides on the H-bonding part of the dimer and randomly orient themselves, resulting in vanishing spontaneous polarization. The orientation of dipole moments orders below 103 K [15], below which the crystal is a weak ferroelectric [16]. The excess entropy involved in this transition is smaller than (N /2)kB ln 2 [14], partly due to the low-dimensionality arising from the dipolar Ising nature [17–19], discussed in Sect. 10.3.3. The smallness of the entropy indicates that the dynamical correlation of the two hydrogen atoms involved in the bonding OH groups is perfect around the transition temperature. The H-bond inside a TCHM dimer starts to break below the melting temperature while accompanying the excess heat capacity [20]. The integration of the excess heat capacity yields the comparable magnitude expected for an independent hydrogen atom in either the H-bonding or the non-bonding state. This magnitude is entirely consistent with the spectroscopic report that reports the almost complete breakage of the H-bond in the liquid state above the melting temperatures [21]. As described, we have the whole evolution of the ceasing the motional correlation between two hydrogen atoms (or two H-bonds) against temperature for the crystalline TCHM. It is noteworthy that we also have the temperature dependence of the correlation time determined through dielectric relaxation [15]. However, as far as the author knows, nobody has sincerely tried its analysis under the view of the motional correlation.

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10 Importance of Molecular Crystals

We may deduce the information concerning the energetic relations of possible states by analyzing the numerical magnitude of relevant entropy in the light of structural details [22]. Further, if the structural feature of the system is restrictive enough, e.g., quasi-one-dimensional, we may correlate the (averaged or statistical) correlation of molecular states and other physical properties by adequately adopting relevant models [23, 24].

10.1.3 Dynamics in Quasi-one-dimensional Systems Although we have not described details, one-dimensional systems have some interesting properties. The impossibility of an ordered state at finite temperatures is a notable example and has a close connection with a correlated dynamics to discuss. Suppose a one-dimensional chain of Ising spins interacting via ferromagnetic interaction (J > 0) between the nearest neighbors. Figure 10.2 depicts a state the next lowest in the energy of the system. A dotted vertical line indicates the location of a domain “wall” or a “surface.” The energy relative for the ground state is written as n J using the number of walls, n. It is interesting to see that the energy relies not on the locations of walls but solely on their number. The flip of one of the spins neighboring to a wall only shifts it while keeping the energy. Since we have (N − 1) choices of the domain location for the N -spin system,2 the entropy of a microcanonical state with this energy is kB ln(N − 1). Note that the averaged magnetization assigned to the whole system in this thermodynamic state is 0. A comparison of free energies of the ground state and the first excited state, F − FGS = J − kB T ln(N − 1), implies that the latter is more stable at finite temperatures. Thus, the ordered state with finite magnetization is plausibly impossible in this system. Although the above discussion is on the ferromagnetic Ising system, the essential characteristics that the number of the domain wall dominates the system property commonly applies to any purely one-dimensional systems with a short-ranged interaction(s). Besides the impossibility of the ordered state, the one-dimensionality brings a particular property. Spin-flip is more accessible at the domain wall than inside a domain because it accompanies no energy increment. Furthermore, the flip of a group of plural spins adjacent to the domain wall shares the property. However, the energy barrier to overcome will become higher in the latter. Molecular complexes consisting of two organic components (co-crystals), phenazine (Phz) and chloranilic or bromanilic acid (H2 ca or H2 ba), exhibit the ferroelectricity at low temperatures [25]. Their crystals share the basic structure and consist of segregated columns of planar molecules. Electronic interaction within a column is expected from the close stacking of π-orbitals. A strong H-bond connects a nitrogen atom of Phz and a hydroxyl group of H2 ca (or H2 ba) in the neighboring columns. The structure is thus characterized by one-dimensional chains of alternate 2A

factor 2 additionally appears if we consider the choices of configurations of two terminal spins.

10.1 Motional Correlation

203

Fig. 10.2 An example of the first excited state of the Ising model with ferromagnetic interaction on a chain. The dotted vertical line indicates the location of a “wall” between ordered domains with opposing magnetizations 2

f / Hz

5 10 8 6 4

z=3 2

1

2

10

4 8 6 4 2

3 10 8 6

0

10

20 -1

T / kK

30

-1

Fig. 10.3 Arrhenius plot of dielectric dispersion modes of the powdered sample of crystalline Phz-H2 ca [28]. The modes designated with z are simultaneously fit with the attempt frequency f 0 = 3.1 · 1012 Hz and the unit activation energy 0 = /z = 119 meV. The other mode yielded f 0 = 1.1 · 1010 Hz and  = 49 meV

H-bonds, the potential energy curve of which is of asymmetric double-well type. The location in which a proton resides can be mapped onto an effective spin variable. The flip of these spins should be responsible for the dielectric dispersion [26]. For powdered Phz-H2 ca, multiple relaxation modes have been reported [27, 28]. Figure 10.3 shows the so-called Arrhenius plot of relaxation modes, which comes from the deconvolution of the temperature dependence of the imaginary dielectric constant (dielectric loss) [28]. Four relaxation modes follow the Arrhenius laws,    , f = f 0 exp − kB T

(10.1)

where f 0 is an attempt frequency, and  is the activation energy. The straight lines are results of the fits under the assumption3 : The relaxation mode with the weakest 3 Although

the data are the same, the fit method is different from that in [28]. Note that the fits treat the frequency as the independent variable considering the experimental reality: the temperature was scanned while measuring the dielectric constant at specified frequencies.

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temperature dependence is independent of others, whereas other modes fulfill the assumption  = z0 (z = 1, 2, 3) with the common f 0 . The fits are successful. The fact that the smallest activation energy (49 meV) is about half of the one with z = 1 (119 meV) implies that the former is of the hopping of individual protons. The latter is, in contrast, the simultaneous hopping of two protons at a single side of the domain wall. Note that the unit repeat distance of the H-bond chain in Phz-H2 ca contains two protons because of the presence of two molecular species. The two-proton transfer causes a simple shift of a domain wall. For the simultaneous hopping of 2z protons, the activation energy larger by a factor z is necessary. Considering these facts in mind, we attribute the relaxation modes with z = 2 and 3 to simultaneous hopping of 2z protons, which again cause a simple shift of a domain. Similar situations, i.e., the existence of many relaxation modes governed by “quantized” activation energies recognized for related compounds [28], indicate the observability of correlated dynamics in quasi-one-dimensional systems. The possibility of structural design, intrinsic to molecular systems, should play a vital role in a related research field.

10.2 Molecular Crystals as Tunable Model System 10.2.1 Basis of Tunability The structure of crystals dominates properties of materials irrespective of molecular or non-molecular ones. If the effective Hamiltonian that describes the property under one’s interest does not contain a coupling term(s) to lattice degrees of freedom, the Hamiltonian parameters primarily depend on molecular properties and the crystal structure. Such situations apply electronic and magnetic properties. In these cases, the crystal structure is the condition given a priori. We can understand physical properties in the light of standard theories of condensed matter physics based on molecular properties, which are mostly calculable for a specified molecular structure based on quantum chemical knowledge. The fundamental basis of the tunability for properties of molecular crystals, thus, lies in the ability of molecular syntheses, which organic and inorganic chemistries achieve. As long as the crystal structure remains unaltered, the crystal properties vary according to the variation of molecular properties. We can find an enormous amount of examples of this kind in the research fields of organic conductors [29, 30] and molecular magnetism [31]. In contrast to the great success of standard theories of condensed matter, the current achievement of crystal engineering is still insufficient. That is, we have no reliable method to not only design but also predict crystal structures. Advancing crystal engineering should be the primary task for utilizing molecular crystals in this line.

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10.2.2 Unified Description of Structural Phase Transitions We have learned that crystalline phases may undergo a transition into other crystalline phases by either a structural instability described in Sect. 5.5.3 or the ordering from a disordered state typically exemplified by the Ising model in Sect. 6.2.1. Indeed, both mechanisms have been proposed and identified for the emergence of the ferroelectricity. The classification is usually into two groups depending on the primary mechanism that brings the ferroelectricity. The order-disorder ferroelectrics is evident from its name, while the term “displacive” is for those derived from the lattice instability. Although the classification implies that the division is concrete because of a fundamental difference between two mechanisms, many experiments have indicated the presence of intermediate cases, which exhibits characteristics of either both or none of the two types. Such a situation prompted theorists to build a model that should describe phase transitions of both types in a unified manner. A representative of such models is as follows [32, 33]: Suppose interacting particles, the mass of which is m, trapped in a single-particle potential V (x), V (x) = Ax 4 + Bx 2 ,

(10.2)

where x is the coordinate, and A (> 0) and B are parameters characterizing the potential. Depending on the sign of B, the potential changes its form between the single minimum form (B > 0) and double minima one with the energy hump V0 = B 2 /4 A at x = 0 (B < 0), as shown in Fig. 10.4. The interaction has a bilinear form, −γxi x j . The exact calculation except for the mean-field treatment on the interaction (i.e., −γxi x j ≈ −γxx) indicates the occurrence of a phase transition, even B > 0 [32]. When the thermal energy at the transition temperature Tc is well below V0 for B < 0, the transition is qualitatively characterized as an order-disorder transition. The different sites where the particle resides correspond to the states of spin. On the other hand, the transition with B > 0 or kB Tc  V0 with B < 0 is of the displacive type accompanying a soft mode, which softens (decreases in frequency) upon approaching the transition temperature from the high-temperature side. This simple model reproduces characteristic features of not only such dynamical behaviors but also thermodynamic characteristics [33]. Although the model is not quantitative but qualitative, it convinces us of the possibility of artificial control of the transition mechanism through tuning material parameter(s). It is noteworthy that the single-particle potential is not easy to identify in general because the dynamics of a “particle” in real systems reflect the entire effect of its environment. The division into the single-particle potential and the relevant interaction is nontrivial. If we overcome this difficulty, we can potentially tune material parameter(s) through designing molecules. Resultant systems are neat and clean under the ambient pressure. These properties are significantly distinct from the tuning by doping and/or mixing, which unavoidably suffer from disorders. The availability under the ambient pressure is crucially preferable for detailed studies, thought the possibility of continuous tuning by compression (change in pressure) might be beneficial unless we mind experimental difficulties.

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Fig. 10.4 Single-particle potential assumed in a unified model of displacive and order-disorder transitions [32]. Depending on the sign of B, the potential has shapes qualitatively different

Biphenyl is the first member (n = 2) of the so-called p-polyphenyls, which is linear oligomers of a phenyl group as represented as H–(C6 H4 )n –H. Its molecule has a characteristic internal mode, the twisting, because of the ease of rotation around the single bond between (flat) phenyl groups. A conjugation of π electrons prefers the planar conformation, whereas the repulsion between hydrogen atoms at the ortho positions hates the planar form. Because of this energetic competition, the molecular energy, which serves as potential energy for the twisting degree of freedom, has a hump at the flat form (θ = 0◦ ) and minima at θ ≈ ±40◦ symmetrically. Upon crystallization, also active is the intermolecular interaction, which prefers the planar form for better crystal packing. The resultant potential curve for the twisting should resemble one shown in Fig. 10.4. Although the molecules are planar at room temperature, the crystal undergoes a displacive phase transition around 40 K (see also Sect. 5.5.3), below which the molecule is twisted [34]. This situation well fits the scenario predicted by the unified model. Remarkably, a single-particle potential like that in Fig. 10.4 is naturally assignable to this case. p-Polyphenyls have very similar crystal structures to each other. Namely, the crystals consist of layers, inside which seemingly planar molecules are nearly normal to the layer [35–39]. Interestingly, the crystals of, at least, the first five members undergo a phase transition at low temperatures [40–44], below which molecules have alternately twisted forms [37, 45–48]. While the transition is of the displacive type in crystalline biphenyl (n = 2) as noted above, those of higher members (n ≥ 3) are of the order-disorder type. It is noteworthy that there exists the energetic competition between the delocalization of π electrons and the repulsion of the ortho hydrogens irrespective of n. For this reason, these twist transitions are, apart from an issue of the intramolecular motional correlation discussed in Sect. 10.1, transitions of entities trapped in single-particle potentials shown in Fig. 10.4. Figure 10.5 shows the temperature of the twist transition in crystalline p-polyphenyls [43]. The difference is significant between the displacive transition of biphenyl (n = 2) and the order-disorder transition p-terphenyl (n = 3). Nevertheless, we can draw a smooth curve as a function of n. This fact implies that we have a real series of compounds that eminently demonstrate the possibility of a continuous change in the transition mechanism. In other words, we should regard the displacive

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SP

300

QQP

Ttrs / K

QP TP

200

DFQP DFTP

100

0

BP DFBP 1

2

3

4 5 number of rings

6

7

Fig. 10.5 Transition temperature of the twist transition in crystalline p-polyphenyls [H–(C6 H4 )n – H, open circle] and their difluoro derivatives [F–(C6 H4 )n –F, open diamond]. The absence of the twist transition in DFBP (n = 2) is represented as Ttrs = 0 K. Arrows indicate an effective decrease in n upon the difluoro substitution

and order-disorder mechanisms as two limiting cases. It is, however, necessary to notice here that we have no detailed information about the points which and how parameters depend on n. A border, even if it is diffuse, between the displacive and the order-disorder mechanism lies between n = 2 and 3, as seen in Fig. 10.5. Here, we can utilize a characteristic feature of molecular crystals. Since the single-particle potential of a double-well type in p-polyphenyls has its origin, not in the crystal packing but molecular properties, appropriate modification of molecules may be possible. The substitution of a fluorine atom for a hydrogen atom slightly changes the degree of delocalization of π electrons. The suitable choice of the substitution sites is the two ends of molecules. Namely, a series of difluoro-substituted molecules, F–(C6 H4 )n –F, is subject to study [49–53]. The substituted compounds are successfully isostructural in crystal structures [51, 54]. This fact implies that the substitution scarcely affects not only the repulsion between the ortho hydrogens but also the intermolecular interaction, which is significant inside the molecular layers [42] while considering distances between interacting molecules. The temperatures of the twist transitions are lower than those of unsubstituted compounds [49, 50, 53], as seen in Fig. 10.5. The twist transition disappears in difluorobiphenyl (DFBP, n = 2) [49, 55]. The transition temperature of DFTP implies that its transition lies in the intermediate region between the displacive and the orderdisorder mechanisms. A rough estimate of the barrier height in the single-particle potential is ca. 10 meV [51],4 which is just close to the thermal energy at the transition temperature, Ttrs = 127.05 K [50]. It is noteworthy that the corresponding estimate 4 The

estimate includes an unknown contribution of the twist-dependent interaction.

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for TP is ca. 50 meV [56] in contrast to its transition temperature, Ttrs = 193.5 K. These facts are consistent with the location of DFTP in Fig. 10.5, because the unified model [32, 33] classifies the transition based on whether kB Ttrs is much smaller than the barrier height or not. We can assign an effective decrease in n, the number of phenyl groups, for the decrease in the transition temperature. Interestingly, the same decrease (an arrow in Fig. 10.5) reasonably reproduces the transition temperatures for DFTP (n = 3) and DFQP (n = 4). The same shift also rationalizes the disappearance of the twist transition in DFBP. These results exemplify the successful and artificial control of the transition mechanism of a structural phase transition. It is noteworthy that the twist transition also disappears in dihydroxybiphenyl [HO–(C6 H4 )2 –OH] [57, 58], for which we expect a similar intramolecular potential for the twisting [58]. In this case, however, the intermolecular interaction enhanced by hydrogen bonds plausibly contributes to stabilizing the planar form of molecules. As mentioned earlier, the pressure can be an external parameter to control the nature of phase transitions. Since the pressure stabilizes the planar conformation of p-polyphenyls in crystal, resulting in the notable change in the single-particle potential, it would effectively decrease n. Indeed, the pressure coefficients of the transition temperature of BP and TP crystals are negative [59, 60]. The incommensurate structure of DFTP crystal [52] is compatible with such instability predicted theoretically for TP crystal under pressure [61]. This compatibility is another support for the conclusion that the twist transition of DFTP crystal is in between the typical displacive and order-disorder transitions [52].

10.2.3 Impurity Effects on Structural Phase Transitions Having learned the systematics of the twist transition in crystalline p-polyphenyls, we can examine impurity effects on structural transitions for this series. This systematic examination is distinct in, at least, two aspects from general studies. One is the possibility of simultaneous examinations on two limiting cases, the displacive type accompanying the soft mode in biphenyl (BP) to the order-disorder one for others in non-substituted p-polyphenyls, in a single series of compounds. The other is that we can utilize well-characterized impurities. For the examination of the impurity effect on an order-disorder transition, utilized as an impurity is a compound that has a similar molecular structure and significantly different single-particle potential. The molecule is a derivative of 1,2,4,5-tetrazine (C2 H2 N4 ), which has a planar hexagon structure (similar to benzene) with two carbon atoms most apart on the ring. The substitution of phenyl groups for two hydrogen atoms yields a molecule (diphenyltetrazine, DPTZ) similar to that of p-terphenyl (TP). Despite the similarity in the molecular shape, however, the absence of hydrogen atoms attached to the central tetrazine ring guarantees the absence of the repulsion between ortho hydrogen atoms. Since the repulsion is the exclusive contribution to the hump in the single-particle potential at the planar molecular form of p-polyphenyls, a

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DPTZ molecule prefers the planar form. Considering that the twist transition vanishes in difluorobiphenyl (DFBP), the single-particle potential of which still has the hump at the planar form in the isolated state, DPTZ is effectively an impurity without the tendency to the twist. We thus expect that the inclusion of DPTZ corresponds to the disappearance of the relevant degree with a minimum perturbation on the crystal lattice. The investigation of solid TP samples doped with 2 and 5% DPTZ indicated a lower temperature of the twist transition and the reduction in the excess heat capacity due to the transition [62]. The lowering is proportional to the concentration (x) of doped DPTZ. That is, ΔTtrs ≈ 23 x Ttrs . The reduction in the excess entropy Δ(ΔS) is compatible with Δ(ΔS)/x = N kB ln 2. These results are consistent with that expected for the so-called mean-field treatment of the Ising model, except for the more modest decrease than ΔTtrs = x Ttrs . This investigation offered a valuable byproduct, the experimental identification of the entity that behaves like the “spin.” The reduction in the entropy of transition by the doping is proportional to the net entropy involved in this transition. It is noteworthy that an operational baseline is arbitrary for this purpose as long as it is common for other doping levels. The deduced net entropy is kB ln 2 per TP molecule. This magnitude asserts that a TP molecule carries only a single degree of freedom for disordering. The situation is possible only when the two conformations of the alternate twist are allowed, but the other twist modes are prohibited. Namely, the motional correlation, discussed in Sect. 10.1, is almost perfect inside a molecule. In contrast to a rather trivial (and anticipated) result in the case of the orderdisorder transition, the study of the displacive transition revealed marked effects depending on the nature of impurity [63]. The compounds chosen as impurities are naphthalene (NA) and DFBP. A molecule of DFBP is twisted in the isolated state but planar in crystal down to low temperatures without the twist transition [49, 55], as described above. The twisting is possible with a low frequency in the crystal accordingly. On the other hand, NA is a condensed aromatic with a planar conformation. The characteristic frequency of the “twisting” vibration should be much higher than those of BP and DFBP. Since the transition temperature of the pure biphenyl crystal is rather low (40.4 K), the heat capacity calorimetry was used to monitor the effects of impurities. While the heat capacity anomaly scarcely exhibited the effect (without the change in the transition temperature) in the 1% NA-doped BP, that disappeared in the DFBP-doped BP with the same concentration. A negligible effect on the transition temperature in the NA-doped BP is remarkably different from the order-disorder transition of TP crystal. Also, the survival and the disappearance of the twist transition are counterintuitive because the NA molecule has effectively no twisting degree, whereas the DFBP molecule has. The counterintuitive behaviors resemble the impurity effect on lattice vibrations [64, 65]. When an atom that has a significantly different mass exists in a crystal lattice, a single vibrational mode localizes on the atom. In contrast, the mild difference in mass causes a weak localization of vibrational modes with a large localization radius. Although the variable parameter in the case of the BP crystal is not the mass but the

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a)

b)

Fig. 10.6 Localization of the soft mode in doped BP crystals (schematic). a weak localization with a large radius, b strong localization with a small radius. Reproduced from J. Phys. Soc. Jpn., 67, 1649 (1998) [63]

single-particle potential, the resemblance seems reasonable if we remember the vital role of the lattice vibration in displacive phase transitions, which takes place as a result of the softening of a relevant vibrational mode. That is, the relevant vibrational mode (twisting mode) scarcely changes in the NA-doped BP crystals at the doping level of 1%. On the other hand, the mode weakly localizes around DFBP molecules. It seems important to remind that the frequency of the twisting vibration of a DFBP molecule in the crystal lattice is higher than BP molecules in the high-temperature phase, in which molecules are planar. The localization, though weak, plausibly prevents the softening because of the higher stability of the planar forms. Then, the overlapping of the affected regions will probably cause the suppression of the softening. That is, the marked difference in the impurity effect originates in a significant difference in the localization radius, as illustrated in Fig. 10.6. Molecular dynamics (MD) simulations mimicking the doped BP crystals reproduced the observations [66].

10.3 Molecular Crystals as Stage for Novelties 10.3.1 Coupling Between Molecular Dynamics and Electronic System Physical properties that originate in the electronic system inside the crystal suffer from the effects of the lattice motional degrees of freedom. For example, electrical resistivity due to the scattering of electric carriers (electrons and holes) by lattice vibrations (phonon) is a ubiquitous example. Also, the superconductivity described well by the BCS theory occurs via the formation of the Cooper pairs of electrons mediated by phonons. When we compare molecular crystals with other kinds of crystals, noteworthy is the presence of the orientation degrees of freedom. We can

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211

imagine not only librational phonons but also other large-amplitude motions and resulting disorders. In this respect, the coupling between such degrees and the electronic system inside the crystal is of interest. The direct coupling may be scarce due to the significant difference in characteristic time scales in electronic and molecular dynamics. However, we can list the following. The degree of conformational disorder in the organic superconductor significantly alters the transition temperature [67, 68]. The electronic/magnetic properties and the instability intrinsic to the onedimensionality of the electronic system in the inorganic MMX complexes offer an example of a systematic study [7, 8, 12]. The description of thermodynamic and magnetic properties by a single statistical model can also be regarded as another example [24].

10.3.2 Phenomena Involving Molecular Flexibility The internal degrees of motional freedom is also intrinsic to molecules within the hierarchy of the material world. In this respect, the phenomena involving molecular flexibility are, in principle, new phenomena characteristic to molecular crystals. The glass transition and phase transitions related to the molecular flexibility, described in Chap. 9, belong to the category. The entropic stabilization of some mesophases seems ready for practical utilization in the phenomena related to the large entropy capacity of alkyl groups.

10.3.3 Dipolar Ising System Spin systems have played an essential role in the study of phase transitions and critical phenomena [69–71]. It is somewhat interesting to see the fact that the exact solutions exist for the cases where the interaction between spins localizes in the neighboring spins (two-dimensional Ising model [72]) and extends to infinitely (mean-field theory). Generally, the analysis becomes difficult with the elongation of the interaction range. The dipolar interaction is one with a physical counterpart decaying as ∝ r −3 and possesses unique anisotropy, ferroic along the dipolar axis, and antiferroic laterally (see Eq. 1.18). The study of spin models with dipolar interaction has a long history [73, 74]. Its Ising version is the dipolar Ising model (DIM), most studies of which mainly have paid attention to critical behaviors [75–78]. Although the DIM intuitively applies to a broad class of crystals consisting of simple polar molecules, they do not fit for studying the physical properties of the DIM because molecules cannot change their orientation free from the crystal lattice. The full clarification of its properties, therefore, requires a physical realization. Since the DIM usually refers to a classical Ising model, real spin systems where spins interact via magnetic dipolar interaction is not ones under discussion. An isolated bistable hydrogen (H) bond (O·H· · · O or O· · · H·O) is also unsuitable because

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of possible quantum effects. On the other hand, a pair of hydroxy groups can serve as an Ising-like spin as O·H· · · OH or HO· · · H·O. We already see this type of correlated dynamics in Sect. 10.1.2. Dimers formed by an H-bond are units of the crystal, and sit on the center of inversion in the disordered state (above Tc = 103 K) [13, 20]. The symmetry of sitting sites guarantees two states of H-bonding are precisely equivalent to each other with reversed electric dipole. The bulkiness of the hydrocarbon moiety [(C6 H11 )3 C–] forces dimers apart compared with the separation of charges in the dipole. Crystalline TCHM is, therefore, a physical realization of the DIM. The estimated magnitude of the dipole moment is 7.9 × 10−30 C m (2.4 D) [20]. Although the crystal is certainly anisotropic with a triclinic cell, the interdimer distances along the three crystallographic axes do not differ significantly. The assumption of isotropy is, therefore, not far from the truth. The system is roughly the system in which Ising spins are arranged on a lattice of about 10 × 10 × 10 Å3 . Experimental temperature dependence of dielectric constant [15] and heat capacity is different from a simple three-dimensional ferro- or antiferroelectrics and compared favorably with that expected by a highly-anisotropic Ising model [20]. This result is partly consistent with existing reports that the spins exhibit a strong correlation along the spin axis () [17–19]. The numerical estimates obtained from the analysis assuming the mean-field model [79, 80] are J /kB ≈ 170 K and z|J⊥ |/kB ≈ 1 ∼ 2 K with z being the number of the nearest neighbor spins (dimers) [20]. Having seen the utility of an isolated H-bonding dimer as an Ising spin with a significant electric dipole moment, we can imagine its modification concerning its separation and spatial arrangements. Such modifications will plausibly contribute to the full clarification of classical DIM.

10.3.4 Exotic Superstructure As mentioned some times previously, the phase transitions between crystalline phases have been subject of extensive studies, because they often accompany a drastic change in material properties of potential applications. The formation of superstructures is an important class of them. The superstructure is a traditional crystalline structure,5 the unit cell of which has the size of an integer multiple of the original lattice. The original lattice is usually stable at the high-temperature side because of its higher symmetry. Since the relationship between two phases is evident, the formation of superstructures is one of the best objects of the Landau’s phenomenology of phase transitions described in Sect. 2.2. It is noteworthy that the low-symmetry phase is fixed before an assessment on a specific transition starts because the Landau theory is essentially one for not predicting but understanding. The theory [1, 81] says that the transition is discontinuous if the third-order term can survive in the expansion of the thermodynamic potential in terms of the 5 See

Sect. 4.2 for the meaning of “traditional.”

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213

relevant (small) order parameter (Sect. 2.2.4). This statement is known as the Landau condition for the continuous transition. There exists the other condition for transitions between crystalline phases. Since the formation of a specific superstructure needs the strongest instability at the specific spatial modulation, this modulation must have a fixed periodicity. Namely, the period should be “locked.” Unless this is fulfilled, there is no reason why the resulting phase is not a general incommensurate phase but a commensurate phase, a kind of superstructures. The condition is known as the Lifshitz condition. The Lifshitz condition implies the importance of the periodicity of the modulation pattern in transitions accompanying the formation of superstructures. The theory of space group says that irreducible representations of space groups characterize the periodicity of modulation patterns. A combination of the reciprocal vector (wavevector) and the set of symmetry operations around a point specifies each of them. Thus, only reciprocal wavevectors with special properties seem to be candidates relevant to the formation of superstructures. Comprehensive analysis [82] indicates that the multiplicity, i.e., the size of superstructures relative to the original, is mostly 2, 4, or 8 in non-hexagonal cases. In hexagonal cases, the multiplicity of 3 and 6 additionally emerge. The possibility of a continuous transition is somewhat limited for the case of 3 because of the Landau condition. The multiplicities mentioned above essentially correspond to cases where one, two, or three of cell constants double by the “softening” of modulations with reciprocal vectors at the Brillouin zone boundary. The appearance of not only 2 but also 4 or 8 originates in the symmetry of structures. Here, we see that a single modulation and symmetrically equivalents drive the formation of superstructures. In this context, it seems natural, apart from the limited scope of the Landau theory, to ask what superstructures we can expect while considering only a single modulation pattern [83]. A trivial family of superstructures is a set of those with a cell constant that is an integer multiple of the original lattice along a single lattice axis. According to the Landau theory, the doubled structure may appear through a continuous transition while others, such as tripled one, emerges only through discontinuous transitions. We see that the coverage of the reciprocal lattice points in the trivial case discussed above completes only if the fundamental modulation wave and its harmonics cooperate. The allowance of the participation of the harmonics of the fundamental modulation introduces other possibilities of superstructures driven by a single modulation [83]. Figure 10.7 shows an example of the relation between reciprocal lattices of the basic structure stable at the high-temperature (HT) side and the low-temperature (LT) superstructure. The multiplicity is 5 in this case. Upon the formation of the superstructure, inside the unit cell of the basic structure shown by black dots, new four reciprocal lattice points (red and green) emerge. Considering the trivial equivalence of inverted wavevectors, we see that the coverage of all lattice points of the LT superstructures completes by successive shifts represented by orange arrows. The successive application is equivalent to harmonics. In this example, the coverage completes only by the fundamental modulation and the second harmonic. This easiness originates in the fact that this is the example of such superstructure with the minimum multiplicity. On the other hand, this example implies that similar

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k LT

b*HT hLT a*HT

Fig. 10.7 Relation between reciprocal lattices of the high-temperature (HT) basic structure and the low-temperature (LT) superstructure. Crossing points of grids, lattice points of the superstructure; black dots, lattice points of the HT structure; dotted arrows, reciprocal lattice vectors of the HT structure; orange arrows, a wave vector of the modulation. Successive shifts by the orange arrow from lattice points of the HT structure completely cover all lattice points of the LT structure. Prepared based on J. Chem. Phys., 146, 074503 (2017) [83]

coverage is possible if new reciprocal lattice points appear as a set of double-lined (2 × j, j ≥ 2) clusters. In this case, the multiplicity is (2 j + 1). Although there is another choice of the modulation wave vector in Fig. 10.7 because of the minimum multiplicity, no arbitrariness remains in cases with j > 2. The consideration in the previous paragraph lapses into a dream without a real counterpart. Indeed, the formation of superstructures with odd multiplicity is uncommon in non-hexagonal cases. However, there exists a compound that undergoes a phase transition accompanying the formation of a superstructure with the multiplicity 5 [83]. Since the transition belongs to the minimum case, the identification of the real example itself seems consistent with the consideration. It suggests that the quest for exotic superstructure deserves to pursue.

10.3.5 Molecular Aggregation in a Gyroid Phase The Gyroid phase (briefly introduced in Sect. 4.3.1) is a bicontinuous cubic phase most widely observed [84] in soft matter ranging from polymers with a cell constant in µm-scale [85] to thermotropic and lyotropic liquid crystals with that in the nmscale [86, 87]. There are two complementary descriptions, which pay emphases on one of the wall (surface) and the jungle gyms. Fig. 4.1a depicted the Gyroid phase in the latter spirit. The two subspaces divided by the G (gyroid) surface (shown in Fig. 10.8), which has extensively been studied in differential geometry, are physically equivalent but have opposing handedness. Among the Gyroid phase formed in

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Fig. 10.8 A unit cell of the Gyroid phase [88]. Two jungle gyms (colored red and blue) are embedded in two subspaces separated by the G surface. Molecules are arranged along rods of jungle gyms with continuous twists with opposing senses. An “expected” ideal molecular arrangement is drawn between two neighboring junctions in the red subspace. Reproduced from J. Phys. Soc. Jpn., 86, 084602 (2017) [89]

various systems, that of thermotropic liquid crystals occupies a particular position. The length scale of the structure is comparable with the length of molecules. This similarity enables us to explore the structures in terms of molecular language. We can thus ask how molecules aggregate in this exotic structure. The clarification should largely contribute to understanding the formation mechanism of this beautiful and complicated structure. Nowadays, the molecular arrangement in the Gyroid phase exhibited by a series of thermotropic mesogens has been revealed to be continuously twisted throughout the space, as schematically shown in Fig. 10.8 [88, 89]. Although the thermotropic Gyroid phase is a crystal even by the traditional definition, the clarification of the molecular aggregation has not been straightforward. First, the structural disorder in the system is very severe, resulting in a minimal number (typically a few tens) of available diffractions. Any standard procedure of structure solution is impractical. We must rely on the description based on the smeared electron density distribution, such as a binary model. Second, by the Babinet principle [90], only the contrast in the density of scatterers (electrons in reality) is responsible for the distribution of diffraction intensities. This difficulty means that there is no way to distinguish two models where aromatic and aliphatic parts, which differ in electron density significantly, interchange their positions. Struggles for the clarification of aggregations were conducted for BABH(n), the molecular structure of which is in Fig. 9.4. BABH(n) exhibits the Gyroid and chiral cubic phases, as in ANBC(n) [91]. The phase diagram is also similar to that of ANBC(n) (Fig. 9.9), including the appearance of two Gyroid phases in short- and long-chain regions. The second difficulty above was overcome by analyzing the chain length dependence of two prominent diffractions [92] following the quasi-binary picture of thermotropics (Sect. 9.4.4.2) [93, 94]. The availability of the data for a wide range of chain lengths was the basis for this strategy. The analysis [92] revealed

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that the distribution of electron density mostly resembles those assuming the finite thickness of the decoration layers at both sides of the ideal G surface [95–97] and that the aromatic core parts form rods of jungle gyms. That is, in turn, the terminal methyl groups decorate the G surface. The successful analysis again indicates the importance of systematics as an indispensable tool in studying complex systems like molecular ensembles (Sect. 9.4.2). The so-called maximum entropy method (MEM) was employed to reveal the molecular aggregation. The MEM is a methodology to estimate the probability distribution p(x) of the system under consideration based on the statistical argument. The entropy I here is not thermodynamic, but information entropy defined like  I =−

p(x) log2 p(x)d x.

(10.3)

The essence of the argument is that the most probable (and consequently reliable) p(x) should realize the maximum uncertainty under the constraint that is compatible with known information (input data). Diffraction crystallography usually treats the electron density as the un-normalized probability density based on its property that cannot be negative as in the usual probability [98, 99]. If no information is available, the most reliable electron density consequently results in the uniform one everywhere. Thus, we can characterize the result of the MEM as the electron density with the least features compatible with the constraint exerted by experimental information. The constraint adopted in the analysis of the Gyroid phase was n 1  (G ci − G oi )2 ≤ 1, n i=1 [σ(G oi )]2

(10.4)

where the indices i distinguish experimental diffractions, and σ is experimentally estimated error for G. The subscript “c” and “o” stand for “calculated” and “observed,” respectively. In the standard use of the MEM in the single-crystal crystallography, the atomic detail of the crystal structure is a priori known, and the distribution of the electron density is precisely refined. On the other hand, in the case of the Gyroid phase, a priori information is only the space group, I a3d. Thus, G was set equal to the structure factors F only for two prominent diffractions and their absolute values |F| for others [88]. In the description of the Gyroid phase based on the G surface, the body diagonal of the cubic unit cell passes four flat points (with vanishing Gaussian curvature) on the surface. The flat points serve as junctions of jungle gyms. Four maxima and minima would exist in this length, accordingly. However, this expectation was correct only for a limited range of chain-length. In reality [88], this holds in the short-chain regime in the phase diagram against the chain length but does not in the long-chain regime. In the latter, the maximum splits into two, with the separation shorter than the length of the molecular core part. Further, the split maxima continue while exhibiting

10.3 Molecular Crystals as Stage for Novelties

217

gyration along the direction of the rod of a jungle gym. The gyration angle is cos−1 31 from a junction to the neighboring junction. These results are compatible with the following model of molecular aggregation. In essence, the molecules locally form a single layer (short-chain cases) or doubled layers (long-chain), the normal of which coincides with a body diagonal, and the long axes of molecules continuously twist from a junction to neighboring junctions along a rod of the jungle gym. It is crucially important that the model is free from the packing frustration at junctions. It is also interesting to note that the formation of doubled core layers brings the recovery of the volume fraction of core parts, possibly resulting in the reentrance to the Gyroid phase [88, 100] in the phase diagrams of BABH(n) [91, 101] and ANBC(n) [102]. Having revealed the average core arrangements in the Gyroid phase, it is in order to consider the packing of alkyl chains [100]. Since molecules locally form layers with normal coinciding the body diagonal of the unit cell, the chain-length dependence of the body diagonal should give useful information as in orthogonal smectic phases discussed in Sect. 9.4.4.4. The analysis yields a smaller increment for long-chain BABH(n) than those discussed there. The increment is compatible with the normal orientation to the layer on average. The smaller increment implies that the chain layer is not bilayer-like but monolayer-like. That is, the single chain-layer consists of chains from both sides. The molecular description enables us to discuss the formation mechanism of the Gyroid phases, which is ubiquitous in the materials world. The sense of the gyration of molecular arrangements on jungle gyms is opposing in two subspaces, following the chirality of two subspaces divided by the G surface. Some mechanisms, which are mostly independent and do not interfere with one another, have been suggested to rationalize the formation of the Gyroid phase [84, 103–109]. If we once accept such mechanisms, it is possible to imagine a virtual lattice structure on jungle gyms. The linear segregation of domains with opposing local chirality with equal volumes is an essential feature of the Gyroid phase. It plausibly originates in the anti-spindle shape of molecules assuming some next-nearest-neighbor interaction, as naïvely and widely assumed [101]. Since constituting molecules are seemingly achiral in real systems, molecules should have the same volume for opposing twists. The Gyroid phase guarantees the volumetric equivalence of two spaces. The equal preference for the two handedness of twisted molecular arrangement is, therefore, another microscopic factor that brings the superior stability of the Gyroid phase, in addition to the factors previously identified by existing treatments [84, 103–109]. The simulations assuming the preference for the twist between neighboring molecules [89] do not give a symptom of a phase exhibiting net chirality (Sect. 7.3). However, the nano segregation of local chirality may emerge. These facts suggest that the chirality of a molecule itself is necessary for such chiral phases. Recently, chiral phases made of seemingly and dynamically achiral molecules were reported [110–114]. The results described suggest that the flexibility of molecules (twisting degrees of freedom around single bonds) and a resulting adaptive chirality responding to the environment play important roles in the formation of such chiral phases, as naïvely assumed without reasoning by the authors of previous reports [112–114].

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