# Ordered phases of a liquid of biaxial particles

TL;DR: In this article, a simple model for an interacting liquid of particles lacking an axis of rotational symmetry is proposed, and the four order parameters necessary to describe an ordered phase are identified.

Abstract: A simple model for an interacting liquid of particles lacking an axis of rotational symmetry is proposed. The four order parameters necessary to describe an ordered phase are identified. An ensemble of such particles is described by a mean field theory. A phase diagram showing both uniaxial and biaxial phases results. The model predicts a phase diagram similar to that of the phenomenological model of Alben.

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TL;DR: This Review will focus on the major classes of columnar mesogens rather than presenting a library of columner liquid crystals, and emphasis will be given to efficient synthetic procedures, and relevant mesomorphic and physical properties.

Abstract: Most associate liquid crystals with their everyday use in laptop computers, mobile phones, digital cameras, and other electronic devices. However, in contrast to their rodlike (calamitic) counterparts, first described in 1907 by Vorlander, disklike (discotic, columnar) liquid crystals, which were discovered in 1977 by Chandrasekhar et al., offer further applications as a result of their orientation in the columnar mesophase, making them ideal candidates for molecular wires in various optical and electronic devices such as photocopiers, laser printers, photovoltaic cells, light-emitting diodes, field-effect transistors, and holographic data storage. Beginning with an overview of the various mesophases and characterization methods, this Review will focus on the major classes of columnar mesogens rather than presenting a library of columnar liquid crystals. Emphasis will be given to efficient synthetic procedures, and relevant mesomorphic and physical properties. Finally, some applications and perspectives in materials science and molecular electronics will be discussed.

1,056 citations

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TL;DR: In this article, the ternary system potassium laurate-1-decanol-D${2}\mathrm{O}$ was studied over concentration ranges where nematic phases are likely to occur.

Abstract: The phase diagram of the ternary system potassium laurate-1-decanol-D$_{2}\mathrm{O}$ was studied over concentration ranges where nematic phases are likely to occur. Two uniaxial nematic phases which are separated by a biaxial nematic phase are found. In limited concentration range the following phase sequence may be observed reversibly on heating and on cooling: isotropic-uniaxial nematic (positive optical anisotropy)-biaxial nematic-uniaxial nematic (negative optical anisotropy)-biaxial nematic-uniaxial nematic (positive optical anisotropy)-isotropic.

629 citations

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TL;DR: A comprehensive overview of phase transition studies can be found in this article, where the authors identify the essential key concepts and points of difficulty associated with the study of phase transitions and discuss the most widely used experimental techniques for measuring these transition properties.

Abstract: Mesogenic materials exhibit a multitude of transitions involving new phases. Studies of these phases are of importance in a wide range of scientific fields and as such have stimulated considerable theoretical and experimental efforts over the decades. This review article presents a comprehensive overview until this date of the developments in this subject. An attempt is made to identify the essential key concepts and points of difficulty associated with the study of phase transitions. The article begins with a brief introduction about the symmetry, structure and types of liquid crystalline phases. This is followed by a discussion of the distribution functions and order parameters which are considered as the basic knowledge essential for the study of ordered phases. A brief discussion of the thermodynamic properties at and in the vicinity of phase transitions, which are required to understand the molecular structure phase stability relationship, is given. The most widely used experimental techniques for measuring these transition properties are critically examined. The remaining parts of the article are concerned with the current status of the theoretical developments and experimental studies in this field. The application of the various theories to the description of isotropic liquid-uniaxial nematic, uniaxial nematic-smectic A, uniaxial nematic-biaxial nematic, smectic A–smectic C phase transitions are reviewed comprehensively. The basic ideas of Landau–de Gennes theory and its applications to study these transitions are discussed. Since the formation of liquid crystals depends on the anisotropy in the intermolecular interactions, questions concerning its role in the mesophase transitions are addressed. The hard particle, Maier-Saupe and van der Waals types of theories are reviewed. The application of density functional theory in studying mesophase transitions is described. A critical assessment of the experimental investigations concerning reentrant phase transitions in liquid crystals is made and the factors which impede its proper understanding are identified. A survey is given of existing computer simulation studies of the isotropic to nematic transition, the nematic to smectic A transition, the smectic A to hexatic S B transition, the smectic A to reentrant nematic transition, and transitions to the discotic phase. The current status of the study of phase transitions involving hexatic smectic, cholesteric, polymeric and ferroelectric liquid crystals is outlined. Finally, a range of unexplored problems and some of the areas which are in greatest need of future attention are identified.

308 citations

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TL;DR: An overview of the current state of research in the field of biaxial nematic liquid crystalline materials is given in this paper, where the major theoretical concepts are outlined, including the classification to different symmetries, the importance of cooperativity and cluster formation for the development of BN order and the conditions for the establishment of field induced and spontaneous BN in nematic phases.

Abstract: An overview over the current state of research in the field of biaxial nematic liquid crystalline materials is given. After a short introduction, providing some general aspects and summarizing the classical approaches, the main part focuses on recent developments of new concepts for designing biaxial nematics. First, the major theoretical concepts are outlined, including the classification to different symmetries, the importance of cooperativity and cluster formation for the development of biaxial order and the conditions for the establishment of field induced and spontaneous biaxiality in nematic phases. These new concepts also require the re-evaluation of the tools used for the identification of phase biaxiality, which are discussed briefly. In the second part, recent progress in the design of potential biaxial nematic materials, especially focussing on bent-core molecules with nematic phases, is reported and, finally, comparisons with phase biaxiality as observed in smectic liquid crystals are made.

299 citations

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TL;DR: A review of the wide variety of predictions that results from a Landau-type of description of the nematic-isotropic phase transition is given in this paper, which includes a discussion of the nature of the order parameter and of the various types of possible phases.

Abstract: A review is given of the wide variety of predictions that results from a Landau-type of description of the nematic-isotropic phase transition. This includes a discussion of the nature of the order parameter and of the various types of possible phases, of the influence of external fields, and of the effect of inclusion of spatial variations of the order parameter. The various predictions are compared with the available experimental results. It is concluded that there is still no clear picture about the nature of the singularity near the nematic-isotropic phase transition. Though the assumption of classical (mean-field) critical behaviour seems to be incorrect, there is no conclusive proof which alternative applies.

291 citations

##### References

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Yale University

^{1}TL;DR: In this article, it was shown that colloids in general are apt to exhibit considerable deviations from Raoult's law and that crystalline phases retaining a fair proportion of solvent may separate from concentrated solutions.

Abstract: Introdzution. The shapes of colloidal particles are often reasonably compact, so that no diameter greatly exceeds the cube root of the volume of the particle. On the other hand, we know many coiloids whose particles are greatly extended into sheets (bentonite), rods (tobacco virus), or flexible chains (myosin, various Iinear polymers). In some instances, a t least, solutions of such highly anisometric particles are known to exhibit remarkably great deviations from Raoult’s law, even to the extent that an anisotropic phase may separate from a solution in which the particles themselves occupy but one or two per cent of the total volume (tobacco virus, bentonite). We shall show in what follows how such results may arise from electrostatic repulsion between highly anisometric particles. Most colloids in aqueous solution owe their stability more or less to electric charges, so that each particle will repel others before they come into actual contact, and effectively claim for itself a greater volume than what it actuaily occupies. Thus, we can understand that colloids in general are apt to exhibit considerable deviations from Raoult’s law and that crystalline phases retaining a fair proportion of solvent may separate from concentrated solutions. However, if we tentatively increase the known size of the particles by the known range of the electric forces and multiply the resulting volume by four in order to compute the effective van der Waal’s co-volume, we have not nearly enough to explain why a solution of 2 per cent tobacco virus in 0.005 normal NaCZ forms two phases.

4,260 citations

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TL;DR: In this paper, the authors describe a mittlere Innere Feld einer nematischen Molekülanordnung abgeleitet, and gezeigt, daß this Innere Field in einem gewissen Temperaturbereich einen nematicalhen Ordnungszustand bewirken muß and daß dieser ordnung sustand was with einer bestimmten Temperatur diskontinuierlich und unter Energieaufnahme in die isot

Abstract: Mit Beschränkung auf den Dipol-Dipol-Anteil der reinen Dispersionswechselwirkungen als dem für die Existenz einer kristallinflüssigen Phase vermutlich maßgebenden Teil der zwischenmolekularen Kräfte wird durch geeignete Mittelungen ein Ausdruck für das auf das Einzelmolekül wirkende mittlere Innere Feld einer nematischen Molekülanordnung abgeleitet. Es wird gezeigt, daß dieses Innere Feld in einem gewissen Temperaturbereich einen nematischen Ordnungszustand bewirken muß und daß dieser Ordnungszustand bei einer bestimmten Temperatur diskontinuierlich und unter Energieaufnahme in die isotrope Molekülanordnung übergehen muß. Die Theorie liefert also einen Umwandlungspunkt 1. Ordnung, wie er experimentell auch beobachtet wird. Die Umwandlungstemperatur wird im wesentlichen durch die Anisotropie der optischen Übergangsmomente der Moleküle bestimmt.

1,328 citations

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TL;DR: Onsager's prediction of a first-order phase transition in a gas of long thin rods is verified by extensive calculations on a simple model as mentioned in this paper, where the molecules are rectangular parallelepipeds, of length l and square cross section d×d.

Abstract: Onsager's prediction of a first‐order phase transition in a gas of long thin rods is verified by extensive calculations on a simple model. The molecules are rectangular parallelepipeds, of length l and square cross section d×d. The long molecular axes can point in only three mutually perpendicular directions. Virial coefficients, up to the seventh, are calculated exactly as functions of orientation, in the limit l→ ∞, d→0, and l2d=constant. The transition predicted by Onsager's theory, based on a second virial approximation to the potential of mean force in the space of relative orientations, is observed also when all virial coefficients up to the seventh are included. The virial expansion appears to converge well for the isotropic phase; but the properties of the anisotropic phase depend sensitively on the order of approximation.

261 citations