Theories of Liquid-Crystalline Ordering

In the current approach, the term 3 cos cj) — 1 is called the nematic orientation parameter. 

Although the Kerr effect is a classical subject well discussed in physics, its significance has gained importance in industry only recently. It is now used to assist in the understanding of the wide apphcation of liquid crystals in modern devices such as computer screens and watch tops. 

Most theories of the polymeric nematic phase are based upon the model in which the liquid crystal is considered as a packed system interacting through its hard-core diameters. Historically, attention was first focused on lyotropic liquid crystals. The binary collision or second virial approximation is taken into account at low volume factions. A slightly different approach is to use a scaling law to describe the fluctuation of nematic order parameters. In this section we select three different methods to represent the three different treatments. 

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Vasilenko, S. V., Khokhlov, A. R., and Shibaev, V. R, Theory of liquid-crystalline ordering in melts of macromolecules with stiff and flexible fragments in the main chain. 1, Macromolecules, 17, 2270-2275 (1984). 

Kelton K (1991) Crystal nucleation in liquids and glasses. Solid State Phys 45 75-177 Khokhlov A, Semenov A (1985) On the theory of liquid-crystalline ordering of polymer chains with limited flexibility. J Stat Phys 38(1-2) 161-182 Kraack H, Sirota E, Deutsch M (2000) Measurements of homogeneous nucleation in normal-alkanes. J Chem Phys 112(15) 6873-6885... 

The first molecular theory of liquid-crystalline ordering was proposed by Onsager on a solution of cylindrical, long, rigid rods. The theory may be described as follows. Let N2 be the number of rods in a volume V. Each rod has length L and cross-sectional area d. The volume faction of rods, V2, in the solution can be calculated as... 

In conclusion, we note that although many problems in the theory of liquid-crystalline ordering in polymer systems have already been solved, this area is still in the initial stage of development on the whole. Among the most important directions of further research (cf. [139] for more detail) are the rheology of thermotropic polymer liquid crystals, the theory of liquid-crystalline elastomers, the statistical physics of the surface in liquid-crystalline polymers, the theory of smectic ordering in polymer systems, and the kinetics of phase transitions in liquid-crystal polymCTS. 

Recent solid state NMR studies of liquid crystalline materials are surveyed. The review deals first with some background information in order to facilitate discussions on various NMR (13C, ll, 21 , I9F etc.) works to be followed. This includes the following spin Hamiltonians, spin relaxation theory, and a survey of recent solid state NMR methods (mainly 13C) for liquid crystals on the one hand, while on the other hand molecular ordering of mesogens and motional models for liquid crystals. NMR studies done since 1997 on both solutes and solvent molecules are discussed. For the latter, thermotropic and lyotropic liquid crystals are included with an emphasis on newly discovered liquid crystalline materials. For the solute studies, both small molecules and weakly ordered biomolecules are briefly surveyed. 

This article reviews the following solution properties of liquid-crystalline stiff-chain polymers (1) osmotic pressure and osmotic compressibility, (2) phase behavior involving liquid crystal phasefs), (3) orientational order parameter, (4) translational and rotational diffusion coefficients, (5) zero-shear viscosity, and (6) rheological behavior in the liquid crystal state. Among the related theories, the scaled particle theory is chosen to compare with experimental results for properties (1H3), the fuzzy cylinder model theory for properties (4) and (5), and Doi s theory for property (6). In most cases the agreement between experiment and theory is satisfactory, enabling one to predict solution properties from basic molecular parameters. Procedures for data analysis are described in detail. 

In this section, we consider the intramolecular liquid-crystalline ordering of the relatively small macromolecules, we will see that the properties of the globules in this case (small globules) are not universal. As already noted in the introduction, the analysis of the structure of small globules requires the use of the formalism of the self-consistent theory of polymer globules proposed by Lifshitz in Ref,23). 

Rod—coil block copolymers have both rigid rod and block copolymer characteristics. The formation of liquid crystalline nematic phase is characteristic of rigid rod, and the formation of various nanosized structures is a block copolymer characteristic. A theory for the nematic ordering of rigid rods in a solution has been initiated by Onsager and Flory,28-29 and the fundamentals of liquid crystals have been reviewed in books.30 31 The theoretical study of coil-coil block copolymer was initiated by Meier,32 and the various geometries of microdomains and micro phase transitions are now fully understood. A phase diagram for a structurally symmetric coil—coil block copolymer has been theoretically predicted as a... 

Some polymers manifest liquid crystalline ordering, which does not have the full long-range three-dimensional periodicity of crystallinity but is far more ordered than amorphicity. Since many excellent books and articles have been published on such polymers and the author does not have much that is new to add to this background information, very little will be said about polymer liquid crystallinity in this book. Van Krevelen [3] has reviewed liquid crystallinity in polymers in a readable manner and discussed its effects on properties for which quantitative structure-property relationships are available. Adams et al [41] have published a valuable compendium of articles covering the theory, synthesis, physical chemistry, processing and properties of liquid crystalline polymers. Woodward [42] has discussed and illustrated liquid crystallinity in polymers with many beautiful micrographs. 

According to the elastic continuum theory of liquid crystals which was introduced in Chapter 1, the three kinds of deformations can be described by three elastic constants, An(splay), / (twist) and / (bend). In the case of small molecular mass liquid crystals, the three constants are mainly determined by the chemical composition of the liquid crystalline molecules. Among them, K22 is the smallest while the other two are approximately close. All three elastic constants are of the order of 10 12 N. The elastic constants of some important liquid crystals are listed in Table 6.1. Each kind of liquid crystals is a mixture of R5-pentyl and R6-hexyl homologues in the ratio of 40 60. The data are obtained at the temperature of T = Tc — 10 °C where Tc is the clear temperature. 

Interesting studies have been reported recently on the rheological properties of liquid-crystalline phases in soap systems. According to a theory of flow in concentrated colloidal systems (Bohlin 1980) it is possible to relate the flow in ordered liquids, like liquid-crystalline phases, to the structure. This theory has been successfully applied to the liquid-crystalline phases of soaps and other amphiphiles, for example phases in the system CTAB or sodium octanoate/decanol/water (Bohlin and Fontell, 1978 Faucompre, 1982). 

SINCE the discovery of liquid crystalline phenomenon for low molecular weight liquid crystals (LMWLCs) more than 100 years ago, anisotropic ordering behaviors of liquid crystals (LCs) have been of considerable interest to academe [1-8], In the 1950s, Hory postulated the lattice model for various problems in LC systems and theoretically predicted the liquid crystallinity for certain polymers [1-3], As predicted by the Hory theory, DuPont scientists synthesized lyotropic LCPs made of rigid wholly aromatic polyamide. Later, Amoco, Eastman-Kodak, and Celanese commercialized a series of thermotropic main-chain LCPs [2]. Thermotropic LCPs have a unique combination of properties from both liquid crystalline and conventional thermoplastic states, such as melt processibility, high mechanical properties, low moisture take-up, and excellent thermal and chemical resistance. Aromatic main-chain LCPs are the most important class of thermotropic LCPs developed for structural applications [2,4-7]. Because they have wide applications in high value-added electronics and composites, both academia and industry have carried out comprehensive research and development. 

However, the number of liquid crystals that have been studied under pressure is very limited. In most cases neither the equation of state nor the pressure dependence of the order parameter is known. Only the mean-field theory of Maier and Saupe was extended to explain the dielectric properties of liquid crystalline phases. However, a recent approach by Photinos et al. analyzed the nematic reentrance and phase stability based on the variational cluster method. The lack of a full theoretical description as well as insufficient experimental data should stimulate further high-pressure investigations in this field. 

In order to synthesize liquid crystals with predicted properties, one of the basic questions for the chemist is Which intermolec-ular interactions are responsible for the stabilization of liquid crystalline phases The known dependence of mesogenity on the molecular geometry prompted the derivation of molecular-statistical theories, based on the shape-induced anisotropy of the repulsion of the molecules [25], Despite the remarkable success of such theories, it is clear that real condensed phases always eon-tain strong attraetive interactions, which are completely neglected by this procedure. Therefore, attempts have been made to... 

In view of the desirable molecular design of liquid crystalline phases, the final goal of microscopic theories is not only to predict the temperature dependence of elastic coefficients, but also to reveal relations between the molecular and elastic properties of the mesophases. Since the original work of Saupe in 1960 [241], a large number of microscopic theories have been developed [181-183, 242-270]. The first molecular statistical theories of the Frank elastic constants were formulated by Priest [242] and Straley [243], and a significant. step was made by Poniewierski and Stecki [244,245] who applied density-functional theory in order to relate in a direct quantitative way the elastic constants to characteristic properties of a statistical ensemble of molecules with... 

A.V. Zhestkov, The Theory of Intra- and Inteimolecular Liquid-Crystalline Ordering in Pcdymer Systems, Dissertation, Moscow (1985). 

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