Liquid crystals theoretical considerations

Liquid crystals have attracted considerable theoretical study. Much of this has been of a secondary nature and has explored the properties of 

Early theoretical treatments of liquid crystals were not surprisingly based on the molecular field approximation. However, it is neccessary to make assumptions about the pair potential employed in the calculation and it is impossible to know whether the predictions of a particular model really arise from the pair potential employed or whether they arise, at least in part, from the deficiencies of the basic approximation employed. The general problem is so complex that a better mathematical treatment of the molecular interactions in a liquid crystal is out of the question. However, with the introduction of ever more powerful computers, it has become possible to carry out meaningful numerical simulations of model liquid crystals. 

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The understanding of the nature of the LC state, the study of their stmaural organization, elucidation of the relationship between the molecular constitution and physical properties of liquid crystals arrived considerably later in the 1940s and 1950s. However, the majority of researchers found nothing more than theoretical interest in the study of the LC state. 

The distinct properties of liquid-crystalline polymer solutions arise mainly from extended conformations of the polymers. Thus it is reasonable to start theoretical considerations of liquid-crystalline polymers from those of straight rods. Long ago, Onsager [2] and Flory [3] worked out statistical thermodynamic theories for rodlike polymer solutions, which aimed at explaining the isotropic-liquid crystal phase behavior of liquid-crystalline polymer solutions. Dynamical properties of these systems have often been discussed by using the tube model theory for rodlike polymer solutions due originally to Doi and Edwards [4], This theory, the counterpart of Doi and Edward s tube model theory for flexible polymers, can intuitively explain the dynamic difference between rodlike and flexible polymers in concentrated systems [4]. 

From thermodynamic investigations invaluable qualitative and quantitative information is provided with regard to the phase transitions and vicinity of transitions of polymers and conventional low molar mass liquid crystals. Furthermore they give information about the kind of transition and phase stability relations, which are necessary to test theories or to evaluate new theoretical considerations. Owing to... 

The rheology of low molecular weight thermotropic compounds has been a subject of considerable theoretical and experimental analysis In general, liquid crystals are easily oriented by surfaces, electromagnetic fields and mechanical stress or shear, and the degree of orientation, in turn, affects their melt viscosity. The rheological behavior of a liquid crystal is known to be greatly dependent on the nature and also on the texture of its mesophase. 

It is beyond the scope of this review to be exhaustive in the field of supercooled liquids that has drawn intense research activities over several decades. Reviews that are exclusive for this field and deal with specific topics in considerable detail are recommended for supplemental reading [9-11]. In view of the scope this chapter, the next section provides the readers with a brief introduction to the systems of interest and associated nomenclature. Section III sets up the background by reviewing experimental results on the dynamics of thermotropic liquid crystals across the I-N transition, then introducing the central issues in the dynamics of supercooled liquids, and finally comparing the dynamics of the two systems in the light of recent experiments. Section IV presents a summary of some of the well-known theoretical approaches to liquid crystals. Section V provides a detailed account of computational efforts. Finally, we conclude in Section VI with a list of problems for future work. 

We have studied the field- and temperature-dependence of the susceptibility of the liquid crystal phase of p-azoxyanisole for fields from 201 to 2550 gauss. The results at the highest fields agree with those of Foex at 8000 gauss, but at lower fields the susceptibility becomes field-dependent. The experimental results may be adequately represented by a simple theoretical expression obtained from statistical considerations. The only adjustable parameter in the theoretical formula is an effective mass which arises from interactions between the molecules and amounts to 2 X 10 n gram. 

Thermotropic liquid crystal polymers (LCI ) are of considerable current interest, because of their theoretical and technological aspects [1-3]. Evidently, a new class of polymers has been developed, combining anisotropic physical properties of the liquid crystalline state with diaracteristic polymer features. This unique combination promises new and interesting material properties with potential ai lications, for example in the field of high modulus fibers [4], storage technology, or non-linear optics [5]. 

A large number of conditions must be fulfilled for the formation of liquid crystals, which particularly concern the geometry of the potential mesogen. Using the example of the hexaphenyls, it can be shown that a high length/diameter ratio is necessary for mesophase formation thus only the linear hexaphenyl is capable of forming the liquid-crystal phase, as was postulated by Paul J. Hory in 1956, on the basis of theoretical considerations. 

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. 

The coupling between the properties of conventional polymer networks and the properties of chiral liquid crystalline phases results in interesting, new opto- and electromechanical effects of the chiral liquid crystalline elastomers, as demonstrated by theoretical considerations and experiments. Knowledge about these new materials is still in its infancy. But the properties analyzed so far for these elastomers indicate promising aspects for application and are the basis for the new syntheses of optimized chiral liquid crystal networks. 

Lyotropic liquid crystals acquire their anisotropic properties from the mixing of two or more components. One component is amphiphilic and contains a polar head group, the second component is usually water. Lyotropic liquid crystals occur abundantly in nature, particularly in all living systems. The most familiar example of a lyotropic liquid crystal is soap in water. During the last few years considerable progress has been made toward understanding the experimental and theoretical aspects of the method reviewed in this book. 

The relationship between macroscopic properties and molecular properties is a major area of interest, since it is through manipulation of the molecular structure of me-sogens, that the macroscopic liquid crystal properties can be adjusted towards paricu-lar values which optimize performance in applications. The theoretical connection between the tensor properties of molecules and the macroscopic tensor properties of liquid crystal phases provides a considerable challenge to statistical mechanics. A key factor is of course the molecular orientational order, but interactions between molecules are also important especially for elastic and viscoelastic properties. It is possible to divide properties into two categories, those for which molecular contributions are approximately additive (i.e. they are proportional to the number density), and those properties such as elasticity, viscosity, thermal conductivity etc. for which intermolecular forces are responsible, and so have a much more complex dependence on number density. For the former it is possible to develop a fairly simple theory using single particle orientational order parameters. 

Calculations of atomic and molecular hyperpolarizabilities usually proceed via time-dependent perturbation theory for the perturbed atomic states. Even for molecules of modest size, the calculation of the complete set of unperturbed wavefunctions, and exact calculation of the hyperpolarizabilities, is prohibitively difficult. Liquid crystals typically consist of organic molecules with aromatic cores, and there is considerable experimental [10] and theoretical [11, 12] evidence to indicate that the dominant contribution to the polarizabilities originates from the delocalized r-electrons in conjugated regions of these molecules. Even considering only r-electrons the calculations rapid-... 

A large variety of phase transitions is present in the mesomorphic state, and for many years, a considerable number of experimental and theoretical studies have been devoted to the understanding of such phase transitions. This understanding is fascinating, but also of great importance from the fundamental point of view (indeed, liquid crystals are very convenient materials for experimental studies in condensed matter physics), as well as for the use of liquid crystals in devices (for example, the temperature dependence of physical parameters such as helical pitch, elastic coefficients, etc.). 

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