Molecular statistical theories nematics

A molecular-statistical theory of the flexoelectric effect in the nematic phase can be derived in a general way using the density-functional approach to the theory of liquid crystals. In this approach, the free energy of a liquid crystal, F, is a functional of the density po(a ) = Po/(w) where /(w) is the orientational distribution fimction. The general structure of the functional F p) is not known, but the functional derivatives are known and are related to the direct correlation functions of the nematic phase. 

Maier and Saupe, in their well-known molecular-statistical theory, described the intermolecular orientational forces by a mean field method. The Maier-Saupe theory successfully predicts the relationship between the molecular orientation parameter S and the nematic potential D as a function of temperature [10,14]. 

W. Maier and A. Saupe, A simple molecular statistical theory for nematic liquid crystal phase. Part H,... 

Ma Y, Zha L, Hu W, Reiter G, Han CC (2008) Crystal nucleation enhanced at the diffuse interface of immiscible polymer blends. Phys Rev E 77(6) 061801 Maier W, Saupe A (1959) A simple molecular statistical theory of the nematic crystalline-liquid phase. Zeitschrift fur Naturforschung 14 882-889 Mandelkem L (1964) Crystallization of polymers. McGraw-Hill, New York Matsuoka S (1962) The effect of pressure and temperature on the speciflc volume of polyethylene. J Polym Sci 57(165) 569-588... 

Molecular-statistical theories are available for several different smectic phase types [25]. In addition to the ingredients of the nematic phases, the theories incorporate parameters responsible for the formation of layers. The clearing temperatures, however. 

S. Chandrasekhar and N. V. Madhusudana, Molecular Statistical Theory of Nematic Liquid Crystals, Acta Crysl., Vol. A27, p. 303 (1971). 

Maier W, Saupe A. A simple molecular statistical theory of the nematic crystalline-liquid phase. Z Naturf 1959 14 882-889. 

Maier, W, and A. Saupe. 1959. A simple molecular-statistics theory of the nematic liquid-crystaUine state. Z. Naturforsch. 13a 564—570. 

The magnitude of surface tension, y, has also been calculated from statistical theory and molecular orientations at the free surface in nematic liquid crystals. ... 

A number of important ideas concerning the N, phase have been discussed theoretically - molecular statistical and phenomenological theories, " " continuum theories, " topological theories of de-fects, - " etc. For example, Saupe and Kini " who used different theoretical approaches, have both concluded that the incompressible orthorhombic nematic has 12 curvature elastic constants (excluding three which contribute only to the surface torque) and 12 viscosity coefficients. 

Liquid crystals manifest a number of transitions between different phases uprm variation of temperature, pressure or a craitent of various compounds in a mixture. All the transitions are divided into two groups, namely, first and second order transitions both accompanied by interesting pre-transitional phenomena and usually described by the Landau (phenomenological) theory or molecular-statistical approach. In this chapter we are going to consider the most important phase transitions between isotropic, nematic, smectic A and C phases. The phase transitions in ferroelectric liquid crystals are discussed in Chapter 13. 

Maier and Saupe [13] developed a statistical theory to describe the liquid crystalline state and the molecular ordering for the nematic phase. In analogy to the treatment of ordering phenomena in ferromagnetics or ferroelectrics, this theory describes the intermolecular orientational forces by a mean field method. Each individual molecule feels a nematic potential D = f (0, S, V) which depends on the momentaneous angle 6 between its long axis and the optic axis, the order parameter S and the molar volume V. S is then given by... 

The majority of the existing molecular theories of nematic liquid crystals are based on simple uniaxial molecular models like sphe-rocylinders. At the same time typical mes-ogenic molecules are obviously biaxial. (For example, the biaxiality of the phenyl ring is determined by its breadth-to-thick-ness ratio which is of the order of two.) If this biaxiality is important, even a very good statistical theory may result in a poor agreement with experiment when the biaxiality is ignored. Several authors have suggested that even a small deviation from uniaxial symmetry can account for important features of the N-I transition [29, 42, 47, 48], In the uniaxial nematic phase composed of biaxial molecules the orientational distri-... 

The principal elastic constants for a nematic liquid crystal have already been defined in Sec. 5.1 as splay (A , j), twist(/ 22) and bend(fc33). In this section we shall outline the statistical theory of elastic constants, and show how they depend on molecular properties. The approach follows that of the generalised van der Waals theory developed by Gelbart and Ben-Shaul [40], which itself embraces a number of earlier models for the elasticity of nematic liquid crystals. Corresponding theories for smectic, columnar and biaxial phases have yet to be developed. 

Lopatina and Selinger recently presented a theory for the statistical mechanics of ferroelectric nanoparticles in liquid crystals, which explicitly shows that the presence of such nanoparticles not only increases the sensitivity to applied electric fields in the isotropic liquid phase (maybe also a possible explanation for lower values for in the nematic phase) but also 7 N/Iso [327]. Another computational study also supported many of the experimentally observed effects. Using molecular dynamics simulations, Pereira et al. concluded that interactions between permanent dipoles of the ferroelectric nanoparticles and liquid crystals are not sufficient to produce the experimentally found shift in 7 N/ so and that additional long-range interactions between field-induced dipoles of nematic liquid crystal molecules are required for such stabilization of the nematic phase [328]. 

The simplest molecular theory of the nematic-isotropic (N-I) transition can be developed in the mean-field approximation. According to the general definition, in the mean-field approximation one neglects all correlations between different molecules. This is obviously a crude and unrealistic approximation but, on the other hand, it enables one to obtain very simple and useful expressions for the free energy. This approximation also appears to be sufficient for a qualitative description of the N-I transition. More precise and detailed theories of the nematic state are based on more elaborate statistical models that will be discussed briefly in Sec. 2.4.3. 

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