Self-consistent anisotropic field theory

Chapters 1 and 2 introduce the main phases and basic properties of liquid crystals and other anisotropic fluids, such as soaps, foams, mono-layers, fluid membranes and fibers. These chapters do not include difficult mafliematical formulas and are probably suitable for imdergraduates or for other professionals, such as K-12 teachers. Chapter 3 describes the nature of phase transitions based on the phenomenological Landau-de Gennes theories, and on the self-consistent mean-field theories that use concepts in statistical physics. 

There are several levels of approximation possible in the consideration of the NA transition. First there is the self-consistent mean field formulation due to Kobayashi and McMillan [8-10]. This is an extension to the smectic-A phase of the self-consistent mean-field formulation for nematics ( Maier-Saupe theory [11]). Kobayashi-McMillan (K-M) theory takes into account the coupling between the nematic order parameter magnitude S with a mean-field smectic order parameter. In Maier-Saupe theory, the key feature of the nematic phase - the spontaneously broken orientational symmetry - is put in by hand by making the pair potential anisotropic. In the same spirit, the K-M formulation puts in by hand a sinusoidal density modulation as well as the nematic-smectic coupling. 

This statistical theory includes contributions from an attractive intermolecular potential. The anisotropic attraction stabilizes the parallel alignment of neighboring molecules, and the theory then considers a mean-field average of the interaction. Solved self-consistently, this theory predicts thermotropic phase transitions consistent with the experiment. 

The basic idea of these theories is to look at the distribution of conformations of a chain molecule attached to the surface. The conformational probability distribution function is written in terms of the non-local interaction field induced by the other chain molecules. This field is anisotropic, i.e., it depends on the direction perpendicular to the surface, because the presence of the surface and the inhomogeneous variation of the density of polymer segments and solvent molecules as a function of the distance from the surface. The non-local mean-field is determined by packing constraints that take into account the fact that the volume (at all distances from the surface) must be filled by polymer segments or solvent molecules. These self-consistent criteria represent the incompressibility assumption at all distances from the surface. 

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