Continuum theory elastic free energy density

One optical feature of helicoidal structures is the ability to rotate the plane of incident polarized light. Since most of the characteristic optical properties of chiral liquid crystals result from the helicoidal structure, it is necessary to understand the origin of the chiral interactions responsible for the twisted structures. The continuum theory of liquid crystals is based on the Frank-Oseen approach to curvature elasticity in anisotropic fluids. It is assumed that the free energy is a quadratic function of curvature elastic strain, and for positive elastic constants the equilibrium state in the absence of surface or external forces is one of zero deformation with a uniform, parallel director. If a term linear in the twist strain is permitted, then spontaneously twisted structures can result, characterized by a pitch p, or wave-vector q=27tp i, where i is the axis of the helicoidal structure. For the simplest case of a nematic, the twist elastic free energy density can be written as ... 

The initial research on electro-optic phenomena in side-chain polymer liquid crystals concentrated on systems that exhibited nematic phases so that a ready comparison could be made with low molar mass mesogens. Such measurements have established that electro-optic devices are feasible and have allowed elastic constants to be deduced from applications of the continuum theory. This theory, originally derived for low molar mass nematic liquid crystals, defines a relationship for the free energy density F in terms of the elastic constants (/ ) and the director n such that ... 

In the continuum theory of liquid crystals, the free energy density (per unit volume) is derived for infinitesimal elastic deformations of the continuum and characterized by changes in the director. To do this we introduce a local right-handed coordinate x, y, z) system with (z) at the origin parallel to the unit vector n (r) and x and y at right angles to each other in a plane perpendicular to z. We may then expand n (r) in a Taylor series in powers of x, y, z, such that the infinitesimal change in n (r) varies only slowly with position. In which case... 

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