Sketch of the mesoscopic description of nematic liquid crystals

Abstract

Liquid crystals are considered in the framework of the mesoscopic description. This mesoscopic approach introduces a more detailed description of the microstructure into a macroscopic theory. Especially in liquid crystal theory the macroscopic director, which represents an averaged orientation in a volume element about a given point in a liquid crystal, is replaced by a microscopic director, describing the orientation of a single uniaxial molecule. Therefore, in the mesoscopic theory the macroscopic director is a derived quantity in contrast to the usual director theories. The orientation of all microscopic directors in a given volume element is described by an orientation distribution function (ODF). This distribution function makes it possible to derive a family of alignment tensors. The macroscopic field of the alignment tensor of second order now replaces the macroscopic director, thus resulting in a more detailed description of the orientation. This first approximation by the alignment tensor of second order can be improved by taking into account alignment tensors of higher order. The case of total alignment of all molecules in a volume element corresponds to the Ericksen–Leslie theory of liquid crystals. In this special case, the macroscopic director is identical to the microscopic one. In the general case, beyond liquid crystal theory, a mesoscopic theory introduces additional mesoscopic variables forming a configuration space, called the mesoscopic space, on which the fields appearing in the balances of the theory are now defined. The orientation distribution function is now replaced by the mesoscopic distribution function, which describes the distribution of the mesoscopic variables at each time and position.