Mesoscopic Theory of Liquid Crystals

Abstract

Liquid crystals of uniaxial and biaxial molecules are considered in the framework of the mesoscopic description which is a general tool of continuum theory. A mesoscopic theory introduces the fields beyond hydrodynamics as additional variables of a configuration space, called mesoscopic space, on which the fields appearing in balances are defined. Besides the mesoscopic space, a mesoscopic distribution function is introduced which describes the distribution of the additional variables at each time and position. It is demonstrated, how the mesoscopic theory can be applied to liquid crystals, and how the Ericksen-Leslie theory and the alignment tensor theory of liquid crystals fit into the mesoscopic framework. general case in constitutive theory. As the following examples show, the additional fields describing complex materials are of various kinds: internal variables ! memory alloys order parameters ! phase transitions damage parameters ! fatigue problems Cosserat triads ! steel, microcrystallites, granular media conformation tensors ! polymers fabric tensors ! composite materials directors, alignment tensors ! liquid crystals.The orientational order in liquid crystals can be di¤erently characterized. Because of thermal fluctuations the molecules are not totally aligned, but have a certain distribution around a ''mean orientation'' which can be described by a normalized macroscopic director field. The name ''macroscopic'' originates from the fact that the macroscopic director field belongs to all molecules of a volume element, whereas a special molecule may be aligned di¤erently. So it is obvious to introduce a microscopic director which describes the alignment of a single molecule and which is different from the local macroscopic director in general.Besides the microscopic and the macroscopic director, other descriptions for alignment are in use which can be found in the following synopsis. Up to now there are five di¤erent phenomenological concepts suitable to describe liquid crystals nonmicroscopically. The first one is the well known Ericksen-Leslie theory [1, 2] whose balance equations are formulated by use of the macroscopic director mentioned above. But in fact this theory is not able to represent a change in the degree of orientational order [3]. In general we need at each point and time a distribution function for describing the macroscopic orientation of the fluid. Therefore the macroscopic director has to be redefined statistically.The second concept describes liquid crystals as micropolar media in the frame of a 3-director theory [4]. Instead of a balance equation for the macroscopic director, the spin balance is taken into account, but no microscopic concepts are introduced.The third concept [5] introduces besides the balance equations of a micropolar medium an additional field, called microinertia tensor field. This field, satisfying its own balance equation, is coupled to the spin balance. The form of this coupling causes all needle-shaped molecules of a volume element to always have the same angul...

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