H. Ehrentraut and W. Muschik
Balance Laws and Constitutive Equations of Microscopic Rigid
Bodies: A Model for biaxial liquid crystals
Mol. Cryst. Liq. Cryst. 262, 1995, 561-568
A reasonably general model of a liquid crystal is achieved by a
continuum consisting of microscopic rigid bodies. Within this model it
is possible to deal with chiral molecules as well as simple rod-like
particles forming a conventional nematic liquid crystal. The
configuration space of a rigid body is the rotation group SO(3); the
configuration space of a "nematic" is - with regard to the head-tail
symmetry - the projective plane P(2). By replacing both manifolds
by their universal coverings (Sü and Sı, resp.)
the internal symmetry
of the fluid can be represented by a generalized (non-normalized)
director. Within this mathematical framework mesoscopic balance
equations are formulated which are applicable in the
biaxial case of chiral molecules and in the uniaxial case of rod-like
particles. Finally it is shown how constitutive equations can be derived by
calculating averages of mesoscopic quantities with respect to an
orientation distribution function which characterizes the
orientational order of the liquid crystal.