H. Ehrentraut

Statistical foundation of macroscopic balances for liquid crystals in alignment tensor formulation

Starting out with the global balance equations of mass, momentum, angular momentum, and energy formulated on the so-called ten-dimensional doubled phase of position, velocity, orientation, and orientation change velocity, the appropriate local balances are derived, which are defined on the five-dimensional half of the doubled phase space including time, position, and the microscopic director. These so-called orientation balance nematic liquid crystals whose alignment need not be uniform as it is presupposed in theories using macroscopic director fields. In R3 we get the usual phenomenological balance equations of micropolar media having the advantage that the balanced quantities are defined statistically. By expanding the orientation distribution function into a series of multipoles we get alignment tensor fields and an additional alignment tensor balance equation on R3.

Orientation-balances for liquid crystals and their representation by alignment tensors

Abstract Up to now there are three different phenomenological concepts suitable to describe liquid crystals: The first one is the well known Ericksen-Leslie theory4 whose balance equations are formulated by use of a macroscopic director for needleshaped molecules with totally relaxed internal degrees of freedom. This macroscopic director is not well defined microscopically: It can not be identified with the orientation of a single molecule because the orientation of the molecules is statistically distributed. Therefore we need a distribution function for describing the orientation in the fluid, or we restrict ourselves to the case of total alignment. In that case the macroscopic director of the Ericksen-Leslie theory coincides with the common alignment of all molecules.

Mesoscopic theory of liquid crystals

The structural roots of the mesoscopic theory of liquid crystals are reviewed and shortly discussed.

Concepts of Mesoscopic Continuum Physics With Application to Biaxial Liquid Crystals

Abstract The mesoscopic concept in continuum mechanics consists of extending the domain of the balance equations by the set of mesoscopic variables and of introducing a local distribution function of these variables as a statistical element. The balance equations defined on this extended domain and an example concerning liquid crystals of biaxial molecules are discussed.

A continuum theory for liquid crystals describing different degrees of orientational order

Abstract Starting out with mesoscopic orientational balance equations for each orientational component of a liquid crystal which is described as a formal mixture, a set of independent macroscopic variables, the state space Z, is induced This set includes a second-order tensorial measure of alignment, called the alignment tensor A, and its derivatives. In terms of these state space variables constitutive equations are proposed by exploiting the dissipation inequality due to Coleman and Noll. The constitutive equations around equilibrium are investigated. The results are compared in the case of total alignment to those of Ericksen and Leslie, who described the alignment in a liquid crystal with only a macroscopic unit director field d(x, t) indicating the ‘mean orientation’ of the media. In a recent paper Ericksen introduced beside the macroscopic director an additional scalar order parameter S(x, t) and its derivatives (Maier-Saupe theory) which turns out to be the uniaxial case in the alignment tensor for...

Ericksen-Leslie Liquid Crystal Theory Revisited from a Mesoscopic Point of View

Starting out with the mesoscopic concept of liquid crystal theory it is proved that the Ericksen-Lesiie theory is valid in two cases: The molecules form a time-independent global planar phase or they are local totally aligned.

Balance Laws and Constitutive Equations of Microscopic Rigid Bodies: A Model for Biaxial Liquid Crystals

Abstract 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 (S3 and S2, 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 ...

The Connection Between Ericksen-Leslie Equations and the Balances of Mesoscopic Theory of Liquid Crystals

Abstract Ericksen-Leslie equations can be derived, if the mesoscopic local balance equations are exploited by use of the orientation distribution function for total alignment.

Nonlinear dynamics of alignment tensor in the presence of electromagnetic fields

An equation of motion for the second order alignment tensor, including the influence of electromagnetic fields is derived. The starting point are the mesoscopic balance equations of mass and of angular momentum. A constitutive equation on the mesoscopic level is discussed. The two balance equations together with the constitutive equation yield a Fokker-Planck type equation of motion for the ODF. This finally leads to a nonlinear partial differential equation for the alignment tensor.