Order Parameters, Irreducible Tensors and the theory of Phase Transitions in Smectic Liquid Crystals

We discuss how various types of orientational and

translational ordering in different liquid crystal phases are

described by macroscopic tensor order parameters. In

particular, we consider a mean-field molecular-statistical

theory of the transition from the orthogonal uniaxial smectic

phase and the tilted biaxial phase composed of biaxial

molecules. The relationship between macroscopic order

parameters, molecular invariant tensors and the symmetry of

biaxial molecules is discussed in detail. Finally we use

microscopic and macroscopic symmetry arguments to consider the

mechanisms of the ferroelectric ordering in tilted smectic

phases determined by molecular chirality.