2. Liquid crystalline phases

The liquid crystal phase is a well-defined and specific phase of matter characterized by a remarkable anisotropy in many of their physical properties as solid crystals do, although they are able to flow. Liquid crystal phases that undergo a phase transition as a function of temperature (thermotropics) exist in relatively small intervals of temperature lying between solid crystals and isotropic liquids. Due to this intermediate nature, sometimes, these states are called also mesophases [32]. In general, liquid crystals are synthesized from organic molecules, some of which are elongated and uniaxial, so they can be represented as rigid rods; others are formed by disc-like molecules [35]. This molecular anisotropy is manifested macroscopically through the anisotropy of the mechanical, optical, and transport properties of these substances. The typical dimensions of the lengths of this type of structures are some tens of angstroms.

Liquid crystals are classified by symmetry. As it is well known, isotropic liquids with spherically symmetric molecules are invariant under rotational, Oð Þ3 , and

### Figure 1. Representation of the average orientation of the molecules of a thermotropic NLC by means of the director vector n^.

Equilibrium and Nonequilibrium Hydrodynamic Modes of a Nematic Liquid Crystal DOI: http://dx.doi.org/10.5772/intechopen.82609

translational, Tð Þ3 , transformations. Thus, the group of symmetries of an isotropic liquid is Oð Þ� 3 Tð Þ3 : However, by decreasing the temperature of these liquids, the translational symmetry Tð Þ3 is usually broken corresponding to the isotropic liquid-solid transition. In contrast, for a liquid formed by anisotropic molecules, by diminishing the temperature, the rotational symmetry Oð Þ3 is broken, which leads to the appearance of a liquid crystal. The mesophases for which only the rotational invariance has been broken are called nematics. As shown, the centers of mass of the molecules of a nematic have arbitrary positions, whereas the principal axes of their molecules are spontaneously oriented along a preferred direction. If the temperature decreases even more, the symmetry Tð Þ3 is also partially broken. The mesophases exhibiting the translational symmetry Tð Þ2 are called smectics [36].

This preferential direction is described by a local unitary vector field, n^, called the director. This vector is easily distorted by the presence of electric and magnetic fields, as well as by the surfaces of the containers of the liquid crystals if they have been prepared properly [32]. With respect to NLC, it is important to point out that the director's orientation does not distinguish between the n^ and �n^ directions (nematic symmetry). A schematic representation of the order presented by the molecules in a nematic is shown in Figure 1 .
