Nematic liquid crystals Landau theory

The Landau theory assumes that the order parameter is small in the vicinity of the transition, so that only the lowest terms required by symmetry and preventing the free energy from diverging are kept in the expansion. In the case of nematic liquid crystals, the order parameter is a tensor and its scalar invariant is its trace. Thus, the Landau free energy reads... 

A qualitative picture, Fig. 10.4, shows the distance dependencies of the orientational order parameter for homeotropically aligned nematic liquid crystal at the solid substrate. The problem is to explain such dependencies [6]. The influence of the surface on the orientational order parameter may be discussed in terms of the modified Landau-de Gennes phase transition theory. Consider a semi-infinite nematic of area A being in contact with a substrate at z = 0 and uniform in the x and y directions. When writing the free energy density a surface term -W8(z)S must be added to the standard expansion of the bulk free energy density ... 

Besides aligning liquid crystals, external electric fields can also change the orientational order and thus the electro-optical properties of liquid crystals. When the long molecular axis of a liquid crystal molecule, whose anisotropy of polarizability is positive, is parallel to the applied field, the potential of the molecule is low. Thus the applied field suppresses the thermal flue-mation and increases the order parameter. Now we discuss how the orientational order of a nematic liquid crystal changes with applied fields. Using the Landau-de Gennes theory, the free energy density of a liquid crystal in an electric field (when the liquid erystal director is parallel to the field) is [4]... 

The phase transition from the isotropic liquid to the nematic liquid crystal in frame of the Landau theory was first described by de Gennes, and is usually referred as Landu-de Gennes theory. The first step of this theory is to find the right order parameter. 

Liquid crystals manifest a number of transitions between different phases uprm variation of temperature, pressure or a craitent of various compounds in a mixture. All the transitions are divided into two groups, namely, first and second order transitions both accompanied by interesting pre-transitional phenomena and usually described by the Landau (phenomenological) theory or molecular-statistical approach. In this chapter we are going to consider the most important phase transitions between isotropic, nematic, smectic A and C phases. The phase transitions in ferroelectric liquid crystals are discussed in Chapter 13. 

Theoretical treatments of liquid crystals such as nematics have proved a great challenge since the early models by Onsager and the influential theory of Maier and Saupe [34] mentioned before. Many people have worked on the problems involved and on the development of the continuum theory, the statistical mechanical approaches of the mean field theory and the role of repulsive, as well as attractive forces. The contributions of many theoreticians, physical scientists, and mathematicians over the years has been great - notably of de Gennes (for example, the Landau-de Gennes theory of phase transitions), McMillan (the nematic-smectic A transition), Leslie (viscosity coefficients, flow, and elasticity). Cotter (hard rod models), Luckhurst (extensions of the Maier-Saupe theory and the role of flexibility in real molecules), and Chandrasekhar, Madhusudana, and Shashidhar (pre-transitional effects and near-neighbor correlations), to mention but some. The devel-... 

The situation with respect to Landau-type phenomenological theories is also contradictory. Drawing an analogy between the smec-tic-A phase of liquid crystals and the superconducting phase of metals, de Gennes22 23 constructed a phenomenological theory from which he concludes that the smectic-A to nematic phase transition can be second order. Halperin and Lubensky, on the other hand, have improved the analogy with superconductors and conclude that the transition will always be at least weakly first order. 

The Rayleigh ratio has been measured in the isotropic phase of both nematic and cholesteric liquid crystals. Agreement between these measurements and the predictions of the Landau-de Gennes theory is excellent in both cases. 

Figure 10.26 shows the temperature dependence of the relaxation time of the fluctuations. It can be seen that it is proportional to the temperature difference firom the phase transition temperature, increases divergently, and shows a critical slowing down, which fits well with the prediction of the Landau theory. In the case of liquid crystal shown in the example, the critical temperature T, estimated by linear extrapolation of the data points, is approximately 1 K below the isotropic-nematic phase transition temperature. 

Such molecrrlar-stmcture-based approaches are clearly extremely complex and often tend to yield contradictory predictions, because of the wide variation in the molecrrlar electronic stmctures and intermolecirlar interactions present. In order to explain the phase transition and the behavior of the order parameter in the vicinity of the phase transition temperature, some simpler physical models have been employed. For the nematic phase, a simple but qirite successful approach was introduced by Maier and Saupe. The liquid crystal molecirles are treated as rigid rods, which are correlated (described by a long-range order parameter) with one another by Corrlomb interactions. For the isotropic phase, deGennes introduced a Landau type of phase transition theory, which is based on a short-range order parameter. 

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