Generalization of the Landau Expansion to Liquid Crystals

We must first identify a suitable order parameter for describing liquid crystal phase transitions. In previous chapters, we have seen that for nematic and cholesteric liquid crystals, the molecular ordering at every spatial point r (where the term point has the meaning defined in Section (2.1)) is characterized by a director h r) pointing along the local axis of uniaxial symmetry, and by a quantity Six) giving the local orientational order of the rod-like molecules. S(r) is defined by 

Let us now consider the Landau free-energy density expression for liquid crystals. All isotropic-nematic (cholesteric) phase transitions 

[27] can be put into a more physically interpretable form by substituting Eq. [26] for Qij f) and noting that 

There are four types of terms in Eq. [30], The first four terms concern only the value of the orientational order S(r). The next two terms account for spatial variation of S(r). Next there is a term concerned with the spatial variation of h(r) we have expressed this term in the familiar form of splay, twist, and bend distortions of the director field n(r). It should be noted that to second order in the Landau expansion there are only two independent elastic constants, L and L2, whereas in the nematic phase there are known to be three independent elastic constants. The last two terms in Eq. [30] represent the interaction between spatial variations of S(f) and spatial variations of n(r). Clearly, the mathematics can be quite complicated if S(r) and n(r) are allowed to vary simultaneously. 

Let us now examine the expansion coefficients B, C, Li, and L2 in more detail. We will consider first the spatially uniform state which can be described by the first four terms of Eq. [30]. Following Landau, we choose C 0, which requires that 0 if the equilibrium value of S is to be positive in the low-temperature phase. We illustrate the behavior of as a function of S for a spatially uniform system in Fig. 4. Next we consider a state for which S(f) is constant but h(r) is allowed to vary from point to point. Since the spatially uniform state must be stable against any distortion, it is clear from Eq. [30] that 

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