Landau-de Gennes

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

The layer and director structure in the bent-core liquid crystal phases can be studied by the Landau-de Gennes type model [32, 34, 35], The layer and director structure is such that the free energy (F = J/dV) has the minimum value. The free energy density (f) is written in terms of ... 

A Landau-de Gennes theory for the Frank constants of long semiflexible worm-Uke chains at low order parameter S gives (Shimada et al. 1988) K = K7, = 0K2 =... 

Landau-de Gennes-t5q)e approach [11] is used. The condensation bulk free-... 

Substituting max into the equation, we obtain the free energy which is a function of the nematic coupling a, the chain rigidity (l/lo) and the temperature T. The transition temperature can be obtained from the form of free energy. The form of free energy is rather complicated. We apply the Landau-de Gennes theory to analyze it. 

Landau used a very simple form of free energy to describe systems such as the para-magnetic system where the magnetic polarization M is the order parameter. The idea has been extended by de Gennes (1973) to deal with nematic liquid crystals. The free energy is given in the extended Landau-de Gennes form as... 

Figure 2.17. Free energy vs. order parameter S for various temperatures according to Landau-de Gennes theory.

Expanding the free energy in Equation 2.89 to the fourth power gives the Landau-de Gennes form. One can evaluate the thermodynamic properties accordingly. 

It is worth pointing out that after expanding the free energy in the Landau-de Gennes form and taking into account the entropic contribution of the free ends of the chain (contrary to circle polymers) and the n = 2 term in... 

We now discuss the orientational dynamics of mesogens. The orientational dynamics in the isotropic phase of thermotropic liquid crystals near the I-N transition have drawn much attention over the years [5-7, 33-40]. The focus was initially on the verification of the Landau-de Gennes (LdG) theory, which predicts a long-time exponential decay with a strongly temperature-dependent... 

There exist pre-transition effects in the isotropic phase heralding the I-N phase transition. Such pre-transition effects, which are consistent with the weakly first-order nature of the I-N transition, can be attributed to the development of short-range orientational order, which can be characterized by a position-dependent local orientational order parameter Q(r), where all component indices have been omitted [2]. In the Landau approximation, the spatial correlation function < G(0)G(r) > has the Omstein-Zemike form < G(0)G(r) exp(—r/ )/r, where is the coherence length or the second-rank orientational correlation length. The coherence length is temperature-dependent and the Landau-de Gennes theory predicts... 

As per the prescription of Li et al., a simple way to get a combined theory is to assume that the total memory function of the correlator of interest is the sum of the mode coupling memory function and the Landau-de Gennes memory... 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model... 

For an experimental check-up of the theoretical considerations about liquid-crystalline elastomers in a mechanical field, Fin-kelmann and coworkers [107, 123] studied, in nematic networks, the evolution of the order parameter and of the transition temperature as a function of the stress. The observed results are in full agreement with the predictions of the Landau-de Gennes theory, since an increasing clearing temperature as well as an increasing order parameter are observed with increasing stress. From their results, it was possible to estimate the crosscoupling coefficient U (see Sec. 3.1.1) between the order parameter and the strain of a nematic elastomer [123]. 

The calculated van der Waals interaction is presented with a dashed line and is nearly temperature independent. On the other hand, it can be clearly seen that the total force is temperature dependent, which can only be a consequence of an additional nematic mean-field contribution. The solid line is a sum of the van der Waals and a nematic mean-field force, derived from the Landau-de Gennes theory. The agreement is quantitatively good and gives us the strengths of the two surface coupling coefficients, which are in the case of DMOAP quite large, i.e. wi = 1.4 x 10 " (1 0.4) J/m and W2 = 7x 10 (1 0.3) J/m [13]. 

The simplest model used to explain the temperature dependence of (Ap) is based on the Landau-de Gennes theory of the isotropic phase. Sluckin and Poniewierski added two surface terms to the free energy density [26]... 

Fig. 3.8. Presmectic interaction in 8CB at three different temperatures above Tni A solid line represents a fit with the Landau-de-Gennes theory (3.6).

Macroscopic Models Phenomenological Landau—de Gennes Theory... 

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