The Landau-de Gennes model

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] ... 

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

Fig. 6.5 (a) Temperature dependence of the order parametru in the Landau-de Gennes model (B) and (C) are coefficients of the expansion. Tjvj Tc is experimental value of the isotropic—nematic phase transition temperature corresponding to equality of free energy densities for the two phases, (b) Experimental dependence of the order parameter for 5CB and the characteristic temperature points Tc, Tc and Tc defined in accordance with the model of panel (a)... 

The Landau-de Gennes model may also be considered for the isothermal pressure case. It leads to a relation analogous to (33)... 

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 ... 

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]... 

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-... 

In a recent work, isotropic-nematic-smectic A phase transitions in thermotropic liquid crystals were also induced by applying an electric field [140]. The liquid crystal investigated (a mixture of 8CB and lOCB) showed a first order isotropic to smectic A transition. When in the isotropic phase and near the spontaneous transition temperature, a field-induced first order transition was observed from a paranematic to a nonspontaneous nematic phase. For higher values of the applied electric field, another first order transition occurred from the nonspontaneous nematic to a phase exhibiting the same order as a smectic A phase. A phenomenological Landau-de Gennes model has been developed to describe these transitions [141],... 

In this model, the properties of the liquid crystal are described by a Landau-de Gennes free energy that contains two contributions first, a contribution of the form... 

In practice, to study the properties of P(r) and especially of f(x), it is convenient to use the correspondence existing between polymer theory and field theory (see Chapter 11). The Green s function (k, — k a) of the Landau-Ginzburg model (zero component field) is connected to the partition function of an isolated chain by the Laplace transform introduced by de Gennes. 

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