Landau-de Gennes theory

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 =... 

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. 

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

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... 

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... 

Equations (10.48) and (10.49) give a special form of the Landau-de Gennes theory equation (10.40) with the phenomenological parameters... 

Priestley, E.B., Sheng, P. Landau-de Gennes theory of liquid crystal phase transitions. In Priestley, E.B., Wojtowicz, R, Sheng, P. (eds.) Introduction to Liquid Crystals, Chapter 10, pp. 143-201. Plenum Press, New York (1975)... 

Landau-de Gennes theory of orientational order in nematic phase... 

There are a few points worth mentioning in Landau-de Gennes theory. It works well at temperatures near the transition. At temperatures far below the transition temperature, the order parameter increases without limit with decreasing temperature, and the theory does not work well because we know that the maximum order parameter should be 1. In Figure 1.6, the... 

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]... 

Figure 12.3. Order parameter as a function of temperature for a specific set of parameters in the Landau-de Gennes theory. T is the nematic-isotropic transition...

The Landau-de Gennes theory for the nematic isotropic transition can be extended to the smectic A-nematic transition. The order parameter for this transition is rl), the amplitude of the density wave describing the formation of layers in the smectic A phase. Since the difference between a value of rlr and -Irlrl only amonnts to a shift of one half layer spacing in the location of all the layers (and therefore no change in the free energy per nnit volume), the expansion in terms of powers of rlr can only contain even powers. Hence the free energy per unit volume in the smectic A phase can be written as... 

A. K. Sen and D. E. Sullivan, Landau-de Gennes theory of wetting and orientational transitions at a nematic liquid-substrate interface, Phys. Rev. A 35,1391 [1987). 

In the isotropic phase not too far from Tc, the molecules are still locally parallel to each other. Clearly, the mean values of all elements of the local order parameter tensor Qotp r ) are zero and this tensor describes local orientational fluctuations in the isotropic phase. The free energy density of the system in the Landau-de Gennes theory [6.28] is given by (neglecting the magnetic field term)... 

One of the most important theories for the nematic-smectic A phase transition is the Landau-de Gennes model. Another is the McMillan model, which was discussed in Section 5.5.2. The Landau-de Gennes theory is applied in the case of a second-order phase transition by combining a Landau expansion (Section 1.5) for the free energy in terms of an order parameter for smectic layering with the elastic energy of the nematic phase (Eq. 5.19). A suitable order parameter for the smectic structure allows both for the layer periodicity and the fluctuations of layer position (r) ... 

Finally, a few words about other liquid-crystal theories the mean-field theory of Maier and Saupe (1959, 1960) has been very successful in describing the behaviour of small-molecule liquid crystals, but it has been much less used for polymeric liquid crystals. Other important theories primarily applied to small-molecule liquid crystals are the Landau theory and its extension, the Landau--de Gennes theory. A detailed presentation of these theories, also including the Maier and Saupe theory, is found in Vertogen and de Jeu (1988). 

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-... 

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