Landau nematics

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] ... 

Unlike ferromagnets, where G((p) = G(-(f), the free energy of nematic liquid crystals exhibits an asymmetry G(S) G(-5), as discussed previously and shown in Fig. 3b. To reproduce this behavior, a term that is an odd power of the order parameter S is needed in the Landau expression for the free energy of a nematic liquid crystal. A term linear in 5 is not allowed since the equilibrium condition dQdS = 0 could not then be satisfied in the disordered isotropic phase where 5=0. The presence of a cubic term will lead to the desired asymmetry in G as a function of 5 and to the emergence on cooling of a second minimum in G at a finite 5 value. 

Thus, one has for the Landau free energy of a nematic liquid crystal... 

The simulation techniques presented above can be applied to all first order phase transitions provided that an appropriate order parameter is identified. For vapor-liquid equilibria, where the two coexisting phases of the fluid have the a similar structure, the density (a thermodynamic property) was an appropriate order parameter. More generally, the order parameter must clearly distinguish any coexisting phases from each other. Examples of suitable order parameters include the scalar order parameter for study of nematic-isotropic transitions in liquid crystals [87], a density-based order parameter for block copolymer systems [88], or a bond order parameter for study of crystallization [89]. Having specified a suitable order parameter, we now show how the EXEDOS technique introduced earlier can be used to obtain in a particularly effective manner for simulations of crystallization [33]. The Landau free energy of the system A( ) can then be related to P,g p( ((/"))... 

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... 

Non-grafted boehmite rods experience a much more complicated interaction potential as positive surface electrical charges now play an important role. In this case, the phase diagram has to be discussed in the frame of both the Onsager model of nematic ordering and the DLVO (named after B.V. Deryagin, L. Landau, E.J.W. Verwey, and J.T.G. Overbeek) theory of colloidal stability, which describes colloidal stabihty as a balance between repulsive electrostatic and attractive van der Waals interactions [67,68]. At low ionic strength, electrostatic repulsion dominates so that the phase stability is essentially described by the... 

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

Johnson D, Allender D, Dehoff D, Maze C, Oppenheim E and Reynolds R 1977 Nematic-smectic A-smectic C polycritical point Experimental evidence and a Landau theory Phys.Revs B 16 470-5... 

Chen J-H and Lubensky TCI 976 Landau-Ginzburg mean-field theory for the nematic to smectic C and nematic to smectic A phase transitions Phys.Rev. A 14 1202-7... 

The occurrence of the BPs can also be described in terms of the Landau theory. " The free energy expansion contains the usual nematic terms... 

If Sc undergoes a transition directly to the nematic phase, 9 is generally found to be temperature independent and usually about 45°. According to the Landau rules, the C-N transition can be continuous, but when fluctuations are taken into account it is predicted to be of first order. Experimentally, only first order C-N transitions have been observed. Some compounds exhibit transitions from Sg to the isotropic phase. Interestingly, a slight increase of 9 with increasing temperature has been reported for two such compounds. ... 

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]. 

By making use of the form of the Landau energy proposed by de Gennes [17], all coefficients in this expression have been evaluated from experimental data [4]. Using these data to calculate U, one obtains a value that is considerably smaller than the one determined from the linearized analysis outlined above. As has been noted, however, a consistent analysis should not only include terms quadratic in the strains and bilinear coupling terms of the order parameter and the strain, but rather also nonlinear effects as well as nonlinear coupling terms between strain and the nematic order parameter [4]. 

As the experimental investigations have focused predominantly on static properties of the nematic-isotropic transition, most of the theoretical papers have used a Ginzburg-Landau description involving an expansion in the nematic order parameter to describe static properties [4, 17, 22-25]. 

The polydomain-monodomain transition in nematic LCE was investigated [26] by incorporating a local anchoring interaction into the Ginzburg-Landau description of the nematic-isotropic transition in LCE [4, 17,25]. 

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]. 

Fig. 8.2. (a) A sketch of the Landau free energy of a typical liquid crystalline material exhibiting a nematic-isotropic transition. Tjvi is the transition temperature and T and T are the temperatures below/above which the corresponding metastable phase can not exist anymore. The dotted line coresponds to the temperature in the nematic phase where the isotopic phase is metastable, thus, a local minimum of the free energy at Qo = 0. As a comparison, the free energy of a system with a continuous phase transition is depicted in (b). Here the transition occurs at Tc and no inetastable solutions are possible. 

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... 

Due to the effect of external fields, the order can vary in space and gradient terms have to be added to the Landau expansion (8.9). Usually, only the terms up to the quadratic order are considered. There are many symmetry allowed invariants related to gradients of the tensorial order parameter [29]. However, in the vicinity of the phase transition, one is not interested in elastic deformations of the nematic director but rather in spatial variations of the degree of nematic order. Therefore, the pretransitional nematic system is described adequately within the usual one-elastic-constant approximation. 

The variation of order parameter with temperature is shown in Figure 18. It gives the equilibrium order parameter of an undistorted nematic for a given set of Landau-de Geimes coefficients. The discontinuity is clearly visible. 

Equations (10.38) and (10.39) give a nonlinear integro-differential equation for W, and its mathematical handling is not easy. A guidance of how to proceed is obtained from the phenomenological theory in nematics. De Gennes showed that the dynamics of nematics is essentially described by the Landau theory of phase transition and proposed a phenomenological nonlinear equation fof the order parameter tensor... 

We can imagine a cholesteric as a smck of nematic quasi-layers of molecular thickness a with the director slightly turned by ( ) from one layer to the next one. In fact it is Oseen model [18]. Such a structure is, to some extent, similar to lamellar phase. Indeed, the quasi-nematic layers behave like smectic layers in formation of defects, in flow experiments, etc. Then, according to the Landau-Peierls theorem, the fluctuations of molecular positions in the direction of the helical axis blur the one-dimensional, long-range, positional (smectic A phase like) helical order but in reality the corresponding scale for this effect is astronomic. 

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. 

---