Order parameters biaxial

The order parameters S for all three molecular axes or alternatively, the combination S plus D describe on the level of the first relevant polynomial term the orientational distribution of a rigid, non-cylindrical molecule in the uniaxial nematic phase. Additional order parameters come into play for biaxial phases (Straley, 1974). A concise overview on the concepts from statistical mechanics relevant to order parameters was given by Zannoni (1979). 

As observed earlier, the assumption that the molecule is cylindrically symmetric is clearly not valid for real systems, and consequently the use of a single order parameter is not adequate. Most molecules are lath-shaped and have a biaxial character. Therefore two order parameters are required to describe the uniaxial nematic phase composed of biaxial molecules. If are the principal axes of the molecule (C defining the molecular long axis), it is necessary to introduce an additional order parameter... 

The local order in a cholesteric may be expected to be very weakly biaxial. The director fluctuations in a plane containing the helical axis are necessarily different from those in an orthogonal plane and result in a phase biaxiality . Further, there will be a contribution due to the molecular biaxiality as well. It turns out that the phase biaxiality plays a significant role in determining the temperature dependence of the pitch. Goossens has developed a general model taking this into account. The theory now involves four order parameters the pitch depends on all four of them and is temperature dependent. However, a comparison of the theory with experiment is possible only if the order parameters can be measured. 

The applicability of homotopic theory becomes much less obvious for liquid crystal phases with more complicated order parameters such as biaxial nematics and cholesterics, which are both locally defined by three directors forming a tripod. This gives rise to a description of the line singularities in terms of the quaternion group, Q. This is particularly interesting because the quaternion group Q is non-Abelian, a property that... 

Because the equilibrium order in heterophase systems is characterized by only one nonzero degree of freedom of the order parameter tensor, the fluctuation modes of all five degrees of freedom are uncoupled. Due to the uniaxial symmetry of the phase the two biaxial modes are degenerate and so are the two director modes. If a nematic layer is bounded by walls characterized by a strong surface interaction and a bulk-like value of the preferred degree of order, the fluctuation modes /3j s are sine waves, and their relaxation rates may be cast into... 

Fig. 8.6. Profiles of the lowest (thick line) and of one of the highly excited modes of biaxial (P i,n) and director (P 2,n) fluctuations in the LC heterophase system in contact with (a) ordering and (b) disordering substrates. Dfished lines correspond to the mean-field scalar order parameter. In all cases T —5- Tni-...

In the bent-director structure the biaxiality can be neglected and the scalar order parameter corresponds to its bulk value at the temperature 0eff — 0+ (37t /4) due to elastic deformation of the director field the temperature of the hybrid system effectively increases [10]. The increase of the ef-... 

The interaction between a solid substrate and a phase boundary consists of three contributions corresponding to three non-degenerated fluctuation modes. The fluctuations of the degree of order and the biaxial fluctuations give rise to a short-range repulsion between the substrate and the phase boundary proportional to exp(—2drv/ ), where dw is the thickness of the wetting layer and or jv, i for order parameter and biaxial fluc-... 

Here Az/g stands for the splitting of the doublet at 0 = 0°. The above measurable quantities determine the orientational order parameter and the biaxiality ... 

In the following, and / q denote the order parameter fluctuations with respect to the nematic director parallel to the x and axis, respectively. The other three fluctuation modes are uncoupled and represent either director fluctuations and low j3 2 modes) or biaxial fluctuations, high /3 2 inodes. 

Fig. 8.10. Spatial dependence of the (a) lowest order parameter fluctuation mode and (b) director (solid) and biaxial (dotted) fluctuation modes in the biaxial structure close to the transition to the bent-director structure. Dashed lines correspond to the equilibrium profiles.

The general expressions for the DF and the order parameters of a biaxial film comprised of molecules with cylindrical symmetry are rather cumbersome [577]. For the TDMs perpendicular to the long molecular axis such as those for VasCH2 and VgCH2 stretching vibrations [517],... 

As a consequence, the refraction index component perpendicular to the director n is larger in case b than in case c, and the component n is smaller. Therefore, the optical anisotropy An = n — n i in case b is smaller. To take the new situation into account, two local order parameters are introduced for the uniaxial nematic phase, one is the same as discussed above for the longitudinal molecular axes (5 = 5 ), and the other describes the local order of the shortest molecular axes that is local biaxiality (D) ... 

Now the biaxial nematic phase has two order parameters Q and and, in general, three different phases can be distinguished, namely, isotropic (Q = Qi = 0), uniaxial nematic Q, Q i = 0) and biaxial nematic (01,02) phases. Note that biaxial molecules may form both biaxial and uniaxial phases the latter appear due, for instance, to free rotation of biaxial molecules around their long molecular axes. As to the uniaxial molecules, they may also form either uniaxial (as a rule) or biaxial phases the latter may be formed by biaxial dimers or other building blocks formed by uniaxial molecules. 

Finally, we can write the tensors of the orientational order parameter Qy in the rotating frame for locally uniaxial and biaxial cholesteric liquid crystal (ChLC) Uniaxial ChLC ... 

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