Biaxial order tensor

We now discuss the orientational order of the LCPs 3,4 and low molecular wei t analogues 6-8 in terms of the familiar oder parameters 22 and Sxx — Syy [34]. They express the ordering of the molecular axis Z and the anisotrc of the orientational order, respectively. Angular dependent lineshapes of the polymers indicate that each repeating unit can be characterized by an order tensor axialfy symnKtrk along Z [10,95], Thus, within experimental error — Syy = 0 for these tems, in contrast to similar combined LCPs, for which a small molecular biaxiality has been observed [114,115]. 

The Cartesian tensor representation can be extended to describe the orientational ordering of biaxial molecules in biaxial phases by introducing [6] a fourth rank ordering tensor ... 

In 1980, the first justified claim for the discovery of a biaxial nematic phase was made for a lyotropic liquid crystal comprised of the ternary system of potassium laurate-l-decanol-D20 [4], In addition to a uniaxial phase (micelles of a bilayer structure), there were two further nematic phases. One of the phases was found to be uniaxial as well, probably corresponding to a phase with cylindrical micelles. Existing in a temperature range in between these two uniaxial nematic phases was a third phase which was found to be biaxial. The phases were classified by microscopic smdies as well as deuterium NMR measurements. Three years later, Galerne and Marcerou studied the same system by conoscopy, leading to a complete determination of the ordering tensor in all three nematic phases [5]. 

To illustrate further the role of the orientational order parameters in a biaxial nematic we consider the case of the second rank ordering tensor, < D >, both because it is available fl-om experiment and because it provides the simplest description of the ordering. In principle < > has twenty-five... 

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... 

In general, however, tensor Qij is biaxial but the biaxiaUty is small, on the order of yPo where is the length corresponding to nematic correlations. This correlation length may be found, for example, from the light scattering in the isotropic phase close to the transition to the nematic phase. Then, at each point, that is locally, the anisotropic part of dielectric susceptibility tensor is biaxial and traceless 8ei + 5e2 + 8e3 = 0 with 8e2 8E3. 

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 ... 

In the biaxial cholesteric phase, choosing the local coordinate with the x axis parallel to the long molecular axis, the tensor order parameter is... 

As noted earlier, Qa/9 may be equally well-defined in terms of other macroscopic properties such as the refractive index or dielectric tensor. However, the simple relation [Eq. (3.6)] cannot be expected to hold for the dielectric anisotropy Ae and electric polarizability aij. This is due to complicated depolarization effects caused by the relatively large near-neighbor electrostatic interaction. The internal field corrections [3.3] are necessary in the electric case. It has been shown that Qa can be used to describe orientational order both in uniaxial and biaxial phases. Furthermore, measurement of Qa/3 is particularly useful when description of flexible molecules using microscopic order parameters becomes problematic. Experimentally, both magnetic resonance and Raman scattering techniques [3.3] may be employed to monitor the orientational order of individual molecules and to determine microscopic order parameters. 

The Maier-Saupe theory assmnes high symmetry for molecules forming liquid crystals. In reahty, this is usually not the case and the theory has been extended [3.18] to lath-like molecules. The order parameter tensor S is given by Eq. (3.8) for a biaxial molecule in a uniaxial phase. In the principal axis x y z) system of 5, only two order parameters, Szz and D = Sxx — Syy, are needed, which are related to the Wigner matrices according to Eq. (2.43) ... 

Now the traceless part of the magnetic susceptibility is a sum of two terms proportional to two tensor order parameters of the biaxial nematic phase and Bap-... 

In the uniaxial nematic phase Bap=0 and the tensor order parameter Q p is uniaxial. By contrast, in the biaxial phase the order parameter Q p can be written as a sum of a uniaxial and a biaxial part ... 

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