Phase 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). 

On the other hand, it has been shown on LMWLCs that the well-known SmC, where the molecules are tilted with respect to the layer normal, is no longer the only possibility to obtain a fluid biaxial phase [63], As a consequence, a strict determination of the chiral smectic phase structure requires not only a careful analysis of the X-ray diagrams obtained on powder as well as on aligned samples, but also a study of the electrooptic response, which allows discrimination between the ferroelectric, the antiferro-electric, and the ferrielectric behavior. 

For the ideal nematic with sim = 0 and %- 1 there is no difference between cases b and c. The locally (microscopically) biaxial nematic phase should not be confused with macroscopically biaxial phases to be discussed in the next section. 

Fig. 3.21 Packing of molecules in a macroscopic nematic biaxial phase of symmetry D2h...

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. 

Fig. 4.11 The optical indicatrix (a) and the microscopic texture (b) of the SmC biaxial phase...

Fig. 4.41 Structure of single non-polar (a) and polar (b) smectic layers formed by bent-shape molecules the longitudinal axes are aligned upright and the plane of the figure is mirror plane. Possible polar three-dimensional smectic biaxial phase (c)...

From this ellipsoid we can find y/z for any direction specified by radius-vector r, see the figure. For example such an ellipsoid corresponds to the biaxial phase of the SmC liquid crystal. In this case all the three semi-axes are different i = 2 = ... 

Liquid crystals can be composed both of polar and apolar molecules. An important fact in connection with polar substances is that in uniaxial phases there is no polar ordering of the molecules. In average the dipole moments aligned in a given direction are compensated by those aligned in the opposite direction. As a consequence no spontaneous macroscopic polarization develops. More generally one can state that rotation of the director by n does not affect the physical state of the liquid crystal. In biaxial phases built of chiral molecules, such as the chiral smectic C phase, the situation is different. In these systems the compensation of the dipole moments is not perfect, a macroscopic polarization appears in the direction perpendicular both to the layer normal and the director. These phases are therefore ferroelectric. Ferroelectric liquid crystals are currently perhaps the... 

Nematic A uniaxial or biaxial phase possessing long-range... 

Scientists have investigated uniaxial-biaxial phase transition for nematic liquid crystal polymers and have tried to describe it through the order parameters and also by considering the terms that account for the energy of elastic deformation and the... 

Corresponding to the different micellar structures and intrinsic order of nematic phases there are three types of chiral nematics the two uniaxial phases ( Nj)) and N, as well as Ng, a biaxial phase. In Figure 14.7 the customary models of the micellar arrangement of the uniaxial phases are sketched. 

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. 

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 ... 

The quantity V X ttax refers to the anisotropy of the tensor for a fully aligned state for which the order parameter is one. For a biaxial phase (i.e. a phase which has different properties along each of the three principal axes), the macroscopic order parameter in principal axes can be written as ... 

If the Saupe ordering matrix is written in terms of the laboratory axis frame, but assuming now that the molecules are uniaxial, then phase biaxiality can be described in terms of the order parameter P, which is nonzero for tilted smectic phases and other intrinsically biaxial phases. For example the diagonal ordering matrix for the molecular long axis z can be written as ... 

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 ... 

If the liquid crystal phase is biaxial, as with SmC phases, then any second rank tensor property has three independent principal components and the average polarizabilities corresponding to the three refractive indices can be expressed in terms of the orientational order parameters introduced for biaxial phases ... 

The principal elastic constants for a nematic liquid crystal have already been defined in Sec. 5.1 as splay (A , j), twist(/ 22) and bend(fc33). In this section we shall outline the statistical theory of elastic constants, and show how they depend on molecular properties. The approach follows that of the generalised van der Waals theory developed by Gelbart and Ben-Shaul [40], which itself embraces a number of earlier models for the elasticity of nematic liquid crystals. Corresponding theories for smectic, columnar and biaxial phases have yet to be developed. 

The second and third order invariants can be regarded as independent in the biaxial phase with the constraint <7 [Pg.315]

The complete phase diagram is obtained with the selection rule (tI (7 and reproduced in Fig. 1 [1]. Note that a direct transition I-Nb occurs at a single point a = b = 0 on the line (Eq. 10) [7]. The biaxial nematic region separates two uniaxial nematics N+ and N of opposite sign. N -Nb transitions are second order since the condition 2 = (7 can be approached continuously from the biaxial phase. 

Figure 7. Mean field phase diagrams with biaxial phases, (a) For weak incommensurability, the antiphase SmA shows up in between the SmA, and SmA2 phases. The SmAj-SmA transition is second order in mean field but first order with fluctuations [47].

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