Directors biaxial nematics

An interesting similarity of what we discussed here appears if one deals with mixtures of rodlike and disklike micelles. These systems could behave very similarly to a truly biaxial nematic, but show interesting differences to them. Whereas for the usual orthorhombic biaxial nematics both directors are perpendicular to each other by construction, in mixtures there is no need to impose this restriction. Pleiner and Brand [70] investigated how mixtures are influenced by an external field (magnetic field or shear flow) and found that the angle between the two directors exhibits a flow aligning behavior similar to the one studied in [42,43],... 

In the simplest liquid-crystalline phase, namely the uniaxial nematic, there is at rest a special direction designated by a unit vector n called the director (see Fig. 10-2). In the plane transverse to the director, the fluid is isotropic. The most common nematics are composed of oblong molecules that tend to point in a common direction, which defines the director orientation. Oblate, or disc-like, molecules can also form uniaxial nematics for these discotic nematics, the director is defined by the average orientation of the short axis of the molecule. Lath-like molecules or micelles (shaped like rectangular slabs), in which all three dimensions of the molecule are significantly different from each other, can form biaxial nematics (Praefcke et al. 1991 Chandrasekhar 1992 Fialtkowski 1997). A biaxial... 

The existence of the biaxial SmA phase (SmAj,), also known as the McMillan phase, has been demonstrated in some charge-transfer complexes (CT complexes) formed with the metallomesogens 126 and TNF by textural observations and X-ray investigations. (Like the biaxial nematic phase mentioned earlier, the SmAt phase has additional long-range order in a direction perpendicular to the principal director, n (Figure 7)). Two novel... 

The difference between the structures of the uniaxial and biaxial nematics is illustrated schematically in fig. 6.6.4. The N phase is depicted here as an orthorhombic fluid whose preferred molecular orientation is described by an orthonormal triad of director fields. (In principle, nematics of lower symmetry are possible, but none of the phases identified to date have been reported to be other than orthorhombic.) The structure, therefore, gives rise to an additional pair of diffuse (liquid-like) X-ray diffraction peaks -" (fig. 6.6.3(A)). 

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... 

Figure 11. The three possible half integer disclination lines in the biaxial nematic phase of Vectra B 950 . (a) Rotation around the X axis, (b) rotation about the Y axis and (c) rotation around the Z axis [53], It is worth noting that the Ey line is perpendicular to the main director Y.

Birefringence is the difference between two principal refractive indices of the polymer. Where it is optically uniaxial, the principal axes are normal to and perpendicular to the director. Of course, some smectic phases are optically biaxial as are some interesting biaxial nematic polymers. It is important to appreciate that birefringence is an optical property of a representative volume of the polymer with dimensions at least those of the wavelength of light. It is not a molecular property but is directly proportional to (P2(cos or)), at least... 

The mesophases differ from each other regarding the positional order of the molecules (Fig. l). In the nematic phase there is no long range positional order at all just as in isotropic liquids. Nematics are normally uniaxial, however biaxial nematics were discovered very recently. In the smectic phases the centre of masses of the molecules are concentrated in layers forming a one-dimensional density wave. In the smectic A and C phases there is no long-range positional order within the layers. The smectic A phase is uniaxial, the director (n) is parallel with the layer normal, 1. In the C phase the director is tilted with respect to the layer normal. This phase is biaxial although the deviation from uniaxiality is usually small. There are further smectic phases in which the molecules form two-dimensional lattices within the layers (ordered smectic phases). The difference between ordered... 

Biaxial nematic or Sa elastomers lack rotational symmetry around their molecular long axes and the orientation of the three axes is described by the additional minor director m. A third director I is thereby automatically defined. Figure 15 gives a schematic representation of a uniaxial and biaxial nematic phase. The orientation methods discussed in the previous section yield polydomains with respect to the... 

Fig. 5.3. A Schematic diagram of the molecular arrangement in a biaxial nematic phase (Nb) comprised of discoid-shaped mesogens. The primary director is designated by the usual letter, n, and the secondary director, o, is orthogonal to n.

In the biaxial nematic phase it is possible to define two orthogonal directors n and m. In this case the magnetic susceptibility tensor can be rewritten as... 

A biaxial nematic phase is characterized by two orthogonal directors a and b of length 1. Introduction of a third orthogonal vector c of length 1 simplifies the notation of the following equations. According to Saupe, Covers and Vertogen the viscous part of the stress tensor of an incompressible biaxial nematic phase is the sum of the sym-... 

Figure 16. Director orientation in a biaxial nematic phase with respect to the flow velocity and the velocity gradient. The y case c z is shown where-grad V as the equations in the text correspond to the general case.

In biaxial nematic mesophases, the rotational symmetry of the phase around its director n is broken. It is commonly understood that the term biaxial nematic refers to an orthorhombic nematic phase. With respect... 

Kini [315, 316] and Fel [317] have discussed the influence of elastic constants of biaxial nematics in several experimental situations. They examined the feasibility of their experimental determinations by means of the simultaneous application of electric and magnetic fields in different geometries. Until now, however, no measurements have been reported that go beyond uniaxial elastic theory that is, there has yet to be an experimental demonstration of elastic deformations connected with the second director in biaxial nematics. 

It is well known that a smectic C liquid crystal which has a monoclinic symmetry is optically biaxial [3]. A biaxial nematic (N ) phase in which the molecules are oriented along the three directions in space, i.e., they have three directors which are perpendicular to one another, is possible. Thus the shape of molecules exhibiting the Nb phase should be different from those exhibiting the uniaxial nematic phase. 

Figure 10.17 shows schematically the biaxial nematic phase Njb and the uniaxial nematic phase Ni [55]. The orientatiOTi of the rodlike molecules in a plane perpendicular to the direction of the major director (n) is random in the uniaxial N1 phase. However, with increasing cmicentration of the rodlike molecules, the rodlike molecides may orient by mutual attraction and the excluded volume effect in a second direction with the minor director (m) as shown in the figure on the right. Thus, the novel biaxial nematic phase Nib may be possible. In the same fashion, in the N2 phase, it may be possible to have a biaxial nematic N2b phase, where the additional ordering of liquid crystals appears in the minor director (in) perpendicular to the major director (h) of rodlike molecules. 

The biaxial order parameter A represents the minor director and defines the degree of orientation along an axis that is perpendicular to the z-axis. In the isotropic phase = A = 0, in the uniaxial nematic phase 0, A = 0, and in the biaxial nematic phase 0, A y 0. 

To see the nature of the orientational order parameters needed to characterise the biaxial nematic we start with the singlet orientational distribution function. Since the molecules are taken to be rigid the distribution is a function of the three Euler angles, a 3y, which we denote by Q connecting the director... 

The first two of these order parameters are non-z o in both the uniaxial and the biaxial nematic phases. They are the major and biaxial order paramet f( - the molecule measured with respect to the director n (see Section C). The renaming two raxto param as vanish in the uniaxial nematic phase but are non-zero in the biaxial nematic. One of these,, measures the biaxiality in die... 

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