Uniaxial Nematic Phase

The above spectral densities can be modified for the occurence of chain flexibility, and for the director being oriented at dLD w.r.t. the external BQ field in the L frame. For CD bonds located in the flexible chain, the effect of DF is reduced due to an additional averaging of the time dependent factor (/f g) by conformational transitions in the chain. Consequently, the spectral densities given in Eqs. (60)-(62) are modified by replacing Soc%0(Pm,q) by the segmental order parameter YCD of the C-D bond at a particular carbon site on the chain.146,147 As observed experimentally,148,149 the spectral densities in a flexible chain show a SqD dependence when DF dominate the relaxation rates. The general expression of Jm(co 0LD) due to DF in uniaxial nematic phases is given by... 

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). 

According to the competition of all the interactions in the side chain nematic polymers, there are three uniaxial nematic phases Ni, Nn, Nm- The three phases, each having a special conformation, may transform each other when the free energies are equal. These phases are shown in Figure 2.26 with the director pointing upward. 

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... 

Fig. 6.6.3. Raw microdensitometer scans of the X-ray intensity plotted against diffracting angle (20) for magnetically aligned nematic samples (a) the uniaxial nematic phase of 80CB at 77 °C (b) the biaxial nematic phase of complex A at 168.5 C, (i) meridional scan (parallel to H), (ii) equatorial scan (perpendicular to H). M represents the diffraction peaks from the mylar film which covered the windows of the sample holder and heater assembly. ...

Fig. 6.6.4. Schematic diagram of the molecular order in (a) the uniaxial nematic phase and (b) the biaxial nematic phase.

Essentially all of the techniques developed to characterize MLCs can be applied to PLCs with the realization that phenomena that are dependent on reorientation processes in the liquid crystal must be considered on considerably longer time scales in a PLC. Underlying most of the physical measurements performed on liquid crystals is the relationship between the observed anisotropic properties of the mesophase and orientational order of the mesogen. For uniaxial nematic phases and idealized low molar mass mesogens (cylindrical molecules), this relationship is embodied in the following equation ... 

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) ... 

N Dooh X T(3) Uniaxial nematic phase possessing long range orientational order and no translational order... 

Let us consider a uniaxial nematic phase (racemic mixture), where there is no twist. Then d = 0 and the reduced free energy density becomes... 

Most of the parameters involved in the hydrodynamic and electrodynamic equations for a nematic have been measured for different substances that show a uniaxial nematic phase. Among these one can mention the elastic constants (Blinov L. M and Chigrinov V. G. 1994) specific heat, the flux alignment parameter X and the viscosities Vj, i=l,2...5, the inverse of the diffusion constant yir the thermal conductivity (Ahlers, Cannell, Berge and Sakurai 1994), and the electric conductivity pijE. 

The uniaxial nematic phase possesses a quadrupole-type symmetry and is characterized by the order parameter Qajj which is a symmetric traceless second-rank tensor ... 

We note that the averaging in Eq. (2) is performed with the one-particle distribution function /, ((a n)) that determines the probability of finding a molecule with a given orientation of the long axis at a given point in space. In the uniaxial nematic phase the distribution function depends only on the angle m between the long axis and the director. Then Eq. (2) can be rewritten as... 

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 majority of the existing molecular theories of nematic liquid crystals are based on simple uniaxial molecular models like sphe-rocylinders. At the same time typical mes-ogenic molecules are obviously biaxial. (For example, the biaxiality of the phenyl ring is determined by its breadth-to-thick-ness ratio which is of the order of two.) If this biaxiality is important, even a very good statistical theory may result in a poor agreement with experiment when the biaxiality is ignored. Several authors have suggested that even a small deviation from uniaxial symmetry can account for important features of the N-I transition [29, 42, 47, 48], In the uniaxial nematic phase composed of biaxial molecules the orientational distri-... 

The only difference between the nematic phase and the isotropic phase is the orientational order. A proper description of this orientational order requires the introduction of a tensor of the second rank [7, 8]. This tensor can be diagonalized and for anisotropic liquids with uniaxial symmetry, the nematic phase can be described by only one scalar order parameter. The thermodynamic behavior in the vicinity of the N-I transition is usually described in terms of the mean-field Landau-de Gennes theory [7]. For the uniaxial nematic phase one can obtain the expansion of the free energy G in terms of the modulus of an order parameter Q. 

Shortly after the first observation of a biaxial nematic phase by Yu and Saupe [56] the hydrodynamic theory of the uniaxial nematic phase was extended to the biaxial case in several papers [57-60]. The following description is similar to that given by Saupe [59] as well as Covers and Vertogen [60], but the notation is different. 

Equations (88) and (89) include the case of the stress tensor for the uniaxial nematic phase [59]. 

In general, liquid crystal molecules do not have the D200 symmetry of the uniaxial nematic phase. Since an interface acts as a field, its presence can provide polar order. Such surface polar ordering, confined to a single molecular layer, has been observed [99]. Surface SHG can also be used to probe the orientational distribution at the surface, and anchoring transitions [100]. 

The molecular theory is similar to Cauchy s description of the elastic theory of solids [1] and utilizes additive local molecular pair interactions to describe elasticity. The latter approach was taken by Oseen [2], who was the first to establish an elastic theory of anisotropic fluids. Oseen assumed short-range intermolecular forces to be the reason for the elastic properties, and he derived eight elastic constants in the expression for the elastic free energy density of uniaxial nematic phases. Finally, he retained only five of them, which enter the Euler-Lagrange equations describing equilibrium deformation states of the nematic mesophase, and omitted the other three. 

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. 

Based on the interaction employed in the Maier-Saupe theory of the nematic state, Preiser [4] was the first to predict the possible existence of an N, phase as an intermediate between two uniaxial nematic phases. Later, a number of other theoretical investigations were carried out [5-7] using various models to predict the possibility of obtaining an liquid crystal. In all these models, a system consisting of hard rectangular plates was considered. These approaches gave the same result, an Nb phase should be obtained between two uniaxial nematics of opposite sign, i.e., those made up of rod-like and plate-like molecules. They also predicted that a transition from a uniaxial nematic to a biaxial nematic would be second order. 

Assuming the phase to be an orthorhombic fluid, X-ray diffraction methods have been used to distinguish it from the Ny phase. The X-ray diffraction patterns of the uniaxial nematic phase of 4 -/i-octyloxy-4-cyanobiphenyl (80CB) and that of the nematic phase exhibited by complex 2 are shown in Figs. 9 and 10, respectively. 

Phase diagram of the potassium laurate (KL)-l-decanol-D20 system, with the Landau point. Nc and Nl are uniaxial nematic phases with cylindrical and lamellar micelle structures, and Nj, denotes the biaxial nematic phase. (Figure reproduced from Reference [8] with permission... 

Consider the dielectric tensor In the uniaxial nematic phase (chosing the z-axis parallel to the nematic axis), has the form... 

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