Order director fluctuations

So far, the discussion has been restricted to isolated motions of single molecules. In LCPs, however, collective motions of a lar number of molecules may occur. For the latter mechanism, known as order director fluctuations, a broad distribution of correlation times is sedicted [%, 37]. In contrast to the isolated modes, discusred above, director order fluctuations are expected to occur only in the mesophase of LCPs, but should be completely absent in the solid and glassy state of these systems. 

Molecular motions in LCs may occur as isolated or collective modes (see Fig. 4). For the latter mechanism, known as order director fluctuations, a continuous distribution of correlation times is expected [36, 37, 170-173]. Recent protean Tjz dispersion measurements of the LCPs 4 and corresponding LMLCs 7 and 8, carried out over a frequency range of five orders of magnitude (10 Hz < t0o/27t < 3 x 10 Hz), clearly show that collective order fluctuations contribute to the relaxation process only at extremely low frequencies in the kHz regime, whereas the conventional MHz range is dominated reorientations of individual molecules [174]. 

Frequency dependence of J2 2uq) in the nematic and smectic A phases has also been observed [6.53, 6.55, 6.10]. For example, the behavior of DEB in 8CB is clearly different from that in 40.8. In particular, J2 shows more frequency dependence in 8CB than in 40.8 [6.10]. An attempt to account for this frequency dependence using higher-order director fluctuations in Eq. (6.39) was not successful. In a BBA/MBBA sample, spectral density data for deuterons in the para and (ortho, meta) positions of the benzyli-dene ring also show a strong frequency dependence [6.58]. BBA is similar in size and shape to its MBBA solvent and unlike DEB can undergo internal... 

Figure 12. Theoretical dependence of the proton Ti on the angle between the nematic director and the direction of the external magnetic field for the nematic order director fluctuations for different ratio between the Larmor frequency and the cut-off frequency tq, [137].

For a nuclear probe consisting of two protons with a fixed separation distance r, the dipole-dipole interactions of which are modulated by fluctuations in the orientation of intemuclear vector r with respect to the external magnetic field H due to nematic order director fluctuations, the spin-lattice relaxation rate is... 

In the nematic phase the spin-lattice relaxation rate at high Larmor frequencies is determined by local reorientations of the molecule and internal molecular motions. The spin-spin relaxation rate, I/T2, on the other hand, is determined by nematic order director fluctuations and rotations induced by translational diffusion [216]. 

Ziherl, P. and Zumer, S. (1996). Nematic order director fluctuations in cylindrical capillaries, Phys. Rev. E 54 1592-1598. 

C and I account for gradients of the smectic order parameter the fifth tenn also allows for director fluctuations, n. The tenn is the elastic free-energy density of the nematic phase, given by equation (02.2.9). In the smectic... 

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. 

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... 

It is considerably larger in the confined liquid crystals above Tni than in the bulk isotropic phase. The additional relaxation mechanism is obviously related to molecular dynamics in the kHz or low MHz frequency range. This mechanism could be either order fluctuations, which produce the well-known low-frequency relaxation mechanism in the bulk nematic phase [3], or molecular translational diffusion. Ziherl and Zumer demonstrated that order fluctuations in the boundary layer, which could provide a contribution to are fluctuations in the thickness of the layer and director fluctuations within the layer [36]. However, these modes differ from the fluctuations in the bulk isotropic phase only in a narrow temperatnre range of about IK above Tni, and are in general not localized except in the case of complete wetting of the substrate by the nematic phase. As the experimental data show a strong deviation of T2 from the bulk values over a broad temperature interval of at least 15K (Fig. 2.12), the second candidate, i.e. molecular translational diffusion, should be responsible for the faster spin relaxation at low frequencies in the confined state. 

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.

Thus, director fluctuations in the small angle approximation can contribute to the spectral density Ji uj) through the Qi q (r) term where Sq = refers to ordering of the molecule with respect to the local director. This small angle approximation is probably not warranted for low orientational orders, where might be as large as 40°. While g o( )... 

Some of the difficulties in tackling director fluctuations in the presence of molecular reorientation have been mentioned in the previous section. The original derivation of spin relaxation in nematics [6.3-6.5] was done by treating an assembly of partially ordered rigid cylindrical molecules. This approach will be followed since the derived expressions can be used directly in more realistic models encompassing both types of motion. First, static distortions in liquid cystals are examined. 

The suppression of director fluctuations near the nematic-smectic A (N-A) transition because of divergence in the twist and bend elastic constants and the twist viscosity 7e [6.6] are now examined. Above the phase transition at T/vaj there are cybotactic smectic A clusters in the nematic phase, whose dimension is measured by a coherence length In fact, it is the coupling between the nematic director and the smectic order parameter that causes the viscoelastic constants to approach infinity at TnA Hence, A22J A33 oc while 7e oc Suppose there is interest in a frequency far below the high-frequency cutoffs such that A >> 1. In this limit,... 

Coupling of Smectic Order with Director Fluctuations... 

Only a few relaxation dispersion measurements have thus far been carried out on smectic phases. However, the influence of smectic order on director fluctuations has been detected. Blinc et al. [6.43] were first to study proton Ti frequency dispersion in the smectic phases of terephthal-bis-p-butylaniline (TBBA) using a field cycling technique. Figure 6.6 shows the... 

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