Director fluctuation

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. 

This is another important example of a successful application of the theory of elasticity. In Section 11.1.3 we shall discuss the nature of strong light scattering by 

We again consider the director no oriented along the z-axis. Its fluctuating part has components (n riy, 0). The total Frank elastic energy, related to the fluctuations in volume V, is given by the integral of free energy density (8.15b)  

The Fourier harmonics and y of the director fluctuating field are represented by volume integrals (q is wavevector)  

In Fig. 8.10b, we see that the fluctuation mode i(q) is a mixture of the splay and bend distortions, and the component 2(q) is a mixture of twist and bend distortions. This may be clarified as follows the splay-bend (SB) mode on the left side of Fig. 8.10b corresponds to realignment of the molecules within the, z-plane as q evolves and there is no twist here. In contrast, on the right side of the same figure the molecules are deflected from the q z-plane of the figure therefore, the twist and bend are present but the splay is absent (TB mode). 

After the transform to the new variables /t x(q) (a = 1,2) the equation for the free energy reads  

---

Zhang Z, Zuckermann M J and Mouritsen O G 1993 Phase transition and director fluctuations in the 3-dimensional Lebwohl-Lasher model of liquid crystals/Mo/. Phys. 80 1195-221... 

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... 

In treating 2D director fluctuations (due to layer undulations) in lamellar phases of biomembranes, Halle27 found a different expression (for oj ojp) when couplings among bilayers were included ... 

In a smectic A liquid crystal one can easily define two directions the normal to the layers p and an average over the molecular axes, the director, h. In the standard formulation of smectic A hydrodynamics these two directions are parallel by construction. Only in the vicinity of phase transitions (either the nematic-smectic A or smectic A-smectic C ) has it been shown that director fluctuations are of physical interest [33, 44, 45], Nevertheless h and p differ significantly in their interaction with an applied shear flow. 

It is often preferable to evaluate 6q by EMD methods because the director fluctuates around the preferred orientation in a shear flow simulation, which makes it hard to obtain accurate estimates. If one performs such a simulation one must fix the director at several alignment angles and calculate the antisymmetric pressure tensor, which, according to Eq. (4.10e), is a linear function of cos 26. One can fit a straight line to the data points and the zero gives... 

Transverse nuclear relaxation due to director fluctuations in liquid crystals for the slow motion regime has been considered this leads to an analysis in detail of the transverse magnetization decay in different kinds of experi-... 

Figure 6.9. The director fluctuation causes light scattering (a) two independent fluctuation modes 8n and <5ri2 (b) two components in 8n splay and bend and (c) two components of <5n2 bend and twist. (Modified from DuPre, 1982.)...

It is noted that the viscosity of liquid crystalline polymers which is much higher than the small molecular mass liquid crystals, affects the amplitude of the director fluctuation, i.e., the overdamped effect. Thus, the scattering... 

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]. 

Thus the director fluctuations make the predominant contribution to the light scattering, as was first pointed out by de Gennes. >... 

We have resolved the director fluctuations into 8n and Sn which from the symmetry of the problem can be seen to be uncoupled. If and are... 

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. 

Comparison between theory and experiment Halperin, Lubensky and Ma have argued that when director fluctuations are taken into account, the A-N transition should be at least... 

In a uniaxial nematic liquid crystal, the spatial orientation of the optical axis is determined by the orientation of the director. Due to thermally excited orientational director fluctuations, the spatial direction of the optical axis is not constant in time. As a result, any light illuminating the sample is... 

Due to the uniaxial symmetry of the nematic phase the two types of fluctuations of biaxiality are degenerated and so are the two types of director fluctuations. 

The correlation length of the director fluctuations is infinite in the whole range of the stable nematic phase and the director excitation with the infinite wavelength is the Goldstone mode. Fluctuations of other degrees of freedom of... 

Biaxial fluctuations (3 2 are degenerated, whereas the eigenfunctions are just mirror images with respect to the middle plane of the cell. The few lowest modes axe expelled from the part of the film where they Ijend the secondary director field and so they effectively represent the director fluctuations (see Fig. 8.10b). Higher modes are spread over the whole film whereas the unfavorable manner of biaxial fluctuations is compensated by the shorter wave vector of a deformation. 

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. 

---