Director inhomogeneous

The damping of the stress oscillations presumably arises from a gradual loss of spatial coherence in the phase of the tumbling orbit across the sample. In a plate-and-plate rheometer, the strain is linearly dependent on the radial distance from the axis of rotation. As a result, the gap-averaged director orientation varies as a function of radial position in the sample. When this source of inhomogeneity in the tumbling orbit accounted for by integrating the torque contributions predicted by Eq. (10-31) over... 

Because nematic liquid-crystalline polymers by definition are both anisotropic and polymeric, they show elastic effects of at least two different kinds. They have director gradient elasticity because they are nematic, and they have molecular elasticity because they are polymeric. As discussed in Section 10.2.2, Frank gradient elastic forces are produced when flow creates inhomogeneities or gradients in the continuum director field. Molecular elasticity, on the other hand, is generated when the flow is strong enough to shift the molecular order parameter S = S2 from its equilibrium value 5 . (Microcrystallites, if present, can produce a third type of elasticity see Section 11.3.6.)... 

In a pioneering viscosity study of polymeric (high-viscosity) liquid crystals Martins et al. considered an extension of eqiudon (9a), a planar LesAe-Ericksen approximation for director motions in a plane (jc, z) with an elastic inhomogeneity term periodic in one dimension (z). This approach. 

In the distorted nematic the director distribution is inhomogeneous, and hence n = n(r) in Eq. (1.46). Taking into account that the length scale of the director variation is much larger than the molecular size we can apply the gradient expansion of the director n(r) with respect to some point Ro inside the molecule ... 

The result obtained has very interesting consequences (i) to have well aligned SmA samples, very flat glasses without corrugation are needed (ii) even small dust particles or other inhomogeneities create characteristic defects in the form of semispheres (see Fig. 8.29b below) and weU seen under an optical microscope (iii) layers are often broken (not bent) by external factors in particular, strong molecular chirality may result in the formatimi of defect phases like twist-grain-boimdary phase (iv) the thermal fluctuations of director in smectic A phase are weak and the smectic samples are not as opaque as nematic samples. In fact there is a critical cell thickness for short-wave fluctuations. 

The problem of light propagation becomes much more complicated in spatially inhomogeneous liquid crystal layers. There is no general method to solve the Maxwell equations for an arbitrary director distribution or, more generally speaking, for an arbitrary spatial dependence of the dielectric tensor. On the other hand in some important special cases exact solutions were found and useful approximations were worked out for other conditions. In this section we survey these results. 

In contrast to the simplest model of the Fredericks transition in static fields (see Refs. 12 and 13), here we take into account the following two facts 1) the amplitude for the deformation of the director above threshold must be found taking into account the distortion of the longitudinal profile of the light wave itself as it propagates in an inhomogeneous anisotropic medium ... 

The EOM effect also provides an important basis for elucidating the correlation between the director rotation and macroscopic deformation. Experimentally, it has often been investigated by mechanically stretching the monodomain nematic elastomers in the direction normal to the initial director [20-25]. However, the mechanical constraint by clamping at the ends sigiuficantly affects the director rotation, and the resultant director reorientation often becomes inhomogeneous [26-28]. In contrast, the EOM effect in Fig. lb enables us to observe unambiguously the characteristic... 

The former reflects the slightly imperfect alignment of the domain directors, whereas the latter is a consequence of a smeared phase transition in the LCEs. Figure 9a,b serves as an example of how the H-NMR spectra are altered by the inhomogeneity of the domain-director aligmnent and of the local order parameter in side-chain LCEs. Both spectra were recorded on LCEs doped with deuterated probes and oriented with their principal aligning direction parallel to the magnetic field. 

Apart from the nonuniform director alignment, the inhomogeneity of the local order parameter is also encountered in LCEs. This is best observed in a H-NMR spectrum (Fig. 9b) recorded in the vicinity of the phase transition. The spread 5v of the spectral intensity between 0 and 20 kHz in this spectrum corresponds to a spread of the local order parameter 85 in the range between approximately 0 and 0.45. Two pronounced peaks can be noted in each half-spectrum, corresponding to the coexisting paranematic (lower S, inner peak at Vpn) and nematic components (higher S, outer peak at v ). 

It appears that the understanding of the local order parameter s inhomogeneity is closely connected with the understanding of the phase transition in LCEs. However, in comparison with the inhomogeneity of the domain-director alignment, the characterization of the heterogeneity of the local order parameter appears much less accessible. This is due to the lack of any general model that predicts the distribution of the local order parameter > 5(5) in LCEs. 

The complication associated with electric fields is due to the large anisotropy of the electric permittivity, which means that above threshold the induced electric polarization is no longer parallel to the applied field. In a deformed sample the director orientation is inhomogeneous through the cell, and as a consequence the electric field is also nonuniform. An additional problem can arise with conducting samples, for which there is a contribution to the electric torque from the conductivity anisotropy. Neglecting this, the expressions for threshold electric fields are similar to those obtained for magnetic fields ... 

Except for the short discussion at the end of the last section, a uniform director alignment has been assumed up to now. Surface alignment and inhomogeneous fields can lead to an inhomogeneous alignment and the occurrence of elastic torques. For a complete description of hydrodynamics of nematic liquid crystals these elastic torques have to be included. 

An externally applied torque on the director can only be transmitted to the surfaces of a vessel without shear, if the director rotation is homogeneous throughout the sample as assumed in Sect. 8.1.2 for the rotational viscosity. Otherwise this transmission occurs partially by shear stresses. The resulting shear flow is called backflow. As there is usually a fixed director orientation at the surfaces of the sample container, a director rotation in the bulk of the sample by application of a routing magnetic field leads to an inhomogeneous rotation of the director and to a backflow [42]. 

The integration constant may still depend on time and can be determined from the boundary condition at the solid surface or in the middle of the cell. Equation (63) shows that every inhomogeneous director rotation is coupled with a velocity gradient. 

Under isothermal conditions the constitutive equations for the description of flow phenomena in nematic and cholesteric liquid crystals are identical [48]. Nevertheless, a series of novel effects are caused by the helical structure of cholesteric phases. They arise firstly because of the inhomogeneous director orientation in the undistorted helix and secondly because of the winding or unwinding of the helix due to viscous torques. 

The viscous part of the stress tensor for the SmC and the ferroelectric chiral smectic (SmC ) phase agree with one another. The flow phenomena with a fixed director orientation discussed in the foregoing section can not be observed due to the inhomogeneous director orientation in the SmC phase. However, there is a large interest in rotational movements of the director in ferroelectric displays. 

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