Director orientation

An orientational order parameter can be defined in tenns of an ensemble average of a suitable orthogonal polynomial. In liquid crystal phases with a mirror plane of symmetry nonnal to the director, orientational ordering is specified. 

If we compare with figure C2.2.I I, we can see that this defonnation involves bend and splay of the director field. This field-induced transition in director orientation is called a Freedericksz transition [9, 106, 1071. We can also define Freedericksz transitions when the director and field are both parallel to the surface, but mutually orthogonal or when the director is nonnal to the surface and the field is parallel to it. It turns out there is a threshold voltage for attaining orientation in the middle of the liquid crystal cell, i.e. a deviation of the angle of the director [9, 107]. For all tliree possible geometries, the threshold voltage takes the fonn [9, 107]... 

Fig. 2.8.16 Director orientation, 0, as a function of shear rate for both flow aligning (solid squares) and tumbling (open squares 325 K, solid circles 328 K and open circles 333 K) nematic polymers. (From Siebert et al. [10].)...

Fig. 2.8.17 Director orientation 0 obtained from 2H NMR spectra from the polysiloxane backbone polymer with liquid-crystal side-chain 4-methoxyphenyl-4 -butenyloxybenzoate, as a function of the apparent extension rate, i. The...

Figure 8.9 Synclinic and anticlinic layer interface configurations, differing in relative azimuthal angle <(> between director orientations for pair of adjacent layers, are illustrated. It appears that many, if not all, SmC and SmC materials possess wells for both synclinic and anticlinic configurations. Materials for which antichnic well is global minimum for system in some temperature...

Spherical droplet that forms during a transition from an isotropic phase to a nematic mesophase. It has characteristic textures that depend on the droplet size and the director orientation at the nematic-isotropic interface. 

Texture consisting of several areas with different director orientations. 

Note 2 The layers are aligned almost normal to the sample surfaces. The regular sets of hyperbolae are called boundaries of Grandjean they serve as limiting surfaces between domains with different director orientations. 

Ratio of the shear stress, a, to the shear velocity gradient, y, for a nematic liquid crystal with a particular director orientation, denoted by /, under the action of an external field ... 

Note 1 The three Miesowicz coefficients (//i, 772, and 773) describe the shear flow of a nematic phase with three different director orientations, (see Fig. 31) namely 771 for the director parallel to the shear-flow axis 772 for the director parallel to the velocity gradient and 773 for the director perpendicular to the shear flow and to the velocity gradient. 

Note 1 The nematic liquid crystal must have a negative dielectric anisotropy (Af < 0), and a positive anisotropy (Aa > 0). The optical texture corresponding to the flow pattern consists of a set of regularly spaced, black and white stripes perpendicular to the initial direction of the director. These stripes are caused by the periodicity of the change in the refractive index for the extraordinary ray due to variations in the director orientation. 

Note 3 The bands are associated with a periodic variation in the director orientation about the flow axis. 

In the absence of other constraints, free energy is minimized by a rotation of the director. There are two possibilities when > (positive anisotropy), the director orients parallel to the field and when > (negative anisotropy), the orientation of the director is perpendicular to the field. 

For an oriented sample with a uniform director orientation the 23Na NMR spectrum at a weak radiofrequency field thus consists of three equally spaced peaks. The relative magnitude of the three resonance lines is governed by the transition probabilities between the different energy levels. It turns out (9) that the central line contributes 40% of the total intensity while each of the satellites contributes 30%. The deuteron NMR spectrum consists of two equally intense peaks. 

Experiments by Muller et al. [17] on the lamellar phase of a lyotropic system (an LMW surfactant) under shear suggest that multilamellar vesicles develop via an intermediate state for which one finds a distribution of director orientations in the plane perpendicular to the flow direction. These results are compatible with an undulation instability of the type proposed here, since undulations lead to such a distribution of director orientations. Furthermore, Noirez [25] found in shear experiment on a smectic A liquid crystalline polymer in a cone-plate geometry that the layer thickness reduces slightly with increasing shear. This result is compatible with the model presented here as well. 

PDLCs comprise LC droplets dispersed in a polymer matrix. Random director orientation leads to the scattering of light due to refractive index mismatch between LC and polymer. A field applied across the film orients the LC so that the refractive indices match and the scattering is reduced, rendering the film transparent. 

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... 

If the director is held in a fixed orientation by a magnetic field strong enough to resist the orienting effects of flow, then shear-rate-independent viscosities can be measured in a simple shearing flow. The three simplest of these, called the Miesowicz viscosities, are obtained in each of the three director orientations shown in Fig. 10-8. These viscosities can be related in a simple way to the or,- s, namely,... 

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... 

As y is increased still further, the tumbling behavior of the director is replaced by a wagging motion wherein the director oscillates back and forth about the flow direction. The frequency of this wagging motion increases with further increases in y and after another transitional shear rate is exceeded, the wagging motion damps out and a steady-state is attained in both 5 and in the director orientation (Larson 1990). 

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