Field director

As in crystals, defects in liquid crystals can be classified as point, line or wall defects. Dislocations are a feature of liquid crystal phases where tliere is translational order, since tliese are line defects in tliis lattice order. Unlike crystals, tliere is a type of line defect unique to liquid crystals tenned disclination [39]. A disclination is a discontinuity of orientation of tire director field. 

Disclinations in tire nematic phase produce tire characteristic Schlieren texture, observed under tire microscope using crossed polars for samples between glass plates when tire director takes nonunifonn orientations parallel to tire plates. In thicker films of nematics, textures of dark flexible filaments are observed, whetlier in polarized light or not. This texture, in fact, gave rise to tire tenn nematic (from tire Greek for tliread ) [40]. The director fields... 

An aligned monodomain of a nematic liquid crystal is characterized by a single director n. However, in imperfectly aligned or unaligned samples the director varies tlirough space. The appropriate tensor order parameter to describe the director field is then... 

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. 22. (a) Identification of the angles and 6 used to describe a disclination. (b) Director arrangement of an 5 = I/2 singularity line. The end of the line attached to the sample surface appears as the point s = + V2 (points P). The director alignment or field does not change along the z direction. The director field has been drawn in the upper and the lower surfaces only. 

In addition, in the absence of the bounding plates, a spontaneous helix develops in the director field about the layer normal along the tilt cone (indicated in the Figure)... 

Throughout this chapter we focus on the extended hydrodynamic description for smectic A-type systems presented in [42,43], We discuss the possibility of an undulation instability of the layers under shear flow keeping the layer thickness and the total number of layers constant. In contrast to previous approaches, Auernhammer et al. derived the set of macroscopic dynamic equations within the framework of irreversible thermodynamics (which allows the inclusion of dissipative as well as reversible effects) and performed a linear stability analysis of these equations. The key point in this model is to take into account both the layer displacement u and the director field ft. The director ft is coupled elastically to the layer normal p = in such a way that ft and p are parallel in equilibrium z is the coordinate perpendicular to the plates. 

Fig. 12 In most parts of the scanned parameter space no possibility for an oscillatory instability was found. If the director field is only very weakly coupled to the layering (in this plot we used Bo = 200 and V = 0.4) a neutral curve for an oscillatory instability (

In Fig. 8 we have illustrated that a small viscosity coefficient V2 facilitates the onset of undulations. In this section we will have a closer look at the effect of an anisotropic viscosity tensor and ask whether undulations can be caused only due to viscosity effects without any coupling to the director field (i.e., we consider standard smectic A hydrodynamics in this section). 

Small-angle light scattering has also been extensively applied to PLCs subject to flow [173]. As in the case of scattering dichroism, SALS patterns arise principally from fluctuations in orientation, and these arc strongest in the vicinity of disclinations, or defects in the director field. The experimental geometries used for SALS in liquid crystals normally use polarizers placed before and after the sample. The arrangements include VV scatter-... 

Ristvet B. (1978) Reverse weathering reactions within recent nearshore marine sediments, Kaneohe Bay, Oahu. Test Directorate Field Command, Kirtland AFB, New Mexico, 314 pp. 

The three elastic constants are the Frank elastic constants, called after Frank, who introduced them already in 1958. They originate from the deformation of the director field as shown in Fig. 15.52. A continuous small deformation of an oriented material can be distinguished into three basis distortions splay, twist and bend distortions They are required to describe the resistance offered by the nematic phase to orientational distortions. As an example, values for Miesowicz viscosities and Frank elastic constants are presented in Table 15.10. It should be mentioned that those material constants are not known for many LCs and LCPs. Nevertheless, they have to be substituted in specific rheological constitutive equations in order to describe the rheological peculiarities of LCPs. Accordingly, the viscosity and the dynamic moduli will be functions of the Miesowicz viscosities and/or the Frank elastic constants. Several theories have been presented that are more or less able to explain the rheological peculiarities. Well-known are the Leslie-Ericksen theory and the Larson-Doi theory. It is far beyond the scope of this book to go into detail of these theories. The reader is referred to, e.g. Aciemo and Collyer (General References, 1996). 

FIG. 15.52 Elastic responses due to the deformation of the director field Frank elastic constants. Kindly provided by Prof. SJ. Picken (2003). 

However, if the director field is not uniform, Frank distortional stresses influence the rate of rotation of the director, and a new equation for h (or, equivalently for N) is required to replace Ericksen s equation (10-3). This is obtained from the balance of angular momentum, which gives... 

As Er is increased further to around 10 in 8CB (at 37°C), there is a roll-cell instability involving (a) a periodic modulation of the director field in the vorticity direction and (b) a cellular flow. The rolls cells are parallel to the primary flow direction (Pieranski and Guyon 1974) (see Fig. 10-19). These transitions in the director field have been both predicted from the Leslie-Ericksen theory (Manneville and Dubois-Violette 1976 Larson 1993) and... 

Here, we shall content ourselves with obtaining an analytic result for the minimum field necessary to induce the first slight reorientation away from uniform alignment. If the director field is only very slightly distorted, then the angle 6 is small, and Eq. (AlO-8) reduces to... 

The transient phenomena discussed above are no doubt produced by distortions of the nematic director field under shear these distortions, known as textures, can be visualized... 

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.)... 

Although we expect for dimensional reasons that the average magnitude of h, and hence the magnitude of ([nh]), will be proportional to pv, the tensorial form of ([nh]) is unknown. To obtain ([nh]), without having to revert back to (an almost impossible) microscopic calculation of the director field, Larson and Doi (1991 Kawaguchi 1996) assumed Aat ([nh]) is a function of the mesoscopic order parameter S— that is, that ([nh]) = Ka f(S). Dimensional reasoning then leads to the ansatz that... 

---