Line singularities

As remarked in chapter 1, the nematic state is named for the threads that can be seen within the fluid under a microscope (fig. 1.1.6(a)). In thin films sandwiched between glass plates these threads can be seen end on. A typical example of the texture in a plane film of thickness about 10 /tm between crossed polarizers - the structures a noyaux or schlieren textures - is given in fig. 1.1.6(6). The black brushes originating from the points are due to line singularities perpendicular to the layer. In analogy with dislocations in crystals, Frank proposed the term disinclinations , which has since been modified to disclinations in current usage. 

Fig. 3.5.14. A line singularity of strength s is located at /I at a distance D from the centre 0 of a cylindrical cavity of radius R.

The first of these two is the equation of zero-isoclinic line, and the second -infinite-isodinic line. Singular points of the velocity field of drops 2 are the points of intersection of these isodinic lines. 

The applicability of homotopic theory becomes much less obvious for liquid crystal phases with more complicated order parameters such as biaxial nematics and cholesterics, which are both locally defined by three directors forming a tripod. This gives rise to a description of the line singularities in terms of the quaternion group, Q. This is particularly interesting because the quaternion group Q is non-Abelian, a property that... 

O = unstable node, = saddle point, dashed line = singular poin curve, solid line = kinetic curve... 

Janecek, V., Nikolayev, V.S. Contact line singularity at partial wetting during evaporation driven by substrate heating. Europhys. Lett. 100, 14003 (2012)... 

It is also possible to have points with 5 = +1/2 joined by line singularities in the nematic phase these 7t disclinations, which are commonly known as threads, pass through the preparation almost perpendicularly with the ends attached to the glass surfaces. Figure 3B shows the topology about s = + 1/2 singularity line the end appears as a point on the... 

For a distillation process not only the stationary point type, but also the behavior of the residue curve in the vicinity of the stationary point is of special importance. If the residue curves in the vicinity of the specific point are tangent to any straight line (singular line) (Fig. 1.4a, b, d, e, g, h), the location of this straight line is of great importance. A special point type and behavior of residue curves in its vicinity are called stationary point local characteristics. 

Carrying this idea over to the helical-isotropic transition, there are two differences. First, we must use disclinations topological line singularities in the director field of the liquid crystal—rather than crystal defects. The second difference is that the helical phase, which has no defects, melts to the blue phase, which is characterized by a stable defect lattice of line disclinations rather than by a random collection of defects. Indeed, there is more than one way to create such a lattice thus BPI and BPII. The helical phase therefore melts to BPI, BPI melts to BPn, and, with a final onset of randomly positioned defects, BPII melts to the isotropic phase. 

It is useful to consider this mesostructure as a packing of monolayer surfaces, or as bilayers. The monolayer picture has it that the aggregates consist of cylindrical, infinitely long micelles. The bilayer picture is less familiar, consisting of a hexagonal honeycomb, with a 2D hexagonal lattice of line singularities, the junctions... 

Vortex distributions. Next, we turn to vortex distributions, which are line singularities used to model flows past shales that support pressure differentials. Let us consider the pressure field associated with a line vortex of strength g(x) from Equation 3-21, the expression for pressure is... 

This type of cylindrical solution may be expected to occur in a cyfindrical container if the orientation of the director n is perpendicular on the cylinder boundary. The line singularity occurring at r = 0 may be observable, given that fine singularities (or defects) are commonly seen in liquid crystal samples. However, this scenario and a possible alternative solution for cyUnders of finite radius are discussed in more detail in Section 3.8.1 below. 

G. F., 1922) inferred from the observation at optical scales of line singularities in the since-called Smectic-A (SmA) phases that theses phases were lamellar. Diffraction methods were not yet invented, and the discovery of the lamellarity, which was some years later confirmed by his son Edmond (X ray diffraction), a student of Maurice de Broglie, was at that time based only on the geometrical properties of these singularities. These singularities were quite remarkable, made of pairs of conics, an ellipse E and one of the branches of an hyperbola... 

Fig. 11 illustrates these characters. The plane which contains the ellipse is always parallel to the layers belonging to the homeotropic region. By changing the focusing of the microscope, it is possible to follow a line singularity, which lies in the vertical plane containing the major axis of the ellipse and passes through the focus of the ellipse. This line is one branch of hyperbola. Between crossed nicols, the focus of the ellipse is the center of four dark brushes (fig.l 1). These observations lead to the conclusion that these domains are FCD s of the first species. 

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