Singular Lines

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. 

The line M2 — H2 is a singularity line of the equation (dotted line, Fig. 5). By defining detonation as a regime in which flame propagation occurs at a velocity greater than the speed of sound in the original gas, we find at the point A c < D c2Jv2 < D2/v2, H2 < M2. After the shock compression, at the point C, as we know, c > D, H2 > M2 the shock compression is accompanied by a jump across the line M = H. 

The reversibility of the adsorption steps in mechanism (4) affects the total number of steady states. As can be seen from Table 1, if two adsorption steps are reversible, boundary steady-state points are absent. Irreversibility of one adsorption step leads to the appearance of one boundary steady-state point in which the concentration of the reversibly adsorbing substance is equal to zero and the irreversibly adsorbing substance occupies all active sites of the catalyst surface. In the case where both adsorption steps are irreversible, there exist two boundary steady-state points (x = 0, y = C2) and (x = Cz, y = 0). In the latter case, at equal kinetic orders of the adsorption steps (n = m) a multiplicity of steady-state solutions is possible, i.e. at pk2 = qk1 (non-rough case) there exists a singular line of steady states connecting two boundary steady-state points. It can manifest itself in the unreproducibility of experimental data in a certain range of the parameters. 

Let us now examine the behaviour of the solutions for the dynamic system (20) in time and analyze the system trajectories in the phase pattern. This analysis permits us to characterize peculiarities of the unsteady-state behaviour (in particular to establish whether the steady state is stable or unstable), to determine its type (focus, node, saddle, etc.) and to find attraction regions for stable steady states, singular lines, etc. 

If two lines OT and QL represent the space-time loci of two material particles, the intercepts OL and OQ of singular lines between these loci have mathematically and physically zero length. Two atoms with such loci are... 

The Oseen theory embraces smectic mesophases, but is not really required for this case. The interpretation of the equilibrium structures assumed by smectic substances under a particular system of external influences may be carried out by essentially geometric arguments alone. The structures are conditioned by the existence of layers of uniform thickness, which may be freely curved, but in ways which do not require a breach of the layering in regions of greater extension than lines. These conditions automatically require the layers to be Dupin cyclides and the singular lines to be focal conics. Nothing, essentially, has been added... 

Before leaving the subject of the smectic state, we may remark that to explain deformations in which the area of individual molecular layers does not remain constant, it is necessary to invoke dislocations of these layers. It is likely that these dislocations are usually combined with the focal conic singularity lines, being then essentially screw dislocations. 

The medium will then relax to a new state owing to the internal interactions. Consequently, a singular line L is left. 

It is known that there is no topological stable singular wall and point but there are topological stable singular lines which are characterized by the Q group, Q C0,C0,C1,C2,C 5). The classification of the cholesteric... 

Such disclinations are closely analogous to nematic wedge disclinations ( 3.5.1). The singular line is along the z axis (parallel to the twist axis) and the director pattern is given by... 

In this case the singular line is perpendicular to the twist axis. On going round this line, one gains or loses an integral number of half-pitches. The director pattern around the -edge disclination was first worked out by de Gennes who proposed a nematic twist disclination type of solution ... 

The cholesteric pitch is altered around the singular line where N is an integer. The pattern for i = j is shown in fig. 4.2.4. Again, the energies and interactions in the one-constant approximation are the same as for nematic twist disclinations. A somewhat more elaborate treatment of this model has been presented by Scheffer and the effect of elastic anisotropy has been investigated by Caroli and Dubois-Violette. ... 

Fig. 5.4.1. (a) Smectic layers in concentric cylinders to form a myelin sheath with a singular line L along the axis (b) the cylinders are closed to form tori there are two singular lines, a circle and a straight line (c) the general case when the smectic layers form Dupin cyclides the circle becomes an ellipse and the straight... 

Fig. 5.4.2. A rare example of a pair of singular lines, one almost straight and the other almost circular in a toric domain in the (a) smectic A and (h) smectic C phases. Additional disclination lines develop near the centre in the smectic C phase for reasons discussed in 5.8.3. (From A. Perez, M. Brunet and O. Parodi, J. de Physique Lettres, 39, 353 (1978)).

So can have lattice disclinations as well, but they are perfect and energetically favoured only along the twofold axis (which is normal to the plane containing the layer normal and the c-director). Focal conic textures are also seen, though they are always accompanied by additional singular lines of disclination, which arise because of the molecular tilt (fig. 5.8.6). 

Fig. 5.8.6. Disposition of the molecules and layers in elliptical domains in (a) smectic A and (6) smectic C. The mismatch of the molecular orientations in smectic C gives rise to additional singular lines. (Bourdon, Sommeria and Kleman. )...

Fig. 3 b shows the director escape at the center of a disclination of strength 5= 1 in a thin capillary of radius R. The arrangement is continuous with no singular line. The deformation involves splay and bend, but no... 

In addition to the above-discussed discli-nations, which are referred to as wedge dis-clinations, there are twist disclinations. The director is always parallel to the xy plane, but the axis of rotation (z-axis) is normal to the singular line (y-axis). Figure 4 shows the director patterns for (a) 5=1/2, 6q=0 and (b) 5=1, 6[)=0. is a linear function of the angle 0=tan (z/x). 

As a rule, thin lines of strength 1/2 or singular lines of strength 1 are seen in the threaded textures of nematic thermotropic MCPs [25,57,58]. Rare cases of thick lines have been observed. As we have already pointed out, the reason for this is that the elastic anisotropy is large in these systems. The disclinations with 5= l/2 were also reported to be the most abundant in the Schlieren textures of nematic copolyesters [59-65]. 

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... 

Fig. 8.21 Structure of the director field around different singular lines (disclinations) in a cholesteric liquid crystal x , X and x, Signs (—) and (+) correspond to different Volterra...

Fig. 9.8. (a) Distribution of the director around a singularity line or disclination, perpendicular to the plane of the figure, (b) Different types of singularity... 

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. 

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