Disclination energy

In a nematic phase, the strength of disclination may have values that are half-numbered or whole-numbered. However because the disclination energy is proportional to s2 (Nehring and Saupe, 1972), it is rare to observe singularities with s > 1. In studies of a series of nematic polymers with two-dimensional mesogenic units Zhou and coworkers (1993) have been... 

The two structures so formed are analogous to a particle and its antiparticle. They can annihilate one another with release of energy but otherwise can only be destroyed at the boundary of the medium. In a similar manner, a suitable injection of energy can create a disclination pair in the body of the material. 

Figure 10. Pairing of two disclination lines of opposite signs (lamellar details are not featured) (top) a less probable model for the core of a dislocation (middle) and focal line appearing on the dislocation in order to release locally deformation energy (bottom)...

In this theory, yielding is produced by local molecular kinks at the nanoscale level. The formation of a kink pair is modeled by a wedge disclination. Permanent yielding is achieved when the surrounding molecules perform a similar process, which can relieve the local stored elastic energy of the initial kink pair. 

The scaling law in Eq. (10-33) was predicted by Marrucci (1984) by assuming that the disclination density at steady state is set by a balance between the viscous energy density r]Y and the Frank elastic energy density K ja. Since the areal density pa is proportional to Pvh a h/a, this balance is... 

Substitute the solution back into Equation (1.28). The energy per unit length of an isolated disclination in a cylindrical sample of a diameter R is... 

An important conclusion obtained from Equation (1.33) is that the deformation energy is proportional to the square of the disclination strength to. The equation is derived for a special case but the above conclusion is universal, with only the coefficient being different. It is expected that a to = 2 disclination is likely to divide into two to = 1 disclinations or so, in order to achieve a more stable configuration. 

If the inequality of the three elastic constants is taken into account, the disclination configurations are slightly different. Even though the energy for same m is different. The integration of defects produces the textures of liquid crystals. Owing to the thermal fluctuation of the molecules and... 

The energy of deformation of an isolated disclination in a circular layer of radius R and of unit thickness... 

The disclination is supposed to have a core whose energy is not known. To allow for this, we postulate a cut-off radius around the disclination and integrate for distances greater than / <, to obtain... 

However, an important parameter that has been ignored in this approach is the surface tension at the interface. The interfadal tension T can be taken into account in an elementary way as is generally done for crystal screw dislocations. The total energy of the disclination in the one-constant approximation, including the energy at the core surface, is... 

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. ... 

Topologically, it turns out that the helical structure of the cholesteric cannot be deformed continuously to produce a cubic lattice without creating defects. Thus BP I and BP II are unique examples in nature of a regular three-dimensional lattice composed of disclination lines. Possible unit cells of such a disclination network, arrived at by minimizing the Oseen-Frank free energy, are shown in fig. 4.8.3. The tubes in the diagram represent disclination lines, whose cores are supposed to consist of isotropic (liquid) material. Precisely which of these configurations represents the true situation is a matter for further study. 

C-A transition near the core of a disclination Let us suppose that the C-A transition is continuous. Near T. , i.e., for small 9, one may write the free energy density as ... 

---