Disclinations cores

R is of the order of the sample dimensions and fc is the radius of the disclination core, i.e., the distance of molecular order over which Hooke-type elasticity is no longer valid. We must add the disclination core energy to (9.9). This is difficult to calculate exactly, but it does not exceed the energy of the disordered nematic [ k T — Tc)], where is the nematic to isotropic transition temperature. 

Cooling the preparation down to 153.5°C, a low-contrast front crosses the sample and we observe a texture almost identical with that of the smectic C phase. As the temperature falls even further below 153.5°C, disclination cores open up into magnificent five-pointed stars, whose description goes beyond the context of the present introduction. The sample is still in the smectic state, for the focal conics are still clearly visible. The exact nature of this phase is more subtle than those described previously. Layers are organised in a different way. Although it is still liquid and molecules are still tilted, the local order is very... 

This term comes from calculating the usual Frank free energy (i.e., with bend, twist, and splay, and equal elastic constants) outside a disclination core of radius Rc but inside a cutoff radius size R. ... 

Vertical disclination lines normal to horizontal layers in smectic C phases also form nuclei. Polarizing microscopy shows that these nuclei have four branches, and when examined in projection onto the layer plane the observed patterns correspond to + withi= l.Ithasbeendemonstrated that the tilt angle of molecules with respect to the normal to layers is variable, but decreases to zero in the vicinity of the disclination core [103]. This also resembles an escape in the third dimension , and is mainly due to the low value of the dilatation modulus B. 

It is also meaningful to discuss the phenomenon of polymer assembly in the disclination from the viewpoint of entropy. As mentioned above, the orientational order of liquid crystals is induced by the entropy of translation, while the random coUs of polymers are due to the entropy of rotation. The coexistence of a liquid crystal and a polymer causes a conflict between the different entropies. The polymers in the liquid crystal phase thus assemble in the discUnation region, where the orientational order is lowered and the polymers can maximize their conformational entropy. A satisfactory coexistence is then self-sufficiently achieved by cooperation between the blue phase (which must have disclinations) and the polymer (which tends to be excluded from the ordered region). The elastic energy of the liquid crystal orientational field that accumulates around the disclination also plays an important role in the diffusion of the polymer. The curvature of the director increases closer to the disclination core. The elastic energy, which is proportional... 

The textures in homeotropic lamellar phases of lecithin are studied in lecithin-water phases by polarizing microscopy and in dried phases by electron microscopy. In the former, we observe the La phase (the chains are liquid, the polar heads disordered)—the texture displays classical FriedeVs oily streaks, which we interpret as clusters of parallel dislocations whose core is split in two disclinations of opposite sign, with a transversal instability of the confocal domain type. In the latter case, the nature of the lamellar phase is less understood. However, the elementary defects (negative staining) are quenched from the La phase they are dislocations or Grandjean terraces, where the same transversal instability can occur. We also observed dislocations with an extended core these defects seem typical of the phase in the electron microscope. 

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)...

At the late stage of lamella orientation, classical topological defects (dislocations and disclinations) dominate [40, 41] (Fig. 8h and Fig. 9), and their movement and annihilation can be followed in Fig. 8h-i and Fig. 9. The latter presents an example of the apparent topological defect interactions and their transformations. Displayed are two dislocations of PMMA, which have an attractive interaction due to their opposite core sign. Therefore, in the next annealing step the dislocation is shifted... 

The most common morphology observed in current mesophase carbon fibers of moderate modulus (55 to 75 Mpsi, 379 to 517 GPa) is a cylindrical filament with a random-structured core and a radial rim (12) Given the fracture section of Figure 3, with its scroll-like features, the core appears to be an array of +2ir and -ir disclinations. The radial rim of heavily wrinkled layers usually constitutes half or more of the cross section. 

Figure 23. Combination of two +tt disclinations may tilt mesophase layers out of fibrous alignment by formation of a continuous core in the +2tt disclination.

Figure 4. Isolated topological defects in a triangular lattice, (a) Isolated -1 and +1 disclinations. A vector aligned along a local lattice direction is rotated by 60° upon parallel transport around a unit strength disclination. (6) An isolated dislocation. The heavy line represents a Burgers circuit around the dislocation, and the Burgers vector of the dislocation is the amount by which the circuit fails to close. The core of the dislocation is a tightly bound pair of +1 and -1 disclinations (Reproduced from [78] by permission of Oxford University Press.)...

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... 

The nature of the core still remains an interesting unsolved problem. We have seen in 3.1.1 that director distortions have stresses associated with them as given by (3.3.4). In the case of a single disclination the stress is a tension which can be expressed as... 

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... 

Fig. 4.8.3. Unit celk of BP disclination lattices. O is simple cubic, O , 0 + and O — are body-centred cubic. The tubes represent disclination lines whose cores are supposed to be isotropic (liquid) material. (From Berreman. )...

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. 

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