Escape in the third dimension

Figure 1.26. A point singularity escape in the third dimension. (From Codings, 1990. Reproduced by permission of Taylor Francis, )...

The projections of the director on the smectic C layer construct a c-vector (or c-director) field. The distortion of the c-director and the displacement of the layer are two parts of the defects in the smectic C phase. Because the c-director is apolar, there are only the integer disclinations. In addition, there is no escape in the third dimension because the c-director is confined in layers. Neither topological stable singular walls nor points exist in the smectic C sample according to the homotopy argument. 

Figure 24. Transverse and meridian patterns of continuous cores in even-indexed (or non-Mobian) screw dislocations in cholesterics, (a-f) Transformations of transverse patterns for Z=-2 (a-c), 2 (d), -4 (e) and -6 (f), by escape in the third dimension , (g) Meridian. section expected iiom (a-d), L representing the axis of the screw line, (h) Another type of meridian section, also compatible with the production of screw dislocations along the axis L. (i, j) Two patterns deduced from (h), where the locus of the vertical directors consists of L and a set of equidistant rings with Z=-2 (i) or a helix with Z=0 (j), the observation axis being either parallel to L or close to it.

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. 

These methods have been refined to give a constant splay/bend ratio in purely planar systems, and they allow one to reproduce the well-known aspects of disclinations in pure bend or pure splay situations. These models have also been extended to three dimensions by taking the twist into account, and one can re-find the geometries of director lines in capillary tubes or other cases of escapes in the third dimension . 

Figures, (a) Wedgedisclinationof strength-t-l ina capillary the arrangement near the center has a discontinuity and must involve a core, (b) Escape of the disclination in the third dimension the arrangement is continuous with no core [34],...

Fig. 9.9. Possible configurations of a nematic liquid crystal in a capillary tube when molecules are constrained to be normal to the glass surface, (a) Molecules remain in the plane normal to the tube axis and a singularity S = - -l appears at the centre (configuration of Fig. 9.8b). (b) Molecules escape into the third dimension and the singularity disappears, (c) Appearance of a singular point separating two regions in which molecules are infiected in opposite directions...

The alternative solution for — mentioned above also has exactly the same energy as that given by equation (3.370). This means that there are two equivalent directions for escape into the third dimension and this can be interpreted as leading to point defects appearing along the axis of the capillary as observed by Williams, Pieranski and Cladis [279] a schematic diagram of such a director orientation is given in Fig. 3.21(c). This topic is beyond our present discussion and interested readers are referred to the review by Cladis and van Saarloos [43]. 

Right, and if the Flatlanders tried to surround you to keep you in one place, you could escape by moving perpendicularly into the third dimension. In their eyes, you would be a God. ... 

In-depth distribution analysis of chemical composition is a special case of local microanalysis, for which the third (axial) dimension is of primary interest. In principle, this task requires the compositional analysis of thin sections (in the ultimate dimension of monatomic layers) defined on a depth scale. It can be obtained either by non-destructive or destructive techniques. Non-destructive techniques are based on an analytical signal parameter (e.g. intensity and/or energy), which has a weU-deflned dependence on its depth of origin. For example, in electron spectroscopy, non-destructive profiling methods are based on either the energy or the emission angle dependence of the mean escape depth of the emitted electrons e.g. ARXPS). Confocal microscopy... 

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