Roll patterns oblique

The original, so called 1-d formula of Carr and Helfrich, was later refined and generalized into a 3-d theory capable of calculating the wavevector and describing real, three dimensional patterns (like normal or oblique rolls), other geometries and the dielectric regime [16]. 

Experiments have been carried out on p-(nitrobenzyloxy)-biphenyl [30] and typical patterns in the conductive range at onset are shown in Fig. 6. At low frequencies disordered rolls without point defects have been observed with a strong zig-zag (ZZ) modulation (see Fig. 6a) which can be interpreted as the isotropic version of oblique rolls. Above a critical frequency, a square pattern is observed which retains the ZZ character because the lines making up the squares are undulated. At onset the structure is disordered however, after a transient period defects are pushed out and the structure relaxes into a nearly defect-free, long-wave modulated, quasi-periodic square pattern (see Fig. 6b). 

Figure 9. Snapshots of electroconvection patterns superposed on the Freedericksz state in case C. a oblique rolls, b normal rolls.

Figure 13. Snapshots of nonstandard electroconvection pattern in case G taken with crossed polarizers, a Oblique rolls, b parallel rolls. Contrast was enhanced hy digital processing. The initial director orientation is horizontal. The depicted image is 0.225 x 0.225mm, d = ll/rm.

Patterns in nematics are easily observed by optical means where the anisotropy of the refractive index is exploited. In this way the stripe patterns in electroconvection in the planar geometry are easily discriminated from flexodomains the angle a between the wave vector q of the EC stripes and the preferred direction no a is small (normal or oblique rolls) in contrast to a = 90° (longitudinal stripes) in flexodomains. 

Fig. 4.8. Light diffraction patterns for flexodomains (left) and for conductive oblique EC rolls (right) at different instants within the same period of the driving voltage, slightly above the onset of the instabilities and at / = 0.1 Hz (close to the transition frequency ft).

Taking into account flexoelectricity, it is possible to explain the appearance of a certain angle a, which the Kapustin-Williams domains form in some cases with the y-axis (the usual domain strips are parallel to the 2/-axis, Fig. 5.5). This oblique roll motion was observed in [90] and cannot be explained within the framework of the usual three-dimensional Carr-Helfrich model with strong anchoring at the boundaries [91]. The angle of the domain pattern a was shown [88, 89] to depend on the flexoelectric moduli eii, 633, the dielectric Ae, and the conductive Aa anisotropy. In certain intervals of the en — 633 and 611/633 values the angle A = 0 (the usual Kapustin-Williams domains) or A = 7t/2 (the longitudinal domains, also seen in experiment near the nematic-smectic A transition [91]). 

From the curve CR in Fig. 13.8b we see that at larger e a short wavelength instability comes into play, i.e. a roll system with a different wave-vector, particular in different orientation, starts growing. This may saturate the often-observed rectangular patterns or, for a non-symmetric superposition, lead to the sometimes-observed oblique modulated structures [17, 105]. In order to examine this possibility one would have to test the stability of such patterns by a suitable Galerkin procedure, which was done for normal rolls. However, since there are other possibilities, in particular turbulent states, this approach is not exhaustive and has to be complemented by simulations of the dynamics. 

The intervertebral disks influence vertebral joint motion in a number of ways. The annulus consists of layers of fibroelastic fibers that are attached to the superior and inferior vertebral end-plates. These fibers intertwine in oblique patterns that permit rolling, rotation, and translation of one vertebra on another. The relative... 

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