Helical phase

Fig. 4. (a) Slater-Koster valence tight-binding and (b) first-principles LDF band structures for [5,5 nanotube. Band structure runs from left at helical phase factor k = 0 to right at K = rr. Fermi level / for Slater-Koster results has been shifted to align with LDF results. 

These coherences will have a helical phase twist in the NMR sample tube and will add to give a net signal of zero in the probe coil during the FID. 

Studies of similar structural transitions for hydrogen within the nanotubes have been carried out by Xia et al. [145], Ying et al. [146], and Ma et al. [147] These results exhibit a wide variety of linear, cylindrical, and helical phases, the existence of which is strongly dependent on the tube radius and the molecular density. Obviously, the transitions between these novel phases are of considerable interest, but space considerations prevent us from discussing the results in detail. Scattering experiments would provide ideal probes of these phases but their interpretation wiU be difficult unless the tubes are aligned. 

Since the discovery of the first liquid crystalline material in 1888, helicity has proven to be one of the most fascinating topics in this field."" Several liquid crystalline phases with helical structure were reported, such as chlolesteric phase, blue phase, ferroelectric and antiferro-electric smectic phases, and helical smectic A phase. In most of these helical phases, at least a fraction of the constituent molecules have an asymmetric carbon, and it was long believed that chirality at a molecular level is a prerequisite to construct chiral architectures at the mesoscopic level. However, Watanabe et al. reported the first example of spontaneous helix formation in liquid crys-... 

Thermal optical microscopy also finds extensive use in the analysis of optically active mesophases where the molecules can form macroscopic helical structures. For instance, the pitch length for helical phases can be determined by measuring the distances between the defect lines formed through the interaction of the helical structure and the surface of the preparation. The microscope can also be used as polarimeter to determine helical twist direction for homeotropically aligned helical mesophases. 

Therefore, at T = T, the relaxation time becomes finite. We meet the same situation in the helical phases as well. 

Rg. 7.37. The exchange function l (q) - 0)]l gj-lf along the c-axis for a number of heavy lanthanide metals in the cone (Er) and helical phases (after Hou-mann and Bjerrum-Mdiler, 1976). 

When the chiral liquid crystal transforms from the unordered isotropic phase to the ordered helical phase, the liquid crystal molecules start to twist with respect to one another. The... 

Carrying this idea over to the helical-isotropic transition, there are two differences. First, we must use disclinations topological line singularities in the director field of the liquid crystal—rather than crystal defects. The second difference is that the helical phase, which has no defects, melts to the blue phase, which is characterized by a stable defect lattice of line disclinations rather than by a random collection of defects. Indeed, there is more than one way to create such a lattice thus BPI and BPII. The helical phase therefore melts to BPI, BPI melts to BPn, and, with a final onset of randomly positioned defects, BPII melts to the isotropic phase. 

Figure 7.3. Heat capacity CpjR versus temperature T in cholesteryl nonanoate (CN). Helical phase (CH), blue phases (BPI, BPII, BPIII), isotropic phase (I) (from Thoen [26]).

The big success of this theory is that cubic phases are shown to indeed be more stable than the helical phase at the helical-isotropic boundary. 

It is well known that the helical phase with a planar texture will selectively reflect light. Selective reflection requires light which has a circularly polar-... 

Once the cubic nature of the blue phase was established, attempts to measure the elastic constants using more sensitive techniques appeared shortly thereafter [25], [96], [97], with those of Kleiman et al. [25] being the most extensive. The latter experiments are very delicate, since the blue phase lattice is both soft (small elastic constants) and weak (small elastic limit). Torsional oscillators configured as cup viscometers were used and the shear distortion was kept to less than 0.02%. Figure 7.12 shows results for both the shear elasticity G and the viscosity rj. These data are taken at various frequencies and must be extrapolated to 0 Hz to obtain the static properties. In the helical phase the extrapolation is somewhat dependent on the model nevertheless, the authors claim that G becomes nearly zero in the helical phase and about 710 dyn cm in BPI. (This figure should be compared to 10" dyn cm 2 in a metal ) However, since BPI also possesses viscosity, its behavior is that of a viscoelastic solid. 

When electric fields are applied to liquid crystals, the molecules tend to align—either parallel to the field (for Sa > 0) or perpendicular (for < 0). For the case of nematics, which already have a preferred direction, the director is simply reoriented without breaking the symmetry. However, the helical phase has two nonequivalent directions the twist axis, and the director, which rotates spatially about the twist axis. If > 0, such a helical director is clearly incompatible with a uniform field. For this case, an increasing field first distorts the helix, then stretches out the pitch, and finally causes the well-known cholesteric-nematic transition [1], If <0, the helical director is only compatible with a uniform field if the twist axis and field are parallel. 

Since the cubic blue phases have three equivalent axes, an applied field breaks the cubic symmetry and creates a preferred axis. But, like the helical phase, blue phases are chiral, being composed of a lattice of double-twist tubes. It is therefore not surprising that applied fields lead to distortion of the lattice (electrostriction) and, for high enough fields, new lower-symmetry phases. These effects occur for both < 0 and a > 0. 

The first experiments with electric fields showed that an increasing field initially lengthens the blue phase lattice parameter and causes birefringence. A theory for weak fields was given by Lubin and Hornreich [102]. With larger fields, the blue phases may transform between themselves, to the helical phase, and ultimately to the nematic phase [23], [103], [104], [105], [106], [107], [108], [109]. Fields also affect the orientation and facetting of blue phase crystallites [81], [82], [110]. 

Figure 7.13. Voltage-temperature phase diagram for a 49.6% mixture of chiral CB15 nematic E9. The helical phase is labeled chol the structure BPE was not yet determined in this article (from Porsch and Stegemeyer [113]).

Figure 7.14. Schematic voltage temperature phase diagram for a 49.8% mixture of CB15 in E9. The shaded region is the coexistence region in which crystals of different shapes appear. C is the helical phase H is the hexagonal BPH phase (from Pier-anski et al. [115]).

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