Helicoidal state

The chains of isotactic polymers (in which the dimensions of R are much greater than those of the hydrogen atom) have the tendency to assume a helicoidal configuration with a pitch which depends on the dimensions of the R group. There is evidence that a helicoidal structure has the tendency to exist also (at least partially) in the amorphous state. It is detectable. 

Liquid crystalline (LC) solutions of cellulose derivatives form chiral nematic (cholesteric) phases. Chiral nematic phases are formed when optically active molecules are incorporated into the nematic state. A fingerprint texture is generally observed under crossed polarizers for chiral nematic liquid crystals when the axis of the helicoidal structure is perpendicular to the incident light (Fig. 2). 

However, in the more general case of an asymmetric particle, the combined action of the lateral and rotational forces may lead to motion along a spatial, for instance, spiral trajectory. At the same time, a steady-state settling trajectory with helicoidal (propeller-like) symmetry remains rectilinear, notwithstanding the body rotation [179]. 

Fig. 1. Left Birefringent fluid liquid crystalline solution of poly-y-benzyl-L-glutamate in /n-cresol. Right Birefringent solid film of poly-y-benzyl-L-glutamate plasticized by 3,3 -dimethyl bisphenyl. Retardation lines characteristic of a helicoidal supramolecular structure are observed in the photomicrographs of both the liquid and solid states of this synthetic polypeptide.

Lyotropic cellulosics mostly exhibit chiral nematic phases, although columnar phases have also been observed. The molecules in the thermotropic state also form chiral nematic order, but it is sometimes possible to align them in such a way that a helicoidal structure of a chiral nematic is excluded. Upon relaxation they show banded textures. Overviews on lyotropic LC cellulosics are... 

The optical properties of chiral nematic phases are closely related to their supermolecular Structures, as stated by the considerations of de Vries. In particular, the planar textures exhibit beautiful colors correlated to the pitch P of the helicoidal structures by Eq. (1), if the selective reflection wavelength lies in the visible range, and many examples are shown in Fig. 2. 

As stated in the introduction, the first LC thermosets were prepared in 1973 by Blumstein and Strzelecki et al. [6,53-56] via the thermal polyaddition of acrylate-terminated schiff-base mesogens within the mesophase. Strzelecki et al. also reported the first preparation of networks with a helicoidal structure. Copolymerization of the mixture of LC diacrylate, LC monoacrylates having a mesogen, and/or a chiral group within the cholesteric mesophase resulted in the LC networks with a characteristic cholesteric texture. 

Fig. 61. Tentative magnetic phase diagram of GdHj (open symbols) and GdDj. (solid symbols) in the relaxed (solid lines) and in the quenched (dotted lines) state obtained from resistivity studies (Vajda et al. 1991a). iC j, incommensurate or magnetically SRO phases hel, helicoidal phase. For the various critical temperatures, see text and table 23b.

One optical feature of helicoidal structures is the ability to rotate the plane of incident polarized light. Since most of the characteristic optical properties of chiral liquid crystals result from the helicoidal structure, it is necessary to understand the origin of the chiral interactions responsible for the twisted structures. The continuum theory of liquid crystals is based on the Frank-Oseen approach to curvature elasticity in anisotropic fluids. It is assumed that the free energy is a quadratic function of curvature elastic strain, and for positive elastic constants the equilibrium state in the absence of surface or external forces is one of zero deformation with a uniform, parallel director. If a term linear in the twist strain is permitted, then spontaneously twisted structures can result, characterized by a pitch p, or wave-vector q=27tp i, where i is the axis of the helicoidal structure. For the simplest case of a nematic, the twist elastic free energy density can be written as ... 

The use of CD as a probe of liquid crystalline properties has been rather limited. A helicoidal structure will induce circular dichroism at an absorption band of a nonchi-ral chromophore, and the magnitude of the induced CD absorption depends on the pitch of the helix, the sign of the CD changing if pitch inversion occurs. This technique has been used to investigate phase transitions between ferrielectric, ferroelectric and anti-ferroelectric smectic C states of MHPOBC... 

This behavior is also known in the solid state, e.g., in the chiral structure NaN02, and the order is called helicoidal antiferroelectric. A shorter useful name is helielec-tric. The helielectric smectic C has zero macroscopic polarization (like an antiferroelectric), no hysteresis, no threshold, and no bistability. However, by an artifice it can be turned into a structure with very different properties. This is illustrated in Fig. 16. If the smectic layers are made perpendicular to the confining glassplates, there is no boundary condition compatible with the... 

This is an important result as it means that the helicoidal C state is a divergence-free vector structure P(r) in space. Hence we have no appearance of space charges anywhere. Thus not only does the helix cancel the macroscopic polarization and thereby any external coulomb fields, although we have a local polarization P every where, but the fact that V P=0 also secures that there are no long range coulomb interactions in... 

As we have repeatedly stressed, flexoelec-tricity is a phenomenon that is a priori independent of chirality. But we have also seen that some flexoelectric deformations do have a tendency to occur spontaneously in a chiral medium. All except the helical C state are, however, suppressed, because they are not space-filling. A flexoelectric deformation may of course also occur spontaneously in the nonchiral case, namely, under exactly the same conditions where the deformation is space-filling and does not give rise to defect structures. In other words, in creating the twist-bend structure which is characteristic of a helielectric. Imagine, for instance, that we have mesogens which have a pronounced bow shape and, in addition, some lateral dipole. Sterically they would prefer a helicoidal structure, as depicted in Fig. 52, which would minimize the elastic... 

Figure 18. Stroboscopic photomicrographs recorded at (a) 0.145 V (253 ras), (b) 0.290 V (256 ms) and (c) 0.339 V (257 ms) by linearly changing the voltage from -48.4 V (0 ms) to 48.4 V (500 ms) in a 4.6 pm thick, homogeneous cell of (5)-MHPOBC at 80°C. One of the ferroelectric states is chosen to be dark, overshooting to which is actually observed as clearly seen in (b) and (c). U1 and Ur refer to ferroelectric states tilted uniformly to the left F(-) and to the right F(+), respectively, and 3rd refers to helicoidal antiferroelectric state (A).

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