Chiral nematics elasticity constants

The application of an electric field above the threshold value results in a reorientation of the nematic liquid crystal mixture, if the nematic phase is of negative dielectric anisotropy. The optically active dopant then applies a torque to the nematic phase and causes a helical structure to be formed in the plane of the display. The guest dye molecules are also reoriented and, therefore, the display appears coloured in the activated pixels. Thus, a positive contrast display is produced of coloured information against a white background. The threshold voltage is dependent upon the elastic constants, the magnitude of the dielectric anisotropy, and the ratio of the cell gap to the chiral nematic pitch ... 

The viscosity of a typical cholesteric made by doping a nematic with a modest amount of chiral nematic is much lower (around 1 P or so) than that of the typical pure eholesteric. Perhaps this is because the pitch of the doped nematic is higher than that of the typical pure cholesteric, or because the twist elastic constant of the doped nematic is much lower. 

For the CB0n0.fSj2MB series with n - 7 and 9 a blue phase was observed but not for n = 6 and 8 thus, the chiral properties of these materials do indeed exhibit an odd-even effect as expected. This was rationalised in terms of the smaller pitch for the odd relative to the even membered dimers which arises from the smaller twist elastic constant of odd dimers and is related to their lower orientational order. Surprisingly, the helical twisting power of the dimers in a common monomeric nematic solvent appears to depend solely on the nature of the chiral group, the 2-methylbutyl chiral centre, and not on its environment. Thus similar helical twisting powers are observed for both odd and even membered dimers. We will return to the nature of the phases exhibited by some of these chiral dimers in Sect. 4.4. 

Thus the pitch of a chiral nematic liquid crystal is determined by the ratio of these two elastic constants. The value of the free energy per unit volume with this value for the pitch is... 

The moduli were calculated from the threshold of the Frederiks transition ((4.9) induced by a magnetic (Ax > 0) and electric (Ae < 0)) field in homeotropically oriented liquid crystal layers. The same order of magnitude (10 -10 dyn), which is typical of conventional nematics, has been found for elastic moduli Kn and for other nematic polymers [233, 234]. Unwinding of the helical structure of chiral nematic polymers allowed the elastic constant K22 to be calculated K22 10" dyn for an arylic comb-like copolymer with cholesterol and cyanobiphenyl side-chair mesogens [229]). 

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 ... 

Torsional distortions can now be written in terms of derivatives of a and c, and it is found [10] that nine torsional elastic constants are required for the smectic C phase. Mention should be made of the biaxial smectic C phase, which has a twist axis along the normal to the smectic layers. This helix is associated with a twist in the c-di-rector, and so elastic strain energy associated with this can be described by terms similar to those evaluated for the chiral nematic phase. 

We will now consider in more detail some of the alignment or director field patterns around different defect structures in chiral nematics. Using the simple one elastic constant approximation (i.e., k as for the nematic case above) and the definition of the chiral director (i.e., n=(cos0, sin0, 0), 6=kz, and 0=0 see Eq. (1)) in the free energy density expression, (Eq. 2) gives... 

The chiral nematic is considered incompressible, i.e., of constant density p with a nonpolar unit director n (i.e., i = 1). This implies that the external forces and fields responsible for the elastic deformation, viscous flow, etc., are much weaker than the intermolecular forces giving rise to the local order, i.e., between the chiral molecules. We will consider a volume of material V bounded by a surface 5 v and O) represent linear velocity and local angular velocity, respectively, i.e.. 

Thus the periodic distortion depends critically on the relationship between the chiral nematic pitch and the cell dimensions. Therefore these phenomena are only observed for cells in which the thickness is considerably greater than the helix pitch [135]. For low threshold fields, the diamagnetic anisotropy should be high with low bend and twist elastic constants. 

E phases, calamities 12 edge disclinations, chiral nematics 354 Ehrlich magic bullet, chromonics 984 elastic constants 63, 79 ff... 

Besides normal alkyl chains, alkenyl chains (-C H2 CH CHCH3) are also important because they exhibit particularly beneficial elastic constants and low rotational viscosities desirable for use in supertwisted nematic (STN) displays. Branched alkyl chains Iowct transition temperatures and increase the viscosity. Thus, chiral liquid crystal compounds, which usually incorporate a branched chain alcohol, e.g., 2-methyl butanol or 2-octanol, are relatively viscous. Compounds with two long alkyl chains favor smectic C phase formation (3, K-S, 48°C S.-S 122°C S -N 128°C N-I 166°C), especially when the core system contains some kind of nonlinear stmcture (such as a lateral substituent and/or este group) which aids the formation of tilted phases. 

The values for the cholesteric elastic constants appearing in equation (2.77) are often comparable to those for nematics given by, for example, the values in equations (2.60) and (2.61), and consequently in theoretical investigations they are often assumed to be of the order 10 N to 10 N. For a cholesteric produced by making a dilute solution of chiral molecules in a conventional nematic it is known that K2 is anticipated to be of the order 10 N [110, p.291]. 

In contrast to the extensive experimental investigations of nematic, cholesteric and chiral SmC phases, comparatively little work has been done on tbe characterization of the physical properties of SmA and SmC elastomers. The elongation, A, has been measured as a function of temperature for constant external load for a number of different loads in SmA sidechain elastomers [6]. It was found that A increases monotonically as a function of temperature for a material, which has a SmA-I transition. In addition it was shown that the elasticity modulus, E, decreases monotonically with temperature. X-ray investigations on SmA phases in side-chain LCE have been performed [70]. It was found that for the family of compounds studied, the orientation of the mesogenic groups was always perpendicular to the direction of stretching. 

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