Chiral elastic constant

Thus, the model predicts that thermal fluctuations in the tilt and curvature change the way that the tubule radius scales with chiral elastic constant— instead of r oc (THp) 1, the scaling has an anomalous, temperature-dependent exponent. This anomalous exponent might be detectable in the scaling of tubule radius as a function of enantiomeric excess in a mixture of enantiomers or as a function of chiral fraction in a chiral-achiral mixture. 

The Helfrich-Prost model was extended in a pair of papers by Ou-Yang and Liu.181182 These authors draw an explicit analogy between tilted chiral lipid bilayers and cholesteric liquid crystals. The main significance of this analogy is that the two-dimensional membrane elastic constants of Eq. (5) can be interpreted in terms of the three-dimensional Frank constants of a liquid crystal. In particular, the kHp term that favors membrane twist in Eq. (5) corresponds to the term in the Frank free energy that favors a helical pitch in a cholesteric liquid crystal. Consistent with this analogy, the authors point out that the typical radius of lipid tubules and helical ribbons is similar to the typical pitch of cholesteric liquid crystals. In addition, they use the three-dimensional liquid crystal approach to derive the structure of helical ribbons in mathematical detail. Their results are consistent with the three conclusions from the Helfrich-Prost model outlined above. 

It is also possible that a membrane might have an even lower symmetry than a chiral smectic-C liquid crystal in particular, it might lose the twofold rotational symmetry. This would occur if the molecular tilt defines one orientation in the membrane plane and the direction of one-dimensional chains defines another orientation. In that case, the free energy would take a form similar to Eq. (5) but with additional elastic constants favoring curvature. The argument for tubule formation presented above would still apply, but it would become more mathematically complex because of the extra elastic constants. As an approximation, we can suppose that there is one principal direction of elastic anisotropy, with some slight perturbations about the ideal twofold symmetry. In that approximation, we can use the results presented above, with 4) representing the orientation of the principal elastic anisotropy. 

Stabilization of BP consisting of bent-core N with chiral dopant was also investigated [29-31]. One of the most dramatic results is that BPIII is easily induced by adding a very small amount of chiral dopant such as 1% [29]. The BPIII temperature range was more than 20°. If BSMs show the N phase at room temperature, the BPIII phase over 20° including room temperature is easily realized [30]. Since BSMs have low compared with Kn [25, 32], the effect of elastic constant on the BP stabilization is confirmed [28]. 

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. 

Calculate the free energy per unit length of the double-twist cylinder as a function of qR from 0 to 71, where q is the chirality of the liquid crystal and R is the radius of the doubletwist cylinder. Use the following elastic constants K22, K33 = 2K22, and K24 = 0.5K22-... 

To investigate the temperature effects quantitatively, here we analyze the temperature and frequency effects on Kerr constant. Based on Gerber s model (Equation (14.2)), Kerr constant is governed by the birefringence (An), average elastic constant (k), dielectric anisotropy (Ae) and pitch length (P) of the chiral LC host. The temperature effects on An, k, and As are described by the following relations [46]... 

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]). 

The applications of liquid crystals have unquestionably added incentive to the quest for new liquid crystal materials with superior properties such as viscosity, elastic constants, transition temperatures, and stability. In recent years this has catalyzed work on chiral materials as dopants for ferroelectric displays and for antiferroelectric materials with structures avoiding the number of potentially labile ester groups that were present in the original materials in which... 

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... 

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. 

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