Elastic chiral nematics

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. 

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... 

In cholesteric (or chiral nematic) liquid crystals the situation is very close to usual nematics. However, due to the chirality of the molecules, the lowest state of elastic energy in cholesterics does no longer correspond to the uniform director orientation, but to the twisted one with a pitch Pq = 27r/yo where yo is the wave vector of cholesteric. Thus for cholesterics the second term in expression (2.24) must be rewritten as... 

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]). 

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. 

External field distortions in SmC and chiral SmC phases have been investigated [38], but the large number of elastic terms in the free-energy, and the coupling between the permanent polarization and electric fields for chiral phases considerably complicates the description. In the chiral smectic C phase a simple helix unwinding Fr6ede-ricksz transition can be detected for the c director. This is similar to the chiral nematic-nematic transition described by Eq. (83), and the result is identical for the SmC phase. Indeed it appears that at least in interactions with magnetic fields in the plane of the layers, SmC and SmC phases behave as two dimensional nematics [39]. 

In achiral nematic phases the first three terms vanish (ki=k =k -0). The (i-[l, m, n ) terms ctm be transformed into surface integrals and, therefore, do not contribute to the equilibrium bulk free energy. The remaining 12 terms correspond to the 12 basic types of bulk elastic deformation of an orthorhombic biaxial non-chiral nematic. 

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 foundations of continuum theory were first established by Oseen [61] and Zocher [107] and significantly developed by Frank [65], who introduced the concept of curvature elasticity. Erickson [17, 18] and Leslie [15, 16] then formulated the general laws and constitutive equations describing the mechanical behavior of the nematic and chiral nematic phases. 

Figure 27. Graphical representation of the splay, twist, and bend elastic deformations in the chiral nematic phase Uj—5], and...

As discussed in Sec. 2.2.2.1, the foundations of the continuum model were laid by Oseen [61] and Zocher [107] some seventy years ago, and this model was reexamined by Frank [65], who introduced the concept of curvature elasticity to describe the equilibrium free energy. This theory is used, to this day, to determine splay, twist, and bend distortions in nematic materials. The dynamic models or how the director field behaves in changing from one equilibrium state to another have taken much longer to evolve. This is primarily due to the interdependency of the director it (r, t) and v (r, /) fields, which in the case of chiral nematics is made much more complex due to the long-range, spiraling structural correlations. The most widely used dynamic theory for chiral... 

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... 

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