Twist director distortions, nematics

We now consider a twist disclination loop in a twisted nematic. The nematic is supposed to have a planar structure with the director parallel to the xy plane and an imposed twist of q per unit length about the z axis, and the disclination loop of radius R is supposed to be in the xy plane. The director distortions are planar, = cos = sin = 0. On going once round the disclination line at any point on the loop, the director orientation changes by 2tis, the sign of which may be either the same as that of q or opposite. 

The elastic coefficient K22 could be measured according to (2.34) either from the threshold of the twist distortion of the homogeneous alignment induced by a magnetic field, or from the threshold of the initially twisted director alignment [58]. It is also possible to measure the unwinding voltage t unw of the cholesteric to nematic transition... 

The three elastic constants are the Frank elastic constants, called after Frank, who introduced them already in 1958. They originate from the deformation of the director field as shown in Fig. 15.52. A continuous small deformation of an oriented material can be distinguished into three basis distortions splay, twist and bend distortions They are required to describe the resistance offered by the nematic phase to orientational distortions. As an example, values for Miesowicz viscosities and Frank elastic constants are presented in Table 15.10. It should be mentioned that those material constants are not known for many LCs and LCPs. Nevertheless, they have to be substituted in specific rheological constitutive equations in order to describe the rheological peculiarities of LCPs. Accordingly, the viscosity and the dynamic moduli will be functions of the Miesowicz viscosities and/or the Frank elastic constants. Several theories have been presented that are more or less able to explain the rheological peculiarities. Well-known are the Leslie-Ericksen theory and the Larson-Doi theory. It is far beyond the scope of this book to go into detail of these theories. The reader is referred to, e.g. Aciemo and Collyer (General References, 1996). 

The spatial and temporal response of a nematic phase to a distorting force, such as an electric (or magnetic) field is determined in part by three elastic constants, kii, k22 and associated with splay, twist and bend deformations, respectively, see Figure 2.9. The elastic constants describe the restoring forces on a molecule within the nematic phase on removal of some external force which had distorted the nematic medium from its equilibrium, i.e. lowest energy conformation. The configuration of the nematic director within an LCD in the absence of an applied field is determined by the interaction of very thin layers of molecules with an orientation layer coating the surface of the substrates above the electrodes. The direction imposed on the director at the surface is then... 

Here u is the position of a layer plane and z is the position coordinate locally parallel to the director n, where n is parallel to the average molecular axis, which is assumed to remain normal to the layer plane, du/dz = e is the compressional (or dilational) strain. Thus, layer bending and layer compression are characterized by a splay (or layer-bend) modulus K and a compression modulus B. Other kinds of distortion present in nematics, such as bend or twisting of the director n, are not compatible with layers that remain nearly parallel, and hence are forbidden. Equation (10-36) is not invariant to rotations of frame, and its validity is limited to weak distortions a rotationally invariant expression has been given by -Grinstein and Pelcovits (1981).---------------------------------------------------------... 

Here K, K2 and iTs are elastic moduli associated with the three elementary types of deformations splay, twist and bend, respectively. Though the three elastic moduli are of the same order of magnitude the ordering K2 < K < K3 holds for most nematics. As a consequence of the orientational elasticity a local restoring torque (later referred to as elastic torque) acts on the distorted director field which tends to reduce the spatial variations. 

Any spatial distortion of n leads to elastic restoring torques, which are determined in the standard continuum description of nematics (exclusively used in this review) by three elastic constants Ki (splay), K2 (twist) and Ks (bend). In addition, the electric field E gives rise to an electric torque on the director. The balance of these torques, reflected in the resulting equilibrium director configuration, corresponds to the minimum of the orientational free energy J- n). For positive dielectric anisotropy (ca = — x) the dielectric torque (oc is destabihzing in the pla-... 

Fig. 7.3. The deviation of the optic axis in a cholesteric (hard-twisted chiral nematic, p <, where p is the cholesteric pitch and A is the wavelength of light) when an electric field E is applied perpendicular to the helical axis. The cholesteric geometry allows a fiexoelectric polarization to be induced in the direction of E. The plane containing the director, which is perpendicular to the page in the middle figure and is shown in the lower figure, illustrates the splay-bend distortion and the corresponding polarization that arises. (After Rudquist, inspired by Meyer and Patel.

The only curvature strains of the director field which must be considered correspond to the splay, bend, and twist distortions (Fig. 2.17). Other types of deformation either do not change the elastic energy (e.g., the above mentioned pure shears) or are forbidden due to the symmetry. In nematic liquid crystals the cylindrical symmetry of the structure, as well as the absence of polarity (head to tail symmetry) must be taken into account. 

The flexoelectric distortion in Hybrid Aligned Nematic (HAN) configuration was studied [190, 191]. In HAN cells the director is homeotropic on one of the substrates and homogeneous on the other. The measme-ments of transmitted intensity in HAN samples allow us to evaluate for MBBA both a nematic anchoring energy W for the homeotropic orientation VF 6 X 10 erg/cm and the sum of the flexoelectric moduli 11 + 33 —4.5 X 10 dyn / [191]- In an electric field normal to the initial director plane, the flexoelectric effect results in the appearance of twist deformation, observed by the rotation of linearly polarized light [190]. 

The polarity of the alignment layer surface does not have much influence on alignment phenomena for nematic liquid crj talline materials. However, in the case of FLC materials, the polarity of the alignment layer surface shows an important effect. This is because the interaction between the spontaneous polarization and the polarity of the surface becomes important. This matter has been approached theoretically [27]. The stable director orientation in the SSFLC device was determined by minimizing the total free energy of the surfaces and the bulk elastic distortion as functions of cell thickness, cone angle, helical pitch, elastic constant and surface interaction coefficient. Because of the tendency of the direction of the spontaneous polarization to point either into or out of the substrate surface due to polar surface interaction, the director of the molecules twists from the top to the bottom surface. Therefore, the uniform state can only be stabilized in the case of a small surface interaction coefficient. 

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. 

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... 

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