Splay-bend constant

The three (positive) elastic constants Kn (splay), K22 (twist), and K33 (bend) are associated to the three principal deformations. In the surface term, fs is the contribution of the two anchorings, k is the unit vector normal to the surface and directed outward, K13 is the splay-bend constant, and K24 is the saddle-splay constant. The two last surface terms play only for thin films the mere existence of the splay-bend constant K13 is a matter of debate. In the framework of Landau-de Gennes analysis, = K33 and the elastic... 

Splay ( bending or curvature ) defined by a splay constant K, or by a curvature elastic modulus = K h [76],... 

Hur S-T, Gim M-J, Yoo H-J, Choi S-W, Takezoe H (2011) Enhanced thermal stability of liquid crystalline blue phase I with decreasing bend and splay elastic constant ratio K33/K11. Soft Matter, in press (DOI 10.1039/clsm06046e)... 

Problem 10.8 You are measuring the elasticities and viscosities of a room-temperature nematic at reduced temperatures and you find that below about 10°C the twist and bend constants K2 and become very large, while the splay constant Ki retains a modest value. Also, the Miesowicz viscosity t], becomes enormous while r) goes up only modestly. What could explain this behavior ... 

Because of the difficulty with which polymeric nematic monodomains are prepared, there are few measurements of Leslie viscosities and Frank constants for LCPs reported in the literature. The most complete data sets are for PBG solutions, reported by Lee and Meyer (1990), who dissolved the polymer in a mixed solvent of 18% dioxane and 82% dichloromethane with a few percent added dimethylformamide. Some of these data, measured by light scattering and by the response of the nematic director to an applied magnetic field, are shown in Figs. 11-19 and 11-20 and in Table 11-1. While the twist constant has a value of around K2 0.6 x 10 dyn, which is believed to be roughly independent of concentration and molecular weight, the splay and bend constants ATj and K3 are sensitive to concentration and molecular weight. 

The strain increases the energy of the solid as a stress is applied. The distortion of the director in liquid crystals causes an additional energy in a similar way. The energy is proportional to the square of the deformations and the correspondent coefficients are defined as the splay elastic constant, K, twisted elastic constant K22 and bend elastic constant Kx, i.e., the respective energies are the half of... 

Here, a = 1,2 denotes the splay-bend and twist-bend mode, respectively, i i,2,3 are the Prank elastic constants, 771 2 are the rotational viscosities, is the component of the fluctuation wave vector parallel to the director and q the component perpendicular to it. 

This chapter is concerned with experimental measurements of flexo-electric coefficients. After a brief introduction to flexoelectricity in nematic liquid crystaJs, some applications exploiting the flexoelectric effect and the influence of this effect on electrohydrodynamic instabilities are pointed out. Flexoelectricity axises in samples with a splay-bend distortion in the director field and as such its measurement is not as direct as for dielectric constants. The theoretical background needed to analyse electro-optic experiments and extract the flexocoefficients is outlined in Section 2.2. Various experimental techniques that have been developed are described in Section 2.3. These involve cells in which the alignment of the nematic director is homeotropic, or planar or hybrid. In the first case, the interdigitated electrode technique is particularly noteworthy, as it has been used to establish several features of flexoelectricity (1) the effect can arise purely from the quadrupolar nature of the medium, and (2) the dipolar contribution relaxes at a relatively low frequency. 

It is clear that the two polar vectors respect the apolar nature of n. There is also an obvious analogy of the above mechanism with the orientation polarization of a liquid dielectric, which was used by Helfrich to relate the two flexocoefficients with molecular properties. The intrinsic splay or bend can be related to an appropriate angle and molecular dimensions. The relevant component of the electric dipole moment and the curvature elastic constant, viz., and the splay constant Ki or /ux and the bend constant It s figure in the estimation of the flexocoefficients. Nematic liquid crystals made of banana-shaped molecules have been studied only recently, and a comparison of the experimental measurements with the Helfrich formula leads to interesting inferences, as will be mentioned later in this chapter, and covered more thoroughly in the companion Chapter 3 by Jakli et al. ... 

Since the flexoelectric effect is associated with curvature distortions of the director field it seems natural to expect that the splay and bend elastic constants themselves may have contributions from flexoelectricity. The shape polarity of the molecules invoked by Meyer will have a direct mechanical influence independently of flexoelectricity and can be expected to lower the relevant elastic constants.The flexoelectric polarization will generate an electrostatic self-energy and hence make an independent contribution to the elastic constants. In the absence of any external field, the electric displacement D = 0 and the flexoelectric polarization generates an internal field E = —P/eo, where eq is the vacuum dielectric constant. Considering only a director deformation confined to a plane, and described by a polar angle 9 z), and in the absence of ionic screening, the energy density due to a splay-bend deformation reads as ... 

In earlier publications3,6 have presented the results of our measurements of the bend and splay elastic constants of PBLG liquid crystals using the Frederik s magneto-optical distortion procedure. The central experimental necessity of this technique is the measurement of the change in the phase retardation of polarized light... 

The oscillations of I (U) are well seen in the experimental plot. Fig. 11.21. The measurements were made at 27°C on 55 nm thick cell filled with a mixture having ta = 22. From the I (U) curve, the field dependence of the phase retardation 8(17) and the Frederiks transition threshold Uc were obtained. In mm, from Ec = UJd and Fq. (11.56) the splay elastic constant Ku was found. The bend modulus "33 was calculated from the derivative dbldU. The same material parameters may be found for the whole temperature range of the nematic phase. 

The cell parameters thickness (d), twist (3o)> capacitive threshold voltage (Vc) and the liquid crystal material parameters birefringence (n ), dielectric anisotropy (Ea)> pitch (P), bend to splay elastic constant ratio (K3/KX) and twist elastic constant (K2) were varied one at a time over a wide range of values. Transmission versus voltage curves were calculated at 0 and 40 incidence in the principal viewing plane and values for M0, My,... 

Table 2. Dependence of figures of merit on the ratio of bend to splay elastic constants K3/K1.

In the toroidal droplet, the liquid crystal director is aligned along concentric circles on planes perpendicular to a diameter, as shown in Figure 11.15(b). There is a line defect along the diameter of the droplet. There is a rotational symmetry around the defect line. The bend elastic deformation is the only one involved. Toroidal droplets exist when bend elastic constant is smaller than the splay elastic constant otherwise the droplets take the bipolar configuration. Toroidal droplets rarely exist because for most liquid crystals the bend elastic constant is usually larger than the splay elastic constant. Nevertheless, toroidal droplets have been... 

Let be the stretching elasticity of the membrane, and let k, k, and Cq be the bending elasticity, the saddle splay bending elasticity, and the spontaneous curvature of the membrane [5,6]. Later on, attention is restricted to a quasispherical flaccid fluctuating vesicle. Only the case of fluctuations with magnitudes much less than the radius of the vesicle are considered. Let the area of the tension-free flat membrane of the vesicle be Sg, and its volume (assumed constant in time) be If Choose a laboratory frame of reference XYZ with origin O inside the vesicle. Let 0, q> be the... 

These methods have been refined to give a constant splay/bend ratio in purely planar systems, and they allow one to reproduce the well-known aspects of disclinations in pure bend or pure splay situations. These models have also been extended to three dimensions by taking the twist into account, and one can re-find the geometries of director lines in capillary tubes or other cases of escapes in the third dimension . 

The elastic moduli AT,-,/= 1,2,3) of low molecular mass nematics are typically of the order of several piconewton. The splay and bend constants 1 and 33 are often of com-... 

In these geometries, the sample cell is usually observed in the transmitted light with crossed polarizers inserted in 45° orientation to the director tilt plane. While the ordinary polarized light in the birefringent medium always experiences the ordinary index of refraction and is unaffected by the deformation, the optical path of the extraordinary wave is sensitive to the director tilt. Both waves are brought to interference at the analyser and the optical interference can be related in a direct way to the director deflection. Saupe first used this magneto-optical method with splay geometry to determine the splay and bend constants of p-azoxyanisol (PAA) [13]. 

De Jeu et al. [70] have discussed the influence of molecular geometry on the elastic constant ratio. From simple geometrical considerations they stated that the splay/ bend ratio should correlate with the molecular length/width ratio for rigid molecules, i.e. Kj, Kii=L W with Kj, >Kn- (Pre-transitional effects of smectic-like ordering are discussed below.) This view has been confirmed by some experimental data [66]. If long, flexible alkyl chains are incorporated in the molecular model structure, the trend is reversed ( r33< Tii). In practice, both cases are observed. For example, the 4 -n-alkyl-4-cyanobiphenyls (nCB) are characterized by the bend/splay... 

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