Bend elastic constant

Here B is again a compressional elastic constant, is a bend elastic constant and tire elastic constant C results from an elliptical defonnation of tire rods (tliis tenn is absent if tire column is liquid). 

These droplet shape fluctuations are governed by both k and k. The problem was theoretically treated by Milner and Safran [42-44] and values of the bending elastic constants k and i< can be obtained by using experimental methods which are able to resolve fluctuations of the surfactant layer in a microemulsion. 

The theoretical description in terms of spherical harmonics also yields a relation between the size polydispersity index p of the microemulsion droplets and the bending elastic constants [43]. The quantity p is accessible by SANS [51, 52, 59-61]. For polydisperse shells as obtained by using deuterated oil and heavy water for the preparation of the microemulsion (contrast variation), one can account for the droplet polydispersity by applying an appropriate form factor, e.g. containing a Gaussian function to model the size distribution [52, 59, 62]. A possible often-used choice is the following form factor... 

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... 

The three elastic constants are reduced by ksT/D (D is the diameter of the rod). They are the functions of the dimensionless parameter Q = cf>L/D with cf) the volume fraction of the liquid crystalline polymers and L the molecular length. It is shown in the Table 6.2 that the three elastic constants increase as Q and the order parameter S increase. Among them the bend elastic constant K33 varies dramatically, and finally becomes infinite as S approaches unity in the perfectly ordered state. 

For an external field pardlel to tlK helicoidal axis, the cholesteric pitch varies depending on the relative magnitudes of 22 and 33 (the bending elastic constant ) 53). 

Here, K, K2, Ks are, respectively, the splay, twist and bend elastic constants... 

Fig. 3.4.1. The experimental Fteedericksz geometries for the determination of the (a) splay, (b) twist and (c) bend elastic constants of a nematic liquid crystal.

We have treated the distortion as a pure bend but this is not exactly true. Since the thickness of the sample is of the same order as the periodicity of the distortion, the orientation of the director will vary in the z direction also. There will therefore be a non-negligible splay component. To allow for this, it has been suggested that the bend elastic constant 33 should be replaced by... 

To discuss the critical behaviour of the twist and bend elastic constants in the nematic phase, we observe that the Frank free energy expression should include the contribution due to smectic short-range order ... 

Fig. 5.5.1. The temperature dependence of the splay and bend elastic constants, (crosses) and (circles) respectively, in the nematic phase of CBOOA prior to the...

We now discuss the fundamental question of fluctuations in the columnar phase. Let us suppose that the liquid-like columns are along the z axis and that the two-dimensional lattice (assumed to be hexagonal) is parallel to the xy plane. The two basic deformations in such a structure are (i) the curvature deformation (or bending) of the columns without distortion of the lattice and (ii) lattice dilatation (or compression) without columnar curvature. There can also be coupling between the two types of distortion but, as shown by Kleman and Oswald, " the coupling term merely rescales the bend elastic constant of the columns. We shall consider only the vibrations of the lattice in its own plane. The free energy may be written... 

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 ... 

Here fj, is the dipole moment perpendicular to the long axis, K3 is the bend elastic constant, 9q is the kink angle (see Fig. 3.6), a is the length, b is the width, N is the number density of the molecules and ksT is the thermal energy. The first measurements of 63 in calamitic nematics yielded higher values than expected, indicating that not only the dipolar mech-... 

All distances will be given in monomer units- The degree of polymerization is denoted by (. Three elastic constants can be used to measure the elastic energy of deformation of the chain stretch, bend and twist"l"l. In the following, we concentrate on molecules for which the bend elasticity (constant B) is dominant. This leads to the wormlike chain model3, with an end-to-end distance = q i (c /1 i—and q = B/kT, is here the persistence length-... 

In the same paper Picken et al [13] also describe a more general two-dimensional lattice model to deal with this additional aspect. In this model only splay and bend deformations are allowed to occur. The longitudinal relaxation (i.e. the band spacing) will be determined mainly by the bend elastic constant K, as used in the one-dimensional model, and the lateral relaxation is driven by the splay modulus K, They find the result of this model to be not completely satisfactory.. .. it is clear that from the present model the aspect ratio of the bands Lj /L l is expected to be a constant (and equal to (iC /K ) ). This is not in agreement with the experimental results where the lateral correlation length seems to depend on the applied preshear rate, (and the longitudinal correlation length does not). They speculate that Possibly, the applied shear rate influences the details of the initial texture that is formed upon cessation of flow. ... 

For most liquid crystals, the twist elastic constant is smaller than the bend elastic constant Therefore it is possible to reduce the total elastie energy by escaping fi om the bend deformation to the twist deformation as shown in Figure 1.21(c). The liquid crystal director is no longer on the r-z plane but twists out of the plane and is given by... 

If the ratio between the twist and bend elastic constants is smaller than 0.34, the total elastic energy is reduced by escaping from the bend deformation to the twist deformation. 

The dielectric constants of a liquid crystal affect the operation voltage, resistivity, and response time. For example, in a vertical alignment (VA) cell the threshold voltage (V, ) is related to dielectric anisotropy (Ae = en — ex) and bend elastic constant ( 33) as [26] ... 

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... 

---