Splay elastic bending constant

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-... 

Since in second-order expansion all terms are preceded by the constant factor S, the Li are order-parameter independent, and all elastic constants are proportional to the square of the order parameter. There are only two invariants in this expansion, and hence this approximation cannot describe correctly the physical situation but leads to equal values for the splay and bend constants. An expansion of the free energy to higher order terms in first derivatives has been performed by, among others, Schiele and Trimper [289], Berreman and Meiboom [290], Poniewierski and Sluckin [291] and Monselesan and Trebin [292]. 

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 ... 

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... 

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...

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... 

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. ... 

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.

Although the dynamics of Freedericksz transition in splay geometry, bend geometry, and twisted geometry is more complicated, the response time is still of the same order and has the same cell thickness dependence. The rotational viscosity coefficient is of the order O.IN - s/m. When the elastic constant is 10 "N and the cell thickness is 10pm, the response time is of the order 100 ms. Faster response times can be achieved by using thinner cell gaps. 

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... 

Fig. 19. Elastic constants ku and (i.e. splay and bend) as a function of concentration for a polysiloxane smectic copolymer of the type shown in Fig. 3(b) dissolved in a low molar mass cyanobiphenyl liquid crystal host.

The behavior of weakly twisted structures depends on the relative values of the elastic constants in (5.3) and (5.4). As we shall see in the next section, splay and bend distortions are often relaxed by twist. It is therefore important to know the elastic constants for different types of deformations these constants are specified by molecular structures and interactions. 

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... 

This suggests that the elastic constants for splay and bend should be equal. The difference between, and 33 has been identified as due to a term in (P2(cos0)) (P4(cos0)), and while this usually leads to 33> i i> particular choices of intermolecu-lar potential can result in 3 3 < j or indeed... 

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