Nematic liquid crystals elastic constants

Allen M P, Warren M A, Wilson M R, Sauron A and Wiliam S 1996 Molecular dynamics calculation of elastic constants in Gay-Berne nematic liquid crystals J. Chem. Phys. 105 2850-8... 

Figure C2.2.11. (a) Splay, (b) twist and (c) bend defonnations in a nematic liquid crystal. The director is indicated by a dot, when nonnal to the page. The corresponding Frank elastic constants are indicated (equation(C2.2.9)).

The theory of nematic liquid crystal deformation, forced by an electric field is well developed and permits to establish the relationship between the threshold voltage U, causing sample orientation, with Ae and elasticity constants of a liquid crystal (Kn). For the main S and B types of deformation the equation is the following27 ... 

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 ... 

Nematic liquid crystals of negative dielectric anisotropy with a low rotational viscosity and a high value of the twist elastic constant k22 are required, see Equations 38-40. Furthermore, a high value for the resistivity and short... 

Assume the nematic liquid crystals to be a homogeneous planar structure, i.e., the directors lie in the x-y plane, making an angle 4> with respect to the x axis. For simplicity, we further suppose that all the three elastic constants are equal, i.e., Ku = K22 = K33 = K. Thus the energy density... 

The energy scale of an elastic distortion around a particle is of order KR, where iC is a typical elastic constant of the nematic liquid crystal [19] and R is the radius of the particle. For a thermotropic liquid crystal, K is approximately 10 N, and for a colloidal particle, i is approximately one micron thus the energy scale is a few thousands k TyWhere is the Boltzmann constant and T the temperature. As a result, the entropy of the particles is negligible compared to the elastic interactions. Under these conditions, the structures formed due to attractive interactions remain stable against thermal fluctuations. 

An important source of error in these calculations is the neglect of short-range order. In particular, the theory fails for the bend and twist elastic constants when smectic-like short-range order is present in the nematic liquid crystal. Under such circumstances these two constants exhibit a critical divergence as the temperature approaches the smectic-nematic point and the light scattering also shows a marked temperature dependence. The present treatment is then inadequate and more elaborate models have been proposed. The phenomenological theory of this aspect of the problem will be discussed in chapter 5. 

The simplest method of measuring the three elastic constants of a nematic liquid crystal is by studying the deformations due to an external magnetic field (Freedericksz and Tsvetkov, Zocher ). The geometry has to be so chosen that the orienting effect of the field conflicts with the orientations imposed by the surfaces with which the liquid crystal is in contact. To develop a static theory of such deformations we apply the equation of... 

In equations (5)-(8), i is the molecule s moment of Inertia, v the flow velocity, K is the appropriate elastic constant, e the dielectric anisotropy, 8 is the angle between the optical field and the nematic liquid crystal director axis y the viscosity coefficient, the tensorial order parameter (for isotropic phase), the optical electric field, T the nematic-isotropic phase transition temperature, S the order parameter (for liquid-crystal phase), the thermal conductivity, a the absorption constant, pj the density, C the specific heat, B the bulk modulus, v, the velocity of sound, y the electrostrictive coefficient. Table 1 summarizes these optical nonlinearities, their magnitudes and typical relaxation time constants. Also included in Table 1 is the extraordinary large optical nonlinearity we recently observed in excited dye-molecules doped liquid... 

A.M. Somoza and P. Tarazona, Density functional theory of the elastic constants of a nematic liquid crystal. Mol. Phys. 72(4), 911-926, (1991). 

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. ... 

M. Majumdar, P. Salamon, A. Jakli, J.T. Gleeson and S. Sprunt, Elastic constants and orientational viscosities of a bent-core nematic liquid crystal, Phys. Rev. E 83(3), 031701/1-8, (2011). doi 10.1103/PhysRevE.83.031701... 

We consider the possible deformations of a nematic liquid crystal confined between two parallel substrates with tangential anchoring condition (parallel to the substrates but no preferred direction on the plane of the substrate). We use the one elastic constant approximation (A n = K22 = Kss = K), the elastic energy is given by... 

J. Wahl and F. Fischer, "Elastic and Viscosity Constants of Nematic Liquid Crystals from a New Optical Method," Mol. Cryst. Liq. Cryst., 22, 359 (1973). 

The Gay-Berne potential has successfully been used for many liquid crystal simulations, and (depending on the parameterisation used and the state points studied) can be used to simulate nematic, smectic-A and smectic-B phases. Variants of the GB potential have also been used to study the biaxial nematic phase (biaxial GB potential) [21] and the smectic C phase (GB with quadrupole) [22]. The GB model has been used also to provide predictions for key material properties, such as elastic constants [23] and rotational viscosities [24], which have an important role in determining how a nematic liquid crystal responds in a liquid crystal display (LCD). 

The initial research on electro-optic phenomena in side-chain polymer liquid crystals concentrated on systems that exhibited nematic phases so that a ready comparison could be made with low molar mass mesogens. Such measurements have established that electro-optic devices are feasible and have allowed elastic constants to be deduced from applications of the continuum theory. This theory, originally derived for low molar mass nematic liquid crystals, defines a relationship for the free energy density F in terms of the elastic constants (/ ) and the director n such that ... 

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... 

The elastic theory of biaxial nematic liquid crystals was developed as the extension of the above-mentioned elastic theory of uniaxial nematics [38]. Fel has discovered a number of volume and surface elastic constants for biaxial nematic liquid crystals in 32-point crystallographic symmetry groups, as well as a number of the corresponding thermodynamic inequalities between these constants, to provide nonnegative values of the elastic energy. 

So, when d = Pq/2, the bands of the domains are parallel to the direction of the rubbing of the glass surfaces, and when d = Pq they are perpendicular to it, i.e., the domains are always perpendicular to the director in the middle of the layer. With thicknesses significantly less than the helical pitch, the planar cholesteric texture is in fact not different from the planar oriented nematic liquid crystal, and the Prederiks transition takes place with a critical field determined by the elastic constant Ku [24]. 

D. Monselesan and H.R. Trebin, Temperature dependence of the elastic constants for biaxial nematic liquid crystals, Phys. Stat Sol. (B) 155, 349 (1989). 

The principal elastic constants for a nematic liquid crystal have already been defined in Sec. 5.1 as splay (A , j), twist(/ 22) and bend(fc33). In this section we shall outline the statistical theory of elastic constants, and show how they depend on molecular properties. The approach follows that of the generalised van der Waals theory developed by Gelbart and Ben-Shaul [40], which itself embraces a number of earlier models for the elasticity of nematic liquid crystals. Corresponding theories for smectic, columnar and biaxial phases have yet to be developed. 

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