Elastic Properties of Liquid Crystals

In nematic phase, the liquid crystal director it is uniform in space in the ground state. In reality, the liquid crystal director it may vary spatially because of confinements or external fields. This spatial variation of the director, called the deformation of the direetor, eosts energy. When the variation occurs over a distance much larger than the moleeular size, the orientational order parameter does not change, and the deformation ean be deseribed by a continuum theory in analogue to the classic elastic theory of a solid. The elastie energy is proportional to the square of the spatial variation rate. 

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Stannarius R, Schuring H, Tolksdorf C, Zentel R (2001) Elastic properties of liquid crystal elastomer baUoims. Mol Cryst Liq Cryst 364 305-312. doi 10.1080/10587250108024999... 

The elastic properties of liquid crystal phases can be modelled using continuum theory. As its name su ests, this involves treating the medium as a continuum at the level of the director, neglecting the structure at the molecular scale. 

Elasticity is a macroscopic property of matter defined as the ratio of an applied static stress (force per unit area) to the strain or deformation produced in the material the dynamic response of a material to stress is determined by its viscosity. In this section we give a simplified formulation of the theory of torsional elasticity and how it applies to liquid crystals. The elastic properties of liquid crystals are perhaps their most characteristic feature, since the response to torsional stress is directly related to the orientational anisotropy of the material. An important aspect of elastic properties is that they depend on intermolecular interactions, and for liquid crystals the elastic constants depend on the two fundamental structural features of these mesophases anisotropy and orientational order. The dependence of torsional elastic constants on intermolecular interactions is explained, and some models which enable elastic constants to be related to molecular properties are described. The important area of field-induced elastic deformations is introduced, since these are the basis for most electro-optic liquid crystal display devices. 

An important aspect of the macroscopic structure of liquid crystals is their mechanical stability, which is described in terms of elastic properties. In the absence of flow, ordinary liquids cannot support a shear stress, while solids will support compressional, shear and torsional stresses. As might be expected the elastic properties of liquid crystals are intermediate between those of liquids and solids, and depend on the symmetry and phase type. Thus smectic phases with translational order in one direction will have elastic properties similar to those of a solid along that direction, and as the translational order of mesophases increases, so their mechanical properties become more solid-like. The development of the so-called continuum theory for nematic liquid crystals is recorded in a number of publications by Oseen [ 1 ], Frank [2], de Gennes and Frost [3] and Vertogen and de Jeu [4] extensions of the theory to smectic [5] and columnar phases [6] have also been developed. In this section it is intended to give an introduction to elasticity that we hope will make more detailed accounts accessible the importance of elastic properties in determining the... 

Physical properties of liquid crystals are generally anisotropic (see, for example, du Jeu, 1980). The anisotropic physical properties that are relevant to display devices are refractive index, dielectric permittivity and orientational elasticity (Raynes, 1983). A nematic LC has two principal refractive indices, Un and measured parallel and perpendicular to the nematic director respectively. The birefringence An = ny — rij is positive, typically around 0.25. The anisotropy in the dielectric permittivity which is given by As = II — Sj is the driving force for most electrooptic effects in LCs. The electric contribution to the free energy contains a term that depends on the angle between the director n and the electric field E and is given by... 

We have omitted discussing such interesting properties of liquid-crystal solutions as the Frank elastic constants, the Leslie viscosity coefficients, cholesteric pitch, textured structure (or defects), and rheo-optics. Some of them are reviewed in recent literature [8,167], but the level of their experimental and theoretical studies still remains largely qualitative. 

Bent-core liquid crystal elastomers have shown to exhibit large values of flexoelectricity as many as three orders of magnitude larger than liquid crystal elastomers containing rod-shaped molecules [44]. These high responses are attributed to a piezoelectric phenomenon. Liquid crystal elastomers combine elasticity and flexibility inherent to rubbers and the optical and electrical properties of liquid crystals, and are promising materials for applications such as electrooptics, flexible electronics, and actuator technologies for biomedical applications. 

Today the electrooptical properties of liquid crystals form well-developed branches both in the physics and technology of liquid crystals. In addition, electrooptical measurements are the basis of a number of precise methods for determining the physical parameters of a material, such as its elastic and viscosity coefficients, optical anisotropy, spontaneous polarization, flexoelectric coefficients, anchoring energies at interfaces, etc. 

The relationship between macroscopic properties and molecular properties is a major area of interest, since it is through manipulation of the molecular structure of me-sogens, that the macroscopic liquid crystal properties can be adjusted towards paricu-lar values which optimize performance in applications. The theoretical connection between the tensor properties of molecules and the macroscopic tensor properties of liquid crystal phases provides a considerable challenge to statistical mechanics. A key factor is of course the molecular orientational order, but interactions between molecules are also important especially for elastic and viscoelastic properties. It is possible to divide properties into two categories, those for which molecular contributions are approximately additive (i.e. they are proportional to the number density), and those properties such as elasticity, viscosity, thermal conductivity etc. for which intermolecular forces are responsible, and so have a much more complex dependence on number density. For the former it is possible to develop a fairly simple theory using single particle orientational order parameters. 

The contribution of translational order parameters to the anisotropy of physical properties of liquid crystals has not been studied in detail. Evidence suggests that there is a very small influence of translational ordering on the optical properties, but effects of translational order can be detected in the measurement of dielectric properties. There are strong effects in both elastic properties and viscosity, but the statistical theories of these properties have not been extended to include explicitly the effects of translational order. 

Equations (4.8)-(4.10) have been solved in simple steady state shear flow using Mathematica software (Leonov and Chen 2010). The stress components are expressed as function of shear rate y with the values of constitutive parameters 00, a,p, r, r2,Xe, and t o. Here Oq and t]o represent relaxation time and zero shear viscosity respectively. The other parameters XgandXv represent the tumbling for elasticity and viscosity. Rest of the characteristic parameters a,p,ri,r2 represent anisotropic properties of liquid crystal polymers. Among the eight parameters, only relaxation time and zero shear viscosity are determined from experimental data. The other six parameters can be obtained from cinve fitting data using the Mathematica software. 

The coupling of the axis of rod-shaped molecules is the basic characteristic that leads to a liquid which displays many of the. properties of a crystalline solid, such as birefringence and electric and diamagnetic anisotropy. The response of liquid crystals to applied forces may be weakly elastic but, in nematics at least, true flow is retained. This combination of anisotropy and liquid flow leads to some interesting properties of liquid crystals, since it becomes possible to reorient the director by imposing an external electric field. 

Generally speaking, we can divide liquid ciystalline phases into two distinctly different types the ordered and the disordered. For the ordered phase, the theoretical framework invoked for describing the physical properties of liquid crystals is closer in form to that pertaining to solids it is often called elastic continuum theory. In this case various terms and definitions typical of solid materials (e.g., elastic constant, distortion energy, torque, etc.) are commonly used. Nevertheless, the interesting fact about liquid crystals is that in such an ordered phase they still possess many properties typical of liquids. In particular, they flow like liquids and thus require hydrody-namical theories for their complete description. These are explained in further detail in the next chapter. 

Stannarius R 1998 Elastic properties of nematic liquid crystals 1998 Handbook of Liquid Crystals Vol 2A. Low Molecular Weight Liquid Crystals led D Demus, J Goodby, G W Gray, Fl-W Speiss and V Vill (New York Wiley-VCH)... 

The earliest approach to explain tubule formation was developed by de Gen-nes.168 He pointed out that, in a bilayer membrane of chiral molecules in the Lp/ phase, symmetry allows the material to have a net electric dipole moment in the bilayer plane, like a chiral smectic-C liquid crystal.169 In other words, the material is ferroelectric, with a spontaneous electrostatic polarization P per unit area in the bilayer plane, perpendicular to the axis of molecular tilt. (Note that this argument depends on the chirality of the molecules, but it does not depend on the chiral elastic properties of the membrane. For that reason, we discuss it in this section, rather than with the chiral elastic models in the following sections.)... 

Chandrasekhar, 1977). This cooperative behavior results in weak elastic properties. Then, the application of an electric field can easily change the molecular orientation, which is initially fixed by the mechanical boundary conditions. The concomitant changes in the optical properties form the basis of liquid crystal displays (LCD). 

The big difference between normal isotropic liquids and nematic liquids is the effect of anisotropy on the viscous and elastic properties of the material. Liquid crystals of low molecular weight can be Newtonian anisotropic fluids, whereas liquid crystalline polymers can be rate and strain dependent anisotropic non-Newtonian fluids. The anisotropy gives rise to 5 viscosities and 3 elastic constants. In addition, the effective flow properties are determined by the flow dependent and history dependent texture. This all makes the rheology of LCPs extremely complicated. 

Althou, in principle, the general theory is superior to the band theory, the appropriate techniques for its application are not yet developed sufficiently well and a unified approach to a quantitative description of the structures and the physical properties of crystals is still lacking. The less generally valid band theory can at present give clearer and more convincing explanations of changes in the physical properties of crystals caused by variations in the temperature, pressure, magnetic and electric fields intensities, impurity concentrations, etc. However, many problems encoimtered in the study of chemical bonds in crystals cannot be considered within the framework of the standard band theory. They include, for example, determination of the elastic, thermal, and thermodynamic properties of solids, as well as the structure and properties of liquid and amorphous semiconductors. 

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