Distortion elastic constant

K and J] are the relevant distortion elastic constant and shear viscosmes, respectively), where the alignment is different form that of the inner area. For this reason, in practice, one always measures a weak shear rate dependence of the apparent viscosity. 

In crystals with the LI2 structure (the fcc-based ordered structure), there exist three independent elastic constants-in the contracted notation, Cn, C12 and 044. A set of three independent ab initio total-energy calculations (i.e. total energy as a function of strain) is required to determine these elastic constants. We have determined the bulk modulus, Cii, and C44 from distortion energies associated with uniform hydrostatic pressure, uniaxial strain and pure shear strain, respectively. The shear moduli for the 001 plane along the [100] direction and for the 110 plane along the [110] direction, are G ooi = G44 and G no = (Cu — G12), respectively. The shear anisotropy factor, A = provides a measure of the degree of anisotropy of the electronic charge... 

All three types of distortions incur an energy penalty, and are defects that often occur near domain walls or boundaries of the enclosing container. The response of the material can then be decomposed into terms based on the elastic constants corresponding to the three types of distortions. 

In this equation, the interaction between active local distortions up to three next nearest neighbour octahedra along the chain was taken into account, while the force constants Km include the effects of interchain interactions. If the interchain contribution is confined by nearest neighbours then only the elastic constant corresponding to the interaction between nearest neighbour octahedra within the chain is renormalized ... 

The three elastic constants are the Frank elastic constants, called after Frank, who introduced them already in 1958. They originate from the deformation of the director field as shown in Fig. 15.52. A continuous small deformation of an oriented material can be distinguished into three basis distortions splay, twist and bend distortions They are required to describe the resistance offered by the nematic phase to orientational distortions. As an example, values for Miesowicz viscosities and Frank elastic constants are presented in Table 15.10. It should be mentioned that those material constants are not known for many LCs and LCPs. Nevertheless, they have to be substituted in specific rheological constitutive equations in order to describe the rheological peculiarities of LCPs. Accordingly, the viscosity and the dynamic moduli will be functions of the Miesowicz viscosities and/or the Frank elastic constants. Several theories have been presented that are more or less able to explain the rheological peculiarities. Well-known are the Leslie-Ericksen theory and the Larson-Doi theory. It is far beyond the scope of this book to go into detail of these theories. The reader is referred to, e.g. Aciemo and Collyer (General References, 1996). 

The spatial and temporal response of a nematic phase to a distorting force, such as an electric (or magnetic) field is determined in part by three elastic constants, kii, k22 and associated with splay, twist and bend deformations, respectively, see Figure 2.9. The elastic constants describe the restoring forces on a molecule within the nematic phase on removal of some external force which had distorted the nematic medium from its equilibrium, i.e. lowest energy conformation. The configuration of the nematic director within an LCD in the absence of an applied field is determined by the interaction of very thin layers of molecules with an orientation layer coating the surface of the substrates above the electrodes. The direction imposed on the director at the surface is then... 

The three independent elastic constants describe the rigidity of the system under uniform distortions but not under arbitrary nonuniform distortions. The elastic energy of the sy.stem,, is a function of the positions of all of the atoms in the crystal, or in particular, of the components , of the displacements of each of the atoms from equilibrium there are 6N such components for the atom pairs. For sufficiently small distortions we can expand that energy for small... 

In our discussion of elastic constants we have imagined uniform distortions of the crystal. An elastic medium can sustain vibrations, which at any instant consist of nonuniform distortions. The normal modes of an elastic continuum arc sound waves (longitudinal and transverse) propagating in the medium, and these normal modes will also exist in the crystal. Indeed, viewing the crystal as an elastic... 

For most distortions, both overlap interactions and Madelung terms contribute to the elastic constants. However, for the distortion associated with C44 in the rocksalt structure, only the Madelung term enters. (There are no changes in nearest-neighbor distance to first order in the strain.) Thus the experimental values can be compared with the Madelung term, and by virtue of the Cauchy relation, c, 2 should take the same value. That contribution has been calculated by Kellerman (1940) and is Values of this expression arc given in Table... 

Timgsten has been of keen theoretical interest for electron band-structure calculations [1.14-1.25], not only because of its important technical use but also because it exhibits many interesting properties. Density functional theory [1.11], based on the at initio (nonempirical) principle, was used to determine the electronic part of the total energy of the metal and its cohesive energy on a strict quantitative level. It provides information on structural and elastic properties of the metal, such as the lattice parameter, the equilibrium volume, the bulk modulus, and the elastic constants. Investigations have been performed for both the stable (bcc) as well as hypothetical lattice configurations (fee, hep, tetragonal distortion). 

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 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. 

In Chapter 7 we discussed changes in energy associated with small uniform changes in the volume of a system the bulk modulus is an elastic constant that, describes the rigidity of the system against such compressions. We now extend the discussion to uniform distortions that lower the symmetry of the system the rigidity of the system against these distortions is described by shear constants. 

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