Название: Liquid Crystals
Автор: Iam-Choon Khoo
Издательство: John Wiley & Sons Limited
Жанр: Техническая литература
isbn: 9781119705796
isbn:
3.2.2. Elastic Constants, Free Energies, and Molecular Fields
Upon application of an external perturbation field, a nematic liquid crystal will undergo deformation just as any solid. There is, however, an important difference. A good example is shown in Figure 3.1a, which depicts a “solid” subjected to torsion, with one end fixed. In ordinary solids, this would create very large stress, arising from the fact that the molecules are translationally displaced by the torsional stress. On the other hand, such twist deformations in liquid crystals, owing to the fluidity of the molecules, simply involve a rotation of the molecules in the direction of the torque; there is no translational displacement of the center of gravity of the molecules, and thus, the elastic energy involved is quite small. Similarly, other types of deformations such as splay and bend deformations, as shown in Figure 3.1b and c, respectively, involving mainly changes in the director axis
Twist, splay, and bend are the three principal distinct director axis deformations in nematic liquid crystals. Since they correspond to spatial changes in
(3.3)
(3.5)
where K1, K2, and K3 are the respective Frank elastic constants.
In general, the three elastic constants are different in magnitude. Typically, they are on the order of 10−6 dyne in centimeter‐gram‐second (cgs) units (or 10−11 N in meter‐kilogram‐second [mks] units). For p‐methoxybenzylidene‐p′‐butylaniline (MBBA), K1, K2, and K3 are, respectively, 5.8 × 10−7, 3.4 × 10−7, and 7 × 10−7 dyne. For almost all nematics K3 is the largest, as a result of the rigid‐rod shape of the molecules.
In general, more than one form of deformation will be induced by an applied external field. If all three forms of deformation are created, the total distortion free‐energy density is given by
This expression, and the resulting equations of motion and analysis, can be greatly simplified if one makes a frequently used assumption, namely, the one‐constant approximation (K1 = K2 = K3 = K). In this case, Eq. (3.6) becomes
Equation (3.6) or its simplified version, Eq. (3.7), describes the deformation of the director axis vector field
(3.8)
where the surface energy term is dependent on the surface treatment. In other words, the equilibrium configuration of the nematic liquid crystal is obtained by a minimization of the total free energy of the system,
Under the so‐called hard‐boundary condition, in which the liquid crystal molecules are strongly anchored to the boundary and do not respond to the applied perturbation fields (see Figure 3.2), the surface energy may thus be regarded as a constant; the surface interactions СКАЧАТЬ