Liquid Crystals. Iam-Choon Khoo
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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 therefore do not enter into the dynamical equations describing the field‐induced effects in nematic liquid crystals.
On the other hand, if the molecules are not strongly anchored to the boundary, that is, the so‐called soft‐boundary condition (Figure 3.3), an applied field will perturb the orientation of the molecules at the cell boundaries. In this case, a quantitative description of the dynamics of the field‐induced effects must account for these surface energy terms. A good account of surface energy interaction may be found in the work of Barbero and Simoni [4], which treats the case of optical‐field‐induced effects in a hybrid aligned nematic liquid crystal cell.
Figure 3.2. A homeotropic nematic liquid crystal with strong surface anchoring: (a) external field off; (b) external field on – only the bulk director axis is deformed.
Figure 3.3. Soft‐boundary condition. The applied field will reorient both the surface and bulk director axis.
From Eq. (3.6) for the free energy, one can obtain the corresponding so‐called molecular fields
where
(3.10)
More explicitly, Eq. (3.9) gives, for the total molecular field associated with splay, twist, and bend deformations,
and torque
(3.12)
(3.14)
with
3.3.