Liquid Crystals. Iam-Choon Khoo

Liquid Crystals

fields, stresses/constraints from the boundary surfaces, the director will also vary spatially. The characteristic length over which significant variation in the order parameter will occur, in most cases, is much larger than the molecular size. Typically, for distortions of the form shown in Figure 3.1a–c, the characteristic length is on the order of 1 μm, whereas the molecular dimension is on the order of at most a few tens of angstroms. Under this circumstance, as in other similar systems or media (e.g. ferromagnets), the continuum theory is valid.

Schematic illustration of (a) twist deformation in a nematic liquid crystal; (b) splay deformation; (c) bend deformation.

Figure 3.1. (a) Twist deformation in a nematic liquid crystal; (b) splay deformation; (c) bend deformation.

The first principle of continuum theory, therefore, neglects the details of the molecular structures. Liquid crystal molecules are viewed as rigid rods; their entire collective behavior may be described in terms of the director axis ModifyingAbove n With ampersand c period circ semicolon left-parenthesis ModifyingAbove r With right harpoon with barb up right-parenthesis , a vector field. In this picture, the spatial variation of the order parameter is described by

(3.2)

In other words, in a spatially “distorted” nematic crystal, the local optical properties are still those pertaining to a uniaxial crystal and remain unchanged; it is only the orientation (direction) of ModifyingAbove n With ampersand c period circ semicolon that varies spatially.

For nematics, the states corresponding to ModifyingAbove n With ampersand c period circ semicolon and – are indistinguishable. In other words, even if the individual molecules possess permanent dipoles (actually, most liquid crystal molecules do), the molecules are collectively arranged in such a way that the net dipole moment is vanishingly small; that is, there are just as many dipoles up as there are dipoles down in the collection of molecules represented by ModifyingAbove n With ampersand c period circ semicolon .

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 ModifyingAbove n With ampersand c period circ semicolon left-parenthesis ModifyingAbove r With right harpoon with barb up right-parenthesis , will incur much less elastic energy change than the corresponding ones in ordinary solids. It is evident from Figure 3.1a–c that the splay and bend deformations necessarily involve flow of the liquid crystal, whereas the twist deformation does not. We will return to these couplings between flow and director axis deformation in Section 3.5.

Twist, splay, and bend are the three principal distinct director axis deformations in nematic liquid crystals. Since they correspond to spatial changes in ModifyingAbove n With ampersand c period circ semicolon left-parenthesis ModifyingAbove r With right harpoon with barb up right-parenthesis , the basic parameters involved in the deformation energies are various spatial derivatives (i.e. curvatures of , such as nabla times ModifyingAbove n With ampersand c period circ semicolon left-parenthesis ModifyingAbove r With right harpoon with barb up right-parenthesis and nabla dot ModifyingAbove n With ampersand c period circ semicolon left-parenthesis ModifyingAbove r With right harpoon with barb up right-parenthesis , etc.). Following the theoretical formalism first developed by Frank [1], the free‐energy densities (in units of energy per volume) associated with these deformations are given by

(3.3) splay colon f 1 equals one half upper K 1 left-parenthesis nabla dot n right-parenthesis squared comma

(3.4) twist colon f 2 equals one half upper K 2 left-parenthesis n dot nabla times n right-parenthesis squared comma

(3.5) bend colon f 3 equals one half upper K 3 left-parenthesis n times nabla times n right-parenthesis squared comma

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

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