Liquid Crystals. Iam-Choon Khoo. Читать онлайн. Hotlib. HOTLIB.NET

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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

(3.7) upper F Subscript d Baseline equals one half upper K left-bracket left-parenthesis nabla dot n right-parenthesis squared plus left-parenthesis nabla times n right-parenthesis squared right-bracket period

Equation (3.6) or its simplified version, Eq. (3.7), describes the deformation of the director axis vector field ModifyingAbove n With ampersand c period circ semicolon left-parenthesis ModifyingAbove r With right harpoon with barb up right-parenthesis in the bulk of the nematic liquid crystal. A complete description should include the surface interaction energy at the nematic liquid crystal cell boundaries. To account for this, the total energy density of the system should be

(3.8) upper F prime Subscript d Baseline equals upper F Subscript d Baseline plus upper F Subscript surface Baseline comma

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, upper F Subscript total Baseline equals integral upper F prime Subscript d Baseline italic d upper V . If external fields (electric, magnetic, or optical) are applied, the corresponding free‐energy terms (see the following sections) will be added to the total free‐energy expression.

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.

Schematic illustration of a homeotropic nematic liquid crystal with strong surface anchoring.

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.

Schematic illustration of soft-boundary condition. The applied field will reorient both the surface and bulk director axis.

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 ModifyingAbove f With right harpoon with barb up using the Lagrange equation [3]. In spatial coordinate component form, we have

(3.9)

where

(3.10) g Subscript italic alpha beta Baseline equals StartFraction partial-differential n Subscript alpha Baseline Over partial-differential x Subscript beta Baseline EndFraction period

More explicitly, Eq. (3.9) gives, for the total molecular field associated with splay, twist, and bend deformations,

(3.11) ModifyingAbove f With right harpoon with barb up equals ModifyingAbove f With right harpoon with barb up Subscript 1 Baseline plus ModifyingAbove f With right harpoon with barb up Subscript 2 Baseline plus ModifyingAbove f With right harpoon with barb up Subscript 3 Baseline comma

and torque ModifyingAbove normal upper Gamma With right harpoon with barb up equals ModifyingAbove n With ampersand c period circ semicolon times ModifyingAbove f With right harpoon with barb up , where

(3.12) splay colon ModifyingAbove f With right harpoon with barb up Subscript 1 Baseline equals upper K 1 nabla left-parenthesis nabla dot n right-parenthesis comma

(3.13) twist colon ModifyingAbove f With right harpoon with barb up Subscript 2 Baseline equals minus upper K 2 left-bracket upper A nabla times n plus nabla times left-parenthesis upper A n right-parenthesis right-bracket comma

(3.14) bend colon ModifyingAbove f With right harpoon with barb up Subscript 3 Baseline equals upper K 3 left-bracket ModifyingAbove upper B With right harpoon with barb up times left-parenthesis nabla times n right-parenthesis plus nabla times left-parenthesis n times ModifyingAbove upper B With right harpoon with barb up right-parenthesis right-bracket comma

with upper A equals ModifyingAbove n With ampersand c period circ semicolon dot left-parenthesis nabla times ModifyingAbove n With ampersand c period circ semicolon right-parenthesis and ModifyingAbove upper B With right harpoon with barb up equals ModifyingAbove n With ampersand c period circ semicolon times left-parenthesis nabla times ModifyingAbove n With ampersand c period circ semicolon right-parenthesis .

3.3.

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