Liquid Crystals. Iam-Choon Khoo

Liquid Crystals

velocity gradient tensor defined by Eq. (3.60).

The viscosity coefficients γ₁ and γ₂ are related to the Leslie coefficient αs by [3]

(3.68a) gamma 1 equals alpha 3 minus alpha 2 comma

(3.68b) gamma 2 equals alpha 2 plus alpha 3 equals alpha 6 minus alpha 5 period

Consider the flow configuration depicted in Figure 3.11. Without the magnetic field and setting ϕ = 0, we have

(3.69a) ModifyingAbove v With right harpoon with barb up equals left-bracket 0 comma 0 comma v left-parenthesis x right-parenthesis right-bracket comma

(3.69b) n equals left-bracket sine theta comma 0 comma cosine theta right-bracket comma

(3.69c) upper A Subscript italic x z Baseline equals one half StartFraction italic d v Over italic d x EndFraction comma

(3.69d) upper N Subscript z Baseline equals omega Subscript y Baseline n Subscript x Baseline equals minus upper A Subscript italic x z Baseline n Subscript x Baseline comma

(3.69e) upper N Subscript x Baseline equals minus omega Subscript y Baseline n Subscript z Baseline equals upper A Subscript italic x z Baseline n Subscript z Baseline period

From Eq. (3.66) the viscous torque along the y‐direction is given by

(3.70)

In the steady state, from which the shear torque vanishes, a stable director axis orientation is induced by the flow with an angle θ_flow given by

(3.71) cosine 2 theta Subscript flow Baseline equals minus StartFraction gamma 1 Over gamma 2 EndFraction period

For more complicated flow geometries, the director axis orientation will assume correspondingly complex profiles.

3.6. FIELD‐INDUCED DIRECTOR AXIS REORIENTATION EFFECTS

We now consider the process of director axis reorientation by an external static or low‐frequency field. Optical field effects are discussed in Chapter 6. The following examples will illustrate some of the important relationships among the various torques and dynamical effects discussed in the preceding sections. We will consider the magnetic field as it does not involve complicated local field effects and other electric phenomena (e.g. conduction). The electric field counterparts of the results obtained here for the magnetic field can be simply obtained by the replacement of Δχ ^m H ² by ΔεE ² (cf. Eqs. [3.26] and [3.29]).

3.6.1. Field‐induced Reorientation Without Flow Coupling: Freedericksz Transition

The following example demonstrates how the viscosity coefficient γ₁ comes into play in field‐induced reorientational effects. Consider pure twist deformation caused by an externally applied field ModifyingAbove upper H With right harpoon with barb up on a planar sample as depicted in Figure 3.12. Let θ denote the angle of deformation. The director axis ModifyingAbove n With ampersand c period circ semicolon is thus given by = (cos θ, sin θ, 0). From this and the preceding equations, the free energy (Eq. 3.4) and elastic torque (Eq. 3.13) are, respectively,

(3.72a) upper F 2 equals StartFraction upper K 2 Over 2 EndFraction left-parenthesis StartFraction partial-differential theta Over partial-differential z EndFraction right-parenthesis squared

and

(3.72b) normal upper Gamma equals upper K 2 StartFraction partial-differential squared theta Over partial-differential z squared EndFraction z period

Schematic illustration of pure twist deformation induced by an external magnetic field H on a planar sample; there is no fluid motion.

Figure 3.12. Pure twist deformation induced by an external magnetic field H on a planar sample; there is no fluid motion.

The viscous torque is given by

(3.73) normal upper Gamma Subscript v i s Baseline equals minus gamma 1 StartFraction italic d theta Over italic d t EndFraction period

The torque exerted by the external field ModifyingAbove upper H With right harpoon with barb up (applied perpendicular to the initial director axis), from Eq. (3.29), becomes

(3.74) normal upper Gamma Subscript e x t Baseline equals normal upper Delta chi Superscript m Baseline upper H squared sine theta cosine theta period

Hence, the torque balance τ equation gives

(3.75a)Скачать книгу