Liquid Crystals. Iam-Choon Khoo

Liquid Crystals

Figure 3.7. (a) Temperature dependence of the refractive indices of 5CB in the visible spectrum. (b) Temperature dependence of the refractive indices of 5CB in the infrared (10.6 μm) region.

Figure 3.8. Plot of dn_||/dT and dn_⊥/dT for the liquid crystal for temperature near T_c (after Khoo and Normandin [14] and Horn [15]). Solid curve is for visual aid only.

3.5. FLOWS AND HYDRODYNAMICS

One of the most striking properties of liquid crystals is their ability to flow freely while exhibiting various anisotropic and crystalline properties. This dual nature of liquid crystals makes them very interesting materials to study; it also makes theoretical formalism very complex.

The main feature that distinguishes liquid crystals in their ordered mesophases (e.g. the nematic phase) from ordinary fluids is that their physical properties are dependent on the orientation of the director axis ModifyingAbove n With right harpoon with barb up left-parenthesis ModifyingAbove r With right harpoon with barb up right-parenthesis ; these orientation flow processes are necessarily coupled, except in very unusual cases (e.g. pure twisted deformation). Therefore, studies of the hydrodynamics of liquid crystals will involve a great deal more (anisotropic) parameters than studies of the hydrodynamics of ordinary liquids.

We begin our discussion by reviewing first the hydrodynamics of an ordinary fluid. This is followed by a discussion of the general hydrodynamics of liquid crystals. Specific cases involving a variety of flow‐orientational couplings are then treated.

3.5.1. Hydrodynamics of Ordinary Isotropic Fluids

Consider an elementary volume dV = dx dy dz of a fluid moving in space as shown in Figure 3.9. The following parameters are needed to describe its dynamics:

position vector: ,

velocity: ,

density: ,

pressure: , and

forces in general: .

In later chapters where we study laser‐induced acoustic (sound, density) waves in liquid crystals, or generally, when one deals with acoustic waves, it is necessary to assume that the density rho left-parenthesis ModifyingAbove r With right harpoon with barb up comma t right-parenthesis is a spatially and temporally varying function. In this chapter, however, we “decouple” such density wave excitation from all the processes under consideration and basically limit our attention to the flow and orientational effects of an incompressible fluid. In that case, we have

Schematic illustration of an elementary volume of fluid moving at velocity v (r, t) in space.

Figure 3.9. An elementary volume of fluid moving at velocity v (r, t) in space.

(3.50) rho left-parenthesis ModifyingAbove r With right harpoon with barb up comma t right-parenthesis equals constant

For all liquids, in fact for all gas particles or charges in motion, the equation of continuity also holds

(3.51) nabla dot left-parenthesis rho ModifyingAbove v With right harpoon with barb up right-parenthesis equals minus StartFraction partial-differential rho Over partial-differential t EndFraction period

This equation states that the total variation of rho ModifyingAbove v With right harpoon with barb up over the surface of an enclosing volume is equal to the rate of decrease of the density. Since ∂ρ/∂t = 0, we thus have, from Eq. (3.51),

(3.52) nabla dot ModifyingAbove v With right harpoon with barb up equals 0 period

The equation of motion describing the acceleration d ModifyingAbove v With right harpoon with barb up slash italic d t of the fluid elements is simply Newton’s law:

(3.53a) rho StartFraction d ModifyingAbove v With right harpoon with barb up Over italic d t EndFraction equals ModifyingAbove f With right harpoon with barb up period

Studies of the hydrodynamics of liquids may be centered around this equation of motion, as we identify all the various origins and mechanisms of forces acting on the fluid elements and attempt to solve their motion in time and space.

We shall start with the left‐hand side of Eq. (3.53a). Since ModifyingAbove v With right harpoon with barb up equals ModifyingAbove v With right harpoon with barb up left-parenthesis ModifyingAbove r With right harpoon with barb up comma t right-parenthesis ,

(3.53b) StartFraction d ModifyingAbove v With right harpoon with barb up Over italic d t EndFraction equals StartFraction partial-differential ModifyingAbove v With right harpoon with barb up Over partial-differential t EndFraction plus left-parenthesis nabla dot ModifyingAbove v With right harpoon with barb up right-parenthesis ModifyingAbove v With right harpoon with barb up period

The force on the right‐hand side of Eq. (3.53a) comes from a variety of sources, including the pressure gradient −Δρ, viscous force ModifyingAbove f With right harpoon with barb up Subscript v i s , and external fields (electric, magnetic, optical, gravitational, etc.). Equation (3.53a) thus becomes

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