Liquid Crystals. Iam-Choon Khoo

Liquid Crystals

equals upper K 2 StartFraction partial-differential squared theta Over partial-differential z squared EndFraction z plus normal upper Delta chi Superscript m Baseline upper H squared sine theta cosine theta period"/>

In the equilibrium situation, γ₁dθ/dt = 0, and Eq. (3.75a) becomes

(3.75b) upper K 2 StartFraction partial-differential squared theta Over partial-differential z squared EndFraction z plus normal upper Delta chi Superscript m Baseline upper H squared sine theta cosine theta equals 0 period

An interesting result from this equation is the so‐called Freedericksz transition [3]. For an applied field strength less than a critical field H_F, θ = 0. For H > H_F, reorientation occurs. The expression for H_F is given by

(3.76) upper H Subscript upper F Baseline equals left-parenthesis StartFraction pi Over d EndFraction right-parenthesis left-parenthesis StartFraction upper K Over normal upper Delta chi Superscript m Baseline EndFraction right-parenthesis Superscript 1 slash 2

assuming that the reorientation obeys the hard‐boundary (strong anchoring) condition (i.e. θ = 0 at z = 0 and at z = d). For H just above H_F, θ is given approximately by

(3.77a) theta equals theta 0 sine left-parenthesis StartFraction italic pi z Over d EndFraction right-parenthesis comma

where

(3.77b) theta 0 tilde 2 StartFraction left-parenthesis upper H minus upper H Subscript upper F Baseline right-parenthesis Superscript 1 slash 2 Baseline Over upper H Subscript upper F Baseline EndFraction period

For the case where H is abruptly reduced from its value above H_F, to 0, Eq. (3.75a) becomes

(3.78) upper A gamma 1 StartFraction italic d theta Over italic d t EndFraction equals upper K 2 StartFraction partial-differential squared theta Over partial-differential z squared EndFraction period

Writing θ(z, t) = θ₀ sin (πz/d) gives

(3.79) theta equals StartFraction minus pi squared upper K 2 Over d squared gamma 1 EndFraction theta comma

that is,

(3.80) theta 0 left-parenthesis t right-parenthesis equals theta 0 e Superscript negative t slash tau Baseline comma

where the relaxation time constant τ is given by

(3.81) tau equals StartFraction gamma 1 d squared Over pi squared upper K 2 EndFraction period

Most practical liquid crystal devices employ ac electric field. Accordingly, the Freedericksz transition field E_F is given by simply replacing Δχ ^m with Δε; i.e. we have

(3.82a) upper E Subscript upper F Baseline equals left-parenthesis StartFraction pi Over d EndFraction right-parenthesis left-parenthesis StartFraction upper K Over normal upper Delta epsilon EndFraction right-parenthesis Superscript 1 slash 2 Baseline comma

(3.82b) upper V Subscript upper F Baseline equals pi left-parenthesis StartFraction upper K Over normal upper Delta epsilon EndFraction right-parenthesis Superscript 1 slash 2 Baseline period

For 5CB [20, 21], k ~ 10⁻¹¹ N, Δε ~ 11 (ε_|| ~ 16, ε_⊥ ~ 5), ε₀ = 8.85 × 10⁻¹² F/m, Δσ/σ_⊥ ~ 0.5, and V_F ~ 1 V.

In Chapters 6 and 7, we discuss these field‐induced nematic director axis reorientations in detail in the context of electro‐optical switching and display applications.

3.6.2. Reorientation with Flow Coupling

Field‐induced director axis reorientation, accompanied by fluid flow, is quite complicated as it involves much more physical parameters. Consider the interaction geometry shown in Figure 3.13. A homeotropically aligned nematic liquid crystal film is acted on by an electric or a magnetic field in the x‐direction. Let ϕ denote the director axis reorientation angle from the original alignment direction z. Assume hard‐boundary conditions at the two cell walls at z = 0 and at z = d. The flow is in the x‐direction, with a z dependence.

The following are the pertinent parameters involved:

(3.83a) director axis colon n equals left-parenthesis sine phi comma 0 comma cosine phi right-parenthesis comma

Schematic illustration of director axis reorientation causing flows.

Figure 3.13. Director axis reorientation causing flows.

(3.83b) velocity field colon ModifyingAbove v With right harpoon with barb up equals left-bracket v left-parenthesis z right-parenthesis comma 0 comma 0 right-bracket comma

(3.83c)

(3.83d) StartLayout 1st Row elastic torques equals left-bracket upper K 1 sine squared phi plus upper K 3 cosine squared phi right-bracket StartFraction d squared phi Over italic d z squared EndFraction 2nd Row plus left-bracket upper K 1 minus upper K 3 right-bracket sine phi cosine phi left-parenthesis StartFraction italic d phi Over italic d z EndFraction right-parenthesis squared comma EndLayout

(3.83e) field hyphen induced torques equals epsilon 0 normal upper Delta epsilon upper E squared sine phi cosine phi

(3.83f) rotation viscous torques equals gamma 1 StartFraction italic d phi Over italic d t EndFraction comma

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