Liquid Crystals - Iam-Choon Khoo

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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"/>

      (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 HF, θ = 0. For H > HF, reorientation occurs. The expression for HF 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 HF, 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 HF, 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

      (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 EF 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 VF ~ 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

      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.

      (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)StartLayout 1st Row free energies colon upper F equals one half upper K 1 left-parenthesis nabla dot n right-parenthesis squared plus one half upper K 3 left-bracket n times left-parenthesis nabla times n right-parenthesis right-bracket squared 2nd Row equals one half upper K 1 sine squared phi left-parenthesis StartFraction italic d phi Over italic d z EndFraction right-parenthesis squared plus one half upper K 3 cosine squared phi left-parenthesis StartFraction italic d phi Over italic d z EndFraction right-parenthesis squared comma EndLayout

      (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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