Liquid Crystals. Iam-Choon Khoo

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)

      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)

      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)

      where

      (3.77b)

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

      (3.78)

Writing θ(z, t) = θ₀ sin (πz/d) gives

      (3.79)

      that is,

      (3.80)

      where the relaxation time constant τ is given by

      (3.81)

      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)

      (3.82b)

      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)

Figure 3.13. Director axis reorientation causing flows.

      (3.83b)

      (3.83c)

      (3.83d)

      (3.83e)

      (3.83f)