gives
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 HF, θ = 0. For H > HF, reorientation occurs. The expression for HF 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 HF, θ is given approximately by
(3.77a)
where
(3.77b)
For the case where H is abruptly reduced from its value above HF, 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 EF 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 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
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)
