href="#ulink_2aafb34d-3c63-51e4-88d3-314bf820d0ea">Figure 3.23. In the field-free state the field of directors is defined by the equilibrium state with minimum free energy. After applying an electric field in the form of a step function the voltage V(t) across the cell has to exceed a threshold Vth before the molecules are able to rotate in order to assume the position imposed by the field. The threshold is caused by the intermolecular forces, which first have to be overcome by the forces of the field. The transition to the new voltage imposed field of directors is called the Fréedericksz transition. The dynamic of this transition (Degen, 1980; Priestley, Wojtowicz and Sheng, 1979) is governed by the interaction between the electric torques forcing the directors into positions parallel for Δε > 0 or perpendicular for Δε < 0 to the electric field and the mechanical torques trying to restore the field-free state. These torques are the only mechanical influences if the molecules do not undergo a translatory movement. A magnetic field is as a rule not applied in LC applications. The transient between the states of the director field is calculated by adding all free energies, and by taking the functional first derivative with respect to the angle Θ in Figure 3.23. The torques are related to splay, twist and bend with the elastic constants K11, K22, K33 and the rotational kinematic viscosity η, as well as to the dielectric torque dependent on Δε. For the derivation of the results we refer to special publications (Labrunie and Robert, 1973; Saito and Yamamoto, 1978). In Saito and Yamamoto (1978), expressions for the rise time Tr and the decay time Td for the reorientation of the LC molecules induced by a voltage step with amplitude V were derived. The results depend upon the tilt angles Θ0 and Θd of the molecules. Tr and Td translate directly in the rise time and the decay time of the luminance as it changes directly with the director field. The results for general angles Θd and Θ0 are:
Figure 3.23 The anchoring of LC molecules at z = 0 and z = d
with
For Θd = Θо = π/2 we obtain the rise time for the Fréedericksz cell as
and for Θd = Θо = 0 the rise time of the DAP cell as
These two results have already been published in Labrunie and Robert (1973).
The threshold voltage in both cases can be detected from the denominators of Tr in the Equations (3.104) and (3.105) as points where Tr becomes infinite. Obviously, Tr increases with the viscosity η and the square of the thickness d independent of Θd and Θ0. Figure 3.24 shows the normalized rise time Trn = Tr/ηd2/π2K11 versus the normalized voltage
Figure 3.24 Normalized rise time Tm versus normalized voltage Vn with various tilt angles θd and θ0. (a) For p-type and (b) n-type nematic LCs
Finally, Figures 3.25(a) and 3.25(b) depict Tm versus Vn with K= (K33− K||)/K|| as a parameter for a p-type and an n-type nematic LC. The Fréedericksz cell in Figure 3.25(a) and the DAP cell in Figure 3.25(b) are independent of K and slower than all the HAN cells with different values of K. The shorter
