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3.2.8 Switching dynamics of untwisted nematic LCDs
ОглавлениеWe assume that the LC molecules are anchored on the surface at z = 0 at an angle Θ0 to the normal and at z = d under the angle Θd, as shown in 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 calculated from Equation (3.103) for a p-type nematic with Δε = 0.55 and K= 0.16 for various angles Θd and Θ0. The Fréedericksz cell (planar cell) with Θd = Θ0 = 90° exhibits a larger rise time than all of the other cells, including the HAN cell with Θ0 = 0 and Θd = π/2. The pronounced decrease of Tm at , as shown in Equation (3.104), is also clearly visible in Figure 3.24(a). Figure 3.24(b) depicts the normalized rise time versus the normalized voltage , again calculated from Equation (3.103), but this time for an n-type nematic LC with Δε = −0.12 and K = 0.43 for various angles Θd and Θ0. In this case, the rise time of the DAP cell with Θd= Θ0 = 0 exceeds the rise time of all other cells. Thus, in both cases, the Fréedericksz cell and the DAP cell are slower than all of the other cells with different combinations of pretilt angles. The decrease of Tm with increasing Vn again takes place only for .
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 rise time of the HAN cell over the other cells can phenomenologically be explained by the fact that half of the molecules are already rotated in the direction imposed by the field, horizontally for Δε < 0 and vertically for Δε > 0. The decay time Td is derived in Saito and Yamamoto (1978) as
(3.106)
Figure 3.25 Normalized rise time Tm versus normalized voltage Vn with the ratio K of elastic constants as parameter (a) for p-type and (b) n-type nematic LCs
which is independent of the applied voltage V and of Δε, and has the same factor outside the magnitude sign as Tr in Equation (3.103). For Θd = Θ0 = π/2 we obtain Td of the Fréedericksz cell as
(3.107)
and for Θd = Θ0 = 0 Td of the DAP cell as
(3.108)
whereas the decay time for the HAN cell is obtained by putting Θd = π/2 and Θ0 = 0, yielding
(3.109)
A comparison between the HAN cell and the Fréedericksz cell which is valid for p-type nematic LCs reveals for the same cell-thickness
Figure 3.26 The ratio Tdn in Equation (3.110) versus K for a p-type nematic LC
For K > 0 the decay time of the HAN cell is shorter, and for − 1 < K< 0 longer than that of the Freedericksz cell, whereas they are equal for K= 0 reached by K11 = K33. Comparing the HAN cell to the DAP cell, which applies for n-type nematic LCs, yields for the same cell thickness
In contrast to the Fréedericksz cell, the decay time of the HAN cell for K > 0 is longer, and for − 1 < K < 0 shorter than that of the DAP cell. Again, for K = 0 the two decay times become equal.
The ratios Tdn in Equations (3.110) and (3.111) are plotted in Figures 3.26 and 3.27 versus K with Θd and Θ0 as parameters. For the p-type nematic in Figure 3.26, Tdn decreases, and for n-type nematics in Figure 3.27 it increases with increasing K. For p-type nematics and for K > 0 the Fréedericksz cell exhibits the longest decay time, whereas for the n-type nematics and for K > 0 the DAP cell has the shortest decay time. The reflective version of the Fréedericksz and the DAP cell with cell gap d/2 have rise times and decay times four times smaller than their transmissive counterparts because of the proportionality to d2. The reflective HAN cell requires the same thickness d as the transmissive Freedericksz and DAP cells, and hence does not share the enhancement of switching speed of the other reflective cells.
Figure 3.27 The ratio Tdn in Equation (3.111) versus K for an n-type nematic LC
