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we obtain the rise time for the Fréedericksz cell as

(3.104)

      and for Θ_d = Θ_о = 0 the rise time of the DAP cell as

(3.105)

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 T_r becomes infinite. Obviously, T_r increases with the viscosity η and the square of the thickness d independent of Θ_d and Θ₀. Figure 3.24 shows the normalized rise time T_rn = T_r/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 Θ_d and Θ₀. The Fréedericksz cell (planar cell) with Θ_d = Θ₀ = 90° exhibits a larger rise time than all of the other cells, including the HAN cell with Θ₀ = 0 and Θ_d = π/2. The pronounced decrease of T_m 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 Θ_d and Θ₀. In this case, the rise time of the DAP cell with Θ_d= 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 T_m with increasing V_n again takes place only for .

Figure 3.24 Normalized rise time T_m versus normalized voltage V_n 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 T_m versus V_n 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 T_m versus normalized voltage V_n 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 T_r in Equation (3.103). For Θ_d = Θ₀ = π/2 we obtain T_d of the Fréedericksz cell as

      (3.107)

      and for Θ_d = Θ₀ = 0 T_d of the DAP cell as

      (3.108)

      whereas the decay time for the HAN cell is obtained by putting Θ_d = 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

(3.110)

Figure 3.26 The ratio T_dn 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

(3.111)

      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

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