Dynamics of Freedericksz transition in twist geometry

In the twist geometry, the rotation of the liquid crystal director is not coupled to the translational motion of the molecules. The rotation of the liquid crystal director is governed by the over-damped dynamics the elastic and electric torques are balanced by the rotational viseosity torque and the inertial term can be neglected [4]. Mathematically we have 

We first eonsider the turn-off time when the applied field is turned off from the distorted state. The anchoring is infinitely strong and the twist angle at the boundary is 0. The initial twist angle can be expanded in terms of all the possible modes  

The turn-off time r is approximately equal to the relaxation time ti of the slowest mode (m = 1), 

Now we consider the mm-on time. When the applied field is not much higher than the threshold, the twist angle is small and m = 1 is the only mode excited because it costs the least energy. The solution to the dynamic equation is 

Using the approximation that sin0 0 and cos0= 1, and substituting Equation (5.108) into Equation (5.103), we have 

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