Theory of Frederiks Transition

Our task now is to find the threshold field strength for the distortion and the distribution of the director n(z) over the cell thickness above the threshold. This time we shall take an initial homeotropic alignment, case (b) in Fig. 11.15. In its modem form, the theory was developed by Saupe [16]. 

11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals 

Our task is to find an analytical expression for 9(z) at different fields. The scheme is as follows. First we shall write a proper integral equation for the free energy. Then, following the variational procedure discussed in Section 8.3, we compose the Euler equation corresponding to the free energy minimum and solve this differential equation for 9(z). To simphfy the problem we use the one-constant approximation Kii= K22 = Ks3= K.ln our geometry, and only one derivative, namely the bend term with dnjdz, is essential in the Frank free energy form (8.15)  

Thus we have an integral equation forF = F(z, 8, d ). The Euler equation of the general form 

Therefore we obtain the differential equation, which express the balance of the elastic and magnetic torques  

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