Dielectric Two-Dimensional Structure in the Frederiks Transition

As we have shown, in the Frederiks effect the nematic director reorients under the action of dielectric torque, defined by the corresponding dielectric contribution 6F to the nematic free energy 

In Chapter 4 we considered the Frederiks transition of the ordinary type, when the director orientation depends only on one coordinate z, measured along the normal to the substrate plane. In some cases, however, the Frederiks transition, leading to modulated structures, seems to be more energetically favorable. 

Consider, for instance, the sign inversion frequency of the dielectric anisotropy As [16, 17]. According to Debye s model the frequency dependence of the dielectric constant s is [2] 

The dielectric anisotropy sign inversion point u = Ui means that 

Let us evaluate the dielectric contribution to the free energy 6Fs at w = Wi, by averaging the free energy density over the period T = 2w/u of the action of an a.c. electric field 

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