Freedericksz Transition Revisited

Case 1 One-elastic-constant approximation. For the simplest case under one-elastic-constant approximation and strong boundary anchoring condition, the free energy of the NLC layer can be obtained by integrating over the sample thickness d 

The relation between the reorientation angle and external field can be derived by integrating Equation (6.17), 

0 j is the maximnm reorientation angle. Constant c can be determined by minimizing the energy G. Then we derive the final expression relating 0 and E, 

The right-hand side of Equation (6.18) is an elliptic integral of the first kind, which is tabulated for values of/c = sin0 j 1. For relatively small reorientation angles, the elliptic integral can be expanded as a series. Equation (6.18) can be approximated to the second-order. 

Equation (6.20) shows that a deformation with 0 0 occurs only if the external fields exceed the threshold E Ep, where 

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