The Fréedericksz-transition is one of the most famous electro-optic switching mechanisms for nematic liquid crystals and the foundation of most modern LC Display devices. The importance of the Fréedericksz-transition arises from the optical properties of the Fréedericksz-cells: When sandwiched between crossed polarizers, the light transmittance through the sample can be controlled electrically!

The LC test cell represents a optical switch, which either allows light to pass through the pixel or block all all light. When miniaturized, this optical switch is perfectly suited for high-resolution display applications. In order add color to the image, colored filters are attached onto single pixels, as can be seen in this microscopic photograph of an old LC-Display with 100x magnification.

### Explanation model

For a single rod-like molecule in free space the minimum electrostatic energy is given by an orientation of the long axis of the molecule parallel to the electric field for molecules with $\epsilon_{l}>\epsilon_{t}$, as this maximizes the electric displacement field $D$. However, the volume of the nematic phase is always confined by substrates, and hence the director field distribution is connected to the boundary conditions at the surfaces. In the equilibrium state, the director field distribution minimizes the free energy $F$ of the volume, which in the absence of external fields is given by equation

Any field-induced reorientation of molecules then creates additional distortions in the director field, which increase the elastic part of the free energy density $f_{elast}$ and therefore give rise to an elastic restoring torque that tends to resist this reorientation. As a result, the equilibrium director field distribution under the influence of electric fields does not only depend on the applied electric field, but also on the elastic properties of the liquid crystal and the boundary conditions at the surfaces.
In a typical experiment, an external electric field is used to distort an uniformly aligned nematic liquid crystal. Deformations in the director field distribution affect the effective birefringence of the sample and therefore the optical properties. Starting with a homogeneously aligned director field configuration between crossed polarizers, the change in birefringene can be used as a light switch in the pixels of LC-Displays.

This field-induced reorientation of the nematic director was at first investigated in detail by V. Fréedericksz and V. Zolina and is therefore referred to as Fréedericksz-transition. There are three basic geometries for the Fréedericksz-transition which correspond to a splay deformation, a bend deformation and a pure twist deformation.

Schematic drawings of the three basic geometries for a Fréedericksztransition for materials with positive dielectric anisotropy. The upper images show the undistorted initial alignment in the eld OFF state, while the lower images show the distorted states, which correspond to a splay deformation (left), a pure twist deformation (middle) and a bend deformation (right).

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