In the following paragraphs, a more detailed explanation of the Fréedericksz-transition is outlined. This derivation is based on the splay-type transition, the respective derivations for the two other geometries are analogue.
The initial equilibrium director field is assumed to be given by an homogeneously planar aligned nematic phase parallel to the y-axis of the sample. Applying electric fields parallel to the z-direction then creates distortions in the director field, but reduces the director distribution to a one-dimensional function depending on z only. Using this geometry, only bend and splay deformations of the director field can occur and the elastic part of the free energy density is given by

f_{elast}=f_0+\frac{1}{2}\left(K_{11}\cos^2\Theta+K_{33}\sin^2\Theta\right)\left(\frac{d\Theta}{dz}\right)^2.

For the undistorted liquid crystal, the slope of the tilt angle \Theta/dz is zero and hence f_{elast} equals the minimum energy f_0. Any deformation of the director field increases the elastic free energy density proportional to the square of the slope of the tilt angle. The elastic torque density originating from such deformations is connected to the elastic free energy density by equation

\tau_{elast}=-\frac{\partial}{\partial \Theta}\Delta f_{elast}+\frac{d}{dz}\frac{\partial}{\left(\frac{d\Theta}{dz}\right)}\Delta f_{elast}.

Tthe resulting elastic restoring torque density is then given by

\tau_{elast}=\left(K_{33}-K_{11}\right)\sin\Theta\cos\Theta \left(\frac{d\Theta}{dz}\right)^2+\left(K_{11}\cos^2\Theta+K_{33}\sin^2\Theta\right)\frac{d^2\Theta}{dz^2}.

On the other hand, the presence of an external electric field tends to minimize the electric part of the free energy density f_{elec} by reorienting the molecules with their long axis parallel to the z-direction. The resulting electrostatic torque density acting on the liquid crystal molecules is given by the relation

\tau_{elec}=-\frac{\partial}{\partial \Theta}\Delta f_{elec}+\frac{d}{dz}\frac{\partial}{\left(\frac{d\Theta}{dz}\right)}\Delta f_{elec},

with the change in the electric field energy density \Delta f_{elec} given by the expression

\Delta f_{elec}=\frac{1}{2}\frac{D^{2}_{z}}{\left(\epsilon_{\parallel}\sin^2\Theta+\epsilon_{\perp}\cos^2\Theta\right)}-\frac{1}{2}\frac{D^{2}_{z}}{\epsilon_{\perp}}.

With the initially chosen uniform alignment in the xy-plane of the test cell, the dielectric tensor \epsilon reduces to a function of z only. As the the electric field within the liquid crystal correlates to the local distribution of \epsilon, it also becomes a function of z. In consequence, the z-component of the displacement field vector D is constant, and hence the second term is independent of the director orientation \Theta\left(z\right). Thereby it follows

\tau_{elec}=\frac{D^{2}_{z}\left(\epsilon_{\parallel}-\epsilon_{\perp}\right)}{\left(\epsilon_{\parallel}\sin^2\Theta+\epsilon_{\perp}\cos^2\Theta\right)^2}\sin\Theta\cos\Theta,

which can be simplified to

\tau_{elec}=\left(\epsilon_{\parallel}-\epsilon_{\perp}\right)E^2\sin\Theta\cos\Theta.

The equilibrium director field distribution under the influence of electric fields is given by an orientation \Theta\left(z\right) where electric and elastic torque sum up and the total torque vanishes

\tau_{elec}+\tau_{elast}=0.

In the simplified geometry treated here, this equilibrium condition is then given by equation

\left(\epsilon_{\parallel}-\epsilon_{\perp}\right)E^2\sin\Theta\cos\Theta+\left(K_{33}-K_{11}\right)\sin\Theta\cos\Theta \left(\frac{d\Theta}{dz}\right)^2+\left(K_{11}\cos^2\Theta+K_{33}\sin^2\Theta\right)\frac{d^2\Theta}{dz^2}=0.

This nonlinear differential equation gives the static equilibrium director distribution \Theta(z) in the liquid crystal cell. Even in this simple geometry, analytical solutions are not available and the equation can only be solved numerically.
Assuming only very small distortions of the director field (\Theta<<1) leads to a pure splay deformation of the director field, while larger displacements are dominated by a splay deformation with an admixture of bend deformations. Thus, the restriction to \Theta<<1 omits the bend deformation and equation can be approximated by the linear differential equation

K_{11}\frac{d^2\Theta}{dz^2}+\epsilon_0\Delta\epsilon E^2\Theta=0.

One solution of this equation is given by

\Theta(z)=\Theta_0\sin\left((\frac{\pi z}{d})\right),

where \Theta_0 is a constant (\Theta_0<<1) and d is the cell gap. Substitution of this solution in equation in the equilibirum condition above and considering only first-order terms yields

\epsilon_0\Delta\epsilon E^2\sin\Theta\cos\Theta-K_{11}\left((\frac{\pi}{d})\right)^2\Theta=0.

For sufficiently small distortions \sin\Theta\cos\Theta<\Theta, so that nontrivial solutions require

E^2>\frac{\pi^2K_{11}}{d^2\epsilon_0\Delta\epsilon}\equiv E^{2}_{Th}.

With the relation E=V/d, the threshold voltage of the deformation of a uniformly aligned nematic liquid crystal is given by

V_{Th}=\pi\sqrt{\frac{K_{11}}{\epsilon_0\Delta\epsilon}}.

Only above this threshold voltage V_{Th} the destabilizing electric torque \tau_{elec}$vercomes the stabilizing restoring elastic torque \tau_{elast} and a deformation of the director field occurs. This field-induced rotation is degenerate, clockwise and anticlockwise rotation are equally probable. In order to avoid domain walls separating regions with different rotation directions within the bulk, rubbed alignment layers which induce small, well-defined pretilt angles to the molecules at the interface can be used. This leads favorable direction of rotation, as the displacement field D degenerates.
The threshold voltage V_{Th} for the Fréedericksz-transition of a nematic material is a simply accessible physical property which allows the determination of the elastic constants the liquid crystal. A larger elastic constant K_{ii} corresponds to a stiffer medium that requires higher electric torques to distort the liquid crystal. On the other hand a large difference between \epsilon_{l} and \epsilon_{t} goes along with a large dielectric anisotropy \Delta\epsilon and therefore increases the electric torque on the molecules. By measuring the threshold voltage V_{Th}, information on the ratio between elastic torque and electric torque is obtained, and for known values of \Delta\epsilon the elastic constant K_{ii} of the respective director deformation can be determined.

Although the relation for the threshold voltage V_{Th} is widely used, it should be noted that it is connected to several restrictions:

  • Restriction to small distortions: The simplifying assumptions made for the electric part of the free energy density are only valid for homogeneously aligned molecules and small distortions. In a strongly deformed director field, the displacement field D is no longer parallel to the applied external field, but non-uniformly distributed over the cell gap. This gives rise to additional electric torques, which can induce a rotation of molecules out of the (y,z)-plane in the geometry stated above. Because of the large anisotropy between \epsilon_{\parallel} and \epsilon_{\perp} compared to the magnetic anisotropy of typical nematic materials, this effect has to be taken into account for a reorientation in electric fields, while it can be neglected for a reorientation in magnetic fields.
  • Conductivity: For conducting samples an additional contribution to the electric torque due to an anisotropy of conductivity has to be considered. Additionally, samples with sufficient amounts of charged dopants can build electric double layers at the interfaces, which can reduce the local field strength E_{loc} and therefore influence on the displacement field D. In consequence, higher external voltages have to be applied to overcome the restoring elastic torque of the liquid crystal, which simulates higher appearing threshold voltages.
  • Boundary Conditions: The derivation of threshold voltage V_{Th} assumes strong boundary conditions with infinite anchoring energy, which is denoted by a director orientation of n=\left(z=0\right)=n\left(z=d\right)=n\left(0,~1,~0\right) for any given field strength E. A finite anchoring energy \left(W_{anchor}<\infty\right) results in an increase in the apparent thickness of the cell, what influences on the critical field strength for the Fréedericksz-transition E_{Th,weak}=\frac{E_{Th,strong}}{d+2b}. Hence, weak anchoring conditions decrease the critical field strength E_{Th}, but do not influence on the elastic properties of the liquid crystal.

It should also be noted that in a strict sence a threshold V_{Th} is only defined for zero pretilt angle of the director at the limiting boundaries. Only an initial director orientation perpendicular to the external field requires a critical electric torque to overcome the restoring elastic torque of the nematic material. For any finite pretilt angle, even small electric torques are sufficient for a reorientation of molecules, thus a continuous distortion of director field instead of a Fréedericksz-transition occurs.

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