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The Fréedericksz transition is a field-driven orientational instability of a confined nematic liquid crystal. Elasticity and surface anchoring resist the electric field or magnetic torque; a lowest allowed distortion mode becomes unstable at a finite threshold. Positive dielectric anisotropy of a nematic permits a normal electric field to tilt a planar anchored director.

The three contributions to the Frank elastic energy penalize distinct slow distortions of the nematic director. Nematic splay measures local spreading or convergence and is governed by $K_1(\nabla \cdot \mathbf{n}{)}^2/2$. Nematic twist measures rotation around an axis perpendicular to the director, with cost $K_2(\mathbf{n}\cdot \nabla \times \mathbf{n}{)}^2/2$. Nematic bend measures curvature of director integral lines; for a unit vector, $(\mathbf{n}\cdot \nabla )\mathbf{n}=-\mathbf{n}\times (\nabla \times \mathbf{n})$, giving cost $K_3|\mathbf{n}\times \nabla \times \mathbf{n}{|}^2/2$. These terms respect the head-tail symmetry of a nematic liquid crystal. Positive elastic constants oppose these distortions.

For the one-dimensional reduction assume lateral uniformity, so $\theta$ and $\varphi$ depend only on $z$. A prime below means $d/dz$. Directly differentiating the nematic director givesThe squared curl is ${\sin }^2\theta {\theta }^{\prime 2}+{\cos }^2\theta {\varphi }^{\prime 2}$. Subtracting the squared twist invariant givesConsequently the one-dimensional twisted nematic energy has elastic coefficientsA normal electric field has $\mathbf{E}\cdot \mathbf{n}=E_0\sin \theta$. There is a genuine factor-of-two inconsistency in the source: its initial functional places the electric term inside the overall factor $1/2$, whereas its subsequent density and threshold use the full electric term. For the requested density and threshold, use the usual orientational coupling $-ϵE_0^2{\sin }^2\theta /(8\pi )$. Here $ϵ$ is the dielectric anisotropy of a nematic, in the printed Gaussian-unit convention. If the first functional is instead read literally, this coupling is halved and the resulting threshold voltage is multiplied by $\sqrt{2}$.

Put $c=ϵE_0^2/(8\pi )$ and $\mathcal{L}=f{\theta }^{\prime 2}/2+g{\varphi }^{\prime 2}/2-c{\sin }^2\theta$. Apply the Euler-Lagrange equations for two fields. Variation in $\varphi$, which does not appear explicitly, givesVariation in $\theta$ gives $(f{\theta }^{\prime }{)}^{\prime }-f_{\theta }{\theta }^{\prime 2}/2-g_{\theta }{\varphi }^{\prime 2}/2+2c\sin \theta \cos \theta =0$, orAt $\theta =0$, $f=K_1$, $g=K_2$, and $f_{\theta }=g_{\theta }=0$. With zero field, ${\theta }^{\prime }=0$ and ${\varphi }^{\prime }=\pi /(2L)$ satisfy both equations and the imposed azimuthal endpoints. Thus the quarter-turn planar texture is a stationary solution. Stationarity alone is not a claim of stability for arbitrary elastic constants.

To find the Fréedericksz transition, impose the additional strong planar anchoring $\theta (0)=\theta (L)=0$. Write $\alpha =\pi /(2L)$. The small-angle expansions areThe conserved $g{\varphi }^{\prime }$ and fixed total twist imply ${\varphi }^{\prime }=\alpha +O({\theta }^2)$ for a pure tilt perturbation. Linearizing the tilt Euler-Lagrange equation therefore givesSubstitute $\theta ={\theta }_0\sin (\pi z/L)$. A nonzero neutral amplitude requiresSince $V_c=E_{c}L$, the quarter-turn nematic instability threshold isThis is the first zero-endpoint mode: higher sine modes replace $K_1$ by $n^2{K}_1$ and cost more elastic energy. The quadratic energy change is $\frac{1}{2}\int [K_1{\theta }^{\prime 2}+\{(K_3-2K_2){\alpha }^2-ϵE_0^2/(4\pi )\}{\theta }^2]dz$, so the first mode has negative stiffness above this voltage. The result assumes $ϵ>0$ and $K_1+(K_3-2K_2)/4>0$. If the latter is negative the planar state is already unstable at zero field; for negative dielectric anisotropy of a nematic, a normal field opposes this tilt. The literal initial electric prefactor would give $V_{c,\mathrm{literal}}=\sqrt{2}{V}_c$.

For reflective operation of the twisted nematic field effect, orient each polarizer's transmission axis along the adjacent planar nematic director. Light entering through the top polarizer initially has $y$ linear polarization. At low voltage, adiabatic optical following in a twisted nematic rotates it through a quarter turn to $x$ as it travels downward, so it passes the bottom polarizer and reaches the mirror. On returning, it follows the reverse twist back to $y$ and exits the top polarizer: the pixel appears bright. At sufficiently high voltage, positive dielectric anisotropy of a nematic aligns most of the director with the normal field. The bulk no longer produces the quarter-turn optical rotation, so the crossed bottom polarizer blocks the incident light and the pixel appears dark. Elastic relaxation restores the bright twisted state after voltage is removed. The adiabatic requirement is more precisely $2\pi |\Delta n|/\lambda \gg |{\varphi }^{\prime }|$, involving birefringence, not thickness alone. Voltage switches director orientation and hence the reflected brightness; it does not require the liquid crystal to emit light.

A nematic liquid crystal has long-range orientational order without a three-dimensional positional lattice. Its nematic director $\mathbf{n}$ identifies the preferred molecular axis, with $\mathbf{n}$ and $-\mathbf{n}$ equivalent for apolar molecules. The scalar nematic order parameter $S=⟨(3{\cos }^2\theta -1)/2⟩$ is zero for isotropic orientations and one for perfectly parallel axes. The tensor $Q_{ij}=⟨u_i{u}_j-{\delta }_{ij}/3⟩$ describes alignment without choosing a sign for the director.

Ordering balances orientational entropy against interactions and packing. For long rods, alignment can increase available translational configurations by reducing mutual excluded volume; thermotropic materials additionally have temperature-dependent energetic preferences. Spatially varying alignment costs Frank elastic energy through splay, twist and bend. Surface anchoring and electric or magnetic fields select the director in a finite specimen.

Crossed polarizers reveal birefringence and defect textures, while scattering tests positional order independently. A particularly direct response experiment is the Fréedericksz transition: confine a planar-aligned layer between anchored plates, apply a transverse field and monitor the transmitted polarization as the director tilts above an elastic threshold. This tests orientational elasticity and dielectric anisotropy rather than presuming that an optically anisotropic fluid has crystalline positional order.

In a nematic liquid crystal, elongated molecules have orientational order around a nematic director $\mathbf{n}$, but lack the long-range positional order of a crystal. Head-tail symmetry identifies $\mathbf{n}$ and $-\mathbf{n}$. Molecular anisotropy and sufficient density favor collective alignment: for long hard rods this can increase translational entropy by reducing excluded-volume constraints, while attractive interactions can also favor alignment. Thermal orientational disorder competes with these effects.

The scalar nematic order parameter convention here isThus $S=0.44$ means $⟨{\cos }^2\theta ⟩=0.6267$. The effective orientation angle from nematic order isThis is not enough information to determine the literal mean angle $⟨\theta ⟩$: different orientation distributions can have the same second moment. Isotropic orientations have $S=0$, and perfect axial alignment has $S=1$.

In the supplied threshold expression, $d$ is the cell thickness; $K_1,K_2,K_3$ are the splay, twist and bend elastic constants of the Frank elastic energy. The dimensionless relative dielectric anisotropy of a nematic is $\Delta ϵ={ϵ}_{\parallel }-{ϵ}_{\perp }$, and ${ϵ}_0$ is the vacuum permittivity. Positive anisotropy favors a director along the applied electric field. Strong surface anchoring fixes orthogonal in-plane directions on the two plates. The low-field director therefore twists by a quarter-turn through the cell. Between crossed polarizers, with the entrance polarization matched to the first surface director, adiabatic optical following in a twisted nematic rotates the polarization and allows transmission. A sufficiently large normal field tilts the interior director towards the field, suppressing that rotation and making the crossed-polarizer cell dark; the anchored boundary layers remain distorted. Reversing the polarizer arrangement can reverse the optical contrast.

The threshold is a Fréedericksz transition: the electric energy gain overcomes the restoring elastic energy of an allowed distortion. Using the expression explicitly supplied by the paper givesThe thickness cancels for this strong-anchoring model. This is the onset of director reorientation, not an assertion that every molecule instantaneously becomes perfectly aligned. The quoted elastic combination is taken as given for this numerical task; a threshold derived from a specified full director-energy model must use that model's own elastic combination consistently. The companion illustration in A2(b) shows the plate anchoring, polarizers and bulk field-aligned state.