 Solution

A liquid crystal flows like a liquid while retaining some molecular order. In a nematic liquid crystal, elongated molecules have long-range orientational order but no ordinary positional lattice. The nematic director $\mathbf{n}$ specifies the preferred axis, with $\mathbf{n}$ and $-\mathbf{n}$ equivalent for head-tail-symmetric molecules. The nematic order parameter $S=⟨(3{\cos }^2\theta -1)/2⟩$ distinguishes an isotropic distribution, $S=0$, from perfect parallel alignment, $S=1$. A smectic phase additionally has a periodic layered density modulation, while molecules remain mobile within layers.

The broken rotational symmetry allows birefringence and distinct elastic costs for nematic splay, nematic twist and nematic bend. Under crossed polarizers, rotating a thin aligned sample changes the transmitted intensity because its two optical polarization components acquire different phases. Heating through a nematic-to-isotropic transition removes that orientational birefringence. An electric field can rotate the director if the dielectric anisotropy of a nematic favors alignment, explaining the optical switching principle of a liquid-crystal cell. Molecular shape, temperature, concentration and surface anchoring determine which ordered phase appears.