Nematic Liquid Crystal — Director Field & Topological Defects
2D Frank elastic energy — ±1/2 defects, order parameter S vs temperature
Director n(x,y) is a headless unit vector (angle θ with π-periodicity).
Frank free energy: F = ½K∫[(∇·n)² + (∇×n)²]dx.
Topological defects: ±1/2 (nematic) where director winds by ±π around a loop.
Order parameter S = ⟨cos(2θ)⟩, S → 0 above T_c (isotropic phase).