Liquid Crystal — Nematic Director Field & Fréedericksz Transition

Observe how rod-like molecules self-organise into a director field n̂. Control temperature and electric field to trigger phase transitions, optical switching, and topological defects.

Director field texture (line = average molecular orientation)

Order parameter S and optical transmittance vs reduced temperature T/T_c

Controls

Fréedericksz transition: Electric field (E > E_c = π√(K/ε₀Δε)/d) realigns molecules from planar to homeotropic configuration, switching cell from opaque to transparent.

Temperature T/T_c 0.85

Electric field E (V/μm) 0.0

Elastic constant K (pN) 7.0

Cell thickness d (μm) 5.0

Physical State

Order param. S— Phase— E_c threshold (V/μm)— Field vs E_c— Director tilt θ— Transmittance T— Birefringence Δn— Elastic energy F—

Frank Elastic Energy

F = ½K₁(∇·n̂)² + ½K₂(n̂·∇×n̂)² + ½K₃|n̂×∇×n̂|²

One-constant approx: K₁=K₂=K₃=K. The order parameter S = ½⟨3cos²θ−1⟩ drops to 0 at T_c (isotropic transition).

Topological defects at ±½ (disclinations) are stable because π/2 rotation around loop returns n̂ to same state.