Nematic Liquid Crystal
A nematic liquid crystal sits between solid crystal and isotropic liquid: rod-like molecules have no positional order but share a common orientation called the director. Watch the director field relax, form topological defects of half-integer winding number (±½), and see defect pairs annihilate as the system minimises its Frank elastic free energy. Drag to disturb; raise temperature past the clearing point to melt order entirely.
F = ½∫[K&sub1;(∇·n)² + K&sub2;(n·∇×n)² + K&sub3;|n×(∇×n)|²] dV
Click & drag on the canvas to disturb the director field and nucleate defects. “Quench” resets to random orientation (like rapid cooling from the isotropic phase).
What is a nematic liquid crystal?
Liquid crystals are phases of matter between crystalline solid and isotropic liquid. In the nematic phase, elongated molecules (calamitic mesogens) have no long-range positional order but align preferentially along a common axis called the director n. Because n and −n are physically equivalent (the molecules have no head or tail), the order parameter is a headless vector.
The elastic free energy (Frank elastic theory) penalises three types of distortion of the director field: splay (∇·n), twist (n·∇×n), and bend (n×≧×n). Minimising this free energy drives the system toward uniform alignment, but thermal fluctuations and boundary conditions create distortions.
Topological defects
Because of the head-tail symmetry, the director can rotate by π (not 2π) around a loop without discontinuity, giving rise to half-integer disclination lines of winding number ±½. These appear as bright or dark spots in polarised-light microscopy. Defects of opposite sign attract and annihilate; same-sign defects repel. The total topological charge in a closed system is conserved.
- +½ defect — director rotates +π around a loop (fan-brush or radial texture)
- −½ defect — director rotates −π (trefoil or hyperbolic texture)
- Schlieren texture — the characteristic four-brush pattern visible between crossed polarisers
Phase transition
The isotropic-nematic (IN) transition is weakly first-order. Raising temperature past the clearing point TNI destroys orientational order (S → 0). The simulation models this via a temperature-dependent effective noise amplitude and a Landau-de Gennes-inspired local restoring force.