2D director field with ±1/2 topological defects, Frank elastic energy, defect annihilation
Nematics minimize the Frank free energy F = ½K∫[(∇·n̂)² + (n̂×∇×n̂)²]dA. Topological defects have winding numbers ±1/2; pairs of opposite charge attract (like charges repel) with a logarithmic potential U∝K ln(r/r₀). The +1/2 defect is polar and self-propels in active nematics; ±1/2 pairs annihilate when they meet, releasing stored elastic energy.