Polymer Brush — Alexander–de Gennes Scaling

Alexander–de Gennes scaling: When polymers are grafted densely enough that chains overlap (σ > σ* = (πR_F²)⁻¹), they stretch away from the surface, forming a brush. The brush height:
H ~ N · a · (v·σ·a²)^(1/3)
This comes from balancing stretching (elastic) energy and excluded-volume repulsion. The scaling exponent 1/3 is characteristic: brush height grows much faster with grafting density than the free Flory radius R_F = a·N^(3/5). The osmotic pressure inside the brush falls as a power law from surface to tip: Π(z) ~ (z/H)^(−9/4) · k_BT/a³.