Polymers tethered to a surface stretch to brush height H ~ N(a/d)^(2/3) due to excluded volume interactions and entropic elastic restoring force
Brush Parameters
Alexander-de Gennes Brush (1977)
Brush height: H ≈ N·a·(a/d)^(2/3)
= N·a·σ^(1/3) (σ = a²/d²)
Free energy per chain:
F/kT ≈ (H/a)²/N + N(a/H)³
[elastic] + [excluded volume]
Osmotic pressure:
Π(D) ~ kT/d³ · (D/H)^(-9/4) for D < H
Condition for brush: σ > N^(-6/5)
When polymers are grafted densely enough that chains overlap (σ > N^(-6/5)), they stretch away from the surface forming a "brush." The competing forces are osmotic pressure (expanding) and elastic entropy (contracting). Polymer brushes provide steric stabilization of colloids and are used in drug delivery and anti-fouling surfaces.