Polymer Excluded Volume & Flory Exponent

Self-avoiding walks: R ~ N^ν with ν = 3/(d+2) — swollen beyond Gaussian

Self-Avoiding Walk (2D)

R vs N scaling (log-log)

Flory Theory

F = k_BT [R²/Nb² + v N²/R^d]
Minimizing: ν = 3/(d+2)
2D: ν = 3/4 = 0.75
3D: ν = 3/5 = 0.60 ≈ 0.588 (exact)
4D: ν = 1/2 (mean field, upper critical dim)

Flory (1949): balance entropic spring force (~R/Nb²) against excluded volume repulsion (~vN²/R^d). The free energy minimum gives R ~ N^ν with ν=3/(d+2).

Ideal vs Real Chain

Ideal (Gaussian): R = b√N, ν = 1/2
SAW (2D): R ~ N^{3/4}
SAW (3D): R ~ N^{0.588}

Ideal chain (Rouse model): monomers can overlap, Gaussian statistics, R ~ N^{1/2}.


Excluded volume: each monomer occupies space — no two monomers can overlap. This swells the chain. The exact 2D exponent ν=3/4 is known; the 3D value ν≈0.5876 comes from renormalization group calculations and simulation.


Flory's mean-field prediction ν=3/5 is remarkably close to the exact 3D value despite being a crude theory — a classic case where errors cancel.