Polymer — Excluded Volume & Flory Exponent
Self-avoiding walk vs. ideal random walk — end-to-end distance ⟨R⟩ ~ N^ν, ν≈3/5 in 3D (Flory 1949)
Chain Parameters
Chain length N:
80
Excluded volume v:
1.0
Ensembles (for scaling):
50
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Display
Show:
Self-Avoiding Walk
Ideal Random Walk
Both (compare)
Flory theory:
F = kT[R²/Nb² + v·N²/R^d]
minimize: R* ~ N^{3/(d+2)}
d=3: ν = 3/5 = 0.600
d=2: ν = 3/4 = 0.750
Exact (d=3): ν = 0.588 (Flory ≈ exact!)
Theory
Ideal RW: ⟨R²⟩ = N·b²
ν = 1/2 (Gaussian chain)
SAW: ⟨R²⟩ ~ N^{2ν}
ν ≈ 0.588 in 3D
ν = 3/4 exact in 2D (Nienhuis)
Excluded volume: each monomer
excludes volume v from others
→ swells chain above ideal
ν(theta solvent) = 1/2
ν(good solvent) ≈ 0.588
ν(rod) = 1.0