Why is ν = ½ called Gaussian or ideal?

For a phantom random walk without excluded volume, ⟨R²⟩ ∝ N so the characteristic size scales as N^{1/2}. Real chains in good solvent swell (ν ≈ 0.588 in 3D) because monomers cannot occupy the same space; the slider illustrates that scaling change schematically.

Is R_g the same as end-to-end distance?

No. End-to-end distance R tracks the vector between chain ends; R_g measures the mass distribution about the center of mass. For ideal linear chains they are proportional (R_g² = R²/6), but they remain distinct observables.

Does changing ν redesign the chemistry of the polymer?

In this lab ν is a pedagogical scaling parameter, not a computed exponent from a specific monomer potential. It stands in for solvent quality and excluded volume. Predicting ν from chemistry requires a fuller statistical-mechanics model.

Polymer Random Coil

Lattice random walk R_g with Flory exponent ν slider (scaling hint vs Gaussian ν = ½).

About this model

A polymer chain is modeled as a lattice random walk whose size is summarized by the radius of gyration R_g. A Flory-exponent slider sets the scaling hint R_g ∼ N^ν, comparing the ideal Gaussian coil ν = ½ with swollen or collapsed regimes suggested by different ν. The visualization is a schematic coil on a lattice, not a full molecular-dynamics force field: no explicit solvent particles, no entanglement dynamics, and no chemical detail beyond the scaling picture. You change chain length and ν to see how excluded-volume ideas enlarge the coil relative to a phantom Gaussian walk — core polymer-physics intuition.

Who it's for: Physical chemistry and soft-matter courses introducing ideal chains, Flory theory, and radius of gyration.

Key terms

Random coil
Radius of gyration
Flory exponent
Self-avoiding walk
Ideal chain
Polymer scaling

How it works

2D square-lattice random walk as a cartoon chain; radius of gyration measures spread about the center of mass. The ν slider only rescales R_g for comparison with typical exponents (Gaussian ν = ½ in 2D; good-solvent 3D SAW is often quoted near ν ≈ 0.588 — not reproduced by this 2D lattice model).

Key equations

R_g² = (1/N) Σᵢ |rᵢ − r_cm|² in 2D. The ν slider does not re-simulate a SAW; it only rescales the displayed R_g for classroom comparison.

Frequently asked questions

Why is ν = ½ called Gaussian or ideal?: For a phantom random walk without excluded volume, ⟨R²⟩ ∝ N so the characteristic size scales as N^{1/2}. Real chains in good solvent swell (ν ≈ 0.588 in 3D) because monomers cannot occupy the same space; the slider illustrates that scaling change schematically.
Is R_g the same as end-to-end distance?: No. End-to-end distance R tracks the vector between chain ends; R_g measures the mass distribution about the center of mass. For ideal linear chains they are proportional (R_g² = R²/6), but they remain distinct observables.
Does changing ν redesign the chemistry of the polymer?: In this lab ν is a pedagogical scaling parameter, not a computed exponent from a specific monomer potential. It stands in for solvent quality and excluded volume. Predicting ν from chemistry requires a fuller statistical-mechanics model.

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