Polymer Chain

Freely-jointed chain model — radius of gyration, end-to-end distance & Flory scaling

Chain Parameters

N (monomers)

b (segment size)

Temperature (kT)

Solvent Quality

FJC: R_g = b√(N/6) Flory: R_g ∝ N^ν
ν = 0.588 (good), 0.5 (theta, FJC), 0.333 (poor/collapsed)

Histogram

Sampling speed

Live Values

R_g (current)

R_g FJC (theory)

R_ee

Samples

ν (Flory exp.)

About — Polymer Chain Physics

Freely-Jointed Chain (FJC) model

The simplest model of a flexible polymer is the freely-jointed chain: N rigid segments of length b connected by freely rotating joints. Each segment orientation is completely uncorrelated with its neighbours (no angular restrictions, no excluded volume). This is equivalent to a random walk in 3D.

The mean-square end-to-end distance is <R²> = Nb², so R_ee = b√N. The radius of gyration R_g = b√(N/6) in 3D.

Flory scaling & solvent quality

Real chains have excluded volume interactions between monomers (no two segments can occupy the same space). In a good solvent, excluded volume swells the coil; in a poor solvent, the chain collapses. Edward-Flory theory gives R_g ∼ N^ν:

ν = 0.588 — good solvent (Flory exponent in 3D)
ν = 0.5 — theta solvent (FJC, ideal chain)
ν = 0.333 — poor solvent (collapsed globule)

Applications

DNA molecule length and packaging in the nucleus
Protein unfolded chain dimensions and folding radius
PEG (polyethylene glycol) hydrogel mesh size design
Polymer solution viscosity (Mark-Houwink equation: [η] ∼ M^a)
Nanoparticle–polymer interactions and colloidal stabilisation