Polymer Chain

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

Chain Parameters

Solvent Quality

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

Histogram

Live Values







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 Ree = b√N. The radius of gyration Rg = 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 Rg ∼ Nν:

  • 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] ∼ Ma)
  • Nanoparticle–polymer interactions and colloidal stabilisation

About this simulation

This simulation models a flexible polymer as a freely-jointed chain (FJC): N rigid segments of length b joined by freely rotating bonds, equivalent to a 3D random walk. A Monte Carlo pivot move perturbs the chain shape each step, and — depending on the chosen solvent quality — an acceptance rule biases the walk toward swollen or collapsed configurations, letting you watch the radius of gyration Rg settle into its theoretical Flory scaling Rg ~ Nν live in a histogram.

🔬 What it shows

A single polymer chain of N beads connected by segments of length b, redrawn after every accepted pivot move, alongside a live histogram of the radius of gyration Rg sampled over many configurations. The green bead marks the chain start, the red bead the free end, and the info box compares the current Rg against the ideal-chain prediction Rg = b·√(N/6).

🎮 How to use

Adjust N (5–120 monomers), segment size b (0.5–3) and temperature kT (0.1–3) to reshape the chain, then pick a solvent quality — Good (ν=0.588), Theta (ν=0.5) or Poor (ν=0.333) — to see the coil swell or collapse. Increase Sampling speed to accumulate histogram statistics faster, and use Reset histogram whenever you change solvent quality to start a fresh distribution.

💡 Did you know?

The three Flory exponents describe real polymer behaviour: ν≈0.588 for a chain swollen by excluded-volume repulsion in a good solvent, exactly ν=0.5 for the ideal freely-jointed chain in a theta solvent where attraction and repulsion cancel, and ν=1/3 for a fully collapsed globule in a poor solvent — the same scaling law that governs how DNA packs inside a cell nucleus.

Frequently asked questions

What is the freely-jointed chain model?

The freely-jointed chain (FJC) represents a flexible polymer as N rigid segments of length b connected by joints that can rotate freely in any direction, with no correlation between neighbouring segment orientations and no excluded-volume interactions. This makes the chain mathematically identical to a random walk in three dimensions, giving a mean-square end-to-end distance of ⟨R²⟩ = Nb² and a radius of gyration Rg = b·√(N/6).

How does the simulation generate and evolve the chain shape?

The chain starts as a 2D random walk of N segments of length b. Each animation step applies a Monte Carlo pivot move: a random point along the chain is chosen as a pivot, and the shorter tail is rotated by a random angle around it. Depending on solvent quality, the move is accepted or rejected using a Metropolis-style rule based on how the radius of gyration changes, biasing the walk toward expansion in good solvent or contraction in poor solvent.

What do the three solvent quality settings actually change?

Good solvent quality biases the pivot-move acceptance rule to favour configurations with a larger radius of gyration, mimicking excluded-volume repulsion between monomers and giving a Flory exponent of ν=0.588. Poor solvent quality favours smaller Rg, mimicking monomer-monomer attraction and collapse toward ν=1/3. Theta solvent applies no such bias — attractive and repulsive effects cancel — reproducing the ideal freely-jointed chain exponent ν=0.5.

Why does the histogram matter, and what should I look for?

Because chain configurations are generated randomly, a single snapshot of Rg is not very informative — the histogram accumulates many sampled configurations to reveal the underlying distribution and its mean. As samples build up, the average Rg should converge toward the value predicted by Rg ~ Nν for the selected solvent quality, letting you verify the Flory scaling law directly from the simulation rather than taking it on faith.

Where does this physics show up in the real world?

Freely-jointed chain and Flory scaling ideas underpin how scientists estimate the size of DNA coils packed inside a cell nucleus, the dimensions of unfolded protein chains before they fold into their native structure, the mesh size of PEG hydrogels used in drug delivery, and the viscosity of polymer solutions through the related Mark-Houwink equation [η] ~ Ma. The same random-walk mathematics also describes colloidal stabilisation, where polymer chains grafted onto nanoparticles prevent them from clumping together.