🧬 Polymer Chain

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 R_g = b·√(N/6).

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.

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 R_g, 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.

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

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] ~ M^a. The same random-walk mathematics also describes colloidal stabilisation, where polymer chains grafted onto nanoparticles prevent them from clumping together.