🧬 Polymer Chain as a Random Walk — Flory Scaling

A polymer chain in solution constantly wiggles due to thermal motion, so its instantaneous shape is essentially random. Modeling it as a random walk — a sequence of fixed-length bonds pointing in random directions — captures the conformational freedom of the backbone without tracking the exact chemistry of every bond.

End-to-end distance R is the straight-line distance between the chain's first and last monomer. Radius of gyration Rg = √((1/N)Σ(r_i − r_cm)²) is the root-mean-square distance of every monomer from the center of mass. Rg is easier to measure experimentally because light, X-ray, and neutron scattering probe the overall mass distribution of the whole coil, not just its two endpoints.

An ideal (freely-jointed) chain allows segments to pass through each other and obeys Gaussian statistics with R ~ N^0.5. A self-avoiding walk (SAW) forbids monomers from overlapping, which swells the coil: in 2D the excluded-volume effect gives R ~ N^ν with ν ≈ 0.75, and in 3D ν ≈ 0.588. The ideal chain is a good approximation only at the theta condition, where attractive and excluded-volume effects cancel.

The Flory exponent ν is the power in the scaling law R ~ N^ν relating chain size to the number of monomers N. Flory derived it from a simple mean-field argument that balances the entropic cost of stretching a chain against the excluded-volume energy of monomers repelling each other, yielding ν = 3/(d+2) in d dimensions — 0.75 in 2D and 0.6 in 3D, remarkably close to the accepted value of 0.588.

In a good solvent, monomers effectively repel each other (excluded volume dominates) and the chain swells into an expanded coil with ν ≈ 0.75 (2D). At the theta condition, monomer-monomer attraction exactly cancels excluded-volume repulsion, so the chain behaves like an ideal random walk with ν = 0.5. In a poor solvent, attraction dominates and the chain collapses into a compact globule with ν ≈ 1/3.

If R = A·N^ν, taking logarithms gives log(R) = log(A) + ν·log(N), a straight line whose slope is exactly the exponent ν. Plotting many (N, R) pairs on log-log axes and fitting a line by linear regression turns a power-law relationship into an easily measured slope.

Random-walk and self-avoiding-walk statistics underlie the physics of plastics and rubber elasticity, the conformational behavior of DNA and unfolded proteins in solution, and gel permeation chromatography, where molecules are separated by their effective coil size as they diffuse through a porous gel.

Because a self-avoiding walk cannot place a new monomer on top of an existing one, a growing chain can wander into a position where every nearby direction is already occupied. The algorithm retries random directions a limited number of times, and if none work it removes the last monomer and tries again from an earlier point.

A single chain's R fluctuates a lot from one random realization to the next, especially at small N. Generating many independent chains at each chain length and averaging R and Rg smooths out this noise, so the log-log plot traces a cleaner straight line and the regression slope converges closer to the true theoretical ν.