the random coil
Drop a long, limp chain — a bead necklace, a length of cooked spaghetti, a garden hose gone slack — onto the floor and it lands as a loose, irregular tangle: not a straight line, not a tight ball, just a shapeless heap. A flexible polymer chain in a melt or in solution does the same thing, and that typical tangled shape is called the random coil. It is the single most important picture in polymer physics, because it is the shape an ordinary chain takes almost all the time.
To make it precise, physicists model the chain as a random walk: start at one end and take N steps, each of fixed length l, but each step pointing in a fresh random direction (the freely-jointed chain). The chain wanders like a drunkard's path. The remarkable result is that although the fully stretched length (the contour length) is N times l, the average straight-line distance between the two ends is far smaller: the root-mean-square end-to-end distance is only l times the square root of N. The square root is the whole story. Take N = 10000 steps of l = 0.25 nm: the contour length is 2500 nm, but the rms end-to-end distance is 0.25 x sqrt(10000) = 0.25 x 100 = 25 nm — the coil is about a hundred times more compact than the stretched-out chain. The end-to-end distances follow a bell-shaped (Gaussian) distribution, so a coil has no single sharp size, only a typical spread.
This matters because a polymer melt, a polymer solution, and the amorphous regions of a solid polymer are all just dense assemblies of interpenetrating random coils. The coil picture explains rubber elasticity (stretching uncoils the chains and lowers their entropy, and the drive to re-coil pulls the rubber back), it sets the viscosity of melts, and it is the baseline every real chain is measured against. Real chains are not perfectly freely jointed — fixed bond angles and bulky side groups stiffen them — so we lump a few real bonds into one effective Kuhn segment and recover the same square-root law with a longer effective step; a stiffer chain simply has a longer Kuhn length and a bigger coil.
Small-angle neutron scattering can pick out one labelled (deuterated) chain in a sea of ordinary chains and measure its size. For a typical polystyrene melt the answer confirms the random-coil law: the chain fills a diffuse blob tens of nanometres wide, its size growing as the square root of molecular weight — proof that even packed shoulder to shoulder in the melt, each chain keeps its own ideal random-coil shape.
A random coil is a chain's random-walk tangle: rms end-to-end distance = l times sqrt(N), far smaller than the contour length.
Random does not mean sizeless or lawless: the coil's size obeys a precise square-root scaling with chain length. And a real chain is not truly freely jointed — fixed bond angles stiffen it — so measure its size in Kuhn segments, not chemical bonds.