Contour length and coil size are different measurements
A loose piece of string can fill a small region even when it would stretch a long way if pulled straight. A flexible polymer molecule presents a similar distinction. Its contour length follows all the bonds along the chain. Its coil size describes the region explored by a chain that bends and continually changes shape in a liquid.
The difference matters when comparing molecules. Add more links, and the chain has more ways to turn back toward itself. Its typical spatial span grows much more slowly than its fully extended length. The useful question is therefore statistical: across many changing conformations, how does a typical size depend on the number of effective segments?
An effective segment is a length over which the chain has mostly forgotten its previous direction. It may contain several chemical bonds. Describing a long molecule by these segments replaces much of its local chemistry with a simpler account of bending and wandering. This is the starting point for the scaling ideas associated with de Gennes.
An ideal walk folds back more freely than a real chain
Begin with an ideal chain whose segment directions are uncorrelated over long distances. Steps taken in opposite directions partly cancel. Its typical span grows roughly as the square root of the number of segments. Making the chain four times as long therefore makes its typical span about twice as large. This simple random-walk result supplies a reference for more realistic chains.
An isolated real chain has another restriction: two segments cannot occupy the same volume. This excluded volume discourages the chain from folding through itself. In a good solvent, where contacts with the solvent favour an expanded chain, self-avoidance swells the coil relative to the ideal random walk.
For a sufficiently long flexible chain in a three-dimensional good solvent, the size grows approximately as chain length to a power near three fifths. More precise calculations give a slightly smaller exponent. The important physical change is the exponent: local exclusion can alter how the size grows across the entire molecule. De Gennes’s 1972 calculation used Wilson’s methods to estimate this nontrivial growth.
Neither growth rule gives a particular chain one fixed shape. At any instant the molecule has a conformation, then thermal motion changes it. The rule describes a typical size across that ensemble. A single drawing of a coil can show a possible shape; many observations are needed to measure the statistical relationship.
The surroundings can change the chain statistics
A solvent changes the balance of interactions. Under theta conditions, attractive and repulsive effects cancel in a way that makes a long chain approximately ideal at large scales. In a dense polymer melt, other chains can also screen the excluded-volume effect of one chain, again allowing nearly ideal large-scale statistics. A dilute good-solvent solution and a melt can therefore contain the same chemical polymer with different chain statistics.
J. P. Cotton and colleagues used neutron scattering to examine polystyrene chains in a good solvent, under theta conditions, and in bulk material in a 1974 study. Scattering at different angles probes structure at different distances. Instead of trying to photograph every bend of a moving molecule, the experiment extracted statistical information about how segments were distributed.
The choice of distances matters. At very short distances one sees local stiffness and chemical structure. At sufficiently large distances a finite chain runs out of length. Between those limits, a scaling interval can reveal how its size changes with the length scale probed. This is where the simple power-law description becomes experimentally useful.
Sources: [4]
De Gennes connected a wandering chain with critical phenomena
The surprising theoretical step was that a self-avoiding chain could be related mathematically to a limiting case of a model for collective ordering. That connection allowed renormalization-group methods, developed for critical phenomena, to calculate chain statistics. The mapping concerned the statistical mathematics; it supplied a new way to solve a physical problem that had looked quite different.
The 1991 Physics Prize recognized de Gennes’s broader achievement of carrying ordering methods into liquid crystals and polymers. The polymer example shows why that transfer mattered. One can ask which features of a molecule affect only the overall size scale and which change the way size grows with chain length. A long, soft molecule becomes a place to study the relation between local restrictions and large-scale behaviour.