In polymer chemistry, a random coil is a conformation of polymers where the monomer subunits are oriented randomly while still being bonded to adjacent units. It is not one specific shape, but a statistical distribution of shapes for all the chains in a population of macromolecules. The conformation's name is derived from the idea that, in the absence of specific, stabilizing interactions, a polymer backbone will "sample" all possible conformations randomly. Many unbranched, linear homopolymers—either in solution, or above their melting temperatures— assume (approximate) random coils.
Random walk model: The Gaussian chain
There are an enormous number of different ways in which a chain can be curled around in a relatively compact shape, like an unraveling ball of twine with much open space, and comparatively few ways it can be more or less stretched out. So, if each conformation has an equal probability or statistical weight, chains are much more likely to be ball-like than they are to be extended –a purely entropic effect. In an ensemble of chains, most of them will, therefore, be loosely balled up. This is the type ofshape any one of them will have most of the time. Consider a linear polymer to be a freely-jointed chain with N subunits, each of length ℓ {\displaystyle \scriptstyle \ell } , that occupy zero volume, so that no part of the chain excludes another from any location. One can regard the segments of each such chain in an ensemble as performing a random walk (or "random flight") in three dimensions, limited only by the constraint that each segment must be joined to its neighbors. This is the ideal chain mathematical model. It is clear that the maximum, fully extended length L of the chain is N × ℓ {\displaystyle \scriptstyle N\,\times \,\ell } . If we assume that each possible chain conformation has an equal statistical weight, it can be shown that the probability P(r) of a polymer chain in the population to have distance r between the ends will obey a characteristic distribution described by the formula
P ( r ) = 4 π r 2 ( 3 2 π ⟨ r 2 ⟩ ) 3 / 2 e − 3 r 2 2 ⟨ r 2 ⟩ {\displaystyle P(r)=4\pi r^{2}\left({\frac {3}{2\;\pi \langle r^{2}\rangle }}\right)^{3/2}\;e^{-\,{\frac {3r^{2}}{2\langle r^{2}\rangle }}}}
where ⟨ r 2 ⟩ {\displaystyle {\langle r^{2}\rangle }} is the mean of r 2 {\displaystyle {r^{2}}} . The average (root mean square) end-to-end distance for the chain, ⟨ r 2 ⟩ {\displaystyle \scriptstyle {\sqrt {\langle r^{2}\rangle }}} , turns out to be ℓ {\displaystyle \scriptstyle \ell } times the square root of N — in other words, the average distance scales with N 0.5.
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