Editing Random coil (section)

==Real polymers==

A real polymer is not freely-jointed. A -C-C- single [[chemical bond|bond]] has a fixed [[alkane#Molecular geometry|tetrahedral]] angle of 109.5 degrees. The value of ''L'' is well-defined for, say, a fully extended [[polyethylene]] or [[nylon]], but it is less than ''N''&nbsp;x&nbsp;''l'' because of the zig-zag backbone. There is, however, free rotation about many chain bonds. The model above can be enhanced. A longer, "effective" unit length can be defined such that the chain can be regarded as freely-jointed, along with a smaller ''N'', such that the constraint ''L''&nbsp;=&nbsp;''N''&nbsp;x&nbsp;''l'' is still obeyed. It, too, gives a Gaussian distribution. However, specific cases can also be precisely calculated. The average end-to-end distance for ''freely-rotating'' (not freely-jointed) polymethylene (polyethylene with each -C-C- considered as a subunit) is ''l'' times the square root of 2''N'', an increase by a factor of about 1.4. Unlike the zero volume assumed in a random walk calculation, all real polymers' segments occupy space because of the [[van der Waals radius|van der Waals radii]] of their atoms, including [[steric effects|bulky substituent groups]] that interfere with [[molecular geometry|bond rotations]]. This can also be taken into account in calculations. All such effects increase the mean end-to-end distance.

Because their polymerization is [[stochastic]]ally driven, chain lengths in any real population of [[chemical synthesis|synthetic]] polymers will obey a statistical distribution. In that case, we should take ''N'' to be an average value. Also, many polymers have random branching.

Even with corrections for local constraints, the random walk model ignores [[steric interference]] between chains, and between distal parts of the same chain. A chain often cannot move from a given conformation to a closely related one by a small displacement because one part of it would have to pass through another part, or through a neighbor. We may still hope that the ideal-chain, random-coil model will be at least a qualitative indication of the shapes and [[dimension]]s of real polymers in [[Solution (chemistry)|solution]], and in the amorphous state, as long as there are only weak [[intermolecular force|physicochemical interactions]] between the monomers. This model, and the [[Flory-Huggins Solution Theory]],<ref>Flory, P.J. (1953) ''Principles of Polymer Chemistry'', Cornell Univ. Press, {{ISBN|0-8014-0134-8}}</ref><ref>Flory, P.J. (1969) ''Statistical Mechanics of Chain Molecules'', Wiley, {{ISBN|0-470-26495-0}}; reissued 1989, {{ISBN|1-56990-019-1}}</ref> for which [[Paul Flory]] received the [[Nobel Prize in Chemistry]] in 1974, ostensibly apply only to [[ideal solution|ideal, dilute solutions]]. But there is reason to believe (e.g., [[neutron diffraction]] studies) that [[steric effects|excluded volume effects]] may cancel out, so that, under certain conditions, chain dimensions in amorphous polymers have approximately the ideal, calculated size <ref>"Conformations, Solutions, and Molecular Weight" from "Polymer Science & Technology"  courtesy of Prentice Hall Professional publications [http://www.informit.com/content/images/chap3_0130181684/elementLinks/chap3_0130181684.pdf]</ref>
When separate chains interact cooperatively, as in forming crystalline regions in [[solid]] thermoplastics, a different mathematical approach must be used.

Stiffer polymers such as [[alpha helix|helical]] polypeptides, [[Kevlar]], and double-stranded [[DNA]] can be treated by the [[worm-like chain]] model.

Even [[copolymer]]s with [[monomers]] of unequal [[length]] will distribute in random coils if the subunits lack any specific interactions. The parts of branched polymers may also assume random coils.

Below their melting temperatures, most [[thermoplastic]] polymers ([[polyethylene]], [[nylon]], etc.) have [[amorphous solid|amorphous]] regions in which the chains approximate random coils, alternating with regions that are [[crystal]]line. The amorphous regions contribute [[elasticity (physics)|elasticity]] and the crystalline regions contribute strength and [[Stiffness|rigidity]].

More complex polymers such as [[protein]]s, with various interacting chemical groups attached to their backbones, [[Molecular self-assembly|self-assemble]] into well-defined structures. But segments of proteins, and [[peptide|polypeptides]] that lack [[secondary structure]], are often assumed to exhibit a random-coil conformation in which the only fixed relationship is the joining of adjacent [[amino acid]] [[residue (chemistry)|residue]]s by a [[peptide bond]]. This is not actually the case, since the [[statistical ensemble (mathematical physics)|ensemble]] will be [[energy]] weighted due to interactions between amino acid [[Side chain|side-chains]], with lower-energy conformations being present more frequently. In addition, even arbitrary sequences of amino acids tend to exhibit some [[hydrogen bond]]ing and secondary structure. For this reason, the term "statistical coil" is occasionally preferred. The [[conformational entropy]] of the random-coil stabilizes the unfolded protein state and represents main free energy contribution that opposes to [[protein folding]].