text: A real polymer is not freely-jointed. A -C-C- single bond has a fixed 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 x 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 = N x 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 2N, 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 radii of their atoms, including bulky substituent groups that interfere with 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 stochastically driven, chain lengths in any real population of 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 dimensions of real polymers in solution, and in the amorphous state, as long as there are only weak physicochemical interactions between the monomers. This model, and the Flory-Huggins Solution Theory, for which Paul Flory received the Nobel Prize in Chemistry in 1974, ostensibly apply only to ideal, dilute solutions. But there is reason to believe (e.g., neutron diffraction studies) that excluded volume effects may cancel out, so that, under certain conditions, chain dimensions in amorphous polymers have approximately the ideal, calculated size 
  When separate chains interact cooperatively, as in forming crystalline regions in solid thermoplastics, a different mathematical approach must be used.
  Stiffer polymers such as helical polypeptides, Kevlar, and double-stranded DNA can be treated by the worm-like chain model.
  Even copolymers 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 regions in which the chains approximate random coils, alternating with regions that are crystalline. The amorphous regions contribute elasticity and the crystalline regions contribute strength and rigidity.
  More complex polymers such as proteins, with various interacting chemical groups attached to their backbones, self-assemble into well-defined structures. But segments of proteins, and 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 residues by a peptide bond. This is not actually the case, since the ensemble will be energy weighted due to interactions between amino acid side-chains, with lower-energy conformations being present more frequently. In addition, even arbitrary sequences of amino acids tend to exhibit some hydrogen bonding 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.