Random Flight Chain

Successive steps in a random flight chain (or random walk) are uncorrelated in their directions and lengths. The absence of any correlation in their directions causes the average of the cosine of the angle between any two steps to be zero. And the average of the dot product of these two bond vectors is also zero.

< li dot lj > = 0

When used with the definition for r2, we obtain the simple expression

<r2>0 = n< l2>

where the zero as a subscript is a reminder that long-range interactions have exerted no affect on the result. (A still simpler expression is obtained when all of the steps are of the same length.) The random flight chain is unperturbed by long-range interactions, and there is no excluded volume. This situation is encountered in real chains in the Theta condition defined long ago by Flory.

Although the random flight chain is an extraordinary simplification of a real polymer, it correctly shows a proportionality between <r2>0 and n (or molecular weight) for unperturbed chains, in the limit where n becomes infinite. However, the proportionality constant, which is < l2> for this simple model, is not correct. For real chains, this proportionality constant depends on the local structure and the short-range intramolecular interactions. Nearly all real chains have a proportionality constant, called the characteristic ratio, that is larger than < l2>.

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June 23, 1999
Wayne L. Mattice: wlm@polymer.uakron.edu