Theory of Excluded Volume in Coiled Polymers: Cumulant Expansion

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Abstract
The excluded-volume effect in a randomly coiled polymer is investigated by the method of cumulants. The spatial expansion factor α2 of a polymer chain is expanded in a power series: α2 − 1 = ∑ n=1∞(− 1)n−1Cnzn, where z is the usual excluded-volume parameter, and a general formula for Cn is given. By means of a “ladder” approximation C4 is calculated to be 1.3438. Contributions of small and giant clusters in a polymer chain to α2 are discussed. The above power series is predicted in the ladder approximation to converge for 0 ≤ z ≤ N, where N is the total number of segments constituting the polymer chain. We are unable, however, to dispose of the possibility that in general Cn for 5 ≤ n ≪ N includes a term proportional to N1/2logN.