A renormalization group calculation of the swelling factor for a non-ideal, randomly jointed polymer chain

A real-space renormalization group method is used to calculate the swelling factor of a three-dimensional, randomly-jointed chain with hard-sphere exc luded-volume interactions. This method differs from more conventional proce dures patterned on the field theoretic approach pioneered by Gell-Mann and Low and Wilson. It is specifically designed to produce estimates of the swe lling factor for finite values of the chain length and for a broad range of [excluded-volume] bare coupling coefficients. In addition to predictions s pecific to chains of finite length, the theory produces two distinct power- law scaling formulas for the asymptotic, long-chain limit of the swelling f actor. One of these is descriptive of an ideal chain and is associated excl usively with very small values of the bare excluded-volume interaction para meter. The other is appropriate to chains with larger values of the interac tion parameter and which exhibit significant deviations from ideality. The "critical exponent" associated with this second class of chains is equal to v = 0.5916, a value which agrees quite well with the results of previous i nvestigations. Our renormalization group calculations are based on a pair o f functional equations, one for an effective coupling function and another for the swelling factor. (C) 1999 Elsevier Science B.V. Ail rights reserved .