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 excluded-volume interactions. This method differs from more conventional procedures patterned on the field theoretic approach pioneered by Gell-Mann and Low and Wilson. It is specifically designed to produce estimates of the swelling factor for finite values of the chain length and for a broad range of [excluded-volume] bare coupling coefficients. In addition to predictions specific to chains of finite length, the theory produces two distinct power-law scaling formulas for the asymptotic, long-chain limit of the swelling factor. One of these is descriptive of an ideal chain and is associated exclusively with very small values of the bare excluded-volume interaction parameter. The other is appropriate to chains with larger values of the interaction 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 investigations. Our renormalization group calculations are based on a pair of functional equations, one for an effective coupling function and another for the swelling factor.