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 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 ν=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.