Universal shape properties of open and closed polymer chains: renormalization group analysis and Monte Carlo experiments

Abstract

We investigate the influence of excluded-volume interaction (EV) on the shape of a long flexible polymer chain. Both open chains and ring polymers are considered. We study probability distributions of shape parameters which are typically ratios of characteristic lengths (such as the principal radii of gyration) of a given conformation. For a class of shape parameters such as the `asphericity' Ad of a chain in d space-dimensions it is shown how mean values or higher moments of the distributions can be evaluated by field theoretic renormalization group methods. The universality of these distributions is shown and the mean asphericity ⟨Ad⟩ is calculated within an ε=4-d expansion. ⟨Ad⟩ is found to be much more sensitive to the EV than a frequently used asphericity-approximant which avoids the ratio-averaging. This is the first analytical confirmation of a result observed by other groups in numerical simulations. We also investigate the complete distribution of A3 and of another (prolate vs. oblate) shape parameter in d=3 by means of Monte Carlo simulations. The dependence on chain length is carefully investigated. This improves the accuracy of previous estimates of universal asymptotic shape distributions. Generally the EV makes the shape more aspherical and prolate. Comparing quantitatively the increase in ⟨A3⟩ due to the EV as implied by the Monte Carlo data with that by the (appropriately extrapolated) first order ε-expansion one finds good [fair] agreement in case of ring polymers [open chains]