On gyration radius distributions of star-like macromolecules

Abstract

Using the path integral approach, we obtain the characteristic functions of the gyration radius distributions for Gaussian star and Gaussian rosette macromolecules. We derive the analytical expressions for cumulants of both distributions. Applying the steepest descent method, we estimate the probability distribution functions (PDFs) of the gyration radius in the limit of a large number of star and rosette arms in two limiting regimes: for strongly expanded and strongly collapsed macromolecules. We show that in both cases, in the regime of a large gyration radius relative to its mean-square value, the PDFs can be described by the Gaussian functions. In the shrunk macromolecule regime, both distribution functions tend to zero faster than any power of the gyration radius. Based on the asymptotic behavior of the distribution functions and the behavior of statistical dispersions, we demonstrate that the PDF for the rosette is more densely localized near its maximum than that for the star polymer. We construct the interpolation formula for the gyration radius distribution of the Gaussian star macromolecule which can help to take into account the conformational entropy of the flexible star macromolecules within the Flory-type mean-field theories.

Toy models of multibranched polymers: opened vs circular structures

“…Recently, both the analytical and numerical estimates for the universal ratios (1) and ( 2) have been obtained in Refs. [26,27,35] for so-called rosette structure, containing single branching point with f r closed rings and f c linear chains, which can be considered as bottlebrush polymers with n = 2. Note that the probability distribution functions of the gyration radii of Gaussian star and rosette structures have been derived analytically in Ref.…”

“…Note that the probability distribution functions of the gyration radii of Gaussian star and rosette structures have been derived analytically in Ref. [35] as well. In this approach, an estimate for the Gaussian bottlebrush polymers containing n branching points of functionality f is found in Ref.…”