On gyration radius distributions of star-like macromolecules

TL;DR

This paper derives analytical expressions for the distribution of gyration radii in Gaussian star and rosette macromolecules, analyzing their behavior in different regimes and providing an interpolation formula for modeling conformational entropy.

Contribution

It introduces new analytical formulas for gyration radius distributions of Gaussian star and rosette macromolecules, including asymptotic behaviors and an interpolation approach.

Findings

Distribution functions are Gaussian in large gyration radius regime.

Distribution functions decay faster than any power in the shrunk regime.

The rosette distribution is more localized near its maximum than the star.

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 the both distributions. Applying the steepest descent method, we estimate the probability distribution functions 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 probability distribution functions 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 probability distribution function 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 for 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