On the shape of Gaussian scale-free polymer networks

TL;DR

This paper investigates the size and shape of Gaussian scale-free polymer networks modeled on hyperbranched structures with power-law degree distributions, revealing how network compactness and symmetry vary with the degree exponent.

Contribution

It introduces a numerical approach to analyze the conformational properties of complex Gaussian networks based on eigenvalue spectra, extending understanding of polymer topology effects.

Findings

Network compactness increases as the degree exponent decreases.

Network symmetry improves with lower alpha values.

Shape characteristics are quantitatively linked to eigenvalue spectra.

Abstract

We consider the model of complex hyperbranched polymer structures formed on the basis of scale-free graphs, where functionalities (degrees) $k$ of nodes obey a power law decaying probability $p(k)\sim{k^{-\alpha}}$. Such polymer topologies can be considered as generalization of regular hierarchical dendrimer structures with fixed functionalities. The conformational size and shape characteristics, such as averaged asphericity $\langle A_3 \rangle$ and size ratio $g$ of such polymer networks are obtained numerically by application of Wei's method, which defines the configurations of any complex Gaussian network in terms of eigenvalue spectra of corresponding Kirchhoff matrix. Our quantitative results indicate, in particular, an increase of compactness and symmetry of network structures with the decrease of parameter $\alpha$.