On the Tree-Like Structure of Rings in Dense Solutions

TL;DR

This paper investigates the complex, tree-like conformations of rings in dense polymer solutions, revealing hierarchical loop structures and the significance of terminal branches, challenging previous lattice animal models.

Contribution

It introduces a novel characterization of ring conformations using local writhing and intra-chain contacts, proposing a relaxed tree-like model with hierarchical loops and terminal branches.

Findings

Hierarchical looped structures increase linearly with ring size

Terminal branches contain about 30% of the ring mass

Ring conformations resemble slip-linked chains with field-theoretic insights

Abstract

One of the most challenging problems in polymer physics is providing a theoretical description for the behaviour of rings in dense solutions and melts. Although it is nowadays well established that the overall size of a ring in these conditions scales like that of a collapsed globule, there is compelling evidence that rings may exhibit ramified and tree-like conformations. In this work I show how to characterise these local tree-like structures by measuring the local writhing of the rings' segments and by identifying the patterns of intra-chain contacts. These quantities reveal two major topological structures: loops and terminal branches which strongly suggest that the strictly double-folded "lattice animal" picture for rings in the melt may be replaced by a more relaxed tree-like structure accommodating loops. In particular, I show that one can identify hierarchically looped structures whose degree increases linearly with the size of a ring, and that terminal branches are found to store about 30% of the whole ring mass, irrespectively of its length. Finally, I draw an analogy between rings in the melt and slip-linked chains, where contact points are enforced by mobile slip-links and for which a field-theoretic treatment can be employed to get some insight into their typical conformations. These findings are ultimately discussed in the light of recent works on the static structure of rings and on the existence of inter-ring threadings.