Analysis of Slow Modes in Ring Polymers: Threading of Rings Controls Long-Time Relaxation

Abstract

To examine the intersection of a ring molecule with another ring, we regard each one of them as a 3d object having the shape of an irregular, solid polygon which spans a surface in 3d space that can be approximated by the sum of surfaces of all successive triangular edges making up the polygon. And then, we use geometric operations on the resulting ensemble of 3d polygons based on vector calculus to identify and quantify situations where a ring threads another ring. In practice, the situation of identifying threading events reveals the occurrence of multiple interpenetrations. An example is shown in Figure S1 below. The snapshot has been taken from the geometric analysis with the PEO-10k melt. The red ring is threaded by both the yellow and the green rings. In turn, the green ring is threaded by the cyan and white rings. Figure S1. Example of multiple threading from the simulations with the PEO-10k melt. The red ring is threaded by both the yellow and the green rings. In turn, the green ring is threaded by the cyan and white rings.

On the tree-like structure of rings in dense solutions

“…with this finding, several very recent works from different groups 12,[16][17][18] reported that rings in dense solutions display largely inter-penetrating configurations: segments of rings double-fold and thread through the contour of their neighbours, eventually leading to strongly overlapping configurations, perhaps best mimicked by the behaviour of ultra-soft colloids 19 rather than by that of polymers in poor solvents.…”

“…Nonetheless, the conjecture described here may suggest that in some circumstances threadings may be able to grow with the size of the threading ring and reach considerable extensions. In these cases, the topological constraints that they generate on the configurations of the threaded neighbours (or of itself, in the case of self-threading 25 ) lead to long-lived correlations that are strong candidates for explaining the "slowing down" in the rings' dynamics observed by several groups 14,[16][17][18]70,71 . The arguments presented here, together with several previous observations 16,18,70 also support the conjecture that in the limit of very large rings, long threadings will populate the system, and may eventually lead to spontaneous topological vitrification 72 .…”

Self-entanglement of a tumbled circular chain

“…Figure 5 shows various self-entanglements on circular chains and how they are characterized by our characterization method. Through the utilization of simple topological descriptors from knot theory, we can characterize complex self-entanglements with many crossings, for example (7 3 , 0 1 , 5) and (9 46 , 0 1 , 0). Furthermore, we find that it is possible to distinguish between circular entanglements that appear to be similar, such as (3 1 , 0 1 , 1) and (3 1 , 0 1 , 2), as well as (5 2 , 0 1 , 1) and (5 2 , 0 1 , 2).…”

Detailed Molecular Dynamics Simulation of the Structure and Self‐Diffusion of Linear and Cyclic n‐Alkanes in Melt and Blends

“…Due to the absence of chain ends and their closed‐loop structure, cyclic polymers exhibit material properties that differ considerably from those of the two other classes of polymers, linear and branched, of the same molecular length. The origin of many of these differences remains still poorly understood even after many years of intense theoretical, computational, and experimental research . Given, in particular, the success of the reptation model in describing the dynamics of linear entangled polymers, particular emphasis has been given recently on understanding the conformational and viscoelastic properties of melts of nonconcatenated ring polymers in the crossover region from unentangled to entangled.…”

“…The main appeal of the MD method is that it provides predictions of the material properties of the system under study, some of which are not easily or directly accessible in an experiment, through Newton's law from the knowledge of the interatomic potential functions. For example, in a recent study, we have used atomistic trajectories accumulated in the course of long MD simulations with model pure ring and mixtures of ring and linear PEO melts in order to geometrically characterize topological interactions and identify ring–ring and ring–linear interpenetration (threading). Our simulations indicated considerable cyclic threading, which can explain the slow relaxation modes observed in melts of high molecular weight ring polymers …”