The rheological properties of polycatenanes were investigated by coarse-grained molecular dynamics simulations using the Kremer−Grest-type bead-spring model. To prevent the combination number from explosively increasing, systematic structural models of [n]catenanes (n = 2, 3, and 4) were generated using a mathematical graph representation. It was confirmed that the behavior of the storage and loss moduli, G′(ω) and G″(ω), respectively, depends approximately on the number of beads (monomer units) per catenane at low frequencies. We found that the crossing numbers affected the behaviors of G′(ω) and G″(ω) in the immediate frequency range. Moreover, the storage modulus at the middle frequency tends to behave as a linear function of the crossing number. For the small rings, an exhaustive study based on mathematics revealed that even if the crossing number is the same, there are cases where the storage modulus at the middle frequency becomes exceptionally large due to the difference in linking rings. For the smaller ring sizes and/or larger crossing numbers, we discovered a sol−gel transition in the G′(ω) and G″(ω) plots. For the Kremer−Grest model of the peptide [6]catenane and peptide [4]catenane structures that have been experimentally prepared by the Fujita group, the threshold ring-size value for this transition was found to be approximately 25 and 23, respectively.
Topological barriers in ring polyethylene (PE) melts of trefoil knots inhibit crystallization due to self-entanglement, as confirmed by united atom molecular dynamics (UAMD) simulations. In this study, we clarified the decrease in the topological barriers of self-entangled knots with increasing polymer chain length (N) and the localization of the knotted segments in the noncrystallized region. In the UAMD simulations, isothermal crystallization processes were performed at T = 300 K for the trefoil knots using a crystallization time t IC = 200 ns. Here, the trefoil knot gives the simplest nontrivial topology with nonzero crossing number. Crystallization was not observed in trefoil PE knots for N ≤ 120; however, crystallization did take place in the ring PE melt of the trivial knot with trivial topology and zero minimal crossing number. For N = 140, suppression of the growth of a crystalline phase (i.e., the polycrystalline phase) was observed in the trefoil PE melts, whereas such suppression was not detected in the trivial PE melts. In contrast, no significant differences in the crystallization behaviors of the trefoil and trivial PE melts were observed at N = 200. These results indicated that the topological barriers decreased with increasing N. Furthermore, to investigate the relationship between the chain conformation and degree of local crystallization, we developed a new mathematical method for searching the knotted segment of a trefoil knot in a crystallized trefoil PE melt. We found that trefoil knots with large subloops (“large-leaves”) exhibited localization of the knotted segment. In addition, the large leaves of the localized knots dominated in the crystallized region, and the knotted segments of the localized knots were located mainly in the noncrystallized region.
We introduce the concept of a handlebody decomposition of a three-manifold, a generalization of a Heegaard splitting, or a trisection. We show that two handlebody decompositions of a closed orientable three-manifold are stably equivalent. As an application to materials science, we consider a mathematical model of polycontinuous patterns and discuss a topological study of microphase separation of a block copolymer melt.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.