Mathematical Classification and Rheological Properties of Ring Catenane Structures

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.