Chain Size and Knots of Ring Polymers in All-Crossing and Intra-Crossing Melts

Using dynamic Monte Carlo simulations based on the bond-fluctuation model, we systematically investigated the size and knots of ring polymers in all-crossing systems and intra-crossing systems. Our results demonstrate that the interchain constraint can increase the knotting probability, but does not alter the scaling relationship between knotting probability and chain length for ring polymers in melts. Having established that, we derived the interchain constraint contribution to the free energy of ring polymers in intra-crossing systems based on the knotting probability and obtained the scaling relationship between the size R and chain length N, i.e., R~N^1/6. And, by calculating the mean-squared radius of gyration of ring polymers in intra-crossing systems, we validated these scaling results. Finally, we analyze the size of knotted ring polymers with different types and compare corresponding scaling exponents for size versus chain lengths of ring polymers with different knotting complexities. These results provide fundamental insights into the static properties of ring polymers in melts.