Effect of Topology on the Conformations of Ring Polymers

The bond fluctuation method is used to simulate both nonconcatenated entangled and interpenetrating melts of ring polymers. We find that the swelling of interpenetrating rings upon dilution follows the same laws as for linear chains. Knotting and linking probabilities of ring polymers in semidilute solution are analyzed using the HOMFLY polynomial. We find an exponential decay of the knotting probability of rings. The correlation length of the semidilute solution can be used to superimpose knotting data at different concentrations. A power law dependence f(n) similar to phi R-2 similar to phi N-0.77 for the average number f(n) of linked rings per ring at concentrations larger than the overlap volume fraction of rings phi* is determined from the simulation data. The fraction of nonconcatenated rings displays an exponential decay P-oo similar to exp(-f(n)), which indicates f(n) to provide the entropic effort for not forming concatenated conformations. On the basis of these results, we find four different regimes for the conformations of rings in melts that are separated by a critical lengths N-OO, N-C, and N*. N-OO describes the onset of the effect of nonconcatenation below which topological effects are not important, N-C is the crossover between weak and strong compression of rings, and N* is defined by the crossover from a nonconcatenation contribution f(n) similar to phi R-2 to an overlap dominated concatenation contribution f(n) similar to phi N-1/2 at N > N*. For N-OO < N < N-C, the scaling of ring sizes R similar to N-2/5 results from balancing nonconcatenation with weak compression of rings. For N-C < N < N*, nonconcatenation and strong compression imply R similar to N-3/8. Our simulation data for noninterpenetrating rings up to N = 1024 are in good agreement with the prediction for weakly compressed rings.