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#### Effect of topology on the conformations of ring polymers

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##### Citation

Lang, M., Fischer, J., & Sommer, J.-U. (2012). Effect of topology on the conformations
of ring polymers.* Macromolecules,* *45*(18), 7642-7648.
doi:10.1021/ma300942a.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0018-18BD-E

##### Abstract

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 ∼ ϕR2 ∼ ϕ0.77N for the average

number f n of linked rings per ring at concentrations larger than the overlap

volume fraction of rings ϕ* is determined from the simulation data. The fraction

of nonconcatenated rings displays an exponential decay POO ∼ 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 NOO, NC, and N*. NOO describes the onset of the effect

of nonconcatenation below which topological effects are not important, NC is the crossover between weak and strong

compression of rings, and N* is defined by the crossover from a nonconcatenation contribution f n ∼ ϕR2 to an overlap

dominated concatenation contribution f n ∼ ϕN1/2 at N > N*. For NOO < N < NC, the scaling of ring sizes R ∼ N2/5 results from

balancing nonconcatenation with weak compression of rings. For NC < N < N*, nonconcatenation and strong compression imply

R ∼ N3/8. Our simulation data for noninterpenetrating rings up to N = 1024 are in good agreement with the prediction for weakly compressed rings.

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 ∼ ϕR2 ∼ ϕ0.77N for the average

number f n of linked rings per ring at concentrations larger than the overlap

volume fraction of rings ϕ* is determined from the simulation data. The fraction

of nonconcatenated rings displays an exponential decay POO ∼ 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 NOO, NC, and N*. NOO describes the onset of the effect

of nonconcatenation below which topological effects are not important, NC is the crossover between weak and strong

compression of rings, and N* is defined by the crossover from a nonconcatenation contribution f n ∼ ϕR2 to an overlap

dominated concatenation contribution f n ∼ ϕN1/2 at N > N*. For NOO < N < NC, the scaling of ring sizes R ∼ N2/5 results from

balancing nonconcatenation with weak compression of rings. For NC < N < N*, nonconcatenation and strong compression imply

R ∼ N3/8. Our simulation data for noninterpenetrating rings up to N = 1024 are in good agreement with the prediction for weakly compressed rings.