Properties of knotted ring polymers. I. Equilibrium dimensions

Area of Science:

• Polymer Physics
• Computational Chemistry
• Statistical Mechanics

Background:

• Knotted ring polymers are complex topological structures.
• Understanding knot localization and scaling behavior is crucial for polymer physics.

Purpose of the Study:

• To investigate knot localization in three classes of ring polymers: SARs, theta-SARs, and ideal Gaussian rings.
• To examine scaling relations and compare computational results with existing conjectures.

Main Methods:

• Lattice self-avoiding ring (SAR) calculations.
• Theta-state SARs with attractive nearest-neighbor interactions.
• Ideal Gaussian ring calculations.
• Extrapolation to large polymerization index (N).

Main Results:

• Knot localization is implied for all three polymer classes at large N.
• Observed scaling relations R(g) ~ N^nu and R(g) ~ p^-0.27 in the crossover regime.
• Results are consistent with conjectures on asymptotic behavior, though the asymptotic domain is not directly sampled.

Conclusions:

• Knot localization is a general feature of these ring polymer systems.
• Current computational methods are limited to the crossover regime.
• The observed scaling relation R(g) ~ p^-0.27 is unlikely to hold in the asymptotic localization regime.