Conjectures on the statistics of ring polymers

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Abstract
We present some conjectures concerning the equilibrium statistics and dynamics of ring polymers. We argue that unconcatenated ring polymers in the melt may have statistics intermediate between those of collapsed and Gaussian chains. An extremely crude (Flory-like) treatment suggests that the radius R of such a ring scales with its polymerization index N as R ∼ N 2/5. In contrast, a ring in a melt of long linear chains (of the same chemical species) should be swollen. Moreover, rings of one chemical species (A) can be compatible with linear chains of another (B), even when linear chains of A and B are not compatible. Rings in a network of fixed obstacles are also discussed. A simple analysis of their dynamics shows that the diffusion constant D of such a ring (in three dimensions) scales with its polymerization index N as D ∼ N-2. This prediction is confirmed by computer simulations. Finally, we consider a knotted ring formed irreversibly in a theta solvent, and argue that, under appropriate formation conditions, such a ring may remain Gaussian when placed in a good solvent