Understanding the dynamics of rings in the melt in terms of the annealed tree model
The dynamical properties of a long polymer ring in a melt of unknotted and unconcatenated rings are calculated. We re-examine and generalize the well known model of a ring confined to a lattice of topological obstacles in light of the recently developed Flory theory of untangled rings which maps every ring on an annealed branched polymer and establishes that the backbone associated with each ring follows self-avoiding rather than Gaussian random walk statistics. We find the scaling of the ring relaxation time and diffusion coefficient with ring length, as well as the time dependence of stress relaxation modulus, zero shear viscosity and the mean square averaged displacements of both individual monomers and the ring's mass centre. Our results agree within error bars with all available experimental and simulation data of the ring melt, although the quality of the data so far is insufficient to make a definitive judgement for or against the annealed tree theory. At the end we review briefly the relation between our findings and experimental data on chromatin dynamics.