Statistics and geometrical picture of ring polymer melts and solutions

We present a detailed account of a recently proposed phenomenological theory for noncatenated ring polymer melts [Sakaue, Phys. Rev. Lett. 106, 167802 (2011)]. A basic assumption lies in the implementation of the noncatenation constraint via the effective excluded-volume effect, from which a geometrical picture of melts emerges. The result captures many of the salient features observed so far, including (i) the overall spatial size of rings, (ii) the coordinate number, i. e., the number of rings surrounding a given ring, (iii) the topological length scale as a function of the molecular weight, and (iv) the effect of the chain stiffness and concentration. We also suggest a geometrical interpretation of the topological length scale, which may shed some light on the entanglement concept in polymeric systems.