  We present a detailed account of a recently proposed phenomenological theory
for noncatenated ring polymer melts (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 spacial 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.
