Ring polymers in melts and solutions: Scaling and crossover

We propose a simple mean-field theory for the structure of ring polymer melts. By combining the notion of topological volume fraction and a classical van der Waals theory of fluids, we take into account many-body effects of topological origin in dense systems. We predict that although the compact statistics with the Flory exponent ν=1/3 is realized for very long chains, most practical cases fall into the crossover regime with the apparent exponent ν2/5 during which the system evolves toward a topological dense-packed limit.