Self-Similar Dynamics of a Flexible Ring Polymer in a Fixed Obstacle Environment: A Coarse-Grained Molecular Model

Abstract

In this contribution we concern ourselves with an interesting problem, namely, the dynamics of ideal flexible ring polymers constrained in an array of fixed obstacles. The fundamental issue in this problem is to understand how a topologically constrained polymer chain is able to relax its conformation in the absence of chain ends. The key physics was provided in an elegant scaling theory by Rubinstein and co-workers (Obukhov, S. P.; Rubinstein, M.; Duke, T. Dynamics of a Ring Polymer in a Gel. Phys. Rev. Lett. 1994, 73, 1263-1267). In this work we develop a coarse-grained mean-field model based on the physical arguments of the scaling theory and derive constitutive relations for rings in fixed obstacle and melt environments. The model is composed of three distinct steps. In the first step the dynamics of an arbitrary section of a ring chain is worked out based on fractal Blob-Spring (BS) dynamics, and the center of mass diffusion and the relaxation spectrum of this section are determined. In the second step the center of mass diffusion obtained using the BS dynamics is used to model the one-dimensional diffusion of the section in a topologically constrained environment. In the final step we invoke the idea of dynamic self-similarity and argue that the dynamics described in the first and the second step, for any arbitrary section of the chain, applies to all sections of the chain. The constitutive relation is obtained consequently as the superposition of dynamic response of all sections of the ring chain.