Self-Similar Conformations and Dynamics in Entangled Melts and Solutions of Nonconcatenated Ring Polymers

Abstract

A scaling model of self-similar conformations and dynamics of nonconcatenated entangled ring polymers is developed. Topological constraints force these ring polymers into compact conformations with fractal dimension df = 3 that we call fractal loopy globules (FLGs). This result is based on the conjecture that the overlap parameter of subsections of rings on all length scales is the same and equal to the Kavassalis–Noolandi number OKN ≈ 10–20. The dynamics of entangled rings is self-similar and proceeds as loops of increasing sizes are rearranged progressively at their respective diffusion times. The topological constraints associated with smaller rearranged loops affect the dynamics of larger loops through increasing the effective friction coefficient but have no influence on the entanglement tubes confining larger loops. As a result, the tube diameter defined as the average spacing between relevant topological constraints increases with time t, leading to “tube dilation”. Analysis of the primitive paths in molecular dynamics simulations suggests a complete tube dilation with the tube diameter on the order of the time-dependent characteristic loop size. A characteristic loop at time t is defined as a ring section that has diffused a distance equal to its size during time t. We derive dynamic scaling exponents in terms of fractal dimensions of an entangled ring and the underlying primitive path and a parameter characterizing the extent of tube dilation. The results reproduce the predictions of different dynamic models of a single nonconcatenated entangled ring. We demonstrate that traditional generalization of single-ring models to multi-ring dynamics is not self-consistent and develop a FLG model with self-consistent multi-ring dynamics and complete tube dilation. This selfconsistent FLG model predicts that the longest relaxation time of nonconcatenated entangled ring polymers scales with their degree of polymerization N as τrelax ~ N7/3, while the diffusion coefficient of these rings scales as D3d ~ N−5/3. For the entangled solutions and melts of rings, we predict power law stress relaxation function G(t) ~ t−3/7 at t < τrelax without a rubbery plateau and the corresponding viscosity scaling with the degree of polymerization N as η ~ N4/3. These theoretical predictions are in good agreement with recent computer simulations and are consistent with experiments of melts of nonconcatenated entangled rings.

Nanoparticle Motion in Entangled Melts of Linear and Nonconcatenated Ring Polymers

“…This prediction is based on the ansatz that the effective viscosity η eff that a NP with a < d < R experiences in the Fickian regime is equal to the melt viscosity of ring polymers with size R ≈ d . A ring polymer with R ≈ d contains n ∼ d 3 monomers, and the melt viscosity 27 η ∼ n 4/3 ∼ d 4 . Therefore, η eff ∼ d 4 and D ∼ k B T /(η eff d ) ∼ d –5 .…”

Flory theory of randomly branched polymers

“…In Statistical Mechanics, the closely related subject of lattice animals 11–13 has deep connections with magnetic spin systems and has been studied by field theoretic methods 14–17 . Our own (renewed) interest in these systems 18–20 is due to the analogy between their behavior and the crumpling of topologically constrained ring polymers 21–25 (Fig. 1c) and, ultimately, chromosomes 26–30 .…”

Detailed Molecular Dynamics Simulation of the Structure and Self‐Diffusion of Linear and Cyclic n‐Alkanes in Melt and Blends

“…Due to the absence of chain ends and their closed‐loop structure, cyclic polymers exhibit material properties that differ considerably from those of the two other classes of polymers, linear and branched, of the same molecular length. The origin of many of these differences remains still poorly understood even after many years of intense theoretical, computational, and experimental research . Given, in particular, the success of the reptation model in describing the dynamics of linear entangled polymers, particular emphasis has been given recently on understanding the conformational and viscoelastic properties of melts of nonconcatenated ring polymers in the crossover region from unentangled to entangled.…”