Threading Dynamics of Ring Polymers in a Gel

We perform large scale three-dimensional molecular dynamics simulations of unlinked and unknotted ring polymers diffusing through a background gel, here a threedimensional cubic lattice. Taking advantage of this architecture, we propose a new method to unambiguously identify and quantify inter-ring threadings (penetrations) and to relate these to the dynamics of the ring polymers. We find that both the number and the persistence time of the threadings increase with the length of the chains, ultimately leading to a percolating network of inter-ring penetrations. We discuss the implications of these findings for the possible emergence of a topological jammed state of very long rings. U the dynamical and rheological properties of solutions of long polymers is of primary importance in several areas of soft matter, material science, and biophysics. The dynamics of linear polymers in the melt is now understood using the tube and reptation models. These models take advantage of the topological constraint represented by the noncrossability of the chains to describe the diffusion of the polymers along their own primitive path, by relaxing the free ends. By contrast, the dynamics of ring polymers, which have no free ends, can differ markedly from those of their linear or branched cousins in the melt, involving fundamentally different modes of stress relaxation, significantly different diffusion constants, and a crossover to free diffusion that occurs only once they traveled many times their own size ⟨Rg 2⟩1/2. Inter-ring penetrations, or “threadings”, have previously been speculated to play some role in ring dynamics, although no methodology to define or identify them yet exists. The goal of this work is to study a system in which we can quantify these threadings and their effect on the long-time dynamics of a concentrated solution of ring polymers. At present its not possible to identify such threadings in the melt. Here we focus our attention on a system that is rather different from a melt of rings: We study a concentrated solution of ring polymers embedded in a physical gel which, for simplicity, we model as a rigid cubic lattice, see Figure 1a. As we show below, this system is well suited to the study of inter-ring penetrations (threadings). It is also highly accessible from an experimental point of view, as it resembles the classical setup used for gel electrophoresis of plasmid DNA rings, except that the polymer concentration is taken above overlap and the gel is prepared in order to have a pore size comparable to the polymers’ Kuhn length. In this Letter we introduce for the first time a quantitative measure of inter-ring threadings for polymers diffusing in a background gel by using a novel algorithm that employs the background gel as a reference frame. We can then study the consequences of this on their dynamics. We show that the number of threadings grows linearly with the degree of polymerization M of the chain and that this leads to the emergence of an extended directed network of threadings that includes of order all rings. This network of threadings is associated with the onset of very slow dynamics and we show that the unthreading process drives the emergence of a significant slowing down of the longest rings we are able to study. Finally, we speculate that such a threadingrich state may be a precursor of a topological jammed state for even longer chains, as these threadings provide long-lived “pinning” sites that represent severe topological constraints in the polymers’ diffusion. We study unlinked and unknotted ring polymers diffusing through a background gel (see Figure 1a,b), formed by a perfect cubic lattice, that is, without dangling ends. The novel aspect of this work is in how we are able to identify the role of inter-ring threadings. By neglecting fluctuations of the gel we can direct computational resources most effectively toward simulating the dynamics of the rings themselves. We use a molecular dynamics engine (LAMMPS) to model the Langevin dynamics at fixed volume and constant temperature of N polymer rings with length M moving inside a three-dimensional cubic lattice of total linear size L and lattice spacing l. The ring monomer density is kept constant for all the systems at ρ = NM/L = 0.1 σ−3 (on top of this, the density of the gel is 0.06 σ−3). The well-established Kremer-Grest model is used to simulate worm-like chains with noncrossability constraint and excluded volume interaction (see SI for simulation details). Received: January 29, 2014 Accepted: February 24, 2014 Published: February 27, 2014 Letter pubs.acs.org/macroletters © 2014 American Chemical Society 255 dx.doi.org/10.1021/mz500060c | ACS Macro Lett. 2014, 3, 255−259 Terms of Use