The topological glass in ring polymers

We study the dynamics of concentrated, long, semi-flexible, unknotted and unlinked ring polymers embedded in a gel by Monte Carlo simulation of a coarse-grained model. This involves the ansatz that the rings compactify into a duplex structure where they can be modelled as linear polymers. The classical polymer glass transition involves a rapid loss of microscopic freedom within the polymer molecule as the temperature is reduced toward Tg. Here we are interested in temperatures well above Tg where the polymers retain high microscopic mobility. We analyse the slowing of stress relaxation originating from inter-ring penetrations (threadings). For long polymers an extended network of quasi-topological penetrations forms. The longest relaxation time appears to depend exponentially on the ring polymer contour length, reminiscent of the usual exponential slowing (e.g., with temperature) in classical glasses. Finally, we discuss how this represents a universality class for glassy dynamics. open access Copyright c © EPLA, 2013 Published by the EPLA under the terms of the Creative Commons Attribution 3.0 License (CC-BY). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. While a great deal of work continues to be published on the glass transition, e.g., in colloids [1–3] and polymers [4–6], a complete understanding of the transition remains elusive. It is generally associated with the suppression of molecular motion due to jamming and/or cooling where, as a result, the system can take an extremely long time to reach equilibrium. The glass transition has several characteristic properties, including a dramatic (exponential) slowing of dynamics as the temperature is reduced towards a glass transition temperature Tg, combined with the lack of any crystalline order or the thermodynamic signatures of a true phase transition. Beyond these broad features the glass transition appears to lack universality in the sense that its properties depend on the microscopic details of each system. There have been recent attempts to understand this transition in terms of the pinning of a fraction of components [7], which has close analogies with the work we outline below. In the present work we investigate a simplified theoretical model of the diffusive dynamics of high-molecularweight ring polymers, above the overlap concentration c!, embedded in a polymer gel. This would correspond to the same experimental system as would be employed for gel electrophoresis of circular plasmid DNA (here without the applied field) [8–11]; DNA electrophoresis is one of the core techniques of molecular biology [12,13]. There is evidence that open circular (ring) DNA can be “trapped” in agarose gel, with exponential slowing-down of its migration. This phenomenon is attributed to the presence of protruding gel fibres, that thread through the ring-shaped DNA [14]. We neglect such ends protruding from the gel in what follows but argue that similar penetrations can occur between two ring polymers when one threads through the other. If these penetrations are numerous, then they may dramatically slow the diffusion of rings. It is this effect that we are most interested in investigating. A few authors have previously conjectured that such interpenetration of rings may occur and that this might significantly slow the dynamics [15,16]. However, no quantitative theory has previously been proposed. It is also possible to synthesise non-DNA ring polymers with few knots and concatenations [17,18]. Their rheological properties are now thought to be rather sensitive to contamination from linear polymers [19,20] and, to a lesser