# A topologically driven glass in ring polymers

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Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved March 31, 2016 (received for review October 27, 2015)

## Significance

The glass transition is commonly associated with a reduction in the temperature of liquids or by an increase in density of granular materials. In this work, we propose a radically different approach to study dynamical arrest that relies on the topology of the components. We find that a concentrated solution of ring polymers can be driven to a kinetically arrested state by randomly pinning a small fraction of rings, a transition not observed in linear polymers. We attribute this jamming to topological interactions, called “threadings,” that populate solutions of rings. Our work provides the first evidence for these threadings and suggests that very long rings may be expected to be kinetically arrested even as the fraction of pinned rings approaches zero.

## Abstract

The static and dynamic properties of ring polymers in concentrated solutions remains one of the last deep unsolved questions in polymer physics. At the same time, the nature of the glass transition in polymeric systems is also not well understood. In this work, we study a novel glass transition in systems made of circular polymers by exploiting the topological constraints that are conjectured to populate concentrated solutions of rings. We show that such rings strongly interpenetrate through one another, generating an extensive network of topological interactions that dramatically affects their dynamics. We show that a kinetically arrested state can be induced by randomly pinning a small fraction of the rings. This occurs well above the classical glass transition temperature at which microscopic mobility is lost. Our work both demonstrates the existence of long-lived inter-ring penetrations and realizes a novel, topologically induced, glass transition.

The physics of ring polymers remains one of the last big mysteries in polymer physics (1). Concentrated systems of ring polymers have been observed, in both simulations and experiments, to display unique features that are not easily reconciled with the standard reptation theory of linear polymers (2⇓⇓⇓–6). The main reason for this is that ring polymers do not possess free terminal segments, or ends, essential for end-directed curvilinear diffusion. In contrast, ring polymers possess a closed contour, which leads to markedly different relaxation and diffusion mechanisms. Recently, there has been much improvement in the production of purified systems of rings (6⇓–8), with the consequent result that more and more experimental puzzling evidence requires a deeper understanding of their motion in concentrated solutions and melts from a theoretical point of view.

Recently, it has been conjectured that ring polymers assume crumpled, segregated conformations in concentrated solution or the melt (5). On the other hand, numerical and experimental findings (5, 6) suggest that rings exhibit strong intercoil correlations, which have proved difficult to address in simplified theoretical models (9⇓⇓–12). Because of this, there have been many recent attempts to rigorously characterize these interchains’ interactions (13⇓⇓–16), although a precise definition and unambiguous identification of these “threadings” in concentrated solutions of rings remains elusive. The primary reason for this is that the rings are assumed to remain strictly topologically unlinked from one another throughout if synthesized in this state.

In the case of concentrated solutions of rings embedded in a gel, a method to identify these interpenetrating threadings has recently been proposed (13). Here it was shown that the number of threadings scales extensively in the polymer length (or mass) and can therefore be numerous for long rings, creating a hierarchical sequence of constraints that can span the entire system. It has also been conjectured that a kinetically frozen state, or a “topological glass” (17) can emerge, because such an extensive network of constraints can eventually suppress the translational degrees of freedom of the rings. However, the molten, or highly concentrated, state does differ from that of polymers embedded in a gel and so whether a similar jamming transition occurs for long enough polymers or even whether threadings are present in the absence of a gel remain open problems and are the main questions addressed in this study. An example of interthreaded ring configuration is shown in Fig. 1. A spherical region (Fig. 1*B*) is carved from the configuration depicted in Fig. 1*A*, which represents a typical system studied in this work. The degree of interpenetration between the different coils is readily appreciable from the figure, and it can be boiled down even further into a network representation (13). The uncrossability constraint between chains transforms the threadings into topological hindrance in the motion of the coils, which we conjecture to form the basis for a dramatic slowing down in the dynamics of long enough coils.

Glass-forming systems exhibit degrees of freedom that become constrained as the temperature, or the density, of the system approaches the glass transition temperature

Recently, a novel and promising theoretical approach to study the glass transition in glass-forming liquids has been advanced: It involves perturbing a system by randomly pinning some fraction of the constituents and by observing the behavior of the unfrozen fraction. This method introduces a field of “quenched” disorder by freezing in space and time a subset of the system (21⇓⇓⇓⇓⇓–27).

Inspired by this approach, we focus our attention on a concentrated solution of rings and apply a similar protocol: We freeze in space and time a fraction of polymers in the system and observe the response of the unfrozen constituents. We find that, although linear polymers are substantially insensitive to this perturbation, ring polymers become irreversibly trapped in a network of intercoil constraints (threadings), which, in the limit of long rings, allows us to drive a kinetically arrested state with only a small fraction of permanently frozen chains. We conjecture that a spontaneous glassy state might therefore emerge in the long chain limit, even as the fraction of explicitly pinned chains goes to zero. Because these constraints are topological in nature, originating from noncrossability of the chains, this glassy state has the potential to be produced at arbitrary temperature or monomer density, provided only that these intercoil topological interactions remain abundant in the system, i.e., the rings are sufficiently long and not too dilute. This system is therefore a candidate for a novel kind of glass transition in systems made of polymers or other elements with nontrivial topology.

## Rings in Solution Assume Crumpled but Largely Overlapping Conformations

The static properties of rings in solution have been studied for some decades, and it is nowadays thought that rings assume crumpled conformations characterized by a scaling law for their gyration radius *M* (11, 28⇓⇓–31). On the other hand, they also have been found to maintain numerous contacts with their neighbors. This is often quantified by measuring the “contact surface” of the polymers, which is defined as the number of segments *SI Appendix*, Figs. S1 and S2) *SI Appendix*, Fig. S3), in light of which the coils can be viewed as interpenetrating ultrasoft colloids of radius

The counterintuitive fact that rings have numerous intercoil interactions while assuming a scaling exponent *γ* near unity (29) (*SI Appendix*, Fig. S3).

These interchain interactions may include threadings that have been conjectured to be intimately related to the slow overall diffusion of the rings’ center of mass (5, 8, 13, 16) (see also *SI Appendix*, Fig. S4), an observation that is in apparent contrast with the very fast stress relaxation (6, 7), characterized by a power-law decay of the relaxation modulus

## Contiguity Is Persistent for Longer Chains

To probe intercoil interactions, we first adopt a definition of surface monomers where the *i*th monomer of chain *I* is a surface monomer of that chain if its distance from a monomer *j* belonging to a different chain *J* is *I* and *J*, by computing a dynamic *I* and initial times. This function quantifies the exchange dynamics of contiguous chains and tracks the time that the chains first become noncontiguous, because it involves the product of *t*. The behavior of **2** gives a more strict measurement of the interchain cooperativity than would be obtained from a standard contiguity correlation function and, unlike the latter, should decay to zero over time scales that are comparable with the time taken for the chains’ centers of mass to diffuse away from one another. Although, for short chains, this is well described by a simple exponential, we observe that its behavior for longer chains can be fitted for about two decades by stretched exponentials*SI Appendix*, Fig. S5). Even more striking is the exponential increase of the typical time to become noncontiguous *Inset*). This is most likely due to the fact that only the long-time tails show deviations from the stretched exponential behavior, and these act to further increase the exchange time of the coils.

These findings are a strong signature that rings display long-lasting intercoil correlations that are present even after they have traveled beyond their own size, in agreement with previous numerical and experimental findings (5, 8). In addition, this is clear evidence that the exchange dynamics of the rings becomes slower, more glass-like (16, 36), as the polymerization *M* increases. The increasingly stretched decay of

It is natural to now ask whether we can understand the nature of these correlations, and the heterogenity in the relaxation dynamics observed in Fig. 2, as being directly related to the threadings recently proved to exist in concentrated solution of rings embedded in gels (13, 14). In the case of pure solutions of rings, such threadings have never been rigorously identified.

Our approach consists in artificially immobilizing (“freezing”), in space and time, a fraction *c* of randomly selected rings from equilibrated configurations and then tracking the dynamics of the “unfrozen” fraction. This protocol is inspired by recent theoretical and experimental studies on the nature of glass transitions and idealizes the case in which a fraction *c* of polymers in the system might be arbitrarily frozen, perhaps by mixing two polymeric species with different

If the rings are mutually threaded, perhaps in a way that resembles the threaded lattice animal in Fig. 1*C*, then one would expect those unfrozen rings that are threaded by frozen ones to have their mobility substantially limited. They would appear to be immobilized within effective “cages,” being the region of space that they can explore limited by the threadings that they experience. In the alternative picture where rings remain unthreaded, as is often envisaged in simplified models, the mobility would be substantially unaffected by the presence of frozen chains.

A primary result of the present work, discussed in *Randomly Pinning Rings Induces a Kinetically Arrested State*, is that we do indeed observe immobilization of the unfrozen fraction. We believe that this represents excellent evidence for the existence of threadings in concentrated solutions of rings, something that has not previously been demonstrated.

## Randomly Pinning Rings Induces a Kinetically Arrested State

Starting from an equilibrated configuration, we perturb the system by randomly freezing, in space and time, a fraction *c* of coils. As a comparison, we first consider a system of linear polymers with one unfrozen linear “probe” chain diffusing through

Because nothing other than the topology of the polymers was changed, this dramatically different dynamical response should be attributed to the presence of topological interactions between ring polymers, which we identify as the threadings. This immediately implies that the equilibrated state of the rings in our systems is one in which threadings constrain the free diffusion of the rings (see sketch in Fig. 1*C*), therefore limiting their motion and, in the extreme case, leading to caged diffusion when neighboring rings that thread them are permanently frozen. The constraints provided by threading between rings in the unperturbed (

It is also worth stressing that the measurement that we perform should be interpreted as carrying a statistical meaning: With enough attempts, one will always be able to find a ring that is not caged, however unlikely that might be. When we interrogate the rings regarding their state, caged or free, we are implicitly drawing from a binomial distribution, and, therefore, we can calculate the probability *p* of observing a noncaged ring in any one test for a given fraction of frozen rings. Having performed 10 tries freezing

Given that, for all systems studied here, we observe a regime in which all of the unfrozen rings are caged in regions smaller than their sizes (

We study this transition by tracking the behavior of each chain’s center of mass diffusion

More practically, one can define *c* at which *SI Appendix*, Fig. S4) and define as caged all of the systems that display a *F*). From this figure, one can notice that the decay of the *F*, *Inset*, one can also appreciate the nearly exponential decay of *c* (27).

Although the diffusion coefficient of the center of mass of the rings informs us about the overall diffusion of the chains, it is also interesting to study the relaxation of the chains at different length scales in response to this external perturbation. This can be done by computing the dynamic scattering function*SI Appendix*, Figs. S6−S8), we report the behavior of this quantity computed for two choices of wave vector *q* probing length scales (*c* and rings’ length (see Fig. 5, and see *SI Appendix* for different choices of

The increasing trend of *c* to significantly slow down the relaxation of the scattering function with respect to systems with shorter chains. This observation is, in turn, in agreement with the fact that threadings are more numerous for systems with longer chains (13), and therefore a smaller fraction of frozen rings is sufficient to achieve a similar slowing down.

Fig. 5 also suggests that although coils are strongly constrained by the presence of frozen rings, the configurations can still partially relax by internal rearrangements, and this allows rings to relax their internal stress. The picture that emerges is rather different from that for linear polymers, where the center of mass diffusion is intimately related to its ability to undergo any and all conformational rearrangements. In the latter case, motion is only arrested by the onset of a microscopically glassy state for

## Phase Diagram of the System

The behavior of both the effective diffusion coefficient *c* as the length of the rings increases (see Figs. 4*F* and 5*B*). This implies that systems made of long chains become extremely sensitive to very small perturbations. Indeed, one can think of these observables as quantifying a form of susceptibility that captures how the dynamic mobility of the system responds to the freezing of (very few) threading constraints by chain immobilization.

Finally, in Fig. 6, we show the transition line *T* is here replaced by the inverse length *p* of finding an uncaged ring in any one sample as obtained from the binomial distribution at 95% confidence interval *c* region. Nonetheless, and somewhat remarkably, our results show that the number of caged chains per explicitly frozen ring is exponential in the ring length (see *SI Appendix*, Fig. S9*B*). This would seem to mean that an arbitrarily large fraction of caged chains can be achieved from an arbitrarily small fraction of frozen rings provided that the rings are long enough. This raises the possibility that a glassy state could emerge spontaneously in the critical regime near *M*.

From these results, it is clear that concentrated solutions of long rings are extremely sensitive to a small external pinning field that can drive the center of mass motion of the rings to become glassy (jammed). We have here traced this phenomenon to the unusual, topological constraints provided by the threadings between ring polymers. Although branched polymers also display exponentially long relaxation times, the origin of this is not cooperative in nature and would therefore be rather insensitive to the freezing of some components. On the other hand, cooperativity, such as the inter-ring threadings we study, is thought to be an essential ingredient of a genuine glass transition (18).

Our work therefore elucidates the conformation of rings in concentrated solutions and unambiguously characterizes, for the first time to our knowledge, the presence of threadings between rings. Furthermore, we show the dynamics of ring polymers to be sensitive to these interactions, and, finally, we provide strong evidence for the emergence of a kinetically arrested state solely driven by topological constraints. This system is therefore a novel instance of a glassy state induced through the topology of the constituents.

## Materials and Methods

We model the system by enclosing *N* semiflexible polymers modeled using the standard Kremer−Grest model (38) and formed by *M* beads of nominal size *σ* in a box with periodic boundary conditions of linear size *L*. The monomer fraction is fixed at *M* as our main control parameter. The simulation time step is denoted with *σ* experiencing a friction *ξ* to diffuse its size. The simulation is carried out in NVT ensemble with the large-scale atomic/molecular massively parallel simulator (LAMMPS) engine. Further computational details are described in *SI Appendix*.

## Acknowledgments

The authors thank Mike Cates, Davide Marenduzzo, Enzo Orlandini, Christos Likos, and Dimitris Vlassopoulos for useful discussions and careful remarks throughout the project. We also acknowledge the support of EPSRC through the Complexity Science Doctoral Training Centre at the University of Warwick (D.M.), Grant EP/E501311, and a Fellowship Grant EP/I005439/1 (to M.S.T.). The computing facilities were provided by the Centre for Scientific Computing of the University of Warwick with support from the Science Research Investment Fund. The authors also gratefully acknowledge the computing time granted by the John von Neumann Institute for Computing and provided on the supercomputer JURECA at Jülich Supercomputing Centre.

## Footnotes

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^{1}To whom correspondence may be addressed. Email: davide.michieletto{at}ed.ac.uk or m.s.turner{at}warwick.ac.uk.

Author contributions: D.M. and M.S.T. designed research; D.M. performed research; D.M. analyzed data; and D.M. and M.S.T. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1520665113/-/DCSupplemental.

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