Glassiness and Heterogeneous Dynamics in Dense Solutions of Ring Polymers

TL;DR

This study investigates how topological constraints influence the dynamics of dense ring polymer solutions, revealing glassy behaviors, universal scaling laws, and the impact of architecture-specific constraints on heterogeneity and vitrification.

Contribution

It provides the first systematic analysis of topological effects in ring polymer solutions, including scaling relations and the connection between dynamics and spatial organization.

Findings

Random pinning induces non-Gaussian, heterogeneous, and glassy dynamics.

Universal scaling relations for density and polymer length at vitrification.

Heterogeneous dynamics are linked to spatial organization and topological constraints.

Abstract

Understanding how topological constraints affect the dynamics of polymers in solution is at the basis of any polymer theory and it is particularly needed for melts of rings. These polymers fold as crumpled and space-filling objects and, yet, they display a large number of topological constraints. To understand their role, here we systematically probe the response of solutions of rings at various densities to "random pinning" perturbations. We show that these perturbations trigger non-Gaussian and heterogeneous dynamics, eventually leading to non-ergodic and glassy behaviours. We then derive universal scaling relations for the values of solution density and polymer length marking the onset of vitrification in unperturbed solutions. Finally, we directly connect the heterogeneous dynamics of the rings with their spatial organisation and mutual interpenetration. Our results suggest that deviations from the typical behaviours observed in systems of linear polymers may originate from architecture-specific (threading) topological constraints.

Tables

Table 1. Table S 1: Measured values for: (1) the average number of overlapping chains per ring, ⟨ m ov ⟩ delimited-⟨⟩ subscript 𝑚 ov \langle m_{\mathrm{ov}}\rangle ; (2) the maximum number of shells in the Bethe-lattice representation of rings solutions, g ¯ ¯ 𝑔 \bar{g} , Eq. ( 9 ) in main text; (3) fraction of caged rings, f c subscript 𝑓 𝑐 f_{c} , at pinning fraction f p = 0.3 subscript 𝑓 𝑝 0.3 f_{p}=0.3 (value chosen for corresponding to the smallest pinning fraction used in this work); (4) “critical” caging probability, p c † ≡ 1 / ( ⟨ m ov ⟩ − 1 ) superscript subscript 𝑝 𝑐 † 1 delimited-⟨⟩ subscript 𝑚 ov 1 p_{c}^{\dagger}\equiv 1/(\langle m_{\mathrm{ov}}\rangle-1) , corresponding to a finite fraction f c subscript 𝑓 𝑐 f_{c} of caged rings in the limit f p → 0 → subscript 𝑓 𝑝 0 f_{p}\rightarrow 0 ; (5) caging probability, p c subscript 𝑝 𝑐 p_{c} , obtained from Eqs. ( 10 )-( 11 ) in main text.

	$N=250$					$N=500$
$\rho {\sigma }^3$	$⟨m_{\mathrm{ov}}⟩$	$\overline{g}$	$f_c\left(f_p=0.3\right)$	$p_c^{†}\equiv 1/\left(⟨m_{\mathrm{ov}}⟩-1\right)$	$p_c$	$⟨m_{\mathrm{ov}}⟩$	$\overline{g}$	$f_c\left(f_p=0.3\right)$	$p_c^{†}\equiv 1/\left(⟨m_{\mathrm{ov}}⟩-1\right)$	$p_c$
$0.1$	$6.098$	$2.874$	$0.129$	$0.196$	$0.053$	$7.520$	$2.175$	$0.283$	$0.153$	$0.080$
$0.2$	$9.958$	$2.213$	$0.312$	$0.112$	$0.063$	$11.184$	$1.804$	$0.556$	$0.098$	$0.094$
$0.3$	$13.383$	$1.953$	$0.515$	$0.081$	$0.070$	$14.186$	$1.641$	$0.623$	$0.076$	$0.086$
$0.4$	$17.032$	$1.785$	$0.606$	$0.062$	$0.065$	$17.549$	$1.519$	$0.649$	$0.060$	$0.076$

Equations

$$f_p^{†}(N)=-f_N\log \left(\frac{N}{N_g}\right),$$

$$g_3(T,\Delta )\equiv \frac{1}{T-\Delta }{\int }_0^{T-\Delta }{\left[{\mathbf{r}}_{CM}(t+\Delta )-{\mathbf{r}}_{CM}(t)\right]}^{2}dt.$$

$$⟨g_3(T,\Delta )⟩\equiv \frac{1}{M_f}\text{sideset}{}^{\prime }\sum g_3(T,\Delta ),$$

$$f_p^{†}(\rho ,N)=-f_{\rho }\log \left(\frac{\rho }{{\rho }_g}\right),$$

$${\rho }_g(N)\sim N^{-\eta }$$

$$N_g(\rho )\sim {\rho }^{-1/\eta },$$

$$P(\Delta x)=⟨\delta (\Delta x-\left|x(t+\Delta )-x(t)\right|)⟩,$$

$$EB(T)\equiv \frac{\left[⟨g_3(T,\overline{\Delta }{)}^2⟩-⟨g_3(T,\overline{\Delta }){⟩}^2\right]}{⟨g_3(T,\overline{\Delta }){⟩}^2},$$

$$\frac{S}{k_{B}T}\sim ⟨m_{\mathrm{ov}}⟩\sim \frac{\rho }{N}{R}_g^3\sim {\rho }^{\alpha },$$

$$\bar{g}=\frac{\log \left(\frac{⟨m_{\mathrm{ov}}⟩-2}{⟨m_{\mathrm{ov}}⟩}(M-1)+1\right)}{\log (⟨m_{\mathrm{ov}}⟩-1)},$$

$$f_c^{\prime }=p_c⟨m_{\mathrm{ov}}⟩\frac{{\left(p_c\left(⟨m_{\mathrm{ov}}⟩-1\right)\right)}^{\bar{g}}-1}{p_c\left(⟨m_{\mathrm{ov}}⟩-1\right)-1}.$$

$$f_c=f_p{f}_c^{\prime }.$$

$$U_{\mathrm{LJ}}(r)=\left\{\begin{array}[]{lr}4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}+\frac{1}{4}\right]&\,r\leq r_{c}\\ 0&\,r>r_{c}\end{array}\right.\,,$$

$$U_{\mathrm{FENE}}(r)=\left\{\begin{array}[]{lcl}-0.5kR_{0}^{2}\ln\left(1-(r/R_{0})^{2}\right)&\ r\leq R_{0}\\ \infty&\ r>R_{0}&\end{array}\right.\,,$$

$$U_{\mathrm{bend}}(\theta )=k_{\theta }\left(1-\cos \theta \right).$$

$$D(\rho ,f_p)\equiv \underset{\Delta \to \infty }{\lim }\frac{⟨g_3(\Delta )⟩}{6\Delta },$$

Full text

Introduction –

The behaviour of unknotted and mutually unlinked ring polymers in dense solutions and melts is a yet unsolved issue in Polymer Physics McLeish (2008), and it has stimulated much theoretical Khokhlov and Nechaev (1985); Grosberg et al. (1988); Obukhov et al. (1994); Müller et al. (2000); Suzuki et al. (2009); Halverson et al. (2011a, b); Rosa et al. (2011); Sakaue (2012); Rosa and Everaers (2014); Grosberg (2014); Narros et al. (2014); Michieletto et al. (2014a); Smrek and Grosberg (2015); Lee et al. (2015); Michieletto and Turner (2016); Ge et al. (2016) and experimental Kapnistos et al. (2008); Gooßen et al. (2015); Brás et al. (2011, 2014); Gooßen et al. (2014); Vlassopoulos (2016) work in last decades. One of the most elusive aspects of this problem is the interplay between topological constraints (TCs) and both, structure and dynamics of the rings, which looks far more intricate than in their linear analogs. In the latter case, TCs induce slow dynamics through the reptative motion of the chain ends De Gennes (1971); Doi and Edwards (1986); Rubinstein and Colby (2003) without affecting the average chain size or gyration radius, $R_{g}$, which remains essentially random-walk-like Flory (1969); De Gennes (1979) and scales with the polymerization index $N$ as $R_{g}\sim N^{\nu}$ with $\nu=1/2$. In the former case, the polymers have no ends to “reptate” McLeish (2008) and global topological invariance requires that all rings remain permanently unlinked at the expense of some entropic loss Cates and Deutsch (1986). As a result, TCs affect both, dynamics and conformations of the rings whose gyration radius is characterized by a non-trivial exponent predicted to be in the range between $\nu=1/4$ Khokhlov and Nechaev (1985) and $\nu=2/5$ Cates and Deutsch (1986).

In recent years, more accurate computational work Suzuki et al. (2009); Halverson et al. (2011a); Rosa and Everaers (2014) has provided evidence that in the limit of large $N$, $\nu\rightarrow 1/3$, in agreement with a picture in which rings fold into “crumpled-globule”-like conformations Grosberg et al. (1993) whose compaction increases with solution density Nahali and Rosa (2016). In spite of this, the surface of each ring, i.e. the fraction of contour length in contact with other chains, is “rough” Nahali and Rosa (2016) and scaling as $N^{\beta}$ with $\beta\lesssim 1$ Halverson et al. (2011a); Halverson et al. (2013); Grosberg (2014); Smrek and Grosberg (2016). In fact, crumpled rings do not fully segregate or expel neighbouring chains from the occupied space Halverson et al. (2011a), rather, they fold into interpenetrating or “threading” conformations Michieletto et al. (2014a, b) that are akin to interacting “lattice animals” Rosa and Everaers (2014) with long-range (loose) loops Ge et al. (2016); Michieletto (2016).

Threadings are architecture-specific TCs that uniquely characterize systems of polymers whose contours display (quenched) closed loops (see Fig. 1(A)). By exploiting the abundance of these peculiar interactions, it has been shown recently Michieletto and Turner (2016) that a novel “topological freezing” can be induced in rings solutions at any temperature $T$. This putative glassy state, inherently driven by the topology of the constituents, is achieved by randomly pinning a fraction of rings, $f_{p}$, above an empirical “critical” value (see Fig. S1(A) in Supplemental Material (SM)):

[TABLE]

where $N_{g}$ is the theoretical length required for spontaneous (i.e., $f_{p}\rightarrow 0$) vitrification and $f_{N}$ a non-universal parameter Pha .

Topological freezing is the consequence of the proliferation of inter-ring constraints Michieletto et al. (2014a); Michieletto and Turner (2016), with the latter depending either on the polymerization index, $N$, or the density of the solution, $\rho$. While it has been shown that longer rings generate more TCs Michieletto and Turner (2016), it remains unclear how they behave if solutions become denser, rings more crumpled Nahali and Rosa (2016) and less space is available to threading.

Motivated by these considerations, in this Letter we study the effect of TCs by “randomly pinning” solutions of semi-flexible ring polymers, and probe the dynamic response of the rings for different solution densities and chain lengths. We show that the threshold pinning fraction $f_{p}^{\dagger}$ obeys an empirical relation akin to Eq. (1) and we derive universal scaling relations for the values of $N_{g}$ and $\rho_{g}$ at which spontaneous ($f_{p}\rightarrow 0$) glassiness is expected. We further discuss the dynamics of rings in terms of ensemble- and time-average observables and report, for the first time, numerical evidence for ergodicity breaking effects and pronounced heterogeneous non-Gaussian dynamics, even in unperturbed ($f_{p}=0$) solutions.

Results –

We present the results of large-scale molecular dynamics (MD) simulations of solutions of semi-flexible ring polymers made of $N=250$ and $N=500$ beads, for monomer densities $\rho\sigma^{3}=0.1,0.2,0.3,0.4$ (see SM for details). For each combination of $N$ and $\rho$, we run a single, independent realisation of the system in which a random fraction $f_{p}$ of rings have been “pinned”, i.e. permanently frozen in space and time.

The dynamics of a single non-frozen ring can then be captured by the mean-square displacement of its centre of mass, $g_{3}(T,\Delta)$, as a function of the lag-time $\Delta$ and measurement time $T$:

[TABLE]

The time-average displacement can be defined as $g_{3}(\Delta)\equiv\overline{g_{3}(T,\Delta)}$ while its ensemble average as

[TABLE]

with $\sum^{\prime}$ indicating that the average is performed over the set of $M_{f}$ “free”, i.e. not explicitly pinned, rings. Accordingly, we indicate the time- and ensemble-average displacement as $\langle g_{3}(\Delta)\rangle$.

Fig. 1(B,C,D) directly compare the behaviour of $\langle g_{3}(\Delta)\rangle$ in response to the random pinning of different fractions $f_{p}$ of rings (see also SM Fig. S2 for more cases). For unperturbed solutions ($f_{p}=0$), the data reproduce the known Halverson et al. (2011b); Brás et al. (2014); Grosberg (2014) crossover from sub-diffusive ($\langle g_{3}(\Delta)\rangle\sim\Delta^{3/4}$) to diffusive ($\langle g_{3}(\Delta)\rangle\sim\Delta$) behaviour. Perturbed systems, instead, display a reduced average diffusion, the more severe the higher the value of $f_{p}$. In particular, for $f_{p}$ larger than $f_{p}^{\dagger}(\rho,N)$, the average displacement remains well below one ring diameter (marked by the horizontal dashed lines) and does not diverge in time, indicating Michieletto and Turner (2016) a solid-like (glassy) behaviour. Furthermore, we find that $f_{p}^{\dagger}(\rho,N)$ decreases as a function of both, ring length $N$ Michieletto and Turner (2016) and, unexpectedly, monomer density $\rho$.

In order to obtain the functional form of $f_{p}^{\dagger}(\rho,N)$, the asymptotic diffusion coefficient $D(\rho,f_{p})\equiv\lim_{\Delta\rightarrow\infty}\langle g_{3}(\Delta)\rangle/6\Delta$ at given ($N$, $\rho$, $f_{p}$) is computed by best fit of the long-time behaviour of the corresponding $\langle g_{3}(\Delta)\rangle$ to a linear function (for details, see SM). Fig. 2(A) ($N=250$) and SM Fig. S1(B) ($N=500$) show plots for $D(\rho,f_{p})/D_{0}(\rho)$ – where $D_{0}(\rho)\equiv D(\rho,f_{p}=0)$ – as a function of $f_{p}$. Corresponding datasets are well fitted by the exponential function $d(f_{p})=\exp{(-kf_{p})}$, in agreement with previous results Michieletto and Turner (2016). We thus extract $f_{p}^{\dagger}(\rho,N)$ by finding the intersection of $d(f_{p})$ with a convenient small value of $0.01$ Michieletto and Turner (2016). The obtained “critical” lines $f_{p}^{\dagger}(\rho,N)$ (see SM Fig. S1(C)) separate regions of the parameter space $(\rho,f_{p})$ with finite (liquid) and vanishing (glassy) diffusion coefficients.

Interestingly, we find that the functional form of $f_{p}^{\dagger}(\rho)$ is akin to Eq. (1), i.e.

[TABLE]

and that our data for $N=250$ and $N=500$ collapse onto a master curve $f_{p}^{\dagger}(x=\rho/\rho_{g}(N))/f_{\rho}=-\log{\left(x\right)}$ with $f_{\rho}=0.43$ (Fig. 2(B)). Given that both, Eqs. (1) and (4), describe the same quantity, one may argue that the right-hand-side of both equations must be equal. By combining them Mas under the assumption that the only dependence on $\rho$ is contained in $N_{g}$, the values of $\rho_{g}$ and $N_{g}$ for spontaneous topological vitrification obey the following universal scaling relations

[TABLE]

with $\eta=f_{N}/f_{\rho}\simeq 0.7$ (using $f_{N}=0.303$ and $f_{\rho}=0.43$). Eqs. (4) and (Results –) provide quantitative predictions that may readily be tested in computer simulations and future experiments on melts of rings.

Having determined the empirical functional form of $f_{p}^{\dagger}$ and the generic behaviours of $N_{g}$ and $\rho_{g}$, we now turn our attention on the role of TCs in the dynamics of single rings. To this end, we consider the distribution of $1d$ displacements $\Delta x$ Not

[TABLE]

which corresponds to the self-part of the van-Hove function Kob et al. (1997); Chaudhuri et al. (2007) at given lag-time $\Delta$. The distribution of rescaled displacements $X\equiv\Delta x/\sqrt{\langle\Delta x^{2}\rangle}$ is expected to be described by the universal Gaussian function with zero mean and unit variance Kob et al. (1997). This is indeed the case for unperturbed solutions of short rings ($f_{p}=0$, Fig. 3(A)), whereas both, perturbed solutions (Fig. 3(B,C)) and even unperturbed ones with long rings and high density (Fig. 3(D)), distinctly deviate from the Gaussian behaviour (see also SM Figs. S3-S5).

Two novel features emerge from these plots: First, a prominence of rings with short “cage-like” displacements, identified by the narrow region centred around $X=0$ where $P(X)$ remains above the Gaussian. Second, the appearance of a population of rings travelling faster than the average ring, giving rise to “fat” exponential tails. Both features are akin to those observed in generic systems of particles close to glass and jamming transitions Chaudhuri et al. (2007) where each component alternates cage-like motion and large-scale rearrangements.

The non-Gaussian behaviour detected in our systems is clearly triggered by pinning perturbations (Fig. 3(C)), arguably through threading TCs. We conjecture that threading configurations may also be at the basis of the (weaker) non-Gaussian behaviour observed in unperturbed ($f_{p}=0$) solutions at large densities and $N=500$ (Fig. 3(D)). In particular, threadings may in general be the reason of the cage-like, non-Gaussian, motion of large ring polymers seen in experiments Brás et al. (2014). Although not permanent as in the case of pinned solutions, threadings between rings may in fact act as transient cages, and our results suggest that they may be the more long-lived the denser the solutions and the longer the rings.

In order to better understand the observed deviations from Gaussian behaviour, we now investigate time-average quantities of single ring trajectories. In Fig. 4(A,B) we report $g_{3}(T,\overline{\Delta})$, i.e. the centre of mass displacement of single rings at fixed lag-time $\Delta=\overline{\Delta}$ and increasing measurement time $T$ (see also SM, Figs. S6-S7). Unperturbed solutions of short rings show that $\lim_{T\rightarrow\infty}g_{3}(T,\overline{\Delta})=\langle g_{3}(\overline{\Delta})\rangle$, i.e. every ring tends to travel at the same average speed (see Fig. 4(A), $f_{p}=0$). Conversely, perturbed systems display heterogeneously-distributed trajectories which can be partitioned into two sub-populations having small and large displacements, respectively. The former reflect the above-mentioned presence of caged rings and we also notice examples of single rings with temporally-heterogeneous dynamics, alternating slow and fast diffusion (see Fig. 4(A), $f_{p}=0.3$). These observations are in agreement with the concept of permanent or transient caging due to threading TCs.

Constraints with diverging life-times have been shown to trigger non-ergodic behaviours Massignan et al. (2014). We quantify these deviations form ergodic diffusion through the ergodicity breaking parameter Jeon et al. (2016)

[TABLE]

which captures how fast the single-ring trajectories $g_{3}(T,\overline{\Delta})$ narrow around the mean $\langle g_{3}(\overline{\Delta})\rangle$. For standard diffusive solutions, $EB(T)\sim T^{-1}$ Metzler et al. (2014); Jeon et al. (2016) whereas non-ergodic systems display $EB(T)\sim T^{0}$ Deng and Barkai (2009). As shown in Fig. 4(C) (see also SM Figs. S6-S7) ergodicity breaking can indeed be triggered by random pinning. Remarkably, even unperturbed ($f_{p}=0$) solutions of rings with $N=500$ and monomer density $\rho=0.4\sigma^{-3}$ (Fig. 4(D)) display slower convergence to ergodic behaviour, thereby suggesting non-standard statistics in the waiting (sojourn) times of diffusing rings Deng and Barkai (2009); Metzler et al. (2014).

Taken together, these results assemble into a picture where pinned rings induce a form of “quenched disorder”, turning into “traps” with very large life-times Massignan et al. (2014), for the other rings. To our knowledge, this is the first instance that spontaneous caging and deviations from standard ergodic behaviour is directly observed in unperturbed solutions of polymers (of any topology).

Having investigated the heterogeneous dynamics of single rings, we conclude by connecting the observed non-Gaussian behaviour to the spatial organisation of the chains. By analogy with Cates and Deutsch (1986), one may argue that a ring of size $R_{g}$ experiences an entropic penalty proportional to the average number of overlapping neighbours $\langle m_{\rm ov}\rangle$

[TABLE]

where we assume that Suzuki et al. (2009); Halverson et al. (2011a); Rosa and Everaers (2014), in the large-$N$ limit, the number of chains in a volume $R_{g}^{3}$ converges to a (density dependent) constant characterized by an exponent $\alpha<1$ Nahali and Rosa (2016), i.e. $R_{g}^{3}/N\sim\rho^{-(1-\alpha)}$. In Eq. (8), $\langle m_{\rm ov}\rangle$ is defined as the average number of chains whose centres of mass are within $2R_{g}$ from the centre of mass of any other ring and we find that $\langle m_{\rm ov}\rangle$ achieves a $N$-independent value with $\alpha\simeq 0.60-0.74$ (see Fig. 5(A)). Importantly, Eq. (8) implies that higher monomer densities lead to a larger number of overlapping neighbours Nahali and Rosa (2016) and, in turn, larger entropic penalties Cates and Deutsch (1986), which consequently drive more compact conformations. On the other hand, results from Figs. 1-2 suggest that denser systems are more susceptible to random pinning, and display glassy behaviour at lower values of $f_{p}$.

This apparent contradiction can be reconciled by resorting to the following picture. We model rings as nodes of an abstract network, and a link between any two nodes indicates that the two corresponding rings overlap for a total time longer than half of the overall simulation runtime. An example of such a network is given in Fig. 5(B), where nodes have been ordered and coloured according to the corresponding single-ring diffusion coefficients, $D\equiv\lim_{\Delta\rightarrow\infty}g_{3}(\Delta)/6\Delta$. This representation intuitively shows that slow rings are connected (overlap) either with other slow rings or with frozen ones. On the other hand, mobile rings (i.e., rings with large diffusion coefficient) have virtually zero degree. This representation thus directly connects static and dynamic features of rings in solution and, in particular, indicates that overlapping rings, arguably exerting TCs on one another, slow down the respective motion.

To obtain then a quantitative estimation of how TCs affect the dynamics of rings, we approximate the network as a Bethe lattice Rubinstein and Colby (2003) of coordination (i.e., number of neighbours per node) $\langle m_{\rm ov}\rangle$. Due to the hierarchical nature of the network, the maximum number of shells, $\bar{g}$, is given by

[TABLE]

where $M$ is the total number of nodes (rings) of the network. We now assume that the effect of pinning a single ring results in the caging of its first neighbours with an unknown probability $p_{c}$, of its second neighbours with probability $p_{c}^{2}$, and so on. The whole process therefore results in a “caging cascade” producing a fraction of trapped rings equal to

[TABLE]

For small $f_{p}$, all pinned rings may be assumed to act independently on their neighbours. Then, the total fraction of caged rings, $f_{c}$, is

[TABLE]

Interestingly, Eqs. (10)-(11) link a measurable quantity (fraction of caged rings, $f_{c}$) to an imposed quantity (fraction of pinned rings, $f_{p}$) and, by inversion, allows to determine the caging (or threading) probability between close-by rings, $p_{c}$ Michieletto et al. (2014a); Michieletto and Turner (2016). In particular, Eq. (10) implies that the system becomes “critical” when $p_{c}=p_{c}^{\dagger}\equiv 1/(\langle m_{\mathrm{ov}}\rangle-1)$, for there exists a finite fraction $f_{c}$ of caged rings even in the limit $f_{p}\rightarrow 0$.

By combining Eqs. (10)-(11) and evaluating $f_{c}$ at $f_{p}=0.3$ as the rings displaying a single-ring diffusion coefficient $D/D_{\rm max}\simeq 0$ (see SM Fig. S8), we can numerically extract values for $p_{c}$ at any given $\rho$ (see Table S1). Interestingly, $p_{c}$ increases with $\rho$ up to where $p_{c}$ is approximately given by the predicted $p_{c}^{\dagger}$. Although we employ a crude approximation, we find that, curiously, the only two cases for which $p_{c}>p_{c}^{\dagger}$ are the ones displaying spontaneous ($f_{p}=0$) deviations from Gaussian behaviour (Fig. 3(D), $N=500$ and $\rho\sigma^{3}\geq 0.3$).

Conclusions –

In this work, we have shown that dense solutions of semi-flexible ring polymers display rich, non-Gaussian behaviours under random pinning perturbations. Glassiness is observed at pinned fractions $f_{p}$ larger than a “critical” value $f_{p}^{\dagger}(\rho,N)$, which obeys an empirical dependence on $\rho$ similar to the one previously reported for $N$ Michieletto and Turner (2016) (Fig. 2). As a consequence, we obtained novel, generic, and surprisingly simple, scaling relations for the threshold $\rho_{g}(N)$ and $N_{g}(\rho)$ marking the onset of spontaneous topological vitrification (Eqs. Results –).

We have reported the first evidence of strong ergodicity breaking in solutions of rings, triggered for any $f_{p}>0$ and non-trivial convergence towards ergodicity has also been found for unperturbed solutions at high density (Fig. 4). These results can be rationalized by assuming that random pinning turn transient threading topological constraints into quenched disorder and permanent cages. Overall, ring solutions appear to cluster into sub-systems with slow/fast diffusivities corresponding to large/little overlaps with other slow or pinned rings (Fig. 5).

An intriguing finding of our work is that, even in the limit $f_{p}\rightarrow 0$, solutions of rings may deviate from standard Gaussian behaviour (Figs. 3(D)-4(D)) and turn into “topological glasses” provided $\rho$ or $N$ are large enough (Fig. 2). We have concluded that a topological glass may form when the probability $p_{c}$ of any pinned ring to cage any of its neighbours is $\geq p_{c}^{\dagger}$, with $p_{c}^{\dagger}$ given by a simple analytical expression for networks in the Bethe lattice approximation.

We argue that the experimentally observed Brás et al. (2014) non-Gaussian behaviour of ring polymers melts may be well reconciled with this picture. At the same time, we hope that the present work will pave the way for future experiments and computer simulations.

Acknowledgments –

The authors thank C. Micheletti and T. Sakaue for insightful remarks on the manuscript and D. Levis for interesting suggestions.

I Model and methods

I.1 The model

Solutions of ring polymers are modelled by resorting to the Kremer and Grest Kremer and Grest (1990) polymer model.

Excluded volume interactions between beads (including consecutive ones along the contour of the chains) are described by the shifted and truncated Lennard-Jones (LJ) potential:

[TABLE]

where $r$ denotes the separation between the bead centers. The cutoff distance $r_{c}=2^{1/6}\sigma$ is chosen so that only the repulsion part of the Lennard-Jones is used. The energy scale is set by $\epsilon=k_{B}T$ and the length scale by $\sigma$. In the course of the paper, we adopt conventional LJ units with $\epsilon=1$ and $\sigma=1$.

Nearest-neighbour monomers along the rings contour length are connected through the finitely extensible nonlinear elastic (FENE) potential:

[TABLE]

where $k=30\epsilon/\sigma^{2}$ is the spring constant and $R_{0}=1.5\sigma$ is the maximum extension of the elastic FENE bond.

In order to maximize mutual chain interpenetration at relatively moderate chain length Müller et al. (2000) and hence reduce the computational effort, we have introduced an additional bending energy penalty between consecutive triplets of neighbouring beads along the chain in order to control polymer stiffness:

[TABLE]

Here, $\theta$ is the angle formed between adjacent bonds and $k_{\theta}=5\,k_{B}T$ is the bending constant. With this choice, the polymer is equivalent to a worm-like chain with Kuhn length $l_{K}$ equal to $10\sigma$ Auhl et al. (2003).

I.2 Simulation details

We consider equilibrated polymer solutions consisting of $M=160$ and $M=80$ ring polymers made of $N=250$ and $N=500$ beads each, respectively. The total number of monomers of each system is then fixed to $40^{\prime}000$ monomer units. We study solutions at four monomer densities, $\rho\sigma^{3}=0.1,0.2,0.3$ and $0.4$. In particular, systems of rings with $\rho\sigma^{3}=0.1$ have been at the center of the previous study Michieletto and Turner (2016), and will be then used as a validation of the present work.

The static and kinetic properties of the chains are studied using fixed-volume and constant-temperature molecular dynamics (MD) simulations (NVT ensemble) with implicit solvent and periodic boundary conditions. MD simulations are performed using the LAMMPS engine Plimpton (1995). The equations of motion are integrated using a velocity Verlet algorithm, in which all beads are weakly coupled to a Langevin heat bath with a local damping constant $\Gamma=0.5\,\tau_{\mathrm{LJ}}^{-1}$ where $\tau_{\mathrm{LJ}}=\sigma(m/\epsilon)^{1/2}$ is the Lennard-Jones time and $m=1$ is the conventional mass unit for monomer and colloid particles. The integration time step is set to $\Delta\tau=0.012\,\tau_{\mathrm{LJ}}$.

I.3 Preparation of the initial configuration and system equilibration

Equilibrated solutions of ring polymers are prepared as described in Ref. Nahali and Rosa (2016). To avoid unwanted linking between close by rings, the chains were initially arranged inside a large simulation box at very dilute conditions. In order to reach the correct monomer density of $\rho\sigma^{3}=0.1$ we performed then a short ($\approx 400\,\tau_{\mathrm{LJ}}$) MD simulation under fixed external pressure which shrinks the simulation box until it reaches the desired value. Similarly, the other densities were reached by compressing the solutions under even higher imposed pressures.

Once any given system was prepared at the correct density, we switched to the NVT ensemble. Then, each system was equilibrated by performing single MD runs up to $1\cdot 10^{9}\Delta\tau=12\cdot 10^{6}\tau_{\mathrm{LJ}}$ (for $N=250$) and $2\cdot 10^{9}\Delta\tau=24\cdot 10^{6}\tau_{\mathrm{LJ}}$ (for $N=500$), during which the center of mass of each chain moves on average a distance comparable to $\approx 3-4$ times its corresponding gyration radius, $R_{g}$ (for more details, see the Supplementary Material of Ref. Nahali and Rosa (2016)).

I.4 Molecular dynamics runs

After equilibration, ring dynamics was studied by performing MD simulations up to $1\cdot 10^{9}\Delta\tau=12\cdot 10^{6}\tau_{\mathrm{LJ}}$ for both $N=250$ and $N=500$. We studied systems with different pinning fractions $f_{p}$ of the total number of rings in the range $f_{p}=0.1-0.7$. For comparison we have also considered unperturbed rings solutions, corresponding to pinning fraction $f_{p}=0$.

I.5 Calculation of diffusion coefficients

The ring asymptotic diffusion coefficient at given ($N$, $\rho$, $f_{p}$) is defined as:

[TABLE]

where $\langle g_{3}(\Delta)\rangle$ is the mean-square displacement of the chain center of mass. By following the same procedure employed in Ref. Michieletto and Turner (2016), $D(\rho,f_{p})$ is computed by standard best fit of the long-time behaviour of the corresponding $\langle g_{3}(\Delta)\rangle$ to a linear function. This is done regardless the long-time behavior of $\langle g_{3}(\Delta)\rangle$ is effectively linear or saturates to a plateau like in frozen set-up’s. In the latter case, the evaluated $D(\rho,f_{p})$ should be then best considered as an upper value to the “true” asymptotic behavior.

II Supplemental Figures

III Supplemental Tables