Glassiness and Heterogeneous Dynamics in Dense Solutions of Ring Polymers

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 nonergodic and glassy behavior. 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 organization and mutual interpenetration. Our results suggest that deviations from the typical behavior observed in systems of linear polymers may originate from architecture-specific (threading) topological constraints.

Figure 1

Random pinning triggers slowing down and glassiness. (a) Typical melt structure for rings of $N=250$ monomers with $f_p=0$ and $\rho =0.2{\sigma }^{-3}$. Inset: two rings isolated from the melt and showing mutual threading. (b)–(d) Mean-square displacement of rings c.m., $⟨g_3(\mathrm{\Delta })⟩$ [Eq. (3)] as a function of lag time $\mathrm{\Delta }$ (see also SM [43]). Rings display glassy behavior ($⟨g_3(\mathrm{\Delta })⟩\sim {\mathrm{\Delta }}^0$) for $f_p>f_p^{†}$ where $f_p^{†}$ decreases with both $N$ and $\rho$. Dashed horizontal lines mark the mean-square ring diameters, $4⟨R_g^2⟩$.

Figure 2

Exponential slowing down and universal phase diagram. (a) $D(\rho ,f_p)/D_0(\rho )$ is compatible with exponential decay (dashed line) in $f_p$. An arbitrarily small (0.01) value is chosen to determine the transition to glassy behavior [20]. (b) Universal master curve [$=-\log (x)$, dashed line] for $f_p^{†}(\rho ,N)/f_{\rho }$ as a function of $\rho /{\rho }_g(N)$ [see Eq. (4)] vs data (symbols) for $N=250$ and $N=500$ with $f_{\rho }(N=250)=0.435\pm 0.015$, $f_{\rho }(N=500)=0.445\pm 0.05$, ${\sigma }^3{\rho }_g(N=250)=0.84\pm 0.05$, and ${\sigma }^3{\rho }_g(N=500)=0.56\pm 0.05$.

Figure 3

Distributions of displacements are non-Gaussian. Distribution functions, $P(X)$, of $1d$ scaled displacements of the c.m. of nonpinned rings, $X\equiv \mathrm{\Delta }x/\sqrt{⟨\mathrm{\Delta }{x}^2⟩}$, at the crossover lag time ${\mathrm{\Delta }}_c$, ($⟨g_3({\mathrm{\Delta }}_c){⟩}_{f_p=0}\equiv 4⟨R_g^2⟩$). (a) Unperturbed solutions of short rings follow a Gaussian function with zero mean and unit variance (dashed lines). (b),(c) Perturbed solutions display caging and fat, exponential tails (solid lines). (d) Deviations from Gaussian behavior are also observed in unperturbed solutions with $N=500$ and $\rho {\sigma }^3\ge 0.3$ since $N\simeq N_g(\rho )$. Deviations become more marked as $\mathrm{\Delta }$ increases (SM [43] Fig. S3).

Figure 4

Heterogeneity and ergodicity breaking. (a),(b) Representative curves for $g_3(T,{\mathrm{\Delta }}_c)/⟨g_3({\mathrm{\Delta }}_c){⟩}_{f_p=0}$ at fixed lag time $\mathrm{\Delta }={\mathrm{\Delta }}_c$ as a function of measurement time $T$ displaying spatial and temporal (gray trace) heterogeneity. (c),(d) Corresponding ergodicity-breaking (EB) parameters [Eq. (6)]. $\mathrm{EB}\sim T^{-1}$ marks standard diffusive processes, whereas $\sim T^0$ is a signature of ergodicity breaking. The system with $N=500$ at the highest density $\rho {\sigma }^3=0.4$ shows weaker convergence $\sim T^{-0.5}$ even at $f_p=0$.

Figure 5

Slowing down of persistently overlapping rings. (a) Average number of overlapping chains per ring, $⟨m_{\mathrm{ov}}(\rho )⟩$. Dashed lines correspond to the fitted power-law behaviors (see Table SII in SM [43]). (b) Abstract network representation: nodes (representing rings) are color coded and arranged clockwise according to their relative diffusion coefficients, $D/D_{\max }$. Edges are drawn only if the nodes have been overlapping for more than 50% of the total run time. Notice that while all rings display the same $m_{\mathrm{ov}}$ at any time, slow-moving and caged nodes display larger degrees indicating more persistent overlaps. In contrast, fast rings show little or no long-time overlaps (see also SM [43] Fig. S8).