Unified understanding of the impact of semiflexibility, concentration, and molecular weight on macromolecular-scale ring diffusion

Significance

Understanding the slow dynamics of dense liquids of large topologically compressed polymer rings, bundled chromosomes, microgels, single-chain nanoparticles, and other compact soft particle systems is a frontier area of macromolecular science, soft condensed-matter physics, and biophysics. By studying ring liquids beyond the traditional focus on molecular weight, we establish the large and universal consequences of polymer semiflexibility and concentration on diffusive transport by performing the largest simulations to date and verifying predictions of an intermolecular force-based theory. Two diffusion regimes are predicted and confirmed, with the slowest providing a fundamental basis for activated transport and macromolecular-scale vitrification. Our findings provide a deeper mechanistic understanding and make testable predictions for guiding future experiments and simulations including syntheses of compact macromolecules.

Abstract

Conformationally fluctuating, globally compact macromolecules such as polymeric rings, single-chain nanoparticles, microgels, and many-arm stars display complex dynamic behaviors due to their rich topological structure and intermolecular organization. Synthetic rings are hybrid objects with conformations that display both ideal random walk and compact globular features, which can serve as models of genomic DNA. To date, emphasis has been placed on the effect of ring molecular weight on their unusual behaviors. Here, we combine simulations and a microscopic force-level theory to build a unified understanding for how key aspects of ring dynamics depend on different tunable molecular properties including backbone rigidity, monomer concentration, degree of traditional entanglement, and molecular weight. Our large-scale molecular dynamics simulations of ring melts with very different backbone stiffnesses reveal unanticipated behaviors which agree well with our generalized theory. This includes a universal master curve for center-of-mass diffusion constants as a function of molecular weight scaled by a chemistry and thermodynamic state-dependent critical molecular weight that generalizes the concept of an entanglement cross-over for linear chains. The key physics is how backbone rigidity and monomer concentration induced changes of the entanglement length, interring packing, degree of interpenetration, and liquid compressibility slow down space-time dynamic-force correlations on macromolecular scales. A power law decay of the center-of-mass diffusion constant with inverse molecular weight squared is the first consequence, followed by an ultraslow activated hopping transport regime. Our results set the stage to address slow dynamics and kinetic arrest in different families of compact synthetic and biological polymeric systems.

Synthetic polymeric materials and condensed phases of biological macromolecules in living cells are generally constructed of linearly connected chains. Long synthetic polymers are random coils at high concentrations, strongly interpenetrate, and become “topologically entangled” which drives their unique slow dynamics and viscoelasticity that is crucially linked to the motion of chain ends (1, 2). The removal of chain ends results in a novel topological constraint which qualitatively changes not only the dynamics but also equilibrium conformation and intermolecular packing (3–27). Specifically, polymer rings form a topologically compressed globule-like conformation on length scales beyond the corresponding linear chain entanglement length scale of order 3 to 10 nm in dense melts (3–19). Other soft polymeric systems, such as intramolecularly cross-linked single-chain nanoparticles (21), many-arm stars (22), micro- and nanogels (25), polymeric micelles (26), glycogens (27), and even folded proteins (23) and two-dimensional polymer chain liquids (24), can also be globular, albeit for different physical reasons. Among this large family of globular-like macromolecules, unconcatenated polymer rings are unique in having a “two-fractal” intramolecular conformational structure in that they are globule-like on large enough length scales, but ideal random walk chain like below an entanglement length. Understanding such systems has been vigorously pursued recently from experimental, simulation, and theoretical perspectives. Beyond synthetic polymer science, ring polymers occur naturally in biology (e.g., circular prokaryotic genomes, plasmid-based DNA vaccines) (28–30), and operationally in the cell nucleus where ultralong linear genomic DNA adopts a compact ring-like conformation (6). Proposed analogies between the physical behavior of dense ring liquids and chromosomes buttress the interest in such macromolecular systems (6, 31, 32).

Neutron scattering (33–35) and mechanical (3, 14, 17, 35) measurements have revealed conformational, dynamic, and viscoelastic behavior of high-molecular-weight synthetic ring melts that distinctively differ from that of linear chains. Very recent experimental advances and simulations have shown that as flexible rings become unprecedently large, their mechanism of stress relaxation undergoes a qualitative change that does not agree with existing theories based on power law in time relaxation (17, 20). This behavior is general and applies even when changing local ring stiffness over a wide range that spans synthetic and biological macromolecules, albeit with a strong shift of the behaviors to lower molecular weights with increasing ring rigidity (17). The emergent physics cannot be explained based solely on the traditional concept of entanglement that is successful for the linear chain analogs. The possibility of so-called interring topological “deep threadings” (5, 36–38) for large-molecular-weight rings has been advanced, but this concept is difficult to quantify for real dense liquids of mobile rings. Overall, our present understanding of the fundamental mechanism of ring liquid dynamics and packing structure remains limited, including for the practical experimental reason (17, 20) that purification and synthesis of rings are difficult.

Given accelerating advances in computational power where ring purity is not an issue, simulations are an increasingly powerful probe of the basic physics of large ring polymer liquids (4, 6, 9–11, 18, 19, 31, 37, 39–43). Striking phenomena identified that are qualitatively distinct from their entangled linear chain counterparts include i) a surprisingly large degree of deep ring–ring interpenetration despite their globally compact globular conformations (5, 6, 9), ii) a puzzling scaling with degree of polymerization N of the ring center-of-mass (CM) self-diffusion constant of $D_{\mathrm{CM}}\propto N^{-2}$ (4, 9, 13), which is essentially identical to that of entangled chains (1, 2) even though conformationally isotropic rings with no free ends cannot diffuse via the anisotropic reptation mechanism, iii) a slower CM diffusive transport compared to internal conformational relaxation on the macromolecular scale in qualitative contrast to entangled chain liquids (4, 5, 9, 11), iv) unusual intermediate time non-Fickian diffusion at the monomer and CM levels (4, 5, 9, 11, 13, 44), and perhaps most dramatically, v) the speculative possibility based on simulations (9, 11) of a second dynamical cross-over with increasing N that stiff rings in semidilute solutions move via rare activated processes on the macromolecular (not monomer) length scale. The latter is proposed to be a dynamic caging process, unrelated to a mechanical jamming transition (45–47). Most recently, simulations have begun to vary ring backbone stiffness over a wide range (16), with unexpected dynamical behaviors observed.

Mechanistic understanding of the above emerging body of molecular dynamics (MD) simulation data, and new experimental observations (14, 17, 20), is a major challenge. Pioneering work modeled liquids as a single mobile ring on a lattice of static obstacles combined with specific dynamic scaling ansatzes (3, 8, 48), most notably the fractal loopy globule (FLG) model (8). Such approaches have led to some understanding when rings are large enough to form compact shapes corresponding to $N> N_{\mathrm{e}}$, where $N_{\mathrm{e}}$ is the critical degree of polymerization for the emergence of dynamic entanglement effects in the corresponding linear chain liquids, but still sufficiently small and flexible that $N/N_{\mathrm{e}}$ only modestly exceeds unity. For example, the ring CM diffusion constant is predicted to obey a power law $D_{\mathrm{C}\mathrm{M}}\propto N^{-y}$ with y varying from 5/3 to 7/3 depending on models, which is close to experimental results [y ~ 2.2 (13)] and early simulations [y ~ 2.0 to 2.4 (4, 9, 11)] performed over a decade or less variation of N. These models are built on self-similar structural and dynamical ansatzes and predict a fractional power-law stress relaxation modulus G(t) on intermediate time and length scales. The latter is consistent with early experiments on polystyrene and polyisoprene ring melts (3, 48, 49) of modest degrees of entanglement, $N/N_{\mathrm{e}}<15$, but does not agree with the very recent experimental and simulation studies (17) that probe flexible ring liquids up to $N/N_{\mathrm{e}}\approx 60-220$ which found a non-power-law stress relaxation. A possible exponentially slow activated transport regime at ultrahigh values of $N/N_{\mathrm{e}}$, and the strong dependence of dynamics on ring backbone stiffness at much higher N recently established (16), are not predicted by such models.

Here, we address the overarching fundamental question of what determines the cross-over degree(s) of polymerization for the emergence of the distinctive dynamical behaviors in ring polymer liquids. Typically, it has been assumed to be $N_{\mathrm{e}}$. For example, the lattice animal and FLG models (3, 8, 48) predict the relaxation time scales as $\tau \approx N^2{\left(N/N_{\mathrm{e}}\right)}^{1/2}$ and $\tau \approx N^2{\left(N/N_{\mathrm{e}}\right)}^{1/3}$ for $N>N_{\mathrm{e}}$, respectively, which can be compared to the de Gennes reptation-tube model result for entangled chain melts (1, 2) of $\tau \approx N^2\left(N/N_{\mathrm{e}}\right)$. As shown here, although these scaling laws are close to the observed N-scaling of flexible ring melts if N is not too high, they qualitatively fail for the $N_{\mathrm{e}}$ dependence in a ring stiffness and monomer concentration-dependent manner. Quantitative predictions and understanding of a possible second cross-over to an activated glassy-like dynamics at a much larger degree of polymerization (9, 11, 50, 51) are essentially nonexistent.

To address the above open questions, we go beyond traditional studies that focus on the role of ring molecular weight and use simulations and theory to establish and understand how slow diffusive dynamics depends on ring backbone stiffness and monomer concentrations spanning the semidilute solution to melt conditions, and its causal connection to the unusual two-fractal ring intramolecular structure and attendant intermolecular packing. Our work builds on the recent statistical mechanical approach of Mei, Dell, and Schweizer (MDS) (44, 51) which combines concepts from polymer, colloid, and glass physics, and qualitatively differs from other theoretical approaches built on tube-like approximations, dynamic scaling ansatzes, or topological free-volume concepts (3, 8, 12, 48). The MDS theory is formulated not in terms of conjectures for the real space trajectories of ring polymers as commonly done in polymer physics models but rather in terms of the correlation in space and time of intersegmental forces experienced by a tagged ring in a dense liquid. Via exact statistical mechanical relations, knowledge of the latter enables prediction of ensemble-averaged dynamical properties. Such an approach was previously favorably compared against a few simulation results for N-dependence of the ring CM diffusion constant (44, 51).

However, the existing MDS theory, and all other theories/models, do not account for the strong influence of ring backbone stiffness and concentration on dynamics and diffusion, and cannot explain the striking simulation findings reported here. The present work identifies the key missing physics and integrates it into the MDS theory framework, thereby providing a unified understanding of both published and our present comprehensive set of MD simulation results. This includes providing evidence for the MDS theory prediction of an activated CM diffusion regime. This cross-over relates to the discussion of “topological glasses” in the literature (9, 11). But in our theoretical approach, it is more akin to an unusual form (due to ring internal structure) of kinetic arrest associated with caging on the macromolecular scale, as occurs in other soft colloidal systems (e.g., microgels, micelles, many arm stars). We believe that our work will help guide the design and execution of future experiments and simulations for cyclic polymers, and also condensed phases of other synthetic and biological fluids composed of compact macromolecules.

Simulation Results and Theory-Motivated Empirical Analysis.

We use the Kremer–Grest (KG) bead-spring model (4, 5, 52) to perform MD simulations. All monomers interact via a truncated and shifted Lennard-Jones repulsive potential and are connected to form rings with the finitely extensible nonlinear elastic bond potential. Initial configurations of dense unconcatenated ring melts are generated via the method developed by Smrek et al. (37). Chain backbone stiffness is varied with a cosine angular bending potential by using a dimensionless stiffness constant $k_{\theta }$. The dynamics of the KG model have been thoroughly characterized in and out of equilibrium (53), and its parameters can be mapped onto large families of synthetic polymers (54), and also structural semiflexible fibers and biopolymers like collagen and DNA. These various chemistries are captured by tuning the backbone stiffness over a wide range (54). To capture this wide range, we have studied KG melts with $k_{\theta }=0,1.5,2.25,3.0,4.0$. All simulation details have been documented previously (4, 5, 52, 53). It is worth mentioning that to compute quantities such as the dimensionless longest relaxation time (see below) ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{LJ}}$, our runs for the largest systems required simulating equilibrium chain dynamics for ∼300 CPU-years.

The Fig. 1A shows snapshots of single ring conformations in the melt which vary dramatically as backbone bending stiffness is changed at fixed $N$ corresponding to variation of the statistical measure of the length scale of backbone rigidity, the Kuhn length $l_{\mathrm{K}}$, from ~1.6 to ~3.5 monomer (bead) diameters (${\sigma }_{\mathrm{L}\mathrm{J}}$, the simulation unit of length). This change of local ring stiffness has large consequences on the degree of interring interpenetration, as can be quantified by a macromolecular volume fraction $\varphi \equiv \left(4\pi /3\right){\rho }_{\mathrm{poly}}{R}_{\mathrm{g}}^3$, where ${\rho }_{\mathrm{poly}}=\rho /N$ is the ring number density, ρ the monomer number density, and R_g the radius of gyration. Although for ideal chain random walk polymers an invariant degree of polymerization $\overline{N}$ can be defined (55, 56) as ${\overline{N}}^{1/2}\equiv {\rho }_{\mathrm{poly}}{R}_{\mathrm{g}}^3$, for rings $\varphi$ does not define a characteristic $\overline{N}$ since the globally compact conformation ($R_{\mathrm{g}}\propto N^{1/3}$) implies it is $N$-independent in the large ring limit, albeit depends on monomer concentration, ring stiffness, and $N_{\mathrm{e}}$. For example, the main frame of Fig. 1B shows that $\varphi$ initially grows with N for both fixed backbone stiffnesses studied, but ultimately saturates, per literature studies (31). However, as stiffness increases, $\varphi$ at fixed N grows rather dramatically (Inset of Fig. 1B). A visual illustration is provided in Fig. 1C for simulation snapshots of the local packing of multiple rings, which reveals their nontrivial, but limited, degree of interpenetration, and roughly globular but highly fluctuating shape.

Fig. 1.

(A) MD snapshots of a single ring polymer conformation taken from a dense melt of N = 1,600 for the four indicated values of bending energy, $k_{\theta }$. Different colors indicate different parts of the same ring. (B) Macromolecular packing fraction $\varphi$ as a function of degree of polymerization at fixed stiffness ($k_{\theta }=0$ and 1.5) (main), and as a function of backbone stiffness k_θ at fixed N = 1,600 (Inset) (17). (C) Snapshots that illustrate the degree of interpenetration of neighboring rings in a melt with N = 6,400 and $k_{\theta }=1.5$. A target ring (red) and its four most intimate neighbors (other colors) are shown. Neighboring rings are shown as opaque (Left) and translucent (Right) to better display ring structures. Upper images are the front view while the lower images are the Bottom view. (D) Time evolution of the segment-MSD $g_1(t)$ (dashed) and CM-MSD $g_3(t)$ (solid) in units of $R_{\mathrm{g}}^2$ for variable bending stiffnesses $k_{\theta }$ at N = 1,600. The orange dot-dashed horizontal line for $g_1\left(t\right)=3R_{\mathrm{g}}^2$ defines the relaxation time ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{L}\mathrm{J}}$. For $k_{\theta }=4$ data, $g_1\left(t\right)$ does not reach $3R_{\mathrm{g}}^2$ at the longest time simulated. However, because the corresponding value of $g_1\left(t\right)$ is close to $3R_{\mathrm{g}}^2$, we adopt a power law to modestly extrapolate the $g_1\left(t\right)$ data to $3R_{\mathrm{g}}^2$ and thus deduce the corresponding value of ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{L}\mathrm{J}}$. The CM-MSD data for $k_{\theta }=2.25$ and 3 do not reach the Fickian regime per strictly linear in time scaling. We have adopted the standard procedure (4) to estimate the long-time CM diffusion constant by plotting $g_1\left(t\right)/6t$ and $g_3\left(t\right)/6t$ versus t^-1 and extrapolating to t⁻¹ → 0. Reassuringly, the extrapolated values based on the two quantities are very similar as they must be, and we use this as an estimate of the long-time CM diffusion constants.

Understanding ring dynamics requires as foundational input specific aspects of the analogous entangled linear chain melts since their globule-like conformation is driven by static consequences of topological entanglements which represent the combined consequences of polymer connectivity and uncrossability due to repulsive excluded volume interactions. There are well-known (57–60) connections between $N_{\mathrm{e}}$, entanglement length scale or tube diameter $d_{\mathrm{T}}$, and the Kuhn length for linear chain melts (SI Appendix, Fig. S1) given by $d_{\mathrm{T}}=\sqrt{N_{\mathrm{e}}}{{(l}_{\mathrm{b}}{l}_{\mathrm{K}})}^{0.5}$ and $\frac{l_{\mathrm{K}}}{l_{\mathrm{b}}}=\frac{1}{0.078}{N}_{\mathrm{e}}^{-0.535}$. For the wide range of ring bending energies studied here ($k_{\theta }=0-4$), $N_{\mathrm{e}}$ varies from ~13 to 45. We use $l_{\mathrm{K}}{l}_{\mathrm{b}}=〈R_{\mathrm{ee}}^2〉/(N_{\mathrm{L}}-1)$ to compute the Kuhn length $l_{\mathrm{K}}$ with $〈R_{\mathrm{ee}}^2〉$ the mean square end-to-end distance of a chain in the melt and $N_{\mathrm{L}}$ is its degree of polymerization. This $l_{\mathrm{K}}$ serves as input to the Z1 method to determine the topological entanglement degree of polymerization $N_{\mathrm{e}}$ and corresponding tube diameter $d_{\mathrm{T}}$ (SI Appendix) (57, 59).

We adopt as a measure of the dimensionless slowest relaxation time, ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{L}\mathrm{J}}$, the time scale for a ring monomer mean-square-displacement (MSD) $g_1\left(t\right)=3<R_{\mathrm{g}}^2>$ (Fig. 1D and SI Appendix, Fig. S2). This practical definition ensures all macromolecules can rejuvenate their local molecular environment. The ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{L}\mathrm{J}}$ data are shown in Fig. 2A as a function of N over a very wide range of N and stiffnesses. We observe an effective power law behavior as a function of N that extends over five decades for the $k_{\theta }=1.5$ data with N = 50 to 6,400 and over three decades for the $k_{\theta }=0$ data with N = 50 to 1,600. Moreover, a very strong slowing down of relaxation extending over two decades is found as ring bending stiffness increases from $k_{\theta }=0\text{to}4$. Importantly, no globally collapsed behavior is found for all these ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{L}\mathrm{J}}$ data.

Fig. 2.

(A) and (B): Nondimensionalized longest relaxation time ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{L}\mathrm{J}}$ as a function of N and N/Ne, respectively, for various ring stiffnesses and hence variable $N_{\mathrm{e}}$. (C) and (D): Same data as in A and B but for the nondimensionalized CM diffusion constant $D_{\mathrm{CM}}/D_{\mathrm{LJ}}$. In C and D, some data for $k_{\theta }=1.5$ are from previous simulations [Halverson et al. (HLGGK) (4) and Rauscher et al. (RSRD) (43)] which agree very well with our results for $k_{\theta }=1.5$. The Inset of D shows experimental results (13) for ring and linear polyethylene oxide (PEO) melts (solid symbols) compared with the simulation data (open symbols) for the $k_{\theta }=1.5$ model [the present results for rings and the literature data for chains (4)]. The simulation data are vertically shifted down by a common identical factor to test whether the simulations can capture the N dependence and large quantitative difference between ring and chain melts.

The simulation data in Fig. 1D also show interesting non-Fickian regimes on intermediate time and length scales. Such anomalous diffusion occurs in diverse soft matter systems, albeit for different physical reasons (61). Our present article is focused on long-time mass transport, and hence we defer a discussion of these rich behaviors in Fig. 1D to future work.

We now empirically explore whether a specific guess for a ring stiffness-dependent critical degree of polymerization might collapse the data in Fig. 2A. Our naive choice is motivated by the fact that linear chain entangled melts display a power-law dependence of the longest relaxation time on N and $N_{\mathrm{e}}$ given by (1, 2) ${\tau }_{\mathrm{d}}={\tau }_{\mathrm{R}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{e}}{\left(N/N_{\mathrm{e}}\right)}^{\alpha }\sim {N^2\left(N/N_{\mathrm{e}}\right)}^{\alpha }$, where $\alpha =1.0\mathrm{and}$ $\sim 1.4$ based on the original reptation-tube model (1, 2) and in experiment (1, 2), respectively. Fig. 2B attempts to collapse our ring melt data based on the ansatz that the entangled chain $N_{\mathrm{e}}$ determines the cross-over degree of polymerization for slow ring relaxation. One sees a reasonable collapse is achieved when ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{L}\mathrm{J}}$ is plotted as a function of $N/N_{\mathrm{e}}$, with relatively modest deviations appearing at very large values of $N/N_{\mathrm{e}}>50$. Moreover, an apparent power law scaling is identified for the collapsed data with an exponent of ~8/3. This result is conceptually very surprising and is qualitatively distinct from the entangled chain behavior and the predictions of extent models for ring dynamics for at least two fundamental reasons: i) cyclic rings do not translate large distances by anisotropic reptation along a curvilinear path in a tube, and ii) the simulation suggested scaling ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{LJ}}\sim {\left(N/N_{\mathrm{e}}\right)}^{8/3}$ is qualitatively distinct from the much weaker $N_{\mathrm{e}}$ dependences predicted by reptation-tube theory (1, 2) for chains, $\tau ={\tau }_{\mathrm{R}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{e}}(N/N_{\mathrm{e}})\propto N^2\left(N/N_{\mathrm{e}}\right)$, and by the FLG dynamic scaling model (8) for rings, ${\tau \propto N}^2{\left(N/N_{\mathrm{e}}\right)}^{1/3}$. To buttress the robustness of our findings we have also analyzed ${\tau }_{\mathrm{CM}}/{\tau }_{\mathrm{L}\mathrm{J}}$ extracted from the CM-MSD based on $〈r_{\mathrm{CM}}^2(t={\tau }_{\mathrm{CM}})〉=B{R}_{\mathrm{g}}^2$ where B is a constant of order unity. Our conclusions remain essentially identical (SI Appendix, Fig. S3).

We now consider global mass transport as quantified by the nondimensionalized Fickian CM diffusion constant $D_{\mathrm{CM}}/D_{\mathrm{LJ}}$ ($D_{\mathrm{LJ}}\equiv {\sigma }_{\mathrm{LJ}}^2/{\tau }_{\mathrm{LJ}}$) determined from the CM-MSD simulation data (SI Appendix). The results are plotted as function of N in Fig. 2C. We do not evaluate $D_{\mathrm{CM}}/D_{\mathrm{LJ}}$ for the N ≥ 3,200 systems due to the dramatic slowing-down of CM mass transport on the computational time scales presently accessible. Recall that the classic unentangled Rouse model (2, 62, 63) for the CM diffusion constant of rings is blind to polymer architecture predicting $D_{\mathrm{CM}}\sim N^{-1}$, corresponding to a total chain friction constant of ${\zeta }_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=N{\zeta }_{\mathrm{s}}$, where ${\zeta }_{\mathrm{s}}$ is the local segmental friction constant. The results in Fig. 2C are qualitatively similar to those in Fig. 2A. Specifically, a power law behavior as a function of N is found for the more flexible rings with $k_{\theta }=1.5$ or $k_{\theta }=0$, but further increase of bending stiffness results in a dramatic decrease of the diffusion constant. Moreover, no collapsed behavior is observed.

One can empirically test whether the CM diffusion constant data collapse in the same manner explored above for ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{L}\mathrm{J}}$. The results are shown in Fig. 2D where N is normalized by $N_{\mathrm{e}}$. A good collapse is obtained. However, the collapsed data are not globally well described as a single power law, although it is a reasonable representation for the significant range of $N/N_{\mathrm{e}}\sim 0.5$ up to $N/N_{\mathrm{e}}\sim 40-50$ with an N-scaling exponent of ~2, i.e., $D_{\mathrm{CM}}/D_{\mathrm{LJ}}\sim {\left(N_{\mathrm{e}}/N\right)}^2$. Such a form might seem akin (surprisingly) to entangled chain melts. However, the latter systems obey (1) $D_{\mathrm{C}\mathrm{M}}\propto D_{\mathrm{R}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{e}}{\left(N_{\mathrm{e}}/N\right)}^{\gamma }\propto N_{\mathrm{e}}^{\gamma }/N^{1+\gamma }$, where $\gamma \approx 1-1.4$, and hence, rings exhibit a different scaling with $N_{\mathrm{e}}$. Moreover, there is no physical basis for rings obeying the reptation-tube model CM diffusion law since the ring has no free ends. Explaining the mystery of these unusual scaling laws, ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{LJ}}\sim {\left(N/N_{\mathrm{e}}\right)}^{8/3}$ and $D_{\mathrm{CM}}/D_{\mathrm{LJ}}\sim {\left(N_{\mathrm{e}}/N\right)}^2$, is one of our central motivations. We shall demonstrate that the theory developed below quantitatively predicts the above scaling laws if $N/N_{\mathrm{e}}$ is not “too large”.

Another important qualitative feature of the data in Fig. 2 is a clear break in apparent power-law scaling indicating a cross-over to a much-slower dynamics beyond $N/N_{\mathrm{e}}\sim 40-50$. These deviations are especially pronounced for stiff rings and for the CM diffusion constant in Fig. 2D where a nearly two decades deviation from the apparent power law scaling is found as $N/N_{\mathrm{e}}\to 100$. These slower than power law behaviors are roughly exponential in form, and are not predicted by existing scaling models (3, 7, 8, 10, 12). The MDS theory (51) has qualitatively predicted such behaviors can occur for flexible rings at sufficient high degrees of polymerization due to the emergence of a “strong-caging” regime defined as when the ring CM becomes transiently localized resulting in non-power-law slowing down activated dynamics. However, to date, no confirmation of this prediction exists in liquids where all rings are mobile. Thus, our simulation data in Fig. 2 provide evidence for molten ring polymers that a cross-over to an exponentially slow dynamical regime, potentially indicative of long-time activated dynamics. The microscopic motional mechanism is not easily deduced from simulation alone, but we believe that its activated nature is strongly supported by our theoretical analysis below.

Concerning experimental validation of our coarse-grained model ring simulations, the Inset of Fig. 2D compares our simulation data for $k_{\theta }=1.5$ with experimental data (13) for ring and entangled chain poly(ethylene oxide) (PEO) melts. Very good agreement is found, not only for the N dependence of the diffusion constants, but also the order of magnitude difference between the ring and chain diffusivities at fixed N. Two other experimental-simulation comparisons for ring melts based on the same simulated model have shown the phenomenon of shear thinning (64) and non-power-law stress relaxation for ultrahigh-molecular-weight flexible rings (17) are well captured by simulation. These findings provide significant support for the experimental relevance of our presented results for even higher ring degrees of polymerization and backbone stiffnesses.

Theoretical Background.

As relevant background, Fig. 3 shows a conceptual schematic of the theoretical approaches. The well-established thread or field theoretic version of PRISM integral equation theory (44, 65) is employed to predict the intermolecular pair correlations and thermodynamics based on input from simulation of the single ring intramolecular structure factor (Left of Fig. 3). This information is combined with methods of time-dependent statistical mechanics to construct a theory for the CM–CM total intermolecular force time correlation function (Middle of Fig. 3) experienced by a tagged ring from which its CM diffusion constant can be predicted in both the weak- and strong-caging regimes. The cross-over between these two dynamical regimes is self-consistently predicted, with the activated barrier hopping characteristic of strong-caging sketched in the Right panel in Fig. 3 (51).

Fig. 3.

Schematic of the equilibrium theory, weak- and strong-caging dynamic theories, and dynamic free energy in the strong-caging regime. The intramolecular ring pair correlation function ω(r) is input to PRISM theory (Left) to predict intermolecular segment–segment pair correlations, $g\left(r\right)$ and thermodynamics. The same ω(r) and $g(r)$ information is then utilized in both the weak- and strong-caging dynamical theories to construct effective forces (F) and their time correlation which are relaxed per the correlation pathway schematically shown in the middle panel. The latter involves the wavevector and time-dependent intraring dynamic structure factor [$\omega (q,t)$] and its collective total density fluctuation analog [$S(q,t)$]. The strong-caging regime emerges when force relaxation and ring CM diffusion occur on the effectively same timescale. Large distance ring displacements are then controlled by a spatially resolved trapping potential, or dynamic free energy, characterized by a transient localization length and an activation barrier (Right).

The CM diffusion constant is rigorously determined in statistical mechanics by the space-time correlations of intermolecular forces exerted by all the surrounding polymers on all the monomers of a tagged polymer (2, 44, 66, 67). For the classic Rouse model, by definition, the time correlations of intermolecular forces between two different monomers on a single polymer, or two different polymers, are assumed to be zero (uncorrelated), resulting in a total CM friction constant of ${\zeta }_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=N{\zeta }_{\mathrm{s}}$. However, in general, ${\zeta }_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ is given by the time integral of the total intermolecular force–force time correlation function associated with a tagged ring and thus must have contributions from the cross-time correlations of interring forces on different monomers. The MDS theory analyzes these “nondiagonal” or “nonlocal” contributions in ring liquids, and predicts the breakdown of the unentangled Rouse model and the emergence of two slower dynamical regimes (44, 51).

The first non-Rouse regime is of a nonactivated or “weak-caging” nature (44). Here, as N grows, the intermolecular kinetic constraints associated with nonrandom packing structure on the macromolecular scale result in nonlocal contributions to the total friction constant which become stronger, larger in number, and sufficiently long lived that a slowly relaxing intermolecular contribution to the CM total force–force time correlation function emerges. Initially, these force correlations relax faster than the timescale for ring CM motion, the physical definition of the weak-caging regime. As a consequence, the total CM friction constant becomes (44) ${\zeta }_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=N{\zeta }_{\mathrm{s}}+(N^2/N_{\mathrm{D}}){\zeta }_{\mathrm{s}}$. Here, $N_{\mathrm{D}}$ is operationally defined as the cross-over degree of polymerization for when the nonlocal friction effects become important relative to the simple uncorrelated Rouse local friction contribution. Importantly, it is neither physically nor mathematically equal to the linear chain $N_{\mathrm{e}}$ (see below). When the ratio $N/N_{\mathrm{D}}$ significantly exceeds unity, the theory predicts the scaling laws (44): $D_{\mathrm{C}\mathrm{M}}/D_{\mathrm{s}}\sim N_{\mathrm{D}}/N^2$ and ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{s}}\propto {R_{\mathrm{g}}^2/D_{\mathrm{C}\mathrm{M}}\propto N}^{8/3}/N_{\mathrm{D}}$, using the compact ring size scaling law $R_{\mathrm{g}}\propto N^{1/3}$. These N-scalings are consistent with the initial non-Rouse scaling regimes found in our melt simulation data of Fig. 2. However, the current MDS theory does not account for the very large effects of ring stiffness nor variable polymer concentration in dense solutions on ${\tau }_{\mathrm{CM}}/{\tau }_{\mathrm{LJ}}$ (or ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{LJ}}$) and $D_{\mathrm{CM}}/D_{\mathrm{LJ}}$, which are of critical importance to understand how $N_{\mathrm{D}}$ relates to the chemistry-specific backbone stiffness, $N_{\mathrm{e}}$, and ring concentration.

Since the MDS weak-caging theory predicts the connection between total CM friction constant, interring packing correlations, and thermodynamic properties [with the latter two known from thread PRISM integral equation theory (44, 65)], an explicit prediction for the cross-over degree of polymerization has been derived (44): $N_{\mathrm{D}}=9\pi \varphi S_0$. Here, as defined above, $\varphi$ depends on ring concentration and R_g, and $S_0$ is the liquid dimensionless isothermal compressibility, which is proportional to the mean-square thermal density fluctuation amplitude (68, 69). Based on thread (44, 65) PRISM theory (70, 71) for ring liquids, it was previously deduced that $S_0\propto {\varphi }^{-2.5}$. However, this result was obtained using simulation data for the intramolecular ring structure factor at only a single value of ring stiffness in semidilute solution (44). Hence, the existing theory cannot address the rich dependence of ring dynamics on backbone stiffness established by our present simulations, nor the effects of variable ring concentration. These issues are the key focus addressed below.

In analogy with the behavior of structural glass-forming liquids and dense colloidal suspensions (72–74), the weak-caging regime of MDS theory is a priori predicted (51) (not assumed) to breakdown at a sufficiently large N due to the relaxation time for total constraining intermolecular forces experienced by a tagged ring approaching the time scale for ring CM Fickian diffusion. Such an emergent lack of time scale separation necessitates a fully self-consistent treatment beyond the weak-caging regime. In analogy with activated barrier hopping dynamics in glass-forming liquids (72–74), one expects a second non-Rouse regime for rings called the strong-caging regime beyond a second cross-over, per $N>N_{\mathrm{o}\mathrm{n}}(>N_{\mathrm{D}}\ne N_{\mathrm{e}})$. A self-consistent dynamical theory (51) for the total friction constant has been formulated for this regime building on successful ideas for molecular and colloidal glass-forming liquids (72–74). The entropic barrier (as shown in Fig. 3) controls the rate of long-time activated CM relaxation and diffusion and is predicted (51) to grow linearly with $N/N_{\mathrm{o}\mathrm{n}}$. The barrier is called “entropic” because it arises from intermolecular hard-core excluded volume interactions and the associated dynamic caging constraints (here on the macromolecular scale), in analogy with activated dynamics in ultradense hard-sphere or colloidal fluids. Thus, this “entropic barrier” is not what is meant in other areas of polymer science, e.g., for the problem of a polymer chain squeezing through a small restricted region of space (e.g., translocation) or in quenched porous media (75, 76).

In the strong-caging regime, the total CM friction constant then acquires a contribution reflecting transient localization and barrier hopping (51), ${\zeta }_{\mathrm{total}}=N{\zeta }_{\mathrm{s}}(1+N/N_{\mathrm{D}}+{\tau }_{\mathrm{hop}}/{\tau }_{\mathrm{Rouse}})$, where the ratio ${\tau }_{\mathrm{h}\mathrm{o}\mathrm{p}}/{\tau }_{\mathrm{R}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{e}}$ (the mean hopping time divided by the Rouse time) varies as $\sim \exp \left(a(N/N_{\mathrm{o}\mathrm{n}}-1)\right)$ with $a$ indicating a numerical constant. This second cross-over begins at a predicted value of ${N\equiv N}_{\mathrm{o}\mathrm{n}}$. Per the middle panel in Fig. 3, the manner that dynamical constraints are constructed from structural correlations is identical in the weak- and strong-caging theories, but their dynamical consequences differ if the constraints become sufficiently strong. As a consequence of this, plus the use of the coarse-grained thread model description of rings, the ratio $N_{\mathrm{o}\mathrm{n}}/N_{\mathrm{D}}$ is predicted to depend only on the dimensionless localization length $R_{\mathrm{L},\mathrm{on}}/R_{\mathrm{g}}$ at the onset of the cross-over to an activated regime, as further discussed in SI Appendix and ref. 51. The theory then predicts the ratio $R_{\mathrm{L},\mathrm{on}}/R_{\mathrm{g}}$ is nearly a constant, and hence $N_{\mathrm{o}\mathrm{n}}$ is proportional to $N_{\mathrm{D}}$ (51). Thus, $N_{\mathrm{D}}$ is the key variable for the emergence of both non-Rouse regimes for CM dynamics, which depend strongly on both ring concentration and stiffness.

Theory Generalizations and Predictions.

We now generalize the entire MDS theoretical approach for ring packing structure, thermodynamics, and CM dynamics to address in a unified manner the effects of local segmental friction, molecular weight, ring stiffness, and entanglement length on the ring CM diffusion constant over a wide concentration range from semidilute solutions to melts. Achieving this requires considering the equilibrium and dynamical consequences of the stiffness parameter defined as the dimensionless length scale ratio ${\lambda }_0\equiv N^{\frac{1}{3}}{d}_{\mathrm{T}}/R_{\mathrm{g}}$ which to leading order is N-independent for compact rings. In the initial formulation of the MDS approaches (44, 51), ${\lambda }_0$ was a unique constant since the focus was limited to rings of fixed stiffness and concentration. Addressing this major limitation is the motivation for our extension of MDS theory to general values of ${\lambda }_0$. The second advance is to incorporate ring stiffness in the “segmental” friction constant ${\zeta }_{\mathrm{s}}$ (or equivalently the elementary short-time scale ${\tau }_{\mathrm{s}}=\beta {\zeta }_{\mathrm{s}}{\sigma }^2$) which sets the basic timescale for all larger scale ring dynamics.

The physical origin of the importance of ${\lambda }_0$ is that rings follow ideal chain conformational statistics on length scales inside the entanglement length but have compact globule conformations on larger length scales up to $R_{\mathrm{g}}$. The interring packing correlations predicted from PRISM theory depend on ${\lambda }_0$ which is a function of the Kuhn length, $N_{\mathrm{e}}$, and polymer concentration. Since large rings are globally compact with (6–8, 58) $R_{\mathrm{g}}\propto \sqrt{l_{\mathrm{b}}{l}_{\mathrm{K}}}{N}^{1/3}{\left(d_{\mathrm{T}}/l_{\mathrm{b}}\right)}^{1/3}$ and (1, 66) $d_{\mathrm{T}}=\sqrt{N_{\mathrm{e}}}{{(l}_{\mathrm{b}}{l}_{\mathrm{K}})}^{0.5}$ (SI Appendix, Fig. S1), one can write ${\lambda }_0\propto \sqrt{N_{\mathrm{e}}}/{\left(d_{\mathrm{T}}/l_{\mathrm{b}}\right)}^{1/3}$ and ${\lambda }_0^3\propto N_{\mathrm{e}}{l}_{\mathrm{b}}/\sqrt{l_{\mathrm{b}}{l}_{\mathrm{K}}}$. The latter expression can be qualitatively interpreted as the number of Kuhn segments between two adjacent entanglement points.

As a technical point, for entangled chain polymers one can write $R_{\mathrm{g}}/d_{\mathrm{T}}\propto {\left(N/N_{\mathrm{e}}\right)}^{\nu }$ with $\nu =1/2$. If one assumes this applies to rings as done in ref. 8, which are fractal globules only on length scales larger than the entanglement length, it has been argued the ring $R_{\mathrm{g}}$ follows from the chain formula by only changing $\nu \to 1/3$, thereby yielding $R_{\mathrm{g}}\propto d_{\mathrm{T}}{\left(N/N_{\mathrm{e}}\right)}^{\frac{1}{3}}=\sqrt{l_{\mathrm{b}}{l}_{\mathrm{K}}}{N}^{\frac{1}{3}}{N}_{\mathrm{e}}^{\frac{1}{6}}$. This expression is modestly different [by a factor of ${\left(l_{\mathrm{K}}/l_{\mathrm{b}}\right)}^{1/6}$] from our above expression $R_{\mathrm{g}}\propto \sqrt{l_{\mathrm{b}}{l}_{\mathrm{K}}}{N}^{\frac{1}{3}}{\left(d_{\mathrm{T}}/l_{\mathrm{b}}\right)}^{\frac{1}{3}}=l_{\mathrm{b}}^{\frac{1}{3}}{l}_{\mathrm{K}}^{\frac{2}{3}}{N}^{\frac{1}{3}}{N}_{\mathrm{e}}^{\frac{1}{6}}$. However, ring conformation is unchanged on scales smaller than the tube diameter. Hence, we do not believe that there is a unique answer concerning which formula is physically more appropriate. From an operational perspective, in SI Appendix, we have explored what these two formulas predict compared to our ring melt simulation data for $R_{\mathrm{g}}$, and also the simulation data for the concentration dependence of $R_{\mathrm{g}}$ of stiff ring solutions. We conclude that our expression $R_{\mathrm{g}}\propto \sqrt{l_{\mathrm{b}}{l}_{\mathrm{K}}}{N}^{1/3}{\left(d_{\mathrm{T}}/l_{\mathrm{b}}\right)}^{1/3}$ is a more accurate choice to understand the consequences of variable ring stiffness, and also is accurate for the stiff ring concentration dependence of $R_{\mathrm{g}}$ in solutions. A theoretical argument for our adopted $R_{\mathrm{g}}$ formula is also given in SI Appendix.

To achieve the desired generalizations discussed above requires using thread (65) PRISM theory (70, 71) to predict the monomer–monomer interring pair correlation function and the relevant thermodynamic properties that enter the dynamical constructions of MDS theory (44, 51). The only required input is the intramolecular ring structure factor $\omega \left(q\right)$, which depends on $N$ and ${\lambda }_0$. In what follows, we take $\omega (q)$ from our MD simulation results for $k_{\theta }=1.5$ ring melts (SI Appendix, Fig. S4 and the SI Appendix) as input to the theories.

We have generalized thread PRISM theory to fully account for the consequences of ${\lambda }_0$ on structure and thermodynamics (44); the detailed technical analysis is presented in SI Appendix, including the asymptotic independence on N of the equation of state property $S_0$. The fundamental dynamical cross-over variable is $N_{\mathrm{D}}=9\pi \varphi S_0$. Numerical results for $S_0$ are shown as a function of ${\varphi }^{-1}$ over a wide range of ${\lambda }_0$ in the Inset of Fig. 4A. The previously derived power law variation with $\varphi$ (44), $S_0\sim {\varphi }^{-2.5}$, remains valid for all values of ${\lambda }_0$. However, a strong dependence on ${\lambda }_0$ emerges, $S_0\sim {\lambda }_0^3{\varphi }^{-2.5}$ (numerical prefactor is known), as indicated in the main frame of Fig. 4A by the good collapse of our numerical calculations. This scaling also holds for ring solutions (SI Appendix, Fig. S5). Importantly, $S_0\sim {\lambda }_0^3{\varphi }^{-2.5}$ now depends on $N_{\mathrm{e}}$, chain stiffness, and monomer concentration. Physically, these PRISM theory results are a consequence of the exact enforcement of the monomer-level intermolecular excluded volume constraint consistent with the internal structure of variable stiffness rings.

Fig. 4.

(A) Theoretical dimensionless compressibility for ring melts predicted by PRISM theory normalized by ${\lambda }_0^3$ as a function of inverse macromolecular volume fraction over a wide range of ${\lambda }_0$. (Inset) Analogous $S_0$ results for the same conditions as the main frame but without the ${\lambda }_0^3$ normalization. (B) Main: Short-time segmental relaxation time ${\tau }_{\mathrm{s}}/{\tau }_0$ for hard chain melts as a function of the contact value g(d) computed based on the Koyama-PRISM-MV structural theory (70, 71, 77, 78) and weak-caging dynamical theory (72–74) for a wide range of stiffness parameters and chain lengths at melt density. We note that ${\tau }_0=16{\varphi }_{\mathrm{seg}}d{\left(\beta m/\pi \right)}^{1/2}$ is typically of order ~1 ps (74, 79–81), with d and m the monomer (site) diameter and mass, respectively. (Inset) Theoretical contact value plotted as a function of the simulation entanglement tube diameter d_T.

The above dependence of $S_0$ on ${\lambda }_0$ is critical for our dynamical predictions as ring stiffness and concentration change. However, to first gain intuition, we note that the dynamical implications of ring stiffness via ${\lambda }_0^3$ can be qualitatively anticipated as follows. Generally, topological dynamical constraints on a ring due to interring interpenetration emerge on length scales that vary from ~d_T to ${2R}_{\mathrm{g}}$. They can be quantified in a dimensionless manner by how much larger the ratio $2R_{\mathrm{g}}/d_{\mathrm{T}}=2N^{\frac{1}{3}}/{\lambda }_0$ is compared to unity. As $N_{\mathrm{e}}$ decreases (e.g., stiffness increases), $R_{\mathrm{g}}$ grows, while $d_{\mathrm{T}}$ decreases. Both trends increase the ratio $2R_{\mathrm{g}}/d_{\mathrm{T}}$, thereby creating a more open conformation, and a larger macromolecular volume fraction $\varphi$ (per Fig. 1 A–C). At a more detailed level, increasing N increases mainly the outer boundary of the strong constraint regime via a larger $R_{\mathrm{g}}$, while increasing ring stiffness modifies both the inner (per d_T/2) and outer boundaries. Of course, $2R_{\mathrm{g}}/d_{\mathrm{T}}<1$ when N is small enough and/or chain stiffness is low, a regime where rings interpenetrate but topological entanglement effects are inoperative (5, 6). Thus, if we take $R_{\mathrm{g}}^3/N$ as effectively N-independent to leading order per a collapsed globule, a critical intrinsic “monomer number” can be defined as, $N_{\mathrm{c}}\equiv d_{\mathrm{T}}^{3}N/8R_{\mathrm{g}}^3={\lambda }_0^3/8$. This quantity is similar to, but different from, the classic entanglement $N_{\mathrm{e}}$ defined for linear chain polymers.

Continuing with the above analysis, dynamical constraints are strong when $N>N_{\mathrm{c}}\propto {\lambda }_0^3$, but the magnitude of interring forces on ring dynamics is also related within MDS theory to the structural correlation in space of the collective density fluctuations of the liquid as quantified by the dimensionless compressibility (44), which directly enters $N_{\mathrm{D}}=9\pi \varphi S_0$. Employing the relations $S_0\sim {\lambda }_0^3{\varphi }^{-2.5}$, $\varphi =\left(4\pi /3\right)\rho R_{\mathrm{g}}^3/N$, and ${\lambda }_0=N^{\frac{1}{3}}{d}_{\mathrm{T}}/R_{\mathrm{g}}$ then yields $N_{\mathrm{D}}\sim {\varphi }^{-1.5}{\lambda }_0^3\sim N_{\mathrm{e}}^{2.12}$. This is a crucial prediction, and $N_{\mathrm{D}}$ is not linearly proportional to $N_{\mathrm{e}}$. The reason is that both $\varphi$ and $S_0$ depend on the Kuhn length and $N_{\mathrm{e}}$, and the Kuhn length can be related to $N_{\mathrm{e}}$ (see below), which allows the prediction to be expressed as $N_{\mathrm{D}}\propto N_{\mathrm{e}}^{2.12}$. We emphasize that the presence of ${\lambda }_0^3=N_{\mathrm{e}}{l}_{\mathrm{b}}/\sqrt{l_{\mathrm{b}}{l}_{\mathrm{K}}}$ is key, and our derivation has employed the well-known relations (1, 6, 57–59):

$$R_{\mathrm{g}}\propto \sqrt{l_{\mathrm{b}}{l}_{\mathrm{K}}}{N}^{\frac{1}{3}}{\left(\frac{d_{\mathrm{T}}}{l_{\mathrm{b}}}\right)}^{\frac{1}{3}},d_{\mathrm{T}}=\sqrt{N_{\mathrm{e}}}(l_{\mathrm{b}}{l}_{\mathrm{K}}{)}^{0.5},l_{\mathrm{K}}^{-1}\sim N_{\mathrm{e}}^{0.535},$$ [1]

which are consistent with ring melt simulation data (SI Appendix, Fig. S1).

All the theoretical analysis above was focused on modification of the correlation hole ($R_{\mathrm{g}}$) scale physics of globular fluctuating rings with deep interpenetrations due to how stiffness (and hence also $N_{\mathrm{e}}$) enters the weak-caging theory via $N_{\mathrm{D}}$. However, to predict relaxation times and diffusion constants also requires the kinetic prefactor associated with the local segmental friction constant which sets the elementary time scale for long-time dynamics. Addressing this defines the second theoretical advance mentioned above. Specifically, the longest ring relaxation time shown in simulation units in Fig. 2 can be written as $\tau /{\tau }_{\mathrm{LJ}}=(\tau /{\tau }_{\mathrm{s}})({\tau }_{\mathrm{s}}/{\tau }_{\mathrm{LJ}})$, where ${\tau }_{\mathrm{s}}/{\tau }_{\mathrm{LJ}}$ is the nondimensionalized segmental relaxation time. On the short segment scale, rings are chain-like. Hence, understanding how changing local stiffness modifies local friction can be achieved based on an analysis of chain polymer melts, a problem that two of us have recently quantitatively analyzed using microscopic liquid state theory (74).

Since our ring simulations are performed far from the monomer scale structural glass transition, the required dynamical theory for the semiflexible hard chain segmental friction constant and timescale is relatively simple. We previously showed (74) that the local segmental relaxation time is ${\tau }_{\mathrm{s}}/{\tau }_{\mathrm{L}\mathrm{J}}\sim \rho g\left(d\right)\left(1+C_{\mathrm{E}}\right)$, where $g\left(d\right)$ is the value at hard-core contact of the intermolecular monomer–monomer radial distribution function, $g\left(r\right)$. The presence of $g\left(d\right)$ is intuitive since it reflects the frequency of binary (two-body) intermonomer collisions (79). The space-time correlation of such local collisional events far from a monomer-scale glassy dynamics regime due to weak local caging correlations is encoded in the factor $C_{\mathrm{E}}$, the expression for which was previously derived (73, 79). Based on the full (not thread) PRISM integral equation theory with monomers of nonzero hard core volume, combined with the accurate modified-Verlet closure (74, 81) for the semiflexible tangent Koyama chain model (77, 78), we have calculated ${\tau }_{\mathrm{s}}/{\tau }_{\mathrm{L}\mathrm{J}}$ for the monomer packing fraction of our MD melt simulations ${\varphi }_{\mathrm{s}\mathrm{e}\mathrm{g}}=\pi \rho {\sigma }_{\mathrm{L}\mathrm{J}}^3/6=0.85\pi /6\sim 0.445$ (SI Appendix). Other input parameters are (SI Appendix, Table S1): bond length $l_{\mathrm{b}}=0.965{\sigma }_{\mathrm{L}\mathrm{J}}$, persistence length $l_{\mathrm{p}}=(l_{\mathrm{b}}+l_{\mathrm{K}})/2$, and N. The obtained numerical results for ${\tau }_{\mathrm{s}}/{\tau }_0$ (${\tau }_0\sim \rho {\tau }_{\mathrm{LJ}}$) are plotted as a function of g(d) in Fig. 4B. Rather remarkably, all the theoretical data collapse onto a power law master curve: $\begin{array}{c}{\tau }_{\mathrm{s}}/{\tau }_0\sim {\tau }_{\mathrm{s}}/\rho \sim g\left(d\right)\left[1+C_{\mathrm{E}}\right]\hfill \end{array}$$\sim {\left[g\right(d\left)\right]}^{3/2}$. Combining this relation with the predicted connection $g\left(d\right)\sim d_{\mathrm{T}}^{-2}$ demonstrated in the Inset of Fig. 4B, one finds ${\tau }_{\mathrm{s}}/{\tau }_0\propto d_{\mathrm{T}}^{-3}\sim N_{\mathrm{e}}^{-0.7}$. Thus, the elementary timescale becomes longer (more local friction) as $N_{\mathrm{e}}$ decreases or ring stiffness (Kuhn length) increases.

We can now combine all the results above with the MDS weak-caging theory to obtain the longest relaxation time:

$$\begin{array}{c}\frac{\tau }{{\tau }_{\mathrm{LJ}}}=\frac{\tau }{{\tau }_{\mathrm{s}}}\frac{{\tau }_{\mathrm{s}}}{{\tau }_{\mathrm{LJ}}}\sim \frac{R_{\mathrm{g}}^2{N}^2}{N_{\mathrm{D}}}\frac{{\tau }_{\mathrm{s}}}{{\tau }_{\mathrm{LJ}}}\sim \frac{N^{\frac{8}{3}}}{N_{\mathrm{e}}^{3.2}} \mathrm{for}N>\; >N_{\mathrm{D}}.\hfill \end{array}$$ [2]

the predicted exponent of 3.2 contains contributions from i) $R_{\mathrm{g}}^2\sim N_{\mathrm{e}}^{-0.38}$ (per Eq. 1), ii) short-time segmental friction ${\tau }_{\mathrm{s}}/{\tau }_{\mathrm{LJ}}\sim N_{\mathrm{e}}^{-0.7}$, and most importantly, iii) $N_{\mathrm{D}}\sim N_{\mathrm{e}}^{2.12}$. For the CM diffusion constant, one similarly has

$$\frac{D_{\mathrm{LJ}}}{D_{\mathrm{CM}}}=\frac{D_{\mathrm{s}}}{D_{\mathrm{CM}}}\frac{D_{\mathrm{LJ}}}{D_{\mathrm{s}}}\sim \frac{N^2}{N_{\mathrm{D}}}\frac{{\tau }_{\mathrm{s}}}{{\tau }_{\mathrm{LJ}}}\sim \frac{N^2}{N_{\mathrm{e}}^{2.82}},$$ [3]

where the predicted exponent 2.82 contains contributions from $D_{\mathrm{s}}/D_{\mathrm{LJ}}={\tau }_{\mathrm{LJ}}/{\tau }_{\mathrm{s}}\sim N_{\mathrm{e}}^{0.7}$ and $N_{\mathrm{D}}\sim N_{\mathrm{e}}^{2.12}$. Our ability to write ${\tau }_{\mathrm{s}}/{\tau }_{\mathrm{LJ}}\sim N_{\mathrm{e}}^{-0.7}$ follows from the known connection between the Kuhn length and $N_{\mathrm{e}}$ of entangled chains (1, 6, 57–59) (SI Appendix, Fig. S1). In contrast, the result $N_{\mathrm{D}}\sim N_{\mathrm{e}}^{2.12}$ is an explicit consequence of the unique ring conformational and packing behavior.

Another interesting aspect of the theory is the prediction of multiple equivalent forms of the scaling relations for $N_{\mathrm{D}}$. For example, the relaxation time can be written in three seemingly different, but conceptually equivalent, ways:

$$\frac{{\tau }_{\mathrm{d}}}{{\tau }_{\mathrm{LJ}}}\propto N^{-\frac{1}{2}}{R}_{\mathrm{g}}^{\frac{19}{2}}{d}_{\mathrm{T}}^{-6}\propto N^{\frac{8}{3}}{l}_{\mathrm{K}}^{\frac{10}{3}}{N}_{\mathrm{e}}^{-\frac{17}{12}}\propto N^{\frac{8}{3}}{l}_{\mathrm{K}}^6.$$ [4]

this “redundancy of expression” arises from the predicted causal interrelationships discussed above per Eq. 1 and other relations summarized here for the benefit of the reader:

$$\frac{{\tau }_{\mathrm{s}}}{{\tau }_{\mathrm{LJ}}}\sim \rho d_{\mathrm{T}}^{-3}\propto N_{\mathrm{e}}^{-0.7},N_{\mathrm{D}}\sim N{d}_{\mathrm{T}}^3{\varphi }^{-1.5}{R}_{\mathrm{g}}^{-3}.$$ [5]

We suggest that the alternative representations in Eq. 4 may be of practical value to experimentalists and simulators as causal relations of ring diffusivity and long-time relaxation to molecular features that can be tuned via chemical synthesis or simulation model selection. They also provide alternative intuitive pictures for interpreting trends in data (SI Appendix, Fig. S6).

In summary, the results in Eqs. 1–5 show that one needs to account for both the local monomeric friction that is not unique to rings, as well as the macromolecular caging physics associated with the topological compaction effects produced by ring polymer conformations at larger scales. We shall demonstrate that it is the latter macromolecular scale effect that dominates the CM diffusion of polymer rings.

Test of Theoretical Predictions: Melts of Variable Stiffness.

We now test the generalized theory by attempting to collapse the MD simulation data in Fig. 2 based on the predictions in Eqs. 1–5. Operationally, the above relations ${\tau }_{\mathrm{s}}/{\tau }_{\mathrm{LJ}}\propto N_{\mathrm{e}}^{-0.7}$ and $D_{\mathrm{s}}/D_{\mathrm{LJ}}\propto N_{\mathrm{e}}^{0.7}$ are employed to vertically normalize the MD data for ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{LJ}}$ and $D_{\mathrm{CM}}/D_{\mathrm{LJ}}$ in both Fig. 2 A and C. The data in Fig. 2A for the longest relaxation time are then replotted in Fig. 5A as a function of N scaled as suggested by Eq. 2, i.e., ${\tau }_{\mathrm{d}}\propto N^{\frac{8}{3}}/N_{\mathrm{e}}^{0.38}{N}_{\mathrm{D}}\propto {\left(N/N_{\mathrm{e}}^x\right)}^{\frac{8}{3}}$ where the theory predicts $x=2.5\ast \frac{3}{8}=0.94$. The analogous replotting of the data of Fig. 2C for the diffusion constant is shown in Fig. 5B guided by the theory prediction of Eq. 3, i.e., $D_{\mathrm{C}\mathrm{M}}\propto N_{\mathrm{D}}/N^2\propto {\left(N_{\mathrm{e}}^y/N\right)}^2$ where $y=2.12\ast 0.5=1.06$.

Fig. 5.

Same displays as in Fig. 2 B and D, respectively, for the longest relaxation time ${\tau }_{\mathrm{d}}/{\tau }_{{\mathrm{s}}^{\prime }}$ (A) and CM diffusion constant $D_{\mathrm{CM}}/D_{{\mathrm{s}}^{\prime }}$ (B) as a function of N, but with both x-axes and y-axes normalized in a theoretically inspired manner with ${\tau }_{{\mathrm{s}}^{\prime }}/{\tau }_{\mathrm{LJ}}=N_{\mathrm{e}}^{-0.7}$ and $D_{{\mathrm{s}}^{\prime }}/D_{\mathrm{LJ}}=N_{\mathrm{e}}^{0.7}$, where ${\tau }_{{\mathrm{s}}^{\prime }}$ ($D_{{\mathrm{s}}^{\prime }}$) is defined based on the derived result ${\tau }_{\mathrm{s}}\propto N_{\mathrm{e}}^{-0.7}$ ($D_{\mathrm{s}}\propto N_{\mathrm{e}}^{0.7}$) but differs by a constant numerical prefactor which does not affect the analyses here. The solid lines in A and B are shown to test the theoretically predicted exponents, an exercise not affected by any vertical or horizontal shifts in the log–log plots.

We find the discrete data points in the format of Fig. 2 A and C collapse well onto master curves over a wide range of stiffnesses and $N$, consistent with the theory. Over the large range where the collapsed data behave as an effective power law in Fig. 5, the corresponding exponents agree well with the theoretical values of 8/3 and −2. However, there are deviations from a collapsed master curve and effective power law scaling for the slowest relaxing and diffusing systems, which will be discussed below. Before doing so, we evaluate the relative importance of the short-time friction correction by directly plotting (no vertical shift) the simulation data for ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{LJ}}$ and $D_{\mathrm{CM}}/D_{\mathrm{LJ}}$ versus $N/N_{\mathrm{e}}^{0.94}$ and $N/N_{\mathrm{e}}^{1.06}$, respectively, in SI Appendix, Fig. S7 A and B. We find a good visual collapse continues to be obtained using the same exponents of 8/3 and −2. This allows us to draw the important conclusion that the rather weak stiffness-dependent short-time friction correction (${\tau }_{\mathrm{s}}/{\tau }_{\mathrm{LJ}}\propto N_{\mathrm{e}}^{-0.7}$), although not negligible, is of secondary importance for ring long-time CM dynamics. By far the most important physics for understanding how stiffness affects ring diffusivity arises from how entanglements and ring interpenetration modify equilibrium ring conformation, which in turn modifies interring packing on the correlation hole scale and S₀.

It is intriguing that the exponents 0.94 and 1.06 obtained from Eqs. 2 and 3, which quantify how $N_{\mathrm{e}}$ appears in the x-axes of Fig. 5, respectively, are close to 1.0. This is not an indication that the physics of ring diffusion is the same as the reptation of entangled linear chains in anisotropic tubes, which is physically implausible. Rather, these close to unity theoretical values arise from the presence of a different power law scaling in the simulation data. Moreover, $D_{\mathrm{C}\mathrm{M}}\propto {\left(N_{\mathrm{e}}/N\right)}^2$ contrasts with the entangled chain analog of $D_{\mathrm{C}\mathrm{M}}\propto N_{\mathrm{e}}/N^2$ and the even weaker ${D_{\mathrm{CM}}\propto N}_{\mathrm{e}}^{2/3}{N}^{-5/3}$ scaling in the FLG ring model.

Thus, our analysis solves the puzzling apparent power law and collapse behaviors empirically found in the simulation results of Fig. 2 B and D. We emphasize that since the MDS weak-caging theory also allows an understanding of the ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{LJ}}$ data obtained from the monomer MSD at long times and distances, a type of simple “slaving” behavior between the long-time monomer and CM motions is suggested, at least in a scaling sense. We have checked the robustness of our conclusions by verifying that ${\tau }_{\mathrm{CM}}/{\tau }_{\mathrm{LJ}}$ defined above behaves similarly as that of ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{LJ}}$ and $D_{\mathrm{CM}}/D_{\mathrm{LJ}}$, as shown in SI Appendix, Fig. S3C.

Finally, we return to the point that simulation results in Figs. 2D and 5B show that for N = 1,600 ring melts the diffusion constant data point for the stiffest rings strongly deviates from power-law behavior by 1.5 to 2 decades. This is objective simulation evidence for the onset of a much slower transport regime. Logically, the computationally unmeasurably small values of D_CM for N = 3,200 and 6,400 would display even larger orders of magnitude deviations from the weak-caging power-law behavior. We emphasize that since N_on is predicted (51) to be proportional to N_D, such a second cross-over to an ultraslow non-power-law regime of mass transport does not affect the qualitative analysis of stiffness effects above.

Tests of Theoretical Predictions: Concentration Effects for Stiff and Flexible Rings in Solutions.

What is the effect of polymer concentration on ring CM diffusivity? Are there interrelations between how concentration, stiffness, and N slow down mass transport? As discussed above, both $N$ and chain stiffness affect the degree of ring interpenetration and entanglement, and so does the monomer density. A decrease of concentration typically leads to some (solvent quality dependent) polymer swelling, increasing N_e, R_g, and d_T. Thus, how the dimensionless stiffness parameter ${\lambda }_0$ changes with monomer concentration could be subtle. Below, we analyze the concentration effect based on the theoretical result $N_{\mathrm{D}}\propto {\lambda }_0^3{\varphi }^{-3/2}$. Given we have shown above that the variable monomer friction constant plays a weak second-order role, we ignore it in our analysis below.

We first recall a recent simulation study that found the entanglement length of semiflexible chain polymer liquids (57, 59) in units of the Kuhn length varies as,

$$\frac{d_{\mathrm{T}}}{l_{\mathrm{K}}}={\left[97.5y^2+17.9y+1.34y^{1/3}\right]}^{1/2},$$ [6]

where $y\equiv \frac{p}{l_{\mathrm{K}}}=\frac{l_{\mathrm{b}}^2}{\rho l_{\mathrm{K}}^2}$ and p is the packing length. For very flexible polymers y > 1, Eq. 6 reduces to the flexible entangled chain result, $d_{\mathrm{T}}\approx 10p$, which scales inversely with monomer number density. Applying this result, the MDS theory expression $N_{\mathrm{D}}\sim {\varphi }^{-1.5}{\lambda }_0^3$ becomes

$$N_{\mathrm{D}}\sim N{d}_{\mathrm{T}}^3{\varphi }^{-1.5}/R_{\mathrm{g}}^3\sim {\rho }^{-3}{\rho }^{-0.915}/{\rho }^{-0.39}\sim {\rho }^{-3.525},$$ [7]

where we adopted the relationships $R_{\mathrm{g}}/N^{1/3}\sim {\rho }^{-0.13}$ and $\varphi \sim {\rho }^{0.61}$ deduced from simulations (SI Appendix, Fig. S8) of stiff rings. Thus, the obtained power law exponent of 3.525 in Eq. 7 might be a bit different for flexible ring systems. However, given the dominant factor in Eq. 7 arises from the relation $d_{\mathrm{T}}^3\sim {\rho }^{-3}$, the power law exponent relating $N_{\mathrm{D}}$ and ρ for flexible rings should be close to the value of 3.525 in Eq. 7. This prediction can be tested in future simulations or experiments. Note the ${N_{\mathrm{D}}\propto \rho }^{-3.525}$ scaling law is much stronger than the scaling for entangled chain polymer solutions of $N_{\mathrm{e}}\propto {\rho }^{-\frac{5}{4}}$ (or ${\rho }^{-2}$) in good (theta) solvents (1, 2). This further emphasizes the crucial quantity $N_{\mathrm{D}}$ is not the same as the entangled linear chain $N_{\mathrm{e}}$.

Systematic simulations of high-molecular-weight flexible rings in solution have not yet been performed. However, Michieletto and Turner (MT) (9) and Micheletto, Nahali, and Rosa (MNR) (11) simulated stiff ring solutions ($k_{\theta }=5$) in semidilute and concentrated solutions, respectively. Their CM diffusion constant data are plotted as $N{D}_{\mathrm{CM}}/D_{\mathrm{L}\mathrm{J}}$ versus N in the Inset of Fig. 6 (note ${\varphi }_{\mathrm{s}\mathrm{e}\mathrm{g}}<0.21$). Notably, the diffusion constants decrease with N more rapidly as ring concentration increases over the same range of N. Our weak-caging theory predicts $N{D}_{\mathrm{CM}}/D_{\mathrm{L}\mathrm{J}}$ is proportional to $N_{\mathrm{D}}/N$, so we replot the simulation data in Fig. 6 guided by the theory with N scaled by the factor $N_{\mathrm{D}}\left(\rho \right)/N_{\mathrm{D}}\left(0.1\right)$. We find a dependence of $N_{\mathrm{D}}\left(\rho \right)\sim {\rho }^{-v}$, as motivated from Eq. 7, does indeed lead to a good collapsed master curve for all the data points if one sets $v\sim 2.0$. The latter exponent differs significantly from $v=3.535$ deduced in Eq. 7 for flexible polymers. However, this is not surprising since the simulations are for very stiff rings. Is this difference understandable?

Fig. 6.

(Inset) $N{D}_{\mathrm{CM}}/D_{\mathrm{L}\mathrm{J}}$ from previous simulations (9, 11) of semidilute and concentrated stiff-ring polymer solutions plotted over a wide range of reduced monomer number densities $\rho {\sigma }_{\mathrm{LJ}}^3$ (0.1 to 0.4) as a function of degree of polymerization N. (Main) Same display as the Inset but as a function of theoretically inspired $N/N_{\mathrm{D}}^{\ast }$ with $N_{\mathrm{D}}^{\ast }=({\rho {\sigma }_{\mathrm{LJ}}^3/0.1)}^{-2.025}$.

To answer this question, we consider the opposite limit of Eq. 6 (57, 59). When polymers become sufficiently stiff, the second term in Eq. 6 dominates, resulting in $d_{\mathrm{T}}\approx l_{\mathrm{K}}{\left(17.9y\right)}^{0.5}\sim l_{\mathrm{b}}/{\rho }^{0.5}$, as also found by Milner (58). This scaling is of the form for entangled rigid rod polymers (2, 57–59). Using this result in our theoretical expression then yields

$$N_{\mathrm{D}}\sim N{d}_{\mathrm{T}}^3{\varphi }^{-1.5}/R_{\mathrm{g}}^3\sim {\rho }^{-2.025}.$$ [8]

This prediction is in very good agreement with the exponent of 2.0 empirically extracted by collapsing the simulation data for stiff polymers in Fig. 6. Hence, our weak-caging theory also captures the diffusive behavior of stiff rings in semidilute and concentrated solutions. We believe that this is highly notable since theories for flexible and stiff polymer dynamics often take on different forms, but the weak-caging theory developed here appears to be sufficiently fundamental to capture both limits within a single framework.

Consistent with the flexible ring melt results in Fig. 5B, Fig. 6 shows that stiff ring solutions display a dramatic break in power-law scaling of $D_{\mathrm{CM}}$ corresponding to a qualitatively more rapid slowdown in diffusion at large $N/N_{\mathrm{D}}$, and hence a second cross-over. This is consistent with the theoretically predicted (51) second cross-over regime to activated transport when N > N_on. Critically, since N_on is proportional to N_D, the achievement of a collapse of all points in Fig. 6 provides more objective support for the fundamental importance of $N_{\mathrm{D}}$, regardless of whether the system is in the weak- or strong-caging regime, and regardless of the precise form of the master curve.

Emergence of Macromolecular-Scale Activated Dynamics, Kinetic Vitrification, and Universal Behavior.

We now address the question of activated dynamics in ring melts and solutions based on MDS theory as self-consistently generalized to the strong-caging regime. Previously discussed technical aspects are recalled in SI Appendix. Fig. 7 shows that choosing N_on ~ 30N_D (justified in SI Appendix) yields predictions for $N{D}_{\mathrm{C}\mathrm{M}}/D_{\mathrm{s}}$ that agree well with the simulation data in the ultraslow regime. The predicted exponential-like decay of $N{D}_{\mathrm{C}\mathrm{M}}/D_{\mathrm{s}}$ with $N/N_{\mathrm{D}}$ provides a theoretical foundation for the concept of a macromolecular-scale kinetic glass transition in ring liquids, and our results are also qualitatively consistent with the simulations of diffusion in solutions of fully mobile stiff rings (9, 11). However, very importantly, the latter simulation work deduction of a so-called “topological glass” cross-over employed random pinning (quenched disorder) of a subset of rings and extrapolated the results to the physical case of a liquid of fully mobile rings. Random pinning is an artificial process and does not directly correspond to the ring liquids of experimental interest. In contrast, our theory is for fully mobile ring liquids and predicts an exponential suppression of diffusion emerges as a consequence of slowly relaxing intermolecular force correlations (on the R_g scale) between rings.

Fig. 7.

Scaled dimensionless CM diffusion constant $N{D}_{\mathrm{CM}}/D_{\mathrm{s}}^{\ast }$ from Figs. 5B and 6 (same colors and symbol codes) as a function of $N/N_{\mathrm{D}}$ over a wide range of N, ring stiffness, $N_{\mathrm{e}}$, and $\rho$. The pink and purple curves represent the predictions of the weak- and strong-caging MDS theories, respectively. The $N_{\mathrm{e}}$ and $N_{\mathrm{D}}$ values for all systems are summarized in SI Appendix, Table S1. The vertical scaling factor associated with monomer friction for MT data of semidilute ring solutions is taken to be $D_{\mathrm{s}}^{\ast }/D_{\mathrm{LJ}}=1$, while for ring melts at $\rho {\sigma }_{\mathrm{L}\mathrm{J}}^3=0.85$, it is $D_{\mathrm{s}}^{\ast }/D_{\mathrm{L}\mathrm{J}}={\left[N_{\mathrm{e}}\left(k_{\theta }\right)/N_{\mathrm{e}}\left(1.5\right)\right]}^{0.7}$. The Inset shows the corresponding theoretical local barrier in thermal energy units as a function of $N/N_{\mathrm{D}}$.

Concerning numbers and the internal consistency of our analysis, we note that Fig. 7 and our calculation of N_on ~ 30 N_D imply that for the $N_{\mathrm{D}}=60$ value (44) relevant to our flexible melt simulations with $k_{\theta }=1.5$, the smooth cross-over from the weak- to strong-caging regime occurs at N_on ~ 1,800 or N/N_e ~ 62. This provides additional post-facto motivation for performing our present simulations of high stiffness ($k_{\theta }=4.0$) ring melts with $N=1600$, and is qualitatively consistent with our observations of deviations from power law scaling of ${\tau }_{\mathrm{d}}/{\tau }_{\mathrm{L}\mathrm{J}}$ and ${\tau }_{\mathrm{CM}}/{\tau }_{\mathrm{L}\mathrm{J}}$ (Fig. 5A and SI Appendix, Fig. S3C) in the high N/N_e regime.

Although the cross-over in the main frame of Fig. 7 to what we interpret as an activated regime formally begins at N_on ~ 30N_D, the barriers (see the Inset of Fig. 7) are low when $N/N_{\mathrm{on}}<3-4$, and hence the CM diffusion constants and relaxation times do not strongly deviate from the power law predictions of the weak-caging regime. Thus, if future experiments and/or simulations wish to search for clear signatures of activated dynamics, probing $N/N_{\mathrm{D}}>100$ should be pursued. For example, as explicitly shown in SI Appendix, Fig. S10, the theory predicts that $N{D}_{\mathrm{CM}}/D_{\mathrm{s}}$ dramatically drops by more than five decades upon increasing $N/N_{\mathrm{D}}$ from 30 to 100. At present, simulations cannot access the long-time Fickian CM diffusion regime at such high values of reduced N, but this may be viable for experiments on semiflexible biopolymers. Finally, from the perspective of the theory, the existence of a second activated regime is general, and is not conceptually tied to the local stiffness of the ring backbone. Of course, the latter impacts the value of $N_{\mathrm{o}\mathrm{n}}$, and hence is very important for practical simulation and experimental studies.

Experimental Implications.

To the best of our knowledge, systematic experimental exploration of the ring stiffness or concentration variables are absent, although it is well known that they play a critical role in understanding the dynamics of entangled linear chains. Thus, we believe that our present work can help guide the formulation of future experimental directions. Typically, in dynamical studies, the degree of polymerization of polymer rings is characterized in terms of the extent of entanglement of the corresponding linear chain liquid, $Z\equiv N/N_{\mathrm{e}}$. Current understanding requires this quantity to be larger than unity for topological effects to modify equilibrium ring conformations. However, our theory indicates that Z is not the fundamentally relevant quantity for ring dynamics, rather $N/N_{\mathrm{D}}$ and $N_{\mathrm{o}\mathrm{n}}/N_{\mathrm{D}}$ are, where $N_{\mathrm{D}}$ does depend on $N_{\mathrm{e}}$, but they are not linearly proportional. Neutron scattering measurements (33–35) up to Z ~ 40 have been performed (see, e.g., the Inset of Fig. 2D), and the findings are qualitatively consistent with our weak-caging theory prediction of $D_{\mathrm{CM}}\sim N^{-2}$. To incisively probe the predicted strong-caging activated regime, diffusion constant measurements at higher Z values are needed. We theoretically estimate a cross-over value of Z ~ 60 or larger for flexible ring melts, and significantly higher values are required for the emergent entropic barrier to become large enough that clear indications of activated hopping transport will be manifest. Perhaps this can be achieved by using biopolymers given our present analysis applies across polymer concentrations and for variable degrees of backbone stiffness.

The believed to be “pure” polystyrene ring melt studies (3, 27, 48, 82) have only been performed up to modest values of $Z\sim 15 \mathrm{or} 20$. Recent experiments by Tu, Schroeder, and the present authors (17) have reported a new ring chemistry system that is highly pure which extends up to $Z\sim 58$ in concentrated solutions, but it remains an open challenge to measure the long-time CM diffusion constant. Even if such measurements were done, given our discussion in the previous section, we expect only modest deviations from the $N^{-2}$ weak-caging scaling for such a value of Z. McKenna and coworkers (14) have employed a different new chemistry and found an intriguing ultralow frequency plateau in the ring melt storage moduli $G\prime$ at extremely large values of Z ~200 to 300. At face value, this seems consistent with the emergence of CM transient localization, a cross-over to an ultraslow activated diffusion regime, and an elastic shear modulus plateau of intermolecular stress origin. However, questions remain concerning the purity and internal structure (not measured to date) of these systems.

In a broad sense, we hope our present work can help guide the design of future experiments that search for the emergence of activated dynamics and a macromolecular-scale glassy dynamics regime by rational adjustment of ring concentration, backbone stiffness, and/or degree of polymerization. In this regard, the rather remarkable “redundancies” in Eq. 4 for the longest relaxation time or diffusion constant associated with different variables ($R_{\mathrm{g}}$, $d_{\mathrm{T}}$, $N_{\mathrm{e}}$, $l_{\mathrm{K}}$, N, and ρ) might help guide the formulation of different systems that can more incisively test our predictions.

Concluding Remarks and Future Outlook

We have extended the segmental scale force-based theory for the CM diffusion constant and the corresponding relaxation time in ring polymer liquids to address the effects of ring stiffness, monomer density, chain entanglement length, and degree of polymerization. The consequences of all these controllable aspects can be embedded in the predicted dynamic cross-over degree of ring polymerization, $N_{\mathrm{D}}$. This provides a unified description/framework for these effects on ring diffusivity that potentially bridges the behavior of synthetic and biological polymers.

The theoretical predictions have been successfully compared to our melt simulations over a wide range of ring stiffness and N, and to prior simulations of flexible rings in melts and also stiff rings in semidilute and concentrated solutions. In our macromolecular dynamic caging approach, the “threading” notion does not literally enter. Rather, the key idea is how strong or deep ring–ring interpenetration, in concert with polymer connectivity and excluded volume constraints, induce strong space-time intermolecular force correlations. The key physics manifested in the parameter N_D depends on ring stiffness, monomer concentrations, Kuhn length, and N_e. Stiffness and concentration-dependent local segmental friction effects are not negligible, but are of secondary importance. Static topological entanglement effects critical to induce the unusual equilibrium conformations of rings enter the structural input to the dynamical theory, and the quantification of $N_{\mathrm{D}}$. The concept of a second cross-over to activated transport and ultimately kinetic vitrification emerges from additional physical effects akin to small molecule and (soft) colloidal glass formers, albeit not on a local segmental scale, but rather on a macromolecular scale mediated by interring packing.

We have also proposed a qualitative physical picture for understanding the combined consequences of topological constraints, interring excluded volume interactions, and ring connectivity on the CM diffusion and long relaxation time of ring melts and solutions based on a measure of dynamically relevant intermolecular “interaction strength”. The obtained insights suggest unique ways to manipulate the interring force time correlations that drive slow dynamics. For example, by introducing special attractions or repulsions between monomers including associating rings and polyelectrolytes, random pinning of rings, or dynamic cross-linking in ring vitrimers. We suggest that our specific results and proposed different way of thinking can help guide the formulation of future experiments and simulations that will broaden the impact of ring polymer physics in soft synthetic and living matter, including realizing the laboratory conditions required to create macromolecular-scale glass-formers by manipulation of variables other than the traditional increase of molecular weight.

More broadly, the present work sets the stage to address CM diffusion in other dense liquids composed of macromolecules of compact, but strongly fluctuating, conformations, such as many arm star polymers (22), soft colloids (25–27), single chain nanoparticles (21), two-dimensional chain melts, (24) and perhaps even folded proteins (23). Extension of the ideas to create a theory of internal conformational mode ring dynamics on intermediate time and length scales is an open problem under study. Progress in this direction will allow predictions to be made for time-dependent dynamical properties such as the monomer MSD and stress relaxation function.

Materials and Methods

We use the classic Kremer–Grest bead-spring model to perform our simulations. All the simulation details are documented in the literature (4, 5, 52, 53), with some technical descriptions specifically relevant to our present work provided in SI Appendix. Germane theoretical backgrounds are also discussed in SI Appendix.

SI Statement of Content.

Additional calculations and analysis that support the conclusions drawn in this article, along with a summary of all parameters employed. Specifically: a) simulation details; b) tables of stiffness-dependent polymer properties, $N_{\mathrm{D}}$, $R_{\mathrm{g}}$, relaxation time, diffusion constant, and contact value of the homopolymer hard chain g(d), for all systems studied; c) figures discussed in the main text (SI Appendix, Figs. S1–S10) that support the conclusions drawn; and d) background for the PRISM theory of equilibrium structure and thermodynamics (65, 70, 71, 74, 78), the MDS weak-caging (44) and strong-caging (51) theories for ring CM dynamics, and a dynamical theory for the elementary time scale and local friction constant in segmental scale of semiflexible chain melts (74).

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix.