Sensing Polymer Chain Dynamics through Ring Topology: A Neutron Spin Echo Study

Simulational Tests of the Rouse Model

An extensive review of literature simulations of quiescent polymer melts is given, considering results that test aspects of the Rouse model in the melt. We focus on Rouse model predictions for the mean-square amplitudes ⟨(Xp(0))2⟩ and time correlation functions ⟨Xp(0)Xp(t)⟩ of the Rouse mode Xp(t). The simulations conclusively demonstrate that the Rouse model is invalid in polymer melts. In particular, and contrary to the Rouse model, (i) mean-square Rouse mode amplitudes ⟨(Xp(0))2⟩ do not scale as sin-2(pπ/2N), N being the number of beads in the polymer. For small p (say, p≤3) ⟨(Xp(0))2⟩ scales with p as p-2; for larger p, it scales as p-3. (ii) Rouse mode time correlation functions ⟨Xp(t)Xp(0)⟩ do not decay with time as exponentials; they instead decay as stretched exponentials exp(-αtβ). β depends on p, typically with a minimum near N/2 or N/4. (iii) Polymer bead displacements are not described by independent Gaussian random processes. (iv) For p≠q, ⟨Xp(t)Xq(0)⟩ is sometimes non-zero. (v) The response of a polymer coil to a shear flow is a rotation, not the affine deformation predicted by Rouse. We also briefly consider the Kirkwood-Riseman polymer model.

Q-dependent collective relaxation dynamics of glass-forming liquid Ca0.4K0.6(NO3)1.4 investigated by wide-angle neutron spin-echo

The relaxation behavior of glass formers exhibits spatial heterogeneity and dramatically changes upon cooling towards the glass transition. However, the underlying mechanisms of the dynamics at different microscopic length scales are not fully understood. Employing the recently developed wide-angle neutron spin-echo spectroscopy technique, we measured the Q-dependent coherent intermediate scattering function of a prototypical ionic glass former Ca_0.4K_0.6(NO₃)_1.4, in the highly viscous liquid state. In contrast to the structure modulated dynamics for Q < 2.4 Å⁻¹, i.e., at and below the structure factor main peak, for Q > 2.4 Å⁻¹, beyond the first minimum above the structure factor main peak, the stretching exponent exhibits no temperature dependence and concomitantly the relaxation time shows smaller deviations from Arrhenius behavior. This finding indicates a change in the dominant relaxation mechanisms around a characteristic length of 2π/(2.4 Å⁻¹) ≈ 2.6 Å, below which the relaxation process exhibits a temperature independent distribution and more Arrhenius-like behavior.

Length scale dependence is important for understanding the collective relaxation dynamics in glass-forming liquids. Here, the authors find in liquid Ca_0.4K_0.6(NO₃)_1.4 a change in the dominant relaxation mechanisms around 2.6 Å, below which the relaxation process exhibits a temperature independent distribution and more Arrhenius-like behavior.

Neutron Spin-Echo Studies of the Structural Relaxation of Network Liquid ZnCl2 at the Structure Factor Primary Peak and Prepeak

Using neutron spin-echo spectroscopy, we studied the microscopic structural relaxation of a prototypical network ionic liquid ZnCl₂ at the structure factor primary peak and prepeak. The results show that the relaxation at the primary peak is faster than the prepeak and that the activation energy is ∼33% higher. A stretched exponential relaxation is observed even at temperatures well-above the melting point T_m. Surprisingly, the stretching exponent shows a rapid increase upon cooling, especially at the primary peak, where it changes from a stretched exponential to a simple exponential on approaching the T_m. These results suggest that the appearance of glassy dynamics typical of the supercooled state even in the equilibrium liquid state of ZnCl₂ as well as the difference of activation energy at the two investigated length scales are related to the formation of a network structure on cooling.

Threading-Unthreading Transition of Linear-Ring Polymer Blends in Extensional Flow

Adding small amounts of ring polymers to a matrix of their linear counterparts is known to increase the zero-shear-rate viscosity because of linear-ring threading. Uniaxial extensional rheology measurements show that, unlike its pure linear and ring constituents, the blend exhibits an overshoot in the stress growth coefficient. By combining these measurements with ex-situ small-angle neutron scattering and nonequilibrium molecular dynamics simulations, this overshoot is shown to be driven by a transient threading-unthreading transition of rings embedded within the linear entanglement network. Prior to unthreading, embedded rings deform affinely with the linear entanglement network and produce a measurably stronger elongation of the linear chains in the blend compared to the pure linear melt. Thus, rings uniquely alter the mechanisms of transient elongation in linear polymers.

Cluster prevalence in concentrated ring-chain mixtures under shear

Semiflexible ring polymers are known to exhibit clustering behavior and form stacks in concentrated solutions. Recently, weak shear was suggested to re-orient these stacks with flow, a phenomenon more easily visible in more concentrated solutions [Liebetreu et al., ACS Appl. Polym. Mater., 2020, 2(8), 3505-3517, DOI: 10.1021/acsapm.0c00522]. In this work, we investigate the impact of mixing linear chains and rings in a similar system under shear, studying clustering in the presence of semiflexible, rod-like chains. We present a correlation between chain monomer fraction and clustering behavior as linear chains take up less space, thus decreasing the system's effective density and, subsequently, clustering. However, we suggest mixtures with a low chain concentration to maintain or potentially enhance clustering at equilibrium while this effect is destroyed under shear. The mixing of chains and rings may therefore be used to create more strongly organized structures susceptible to reorientation via weak shear.

Local Effects of Ring Topology Observed in Polymer Conformation and Dynamics by Neutron Scattering-A Review

The physical properties of polymers depend on a range of both structural and chemical parameters, and in particular, on molecular topology. Apparently simple changes such as joining chains at a point to form stars or simply joining the two ends to form a ring can profoundly alter molecular conformation and dynamics, and hence properties. Cyclic polymers, as they do not have free ends, represent the simplest model system where reptation is completely suppressed. As a consequence, there exists a considerable literature and several reviews focused on high molecular weight cyclics where long range dynamics described by the reptation model comes into play. However, this is only one area of interest. Consideration of the conformation and dynamics of rings and chains, and of their mixtures, over molecular weights ranging from tens of repeat units up to and beyond the onset of entanglements and in both solution and melts has provided a rich literature for theory and simulation. Experimental work, particularly neutron scattering, has been limited by the difficulty of synthesizing well-characterized ring samples, and deuterated analogues. Here in the context of the broader literature we review investigations of local conformation and dynamics of linear and cyclic polymers, concentrating on poly(dimethyl siloxane) (PDMS) and covering a wide range of generally less high molar masses. Experimental data from small angle neutron scattering (SANS) and quasi-elastic neutron scattering (QENS), including Neutron Spin Echo (NSE), are compared to theory and computational predictions.

On-line monitoring of the sol-gel transition temperature of thermosensitive chitosan/β-glycerophosphate hydrogels by low field NMR

A temperature controlled low field nuclear magnetic resonance (LF-NMR) T₂ relaxometry technique based on the mobility changes of water trapped in hydrogels, was successfully used for on-line determination of the sol-gel transition temperature for chitosan/β-glycerophosphate (CS/GP) hydrogels in real time. The LF-NMR results indicated that the gelation temperature decreased gradually with increasing GP concentration, and the results were supported by both thermogravimetric differential scanning calorimetry (DSC) and rheological findings; however, LF-NMR allows non-destructive monitoring of samples during continuous heating. Moreover, as the mobility of water molecules varies greatly during the sol-gel phase transition, the LF-NMR measurement was more sensitive and accurate (RSD ≤ 0.1 %, n = 5) compared with DSC (RSD: 1.2 %-3.7 %, n = 5) and rheology (RSD: 1.1 %-2.3 %, n = 5).

Effect of molecular architecture on ring polymer dynamics in semidilute linear polymer solutions

Understanding the dynamics of ring polymers is a particularly challenging yet interesting problem in soft materials. Despite recent progress, a complete understanding of the nonequilibrium behavior of ring polymers has not yet been achieved. In this work, we directly observe the flow dynamics of DNA-based rings in semidilute linear polymer solutions using single molecule techniques. Our results reveal strikingly large conformational fluctuations of rings in extensional flow long after the initial transient stretching process has terminated, which is observed even at extremely low concentrations (0.025 c*) of linear polymers in the background solution. The magnitudes and characteristic timescales of ring conformational fluctuations are determined as functions of flow strength and polymer concentration. Our results suggest that ring conformational fluctuations arise due to transient threading of linear polymers through open ring chains stretching in flow.

Viscoelastic Properties of Unentangled Multicyclic Polystyrenes

We report on the viscoelastic properties of linear, monocyclic, and multicyclic polystyrenes with the same low molecular weight. All polymers investigated were found to exhibit unentangled dynamics. For monocyclic polymers without inner loops, a cyclic-Rouse model complemented by the contribution of unlinked chains (whose fraction was determined experimentally) captured the observed rheological response. On the other hand, multicyclic polymers with inner loops were shown to follow a hierarchical cyclic-Rouse relaxation with the outer loops relaxing first, followed by the inner loop relaxation. The influence of unlinked linear chains was less significant in multicyclic polymers with inner loops. The isofrictional zero-shear viscosity decreased with increasing number of constrained segments on the coupling sites, which was attributed to the decreasing loop size and the dilution effect due to the hierarchical relaxation.

Dynamical structure of entangled polymers simulated under shear flow

The non-linear response of entangled polymers to shear flow is complicated. Its current understanding is framed mainly as a rheological description in terms of the complex viscosity. However, the full picture requires an assessment of the dynamical structure of individual polymer chains which give rise to the macroscopic observables. Here we shed new light on this problem, using a computer simulation based on a blob model, extended to describe shear flow in polymer melts and semi-dilute solutions. We examine the diffusion and the intermediate scattering spectra during a steady shear flow. The relaxation dynamics are found to speed up along the flow direction, but slow down along the shear gradient direction. The third axis, vorticity, shows a slowdown at the short scale of a tube, but reaches a net speedup at the large scale of the chain radius of gyration.

Scaling Laws for the Conformation and Viscosity of Ring Polymers in the Crossover Region around Me from Detailed Molecular Dynamics Simulations

We present results from detailed, atomistic molecular dynamics (MD) simulations of pure, strictly monodisperse linear and ring poly(ethylene oxide) (PEO) melts under equilibrium and nonequilibrium (shear flow) conditions. The systems examined span the regime of molecular weights (M_w) from sub-Rouse (M_w < M_e) to reptation (M_w ∼ 10 M_e), where M_e denotes the characteristic entanglement molecular weight of linear PEO. For both PEO architectures (ring and linear), the predicted chain center-of-mass self-diffusion coefficients D_G as a function of PEO M_w are in remarkable agreement with experimental data. From the flow simulations under shear, we have extracted and analyzed the zero-shear viscosity of ring and linear PEO melts as a function of M_w.

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.

Evidence and Limits of Universal Topological Surface Segregation of Cyclic Polymers

If you mix lines and circles, what happens at the edge of the mixture? The problem is simply stated, but the answer is not obvious. Twenty years ago it was proposed that a universal topological driving force would drive cyclic chains to enrich the surface of blends of linear and cyclic chains. Here such behavior is demonstrated experimentally for sufficiently long chains and the limit in molecular weight where packing effects dominate over the topological driving force is identified.

Glass transition of polymers in bulk, confined geometries, and near interfaces

When cooled or pressurized, polymer melts exhibit a tremendous reduction in molecular mobility. If the process is performed at a constant rate, the structural relaxation time of the liquid eventually exceeds the time allowed for equilibration. This brings the system out of equilibrium, and the liquid is operationally defined as a glass-a solid lacking long-range order. Despite almost 100 years of research on the (liquid/)glass transition, it is not yet clear which molecular mechanisms are responsible for the unique slow-down in molecular dynamics. In this review, we first introduce the reader to experimental methodologies, theories, and simulations of glassy polymer dynamics and vitrification. We then analyse the impact of connectivity, structure, and chain environment on molecular motion at the length scale of a few monomers, as well as how macromolecular architecture affects the glass transition of non-linear polymers. We then discuss a revised picture of nanoconfinement, going beyond a simple picture based on interfacial interactions and surface/volume ratio. Analysis of a large body of experimental evidence, results from molecular simulations, and predictions from theory supports, instead, a more complex framework where other parameters are relevant. We focus discussion specifically on local order, free volume, irreversible chain adsorption, the Debye-Waller factor of confined and confining media, chain rigidity, and the absolute value of the vitrification temperature. We end by highlighting the molecular origin of distributions in relaxation times and glass transition temperatures which exceed, by far, the size of a chain. Fast relaxation modes, almost universally present at the free surface between polymer and air, are also remarked upon. These modes relax at rates far larger than those characteristic of glassy dynamics in bulk. We speculate on how these may be a signature of unique relaxation processes occurring in confined or heterogeneous polymeric systems.

What Happens When Threading is Suppressed in Blends of Ring and Linear Polymers?

Self-diffusivity of a large tracer ring polymer, D r , immersed in a matrix of linear polymers with N l monomers each shows unusual length dependence. D r initially increases, and then decreases with increasing N l . To understand the relationship between the nonmonotonic variation in D r and threading by matrix chains, we perform equilibrium Monte Carlo simulations of ring-linear blends in which the uncrossability of ring and linear polymer contours is switched on (non-crossing), or artificially turned off (crossing). The D r ≈ 6 . 2 × 10 - 7 N l 2 / 3 obtained from the crossing simulations, provides an upper bound for the D r obtained for the regular, non-crossing simulations. The center-of-mass mean-squared displacement ( g 3 ( t ) ) curves for the crossing simulations are consistent with the Rouse model; we find g 3 ( t ) = 6 D r t . Analysis of the polymer structure indicates that the smaller matrix chains are able to infiltrate the space occupied by the ring probe more effectively, which is dynamically manifested as a larger frictional drag per ring monomer.

Influence of chain topology on polymer crystallization: poly(ethylene oxide) (PEO) rings vs. linear chains

The absence of entanglements, the more compact structure and the faster diffusion in melts of cyclic poly(ethylene oxide) (PEO) chains have consequences on their crystallization behavior at the lamellar and spherulitic length scales. Rings with molecular weight below the entanglement molecular weight (M < M_e), attain the equilibrium configuration composed from twice-folded chains with a lamellar periodicity that is half of the corresponding linear chains. Rings with M > M_e undergo distinct step-like conformational changes to a crystalline lamellar with the equilibrium configuration. Rings melt from this configuration in the absence of crystal thickening in sharp contrast to linear chains. In general, rings more easily attain their extended equilibrium configuration due to strained segments and the absence of entanglements. In addition, rings have a higher equilibrium melting temperature. At the level of the spherulitic superstructure, growth rates are much faster for rings reflecting the faster diffusion and more compact structure. With respect to the segmental dynamics in their semi-crystalline state, ring PEOs with a steepness index of ∼34 form some of the "strongest" glasses.

DNA as a Model for Probing Polymer Entanglements: Circular Polymers and Non-Classical Dynamics

Double-stranded DNA offers a robust platform for investigating fundamental questions regarding the dynamics of entangled polymer solutions. The exceptional monodispersity and multiple naturally occurring topologies of DNA, as well as a wide range of tunable lengths and concentrations that encompass the entanglement regime, enable direct testing of molecular-level entanglement theories and corresponding scaling laws. DNA is also amenable to a wide range of techniques from passive to nonlinear measurements and from single-molecule to bulk macroscopic experiments. Over the past two decades, researchers have developed methods to directly visualize and manipulate single entangled DNA molecules in steady-state and stressed conditions using fluorescence microscopy, particle tracking and optical tweezers. Developments in microfluidics, microrheology and bulk rheology have also enabled characterization of the viscoelastic response of entangled DNA from molecular levels to macroscopic scales and over timescales that span from linear to nonlinear regimes. Experiments using DNA have uniquely elucidated the debated entanglement properties of circular polymers and blends of linear and circular polymers. Experiments have also revealed important lengthscale and timescale dependent entanglement dynamics not predicted by classical tube models, both validating and refuting new proposed extensions and alternatives to tube theory and motivating further theoretical work to describe the rich dynamics exhibited in entangled polymer systems.

Microscopic Dynamics and Topology of Polymer Rings Immersed in a Host Matrix of Longer Linear Polymers: Results from a Detailed Molecular Dynamics Simulation Study and Comparison with Experimental Data

We have performed molecular dynamics (MD) simulations of melt systems consisting of a small number of long ring poly(ethylene oxide) (PEO) probes immersed in a host matrix of linear PEO chains and have studied their microscopic dynamics and topology as a function of the molecular length of the host linear chains. Consistent with a recent neutron spin echo spectroscopy study (Goossen et al., Phys. Rev. Lett. 2015, 115, 148302), we have observed that the segmental dynamics of the probe ring molecules is controlled by the length of the host linear chains. In matrices of short, unentangled linear chains, the ring probes exhibit a Rouse-like dynamics, and the spectra of their dynamic structure factor resemble those in their own melt. In striking contrast, in matrices of long, entangled linear chains, their dynamics is drastically altered. The corresponding dynamic structure factor spectra exhibit a steep initial decay up to times on the order of the entanglement time τe of linear PEO at the same temperature but then they become practically time-independent approaching plateau values. The plateau values are different for different wavevectors; they also depend on the length of the host linear chains. Our results are supported by a geometric analysis of topological interactions, which reveals significant threading of all ring molecules by the linear chains. In most cases, each ring is simultaneously threaded by several linear chains. As a result, its dynamics at times longer than a few τe should be completely dictated by the release of the topological restrictions imposed by these threadings (interpenetrations). Our topological analysis did not indicate any effect of the few ring probes on the statistical properties of the network of primitive paths of the host linear chains.

Minimal Surfaces on Unconcatenated Polymer Rings in Melt

In order to quantify the effect of mutual threading on conformations and dynamics of unconcatenated and unknotted rings in the melt we computationally examine their minimal surfaces. We found a linear scaling of the surface area with the ring length. Minimal surfaces allow for an unambiguous algorithmic definition of mutual threading between rings. Based on it, we found that, although ring threading is frequent, majority of cases correspond to short loops. These findings explain why approximate theories that neglect threading are so unexpectedly successful despite having no small parameter justification. We also examine threading dynamics and identify the threading order parameter that reflects the ring diffusivity.

Analysis of Slow Modes in Ring Polymers: Threading of Rings Controls Long-Time Relaxation

Atomistic configurations of pure, precisely monodisperse ring poly(ethylene oxide) (PEO) melts accumulated in the course of very long molecular dynamics (MD) simulations at T = 413 K and P = 1 atm have been subjected to a detailed geometric analysis involving three steps (reduction to ensembles of coarse-grained paths, triangulation of the resulting three-dimensional polygons, and analysis of interpenetrations using vector calculus) in order to locate ring-ring threading events and quantify their strength and survival times. A variety of threading situations have been identified corresponding to single and multiple penetrations. The percentage of inter-ring threadings that correspond to full penetrations has also been quantified. By repeating the analysis for several PEO melts, the dependence of the degree of inter-ring threading on molecular weight (MW) has been obtained. Simulations with MWs up to 10 times the reported entanglement molecular weight (M_e) of linear PEO have revealed several multiple threading events in all systems, with their relative number increasing with increasing MW. Our analysis indicates the existence of strong ring-ring topological interactions, which can last up to several times the corresponding average orientational ring polymer relaxation time. We show that these ring-ring interactions, together with the additional ring-linear threadings due to the remaining linear impurities, can explain the appearance of slow relaxation modes observed experimentally in entangled rings.