Non-concatenated Ring Research Articles

Molecular Simulations Revealing Effects of Non-concatenated Ring Topology on Phase Behavior of Symmetric Diblock Copolymers

Molecular Simulations Revealing Effects of Non-concatenated Ring Topology on Phase Behavior of Symmetric Diblock Copolymers

Entropic Mixing of Ring/Linear Polymer Blends.

The topological constraints of nonconcatenated ring polymers force them to form compact loopy globular conformations with much lower entropy than unconstrained ideal rings. The closed-loop structure of ring polymers also enables them to be threaded by linear polymers in ring/linear blends, resulting in less compact ring conformations with higher entropy. This conformational entropy increase promotes mixing rings with linear polymers. Here, using molecular dynamics simulations for bead-spring chains, ring/linear blends are shown to be significantly more miscible than linear/linear blends and that there is an entropic mixing, negative χ, for ring/linear blends compared to linear/linear and ring/ring blends. In analogy with small angle neutron scattering, the static structure function S(q) is measured, and the resulting data are fit to the random phase approximation model to determine χ. In the limit that the two components are the same, χ = 0 for the linear/linear and ring/ring blends as expected, while χ < 0 for the ring/linear blends. With increasing chain stiffness, χ for the ring/linear blends becomes more negative, varying reciprocally with the number of monomers between entanglements. Ring/linear blends are also shown to be more miscible than either ring/ring or linear/linear blends and stay in single phase for a wider range of increasing repulsion between the two components.

Entanglement Length Scale Separates Threading from Branching of Unknotted and Non-concatenated Ring Polymers in Melts.

Current theories on the conformation and dynamics of unknotted and non-concatenated ring polymers in melt conditions describe each ring as a tree-like double-folded object. While evidence from simulations supports this picture on a single ring level, other works show pairs of rings also thread each other, a feature overlooked in the tree theories. Here we reconcile this dichotomy using Monte Carlo simulations of the ring melts with different bending rigidities. We find that rings are double-folded (more strongly for stiffer rings) on and above the entanglement length scale, while the threadings are localized on smaller scales. The different theories disagree on the details of the tree structure, i.e., the fractal dimension of the backbone of the tree. In the stiffer melts we find an indication of a self-avoiding scaling of the backbone, while more flexible chains do not exhibit such a regime. Moreover, the theories commonly neglect threadings and assign different importance to the impact of the progressive constraint release (tube dilation) on single ring relaxation due to the motion of other rings. Despite that each threading creates only a small opening in the double-folded structure, the threading loops can be numerous and their length can exceed substantially the entanglement scale. We link the threading constraints to the divergence of the relaxation time of a ring, if the tube dilation is hindered by pinning a fraction of other rings in space. Current theories do not predict such divergence and predict faster than measured diffusion of rings, pointing at the relevance of the threading constraints in unpinned systems as well. Revision of the theories with explicit threading constraints might elucidate the validity of the conjectured existence of topological glass.

Conformation and structure of ring polymers in semidilute solutions: A molecular dynamics simulation study

Scaling is one of the most fundamental concepts in theoretical polymer physics. For linear polymers in semidilute solutions under a good solvent condition, there has been a well established scaling behavior of mean-square radius of gyration of chains with the concentration, i.e., R g 2 ~ ϕ β , with β = −0.25, which receives strong support from experiments, computer simulations and scaling arguments. However, a clear-cut conclusion about the scaling behavior of dimensions of nonconcatenated and unknotted ring polymers in the semidilute solution with the concentration has not been achieved. Values of the corresponding scaling exponent β are diversely reported in literature, ranging from −0.25 to −0.59 and in contradiction within experimental, simulation and theoretical studies. As an endeavour for clarifying this issue, we carried out extensive molecular dynamics (MD) simulations based on the Kremer-Grest model of ring polymers with chain lengths N up to 5120 in semidilute solutions, and compared our results about ring conformations with almost all of experimental, simulation and theoretical data available in literature. Our MD simulation results lead to a conclusion that the mean-square radius of gyration of ring polymers scales with the concentration in semidilute regime as R g 2 ∼ φ − 0.59 ± 0.01 . This conclusion is in a good agreement with a previous finding by a lattice Monte Carlo simulation. Surprisingly, by looking into the experimental and other simulation results reported previously with sufficient caution, we tend to conclude that in reality almost all of their data can be described reasonably well by the same scaling law, R g 2 ~ ϕ − 0.59 . Furthermore, the scaling exponent obtained (−0.59) can be successfully explained by a scaling argument based on concept of correlation length. Finally, effects of the concentration on shape properties and packing behavior of ring polymers in the semidilute solutions have also been illuminated within our MD simulations. • Scaling behavior of ring dimensions in semidilute solutions with the concentration has been revisited. • Comparison of our results with experimental, and other simulation and theoretical results has been conducted. • Effect of concentration on shape and packing behavior of ring polymers in semidilute solutions has been illuminated.

Dynamical Entanglement and Cooperative Dynamics in Entangled Solutions of Ring and Linear Polymers.

Understanding how entanglements affect the behavior of polymeric complex fluids is an open challenge in many fields. To elucidate the nature and consequence of entanglements in dense polymer solutions, we propose a novel method: a "dynamical entanglement analysis" (DEA) to extract spatiotemporal entanglement structures from the pairwise displacement correlation of entangled chains. By applying this method to large-scale molecular dynamics simulations of linear and unknotted, nonconcatenated ring polymers, we find a strong and unexpected cooperative dynamics: the footprint of mutual entrainment between entangled chains. We show that DEA is a powerful and sensitive probe that reveals previously unnoticed and architecture-dependent spatiotemporal structures of dynamical entanglement in polymeric solutions. We also propose a mean-field approximation of our analysis that provides previously under-appreciated physical insights into the dynamics of generic entangled polymers. We envisage DEA will be useful to analyze the dynamical evolution of entanglements in generic polymeric systems such as blends and composites.

Conformation and dynamics of ring polymers under symmetric thin film confinement.

Understanding the structure and dynamics of polymers under confinement has been of widespread interest, and one class of polymers that have received comparatively little attention under confinement is that of ring polymers. The properties of non-concatenated ring polymers can also be important in biological fields because ring polymers have been proven to be a good model to study DNA organization in the cell nucleus. From our previous study, linear polymers in a cylindrically confined polymer melt were found to segregate from each other as a result of the strong correlation hole effect that is enhanced by the confining surfaces. By comparison, our subsequent study of linear polymers in confined thin films at similar levels of confinements found only the onset of segregation. In this study, we use molecular dynamics simulation to investigate the chain conformations and dynamics of ring polymers under planar (1D) confinement as a function of film thickness. Our results show that conformations of ring polymers are similar to the linear polymers under planar confinement, except that ring polymers are less compressed in the direction normal to the walls. While we find that the correlation hole effect is enhanced under confinement, it is not as pronounced as the linear polymers under 2D confinement. Finally, we show that chain dynamics far above Tg are primarily affected by the friction from walls based on the monomeric friction coefficient we get from the Rouse mode analysis.

Unexpected Stretching of Entangled Ring Macromolecules.

In the melt state at equilibrium, entangled nonconcatenated ring macromolecules adapt more compact conformations compared to their linear analogs and do not form an entanglement network. We show here that, when subjected to uniaxial stretching, they exhibit a unique response, which sets them apart from any other polymer. Remarkably, whereas both linear and ring polymers strain-harden, the viscosity of the rings increases dramatically (the melt thickens) at very low stretch rates due to the unraveling of their conformations along the stretching direction. At high rates, stretching leads to viscosity thinning similar to that of entangled linear polymers, albeit with subtle differences.

Simulation of ring polymer melts with GPU acceleration

We implemented the elastic lattice polymer model on the GPU (Graphics Processing Unit), and show that the GPU is very efficient for polymer simulations of dense polymer melts. The implementation is able to perform up to 4.1⋅109 Monte Carlo moves per second. Compared to our standard CPU implementation, we find an effective speed-up of a factor 92. Using this GPU implementation we studied the equilibrium properties and the dynamics of non-concatenated ring polymers in a melt of such polymers, using Rouse modes. With increasing polymer length, we found a very slow transition to compactness with a growth exponent ν≈1/3. Numerically we find that the longest internal time scale of the polymer scales as N3.1, with N the molecular weight of the ring polymer.

Topological free volume and quasi-glassy dynamics in the melt of ring polymers

Motivated by recent observations that non-concatenated ring polymers in their dense solution exhibit a glass-like dynamics, we propose a free volume description of the motion of such rings based on the notion of topological volume. We first construct a phenomenological free energy which enables one to quantify the degree of topological crowding measured by the coordination number. Then we pinpoint a key role of the cooperative dynamics of neighboring rings, which is responsible for an anomalous dependence of the global structural relaxation (diffusion) time on ring length. Predictions on molecular weight dependence of both static (ring size, coordination number) and dynamic (relaxation time, diffusion coefficient) quantities are in very good agreement with reported numerical simulations. Throughout the discussion, the entanglement length Ne is assumed to be a unique characteristic length for the topological constraint, and hence, all the physical quantities are universally described in terms of the rescaled chain length N/Ne. Finally, we discuss how the dense solution of rings is analogous to yet different from ordinary glassy systems.

Nanoparticle Motion in Entangled Melts of Linear and Nonconcatenated Ring Polymers.

The motion of nanoparticles (NPs) in entangled melts of linear polymers and nonconcatenated ring polymers are compared by large-scale molecular dynamics simulations. The comparison provides a paradigm for the effects of polymer architecture on the dynamical coupling between NPs and polymers in nanocomposites. Strongly suppressed motion of NPs with diameter d larger than the entanglement spacing a is observed in a melt of linear polymers before the onset of Fickian NP diffusion. This strong suppression of NP motion occurs progressively as d exceeds a and is related to the hopping diffusion of NPs in the entanglement network. In contrast to the NP motion in linear polymers, the motion of NPs with d > a in ring polymers is not as strongly suppressed prior to Fickian diffusion. The diffusion coefficient D decreases with increasing d much slower in entangled rings than in entangled linear chains. NP motion in entangled nonconcatenated ring polymers is understood through a scaling analysis of the coupling between NP motion and the self-similar entangled dynamics of ring polymers.

Macromolecular topology and rheology: beyond the tube model

The motion of entangled polymers is marked by their ability to make conformational adjustments, which is mediated by their free ends. A credible account of the basic features of entanglement release in linear and nonlinear rheology is offered by the tube model which, despite its limitations and shortcomings, is considered as the “standard” model in the field, accounting for the dynamics of linear and branched polymers with homogeneous monomer density. Here, we challenge the two central elements of the molecular picture of entanglements by exploiting the consequences of absence of free ends and monomer density distribution. Non-concatenated ring polymers of high molar mass do not form an entanglement network with plateau modulus, but instead relax stress self-similarly, while they deform much less than their linear counterparts in nonlinear shear flow. Their rheology is extremely sensitive to the presence of unlinked linear chains. On the other hand, star polymers with many arms have a dual nature: polymeric, which governs arm relaxation, and colloidal, which controls their subsequent center-of-mass motion and completes the stress relaxation process. Appropriate choice of number and size of arms allows to tune their dynamic and structural properties, and therefore bridge the gap between polymers and colloids. These examples demonstrate a different manifestation of topological interactions, with distinct linear and nonlinear rheology, which cannot be described in full by the tube model. They also provide an avenue for taking advantage of macromolecular architecture in order to engineer the rheology of polymeric structures and soft composites. Still, a number of outstanding issues remain and we outline some perspectives in this exciting field of molecular rheology.

Self-Similar Conformations and Dynamics in Entangled Melts and Solutions of Nonconcatenated Ring Polymers.

A scaling model of self-similar conformations and dynamics of nonconcatenated entangled ring polymers is developed. Topological constraints force these ring polymers into compact conformations with fractal dimension df = 3 that we call fractal loopy globules (FLGs). This result is based on the conjecture that the overlap parameter of subsections of rings on all length scales is the same and equal to the Kavassalis-Noolandi number OKN ≈ 10-20. The dynamics of entangled rings is self-similar and proceeds as loops of increasing sizes are rearranged progressively at their respective diffusion times. The topological constraints associated with smaller rearranged loops affect the dynamics of larger loops through increasing the effective friction coefficient but have no influence on the entanglement tubes confining larger loops. As a result, the tube diameter defined as the average spacing between relevant topological constraints increases with time t, leading to "tube dilation". Analysis of the primitive paths in molecular dynamics simulations suggests a complete tube dilation with the tube diameter on the order of the time-dependent characteristic loop size. A characteristic loop at time t is defined as a ring section that has diffused a distance equal to its size during time t. We derive dynamic scaling exponents in terms of fractal dimensions of an entangled ring and the underlying primitive path and a parameter characterizing the extent of tube dilation. The results reproduce the predictions of different dynamic models of a single nonconcatenated entangled ring. We demonstrate that traditional generalization of single-ring models to multi-ring dynamics is not self-consistent and develop a FLG model with self-consistent multi-ring dynamics and complete tube dilation. This selfconsistent FLG model predicts that the longest relaxation time of nonconcatenated entangled ring polymers scales with their degree of polymerization N as τrelax ~ N7/3, while the diffusion coefficient of these rings scales as D3d ~ N-5/3. For the entangled solutions and melts of rings, we predict power law stress relaxation function G(t) ~ t-3/7 at t < τrelax without a rubbery plateau and the corresponding viscosity scaling with the degree of polymerization N as η ~ N4/3. These theoretical predictions are in good agreement with recent computer simulations and are consistent with experiments of melts of nonconcatenated entangled rings.

Multi-blob coarse graining for ring polymer solutions.

We present a multi-scale molecular modeling of concentrated solutions of unknotted and non-concatenated ring polymers under good solvent conditions. The approach is based on a multi-blob representation of each ring polymer, which is capable of overcoming the shortcomings of single-blob approaches that lose their validity at concentrations exceeding the overlap density of the solution [A. Narros, A. J. Moreno, and C. N. Likos, Soft Matter, 2010, 6, 2435]. By means of a first principles coarse-graining strategy based on analytically determined effective pair potentials between the blobs, computed at zero density, we quantitatively reproduce the single molecule and solution properties of a system with well-defined topological constraints. Detailed comparisons with the underlying, monomer-resolved model demonstrate the validity of our approach, which employs fully transferable pair potentials between connected and unconnected blobs. We demonstrate that the pair structure between the centers of mass of the rings is accurately reproduced by the multi-blob approach, thus opening the way for simulation of arbitrarily long polymers. Finally, we show the importance of the topological constraint of non-concatenation on the structure of the concentrated solution and in particular on the size of the correlation hole and the shrinkage of the rings as melt concentrations are approached.

Cluster Glasses of Semiflexible Ring Polymers.

We present computer simulations of concentrated solutions of unknotted nonconcatenated semiflexible ring polymers. Unlike in their flexible counterparts, shrinking involves a strong energetic penalty, favoring interpenetration and clustering of the rings. We investigate the slow dynamics of the centers-of-mass of the rings in the amorphous cluster phase, consisting of disordered columns of oblate rings penetrated by bundles of prolate ones. Scattering functions reveal a striking decoupling of self- and collective motions. Correlations between centers-of-mass exhibit slow relaxation, as expected for an incipient glass transition, indicating the dynamic arrest of the cluster positions. However, self-correlations decay at much shorter time scales. This feature is a manifestation of the fast, continuous exchange and diffusion of the individual rings over the matrix of clusters. Our results reveal a novel scenario of glass formation in a simple monodisperse system, characterized by self-collective decoupling, soft caging, and mild dynamic heterogeneity.

Response to “Comment on ‘Molecular dynamics simulation study of nonconcatenated ring polymers in a melt. I. Statics”' [J. Chem. Phys. 139, 217101 (2013)

Response to “Comment on ‘Molecular dynamics simulation study of nonconcatenated ring polymers in a melt. I. Statics”' [J. Chem. Phys. 139, 217101 (2013)

Comparing the results of lattice and off-lattice simulations for the melt of nonconcatenated rings

To study the conformational properties of unknotted and nonconcatenated ring polymers in the melt, we present a detailed qualitative and quantitative comparison of simulation data obtained by molecular dynamics simulation using an off-lattice bead-spring model and by Monte Carlo simulation using a lattice model. We observe excellent, and sometimes even unexpectedly good, agreement between the off-lattice and lattice results for many quantities measured including the gyration radii of the ring polymers, gyration radii of their subchains, contact probabilities, surface characteristics, number of contacts between subchains, and the static structure factors of the rings and their subchains. These results are, in part, put in contrast to Moore curves, and the open, linear polymer counterparts. While our analysis is extensive, our understanding of the ring melt conformations is still rather preliminary.

Molecular dynamics simulation study of nonconcatenated ring polymers in a melt. I. Statics

Molecular dynamics simulations were conducted to investigate the structural properties of melts of nonconcatenated ring polymers and compared to melts of linear polymers. The longest rings were composed of N = 1600 monomers per chain which corresponds to roughly 57 entanglement lengths for comparable linear polymers. For the rings, the radius of gyration squared, [linear span]R(g)(2)[linear span], was found to scale as N(4/5) for an intermediate regime and N(2/3) for the larger rings indicating an overall conformation of a crumpled globule. However, almost all beads of the rings are "surface beads" interacting with beads of other rings, a result also in agreement with a primitive path analysis performed in the next paper [J. D. Halverson, W. Lee, G. S. Grest, A. Y. Grosberg, and K. Kremer, J. Chem. Phys. 134, 204905 (2011)]. Details of the internal conformational properties of the ring and linear polymers as well as their packing are analyzed and compared to current theoretical models.

Molecular dynamics simulation study of nonconcatenated ring polymers in a melt. II. Dynamics

Molecular dynamics simulations were conducted to investigate the dynamic properties of melts of nonconcatenated ring polymers and compared to melts of linear polymers. The longest rings were composed of N = 1600 monomers per chain which corresponds to roughly 57 entanglement lengths for comparable linear polymers. The ring melts were found to diffuse faster than their linear counterparts, with both architectures approximately obeying a D ∼ N(-2.4) scaling law for large N. The mean-square displacement of the center-of-mass of the rings follows a sub-diffusive behavior for times and distances beyond the ring extension [linear span]R(g)(2)[linear span], neither compatible with the Rouse nor the reptation model. The rings relax stress much faster than linear polymers, and the zero-shear viscosity was found to vary as η(0) ∼ N(1.4 ± 0.2) which is much weaker than the N(3.4) behavior of linear chains, not matching any commonly known model for polymer dynamics when compared to the observed mean-square displacements. These findings are discussed in view of the conformational properties of the rings presented in the preceding paper [J. D. Halverson, W. Lee, G. S. Grest, A. Y. Grosberg, and K. Kremer, J. Chem. Phys. 134, 204904 (2011)].

GPU Based Molecular Dynamics Simulations of Polymer Rings in Concentrated Solution: Structure and Scaling

We report on equilibrium properties of a concentrated solution of non-concatenated ring polymers by Molecular dynamics simulations using HooMD-blue, a fast implementation on graphics processor units (GPUs). We are able to identify the intermediate scaling regime for the radius of gyration Rg ∝ N as well as indication for a crossover to Rg ∝ N for rings with chain length N in our fully flexible off-lattice polymer model. This crossover takes place between a ring size of 2500 and 7500 monomers for monomer density ρ = 0.5. Our results are in agreement with recent studies for lattice and stiff off-lattice models and show once again that this scaling is not model dependent at all. Furthermore this study demonstrates that GPUs are suited to tackle large scale problems in computational polymer science. To this end we present performance measurements relating the computing performance of a single NVidias GTX580 consumer graphic card to conventional state of the art supercomputers.

Self‐diffusion coefficient of ring polymers in semidilute solution

AbstractIn a topologically constraining environment the size of a flexible nonconcatenated ring polymer (macrocycles) and its dynamics are known to differ from that of linear polymers. Hence, the diffusion coefficient of ring polymers can be expected to be different from linear chains. We present here scaling arguments for the concentration and molecular weight dependence of self‐diffusion coefficient of ring polymers in semidilute solutions, and show that contrary to expectations these scaling relations are identical to what is known for linear polymers. At higher concentrations excluded volume interactions arising from possibilities of segmental overlap can become effective for large ring polymers. In this regime the diffusion coefficient of large ring polymers shows a relatively weaker dependence on concentration and molecular weight. ©2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 2370–2379, 2008