Semiflexible Ring Polymers Research Articles

Size-dependent nanoparticle dynamics in semiflexible ring polymer nanocomposites

By employing molecular dynamics simulations, we explored different size-dependent dynamics of NPs in semiflexible linear and ring polymer nanocomposites. Large NPs diffuse faster slightly in flexible ring polymer nanocomposites than in flexible linear polymer ones. However, the motions of large NPs in semiflexible ring polymer nanocomposites with large bending energies are very slow and NPs exhibit the glassy state in which the NPs are all caged by the neighbouring semiflexible ring polymers, while large NPs in semiflexible linear polymer nanocomposites transport much faster than those in semiflexible ring polymer ones because rod-like structures of semiflexible linear polymers can favour the motions of NPs. Meanwhile, the diffusion coefficients Dpolymer of semiflexible ring polymers decrease monotonously with increasing bending energy Kb, while the minimum diffusion coefficient for semiflexible linear polymers occurs at the moderate bending energy such as Kb = 6, in which the semiflexible linear polymers hold the maximum entanglement density. NP size, polymer topological constraint as well as chain stiffness affect the dynamics of NPs in polymer nanocomposites seriously, and this investigation can help us understand the complicated dynamical behaviours for NPs in polymer nanocomposites.

Ordered quasi-two-dimensional structure of nanoparticles in semiflexible ring polymer brushes under compression

Molecular dynamics simulations of a coarse-grained bead-spring model of ring polymer brushes under compression are presented. Flexible polymer brushes are always disordered during compression, whereas semiflexible polymer brushes tend to be ordered under sufficiently strong compression. Further, the polymer monomer density of the semiflexible polymer brush is very high near the brush surface, inducing a peak value of the free energy near the surface. Therefore, when nanoparticles are compressed in semiflexible ring polymer brushes, they tend to exhibit a closely packed single-layer structure between the brush surface and the impenetrable wall, and a quasi-two-dimensional ordered structure near the brush surface is formed under strong compression. These findings provide a new approach to designing responsive applications.

Glassy dynamics of nanoparticles in semiflexible ring polymer nanocomposite melts

By employing molecular dynamics simulations, we explore the dynamics of NPs in semiflexible ring polymer nanocomposite melts. A novel glass transition is observed for NPs in semiflexible ring polymer melts as the bending energy (Kb) of ring polymers increases. For NPs in flexible ring polymer melts (Kb = 0), NPs move in the classic diffusive behavior. However, for NPs in semiflexible ring polymer melts with large bending energy, NPs diffuse very slowly and exhibit the glassy state in which the NPs are all irreversibly caged be the neighbouring semiflexible ring polymers. This glass transition occurs well above the classical glass transition temperature at which microscopic mobility is lost, and the topological interactions of semiflexible ring polymers play an important role in this non-classical glass transition. This investigation can help us understand the nature of the glass transition in polymer systems.

Aggregation-Dispersion Transition for Nanoparticles in Semiflexible Ring Polymer Nanocomposite Melts.

By employing molecular dynamics simulations, we explored the conformation transition of nanoparticles (NPs) in semiflexible ring polymer nanocomposite melts. A novel aggregation-dispersion transition for NPs in ring polymer nanocomposites occurs when the bending energy of ring chains increases. The conformations of flexible ring chains near NPs are radial distribution, and the entropic depletion interactions between a pair of NPs in flexible ring polymer nanocomposite melts are attractive, however, the rod-like ring chains wrap around the NPs and the entropic depletion interactions between NPs in rod-like ring polymer melts are repulsive. The aggregation-dispersion transition for NPs induced by chain topology in polymer nanocomposites can provide a new access to achieve miscibility in producing high-performance polymer-nanoparticle composites by simply varying the topological structure of polymers.

SEMI-FLEXIBLE COMPACT POLYMERS IN A DISORDERED ENVIRONMENT

Hamiltonian cycles with bending rigidity are studied on the first three members of the fractal family obtained by generalization of the modified rectangular (MR) fractal lattice. This model is proposed to describe conformational and thermodynamic properties of a single semi-flexible ring polymer confined in a poor and disordered (e.g. crowded) solvent. Due to the competition between temperature and polymer stiffness, there is a possibility for the phase transition between molten globule and crystal phase of a polymer to occur. The partition function of the model in the thermodynamic limit is obtained and analyzed as a function of polymer stiffness parameter s (Boltzmann weight), which for semi-flexible polymers can take on values over the interval (0,1). Other quantities, such as persistence length, specific heat and entropy, are obtained numerically and presented graphically as functions of stiffness parameter s.

Mean-Square Radius of Gyration and Scattering Function of Semiflexible Ring Polymers of the Trefoil Knot.

A Monte Carlo study of the mean-square radius of gyration and scattering function with k the magnitude of the scattering vector for semiflexible ring polymers of the trefoil knot was conducted by the use of the discrete version of the Kratky–Porod (KP) wormlike ring model. The behavior of and as functions of the reduced contour length , defined as the total contour length L divided by the stiffness parameter , is clarified. A comparison is made of the results for the KP ring of the trefoil knot with those for the KP ring of the trivial knot and for the phantom KP ring without the topological constraints.

Anisotropic effective interactions and stack formation in mixtures of semiflexible ring polymers.

By means of extensive computer simulations, we investigate the formation of columnar structures (stacks) in concentrated solutions of semiflexible ring polymers. To characterize the stacks we employ an algorithm that identifies tube-like structures in the simulation cell. Stacks are found both in the real system and in the fluid of soft disks interacting through the effective anisotropic pair potential derived for the rings [P. Poier et al., Macromolecules, 2015, 48, 4983-4997]. Furthermore, we investigate binary mixtures of cluster-forming and non-cluster-forming rings. We find that monodispersity is not a requirement for stack formation. The latter is found for a broad range of mixture compositions, though the columns in the mixtures exhibit important differences to those observed in the monodisperse case. We extend the anisotropic effective model to mixtures. We show that it correctly predicts stack formation and constitutes a significant improvement with respect to the usual isotropic effective description based only on macromolecular centers-of-mass.

An Anisotropic Effective Model for the Simulation of Semiflexible Ring Polymers.

We derive and introduce anisotropic effective pair potentials to coarse-grain solutions of semiflexible ring polymers of various lengths. The system has been recently investigated by means of full monomer-resolved computer simulations, revealing a host of unusual features and structure formation, which, however, cannot be captured by a rotationally averaged effective pair potential between the rings’ centers of mass [BernabeiM.; Soft Matter2013, 9, 1287]. Our new coarse-graining strategy is to picture each ring as a soft, penetrable disk. We demonstrate that for the short- and intermediate-length rings the new model is quite capable of capturing the physics in a quantitative fashion, whereas for the largest rings, which resemble flexible ones, it fails at high densities. Our work opens the way for the physical justification of general, anisotropic penetrable interaction potentials.

A Monte Carlo study of the intrinsic viscosity of semiflexible ring polymers

The intrinsic viscosity [η] of the Kratky–Porod (KP) wormlike rings is evaluated by Monte Carlo simulations. The behavior of the ratio of [η] of the rings of the trivial knot ([η]t.k.) to that of the rings without the topological constraint ([η]mix) is examined as a function of the reduced contour length λL, where λ −1 is the stiffness parameter of the KP ring and L is the total contour length.

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.

Dilute solution properties of semiflexible star and ring polymers

Our recent theoretical and/or Monte Carlo (MC) studies of dilute solution properties of semiflexible stars and rings are briefly summarized. The theoretical results for the intrinsic viscosity [η] of the Kratky–Porod (KP) wormlike three- and four-arm stars are shown, and effects of chain stiffness on [η] of the stars are examined. A comparison of the results for [η] with those for the effective hydrodynamic radius and the second virial coefficient A2 in a good solvent was made for the semiflexible three-arm stars. It was found that [η] is the most suitable object of study to examine the effects of chain stiffness on average chain dimensions of the stars. As for the rings, the MC results for A2 of the ideal KP rings, which is related to the intermolecular topological interactions, are presented and then compared with the data in the literature for ring atactic polystyrene (a-PS) at Θ for large molecular weight M (1 × 104−6 × 105). Even for ring a-PS in such a range of M, the effects of chain stiffness were still remarkable. The effects of the intramolecular topological constraints on the mean-square radius of gyration and the scattering function of the KP rings are also discussed. Theoretical and/or Monte Carlo studies are made of dilute solution properties of semiflexible stars and rings based on the Kratky–Porod wormlike chain model. The behavior of the properties in the range of the crossover from the rigid limit to the random-coil one is clarified. It is shown that effects of chain stiffness affect largely the dilute solution behavior not only of linear polymers but also of stars and rings, and are still remarkable even for typical flexible polymers with large molecular weight (∼105).

Effects of Chain Stiffness on Conformational and Dynamical Properties of Individual Ring Polymers in Shear Flow

Individual semiflexible ring polymers in a steady shear flow are studied by the multiparticle collision dynamics method combined with molecular dynamics simulations. We report the effects of chain stiffness on the conformational, dynamical, and rheological properties of ring polymers. When the Weissenberg numbers are smaller than unity, the behavior of semiflexible ring polymers is consistent with that of flexible ones. For larger Weissenberg numbers, the effects of chain stiffness are observed. We find that the radius of gyration tensor elements, the orientation resistance parameter, and the alignment distribution functions show strong stiffness dependences. The scaling behavior of tumbling motion corresponding to large conformational changes is found independent of chain stiffness, while the scaling behavior of tank-treading motion corresponding to the motion of monomers moving along the contour of the chain exhibits a chain stiffness dependence in the crossover regime from a flexible to a rigid ring polymer. Chain rigidities are found to have negligible effects on the polymer shear viscosity. The simulations reveal the similarities and differences in the nonequilibrium behavior of flexible and semiflexible ring polymers.

The topological glass in ring polymers

We study the dynamics of concentrated, long, semi-flexible, unknotted and unlinked ring polymers embedded in a gel by Monte Carlo simulation of a coarse-grained model. This involves the ansatz that the rings compactify into a duplex structure where they can be modelled as linear polymers. The classical polymer glass transition involves a rapid loss of microscopic freedom within the polymer molecule as the temperature is reduced toward Tg. Here we are interested in temperatures well above Tg where the polymers retain high microscopic mobility. We analyse the slowing of stress relaxation originating from inter-ring penetrations (threadings). For long polymers an extended network of quasi-topological penetrations forms. The longest relaxation time appears to depend exponentially on the ring polymer contour length, reminiscent of the usual exponential slowing (e.g., with temperature) in classical glasses. Finally, we discuss how this represents a universality class for glassy dynamics.

Fluids of semiflexible ring polymers: effective potentials and clustering

We present a computational investigation of the structural properties of a fluid of semiflexible ring polymers. Stiffness is introduced by implementing intramolecular barriers. Because of these barriers, shrinkage of the rings is energetically unfavourable, and the ring size can exhibit a non-monotonic density dependence. At high concentrations the rings can swell and adopt open configurations that facilitate interpenetration and clustering. We obtain effective potentials between the centers-of-mass of the rings at infinite dilution, and explore their validity over the whole range of concentrations. Except for the limit of small rings, the effective fluid of ultrasoft particles provides a good description of the real system over a considerable range of densities, even above the overlap concentration. In particular the clustering behaviour predicted by the effective description is observed in the real system for a certain range of molecular masses. However, the effective description is incomplete. Inspection of the clusters of real rings reveals that these can arrange in a complex disordered phase formed by long columns of oblate rings, which are penetrated by bundles of elongated prolate rings. These anisotropic features of the real system are not captured by the standard effective approach, which only considers macromolecular centers-of-mass. This suggests the need to include the relative orientation between rings in the effective potentials.

Maximum entropy principle applied to semiflexible ring polymers

Based on the success of the maximum entropy principle (MEP) in the study of semiflexible treelike polymers [M. Dolgushev and A. Blumen, J. Chem. Phys. 131, 044905 (2009)], it is of much interest to establish MEP's potential for general semiflexible polymers which contain loops. Here, we embark on this endeavor by considering discrete semiflexible polymer rings in a Rouse-type scheme. Now, for treelike polymers a beads-and-bonds (i.e., a discrete) picture is essential for an easy inclusion of branching points. Moreover, one may envisage (similar to our former work [M. Dolgushev and A. Blumen, J. Chem. Phys. 131, 044905 (2009)]) to impose for each angle between two bonds a distinct stiffness condition. Working in this way leads already for a polymer ring to a complicated problem. Hence, we follow a reduced variational approach as applied earlier to polymer chains, in which a single Lagrange multiplier is used for each set of identical conditions imposed on topologically equivalent bonds and bonds' orientations. In this way, we obtain for the discrete ring an analytically closed form which involves Chebyshev polynomials. This expression turns out to lead to a series of solutions: Apart from the regular solution, several other solutions appear. One may be tempted to discard the other solutions, since for them the potential energy matrix is not positive definite. A more careful analysis based on topological features suggests, however, that such solutions can be assigned to rings displaying knots. Monte Carlo simulations which take excluded volume interactions into account agree with our interpretation.

Confinement driven spatial organization of semiflexible ring polymers: Implications for biopolymer packaging

We investigate the conformational properties of a semiflexible ring polymer in confined spaces. Taking into account the competing interplay between configurational entropy, bending energy and excluded volume, we elucidate the role that different geometrical constraints can play in shaping the spatial organization of biopolymers. While elongated, rod-like geometries reduce the amount of chain overcrossings and induce a pronounced ordering with respect to the long axis of the surrounding envelope, there exists no preferred orientational axis in the case of spherical confinement. Upon increasing the system density and the rigidity of the chain, the polymer migrates from the center of the accessible space towards the surrounding surface, forming a spool-like structure known for DNA condensation within viral capsids. The existence of distinguished loop sizes for different confining geometries might influence co-localization in biopolymers necessary for the genome-wide coordination of gene expression. Thus, the advantages of certain geometric constraints, such as spherical confinement of viral DNA in a capsid or the rod-shaped envelope of the circular chromosome in Escherichia coli could be one driving force for controlling proper biological functioning.

A Monte Carlo study of the second virial coefficient of semiflexible ring polymers

A Monte Carlo (MC) study was made of the second virial coefficient A2 of the ideal Kratky–Porod (KP) worm-like ring using a model composed of infinitely thin bonds with harmonic bending energy between successive bonds. Two kinds of statistical ensembles were generated: one composed of configurations of all kinds of knots with the Boltzmann weight, called the mixed ensemble, and the other composed of only those of the trivial knot, called the trivial-knot ensemble. The effective volume VE excluded to one ring by the presence of another, resulting only from a topological interaction, and also the mean-square radius of gyration 〈S2〉 were evaluated for each ensemble. The dimensionless quantity λVE/L2 proportional to A2 was found to be a function only of the reduced total contour length λL, as in the case of λ〈S2〉/L, where λ−1 is the stiffness parameter of the KP ring and L is its total contour length. The quantity λVE/L2 first increased and then decreased after passing through a maximum at λL≃5, as λL was increased. A comparison with literature data for ring atactic polystyrene in cyclohexane at Θ shows that the present MC results may qualitatively explain the behavior of the data. The effective volume VE excluded to a Kratky–Porod (KP) worm-like ring by the presence of another, resulting only from a topological interaction, was evaluated by Monte Carlo simulations. The quantity λVE/L2 proportional to the second virial coefficient A2 was shown to be a function only of λL with λ−1 the stiffness parameter of the KP ring and L its total contour length.

Excluded Volume Effects on Semiflexible Ring Polymers

Two-dimensional semiflexible polymer rings are studied both by imaging circular DNA adsorbed on a mica surface and by Monte Carlo simulations of phantom polymers as well as of polymers with finite thickness. Comparison of size and shape of the different models over the full range of flexibilities shows that excluded volume caused by finite thickness induces an anisotropic increase of the main axes of the conformations, a change of shape, accomplished by an enhanced correlation along the contour.

Opening of a weak link in a semiflexible ring polymer

The dynamics of contact formation between different parts of a long chain molecule is of considerable interest in biology. The related processes of opening of a loop or closing to form a loop also are of considerable interest and have attracted the attention of experimentalists/theorists. For closing, results are available in the completely flexible limit. However, this limit is not realized in many cases. Recently, there have been investigations for the semiflexible case too. We develop an approach, which leads to an easy description of the dynamics, incorporating semiflexibility rigorously into account. With this approach, the dynamics of a semiflexible polymer ring formed by a weak bond between the two ends can be modeled as the escape of a particle over a barrier in a multidimensional potential energy surface. We then calculate the rate of opening using a multidimensional transition state theory. Effects of friction on the rate are also taken into account using the standard coupling to a bath of harmonic oscillators. We find that for shorter chains (i.e., semiflexible), the rate of opening is strongly length dependent and is well described by the equation A(L/lp)v exp(Blp/L), with L as the length, and lp as the persistence length, A, B as the constants, and v approximately 1.2.

Barrier crossing of a semiflexible ring polymer.

Motivated by the dynamics of a membrane in response to an external force, we study the thermally activated crossing of a semiflexible ring polymer over a potential barrier. For the bistable potential of Kramers type smoothly varying over a long length scale, we calculate the crossing rate using the multidimensional Kramers' rate theory and the functional integral method. We find that, due to its conformational fluctuation, the rate for a flexible ring is much larger than that for a stiff ring. For a sufficiently long chain length or a sufficiently weak bending modulus, the ring undergoes a compact-stretch transition. The stretched conformation of the chain results in a decrease of the activation energy and so the further increase of the rate. This result implies that the soft matter conformational flexibility and adaptability facilitate the barrier crossing.