Semiflexible Ring Research Articles

Linking of Ring Polymers in Slit-Like Confinement

Stochastic simulations are used to study the linking properties of solutions of circular polymers in slit confinement. Specifically, we consider dispersions of semiflexible rings at various densities ϕ and slit height, H. The competing length scales in the system have significant effects on the interchain entanglement. We observe that the linking probability is largest for a specific slit height that is about independent of solution density. However, when ϕ is large, links with given number of components can significantly depart from the overall linking trend with H. In this case, binary links are found to be least probable when the overall incidence of links is maximum. We show that this intriguing dichotomy and other properties, including upper bounds on links abundance, can be quantitatively captured with an approximate model based on continuum percolation theory. Our results suggest that slit-like confinement could be used in applicative contexts to control independently both the average abundance and...

Non-monotonic knotting probability and knot length of semiflexible rings: the competing roles of entropy and bending energy.

We consider self-avoiding rings of up to 1000 beads and study, by Monte Carlo techniques, how their equilibrium knotting properties depend on the bending rigidity. When the rings are taken from the rigid to fully-flexible limit, their average compactness increases, as expected. However, this progressive compactification is not parallelled by a steady increase of the abundance of knots. In fact the knotting probability, Pk, has a prominent maximum when the persistence length is a few times larger than the bead size. At similar bending rigidities, the knot length has, instead, a minimum. We show that the observed non-monotonicity of Pk arises from the competition between two effects. The first one is the entropic cost of introducing a knot. The second one is the gain in bending energy due to the presence of essential crossings. These, in fact, constrain the knotted region and keep it less bent than average. The two competing effects make knots maximally abundant when the persistence length is 5-10 times larger than the bead size. At such intermediate bending rigidities, knots in the chains of 500 and 1000 beads are 40 times more likely than in the fully-flexible limit.

Coil-helix-globule transition for self-attractive semiflexible ring chains

An off-lattice dynamics Monte Carlo method is used to study the conformations of self-attractive semiflexible ring chains. The conformations depend mainly on the bending energy b and the self-attractive interaction ε of ring chains, and perfect helical structures are found in self-attractive semiflexible ring chains with the moderate attractive interaction ε. Some monomers of the semiflexible ring chains wrap around the linearly aligned monomers and the size of helix is independent of chain length of ring chains. Coil-helix-globule transition occurs when the self-attractive interaction increases within semiflexibe ring chains, and the phase transition is attributed to the competition among the configurational entropy, the bending energy, and the self-attractive interaction. This study can help us understand the effects of topological constraints on the conformations and dynamical behaviors of polymer chains, and the importance of chain stiffness in biological systems and in the dynamics of DNA and protein folding.

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.

Energy localization and shape transformations in semiflexible polymer rings.

Shape transformations in driven and damped molecular chains are considered. Closed chains of weakly coupled molecular subunits under the action of spatially homogeneous time-periodic external field are studied. The coupling between the internal excitations and the bending degrees of freedom of the chain modifies the local bending rigidity of the chain. In the absence of driving the array takes a circular shape. When the energy pumped into the system exceeds some critical value the chain undergoes a nonequilibrium phase transition: The circular shape of the aggregate becomes unstable and the chain takes the shape of an ellipse or, in general, of a polygon. The excitation energy distribution becomes spatially nonuniform: It localizes in such places where the chain is more flat. The weak interaction of the chain with a flat surface restricts the dynamics to a flat manifold.

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.

Gaussian semiflexible rings under angular and dihedral restrictions.

Semiflexible polymer rings whose bonds obey both angular and dihedral restrictions [M. Dolgushev and A. Blumen, J. Chem. Phys. 138, 204902 (2013)], are treated under exact closure constraints. This allows us to obtain semianalytic results for their dynamics, based on sets of Langevin equations. The dihedral restrictions clearly manifest themselves in the behavior of the mean-square monomer displacement. The determination of the equilibrium ring conformations shows that the dihedral constraints influence the ring curvature, leading to compact folded structures. The method for imposing such constraints in Gaussian systems is very general and it allows to account for heterogeneous (site-dependent) restrictions. We show it by considering rings in which one site differs from the others.

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).

Dynamics of a semiflexible polymer or polymer ring in shear flow

Polymers exposed to shear flow exhibit a remarkably rich tumbling dynamics. While rigid rods rotate on Jeffery orbits, a flexible polymer stretches and coils up during tumbling. Theoretical results show that in both of these asymptotic regimes the corresponding tumbling frequency f(c) in a linear shear flow of strength γ scales as a power law Wi(2/3) in the Weissenberg number Wi = γτ, where τ is a characteristic time of the polymer's relaxational dynamics. For a flexible polymer these theoretical results are well confirmed by a large body of experimental single molecule studies. However, for the intermediate semiflexible regime, especially relevant for cytoskeletal filaments like F-actin and microtubules, the situation is less clear. While recent experiments on single F-actin filaments are still interpreted within the classical Wi(2/3) scaling law, theoretical results indicated deviations from it. Here we perform extensive Brownian dynamics simulations to explore the tumbling dynamics of semiflexible polymers over a broad range of shear strength and the polymer's persistence length l(p). We find that the Weissenberg number alone does not suffice to fully characterize the tumbling dynamics, and the classical scaling law breaks down. Instead, both the polymer's stiffness and the shear rate are relevant control parameters. Based on our Brownian dynamics simulations we postulate that in the parameter range most relevant for cytoskeletal filaments there is a distinct scaling behavior with f(c) τ* = Wi(3/4)f(c)(x) with Wi = γτ* and the scaling variable x = (l(p)/L)(Wi)(-1/3); here τ* is the time the polymer's center of mass requires to diffuse its own contour length L. Comparing these results with experimental data on F-actin we find that the Wi(3/4) scaling law agrees quantitatively significantly better with the data than the classical Wi(2/3) law. Finally, we extend our results to single ring polymers in shear flow, and find similar results as for linear polymers with slightly different power laws.

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.

Multiscale Entanglement in Ring Polymers under Spherical Confinement

The interplay of geometrical and topological entanglement in semiflexible knotted polymer rings confined inside a spherical cavity is investigated by using advanced numerical methods. By using stringent and robust algorithms for locating knots, we characterize how the knot length l(k) depends on the ring contour length L(c) and the radius of the confining sphere R(c). In the no- and strong-confinement cases, we observe weak knot localization and complete knot delocalization, respectively. We show that the complex interplay of l(k), L(c), and R(c) that seamlessly bridges these two limits can be encompassed by a simple scaling argument based on deflection theory. The same argument is used to rationalize the multiscale character of the entanglement that emerges with increasing confinement.

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.

Branched Semiflexible Polymers: Theoretical and Simulation Aspects

AbstractRecent approaches for taking stiffness effects into account when modeling multifunctional polymer structures are reported. The theoretical part is focused on arbitrary tree‐like polymers, modeled through extensions of the GGS scheme that allows the inclusion of local constraints. The theoretical findings are compared to the results of numerical simulations in which stiffness is included into the bond fluctuation model with the help of additional bending potentials. The procedure is illustrated by evaluating the static properties of semiflexible linear chains, stars, cospectral polymers and rings. The dynamical properties are exemplified by the mechanical relaxation of semiflexible polymers, including situations in which the local conditions at different sites are the same or different.magnified image