Monte Carlo simulation of a single ring among linear chains: Structural and dynamic heterogeneity

Chain Size and Knots of Ring Polymers in All-Crossing and Intra-Crossing Melts

Using dynamic Monte Carlo simulations based on the bond-fluctuation model, we systematically investigated the size and knots of ring polymers in all-crossing systems and intra-crossing systems. Our results demonstrate that the interchain constraint can increase the knotting probability, but does not alter the scaling relationship between knotting probability and chain length for ring polymers in melts. Having established that, we derived the interchain constraint contribution to the free energy of ring polymers in intra-crossing systems based on the knotting probability and obtained the scaling relationship between the size R and chain length N, i.e., R~N1/6. And, by calculating the mean-squared radius of gyration of ring polymers in intra-crossing systems, we validated these scaling results. Finally, we analyze the size of knotted ring polymers with different types and compare corresponding scaling exponents for size versus chain lengths of ring polymers with different knotting complexities. These results provide fundamental insights into the static properties of ring polymers in melts.

Impacts of External Electric Fields on Structures and Alignments of Ring Molecules

Ring molecules, which lack free ends, exhibit unique chemical and physical properties, making them promising candidates for nanodevice applications. Unlike their linear counterparts with two free ends, the behavior of ring molecules in water under external electric fields (EF) is not well understood. In this research, we employ molecular dynamics (MD) simulations to explore the structural and alignment behavior of two ring molecules of different sizes─C30H60 and C60H120─in water, under 300 K, 1 bar and various EF conditions, including direct current EF (DC EF), alternating current EF (AC EF), and circular polarized EF (CP EF) at different frequencies. Our findings reveal the following: (1) both large and small rings exhibit two free energy minima. For C60H120, these correspond to collapsed and stretched configurations, while for C30H60, they represent open and closed configurations. (2) The applied EF can regulate the depth of these free energy minima. For C60H120, no EF, AC EF, and high-frequency CP EF favor the collapsed state, while DC EF and low-frequency CP EF promote the stretched configuration. In the case of C30H60, no EF and high-frequency CP EF favor the open-ring state, whereas all other EF conditions tend to close the ring. (3) Both ring molecules align with the directional EF to minimize disruption of the hydrogen-bond network, with C60H120 showing a stronger alignment effect than C30H60 due to its longer structure. (4) Under CP EF, ring molecules exhibit rotation driven by the rotating EF, but there is a lag in the angle between the EF vector and the molecule's elongation. Higher frequency CP EF shows less ability to capture and align the molecule. This research enhances our understanding of how ring molecules behave in water under external EF and provides a theoretical foundation for future engineering applications involving controlled manipulation of these molecules.

Shaping membrane vesicles by tuning the activity of confined active polymer chains

Semi-flexible polymers, such as actin filaments, can deform the shape of membrane when confined in a membrane vesicle, playing an important role in biological processes. Here, we use dynamic Monte Carlo simulations to study an active polymer chain confined in a membrane vesicle. For flexible polymer chains, the membrane shape is governed by the competition between membrane bending rigidity and polymer activity. Stiff membrane is unaffected by small active forces, but moderate forces cause the polymer to alternate between stretched and disordered configurations, increasing the asphericity of both the polymer and the vesicle. For semi-flexible polymer chains, their stiffness can significantly impact both the vesicle and polymer shapes. We identify distinct classes of configurations that emerge as a function of polymer stiffness, membrane bending rigidity, and polymer activity. A weak polymer activity can cause the polymer to align along its contour, effectively increasing its stiffness. However, a moderate polymer activity softens the polymer chain. For membranes with low bending rigidities κ, large-scale deformations, such as wormlike or tadpole-shaped vesicles, appear at a weak polymer activity and high polymer stiffness. In the wormlike configuration, the polymer chain adopts a hairpin configuration to minimize the polymer bending energy. As the polymer stiffness increases, a tadpole-like vesicle forms, with part of the polymer deforming the membrane into a protrusion while the rest remaining confined in a bud-like structure. For stiffer membranes, we observe oblate vesicles containing toroidal polymer chains, resulting from the high cost of membrane bending energy. A moderate polymer activity causes the softening of the polymer chain, leading to a nearly spherical vesicle with slight shape fluctuation. We further characterize the order parameter of toroidal polymer chains in oblate vesicles and reveal that a slight increase in polymer activity leads to a more ordered helical structure of polymer chains.

High-Performance Flexible Sulfur Cathodes with Robust Electrode Skeletons Built by a Hierarchical Self-Assembling Slurry

The electrochemical performance of lithium-sulfur batteries is fundamentally determined by the structural and mechanical stability of their composite sulfur cathodes. However, the development of cost-effective strategies for realizing robust hierarchical composite electrode structures remains highly challenging due to uncontrollable interactions among the components. The present work addresses this issue by proposing a type of self-assembling electrode slurry based on a well-designed two-component (polyacrylonitrile and polyvinylpyrrolidone) polar binder system with carbon nanotubes that forms hierarchical porous structures via optimized water-vapor-induced phase separation. The electrode skeleton is a highly robust and flexible electron-conductive network, and the porous structure provides hierarchical ion-transport channels with strong polysulfide trapping capability. Composite sulfur cathodes prepared with a sulfur loading of 4.53 mg cm-2 realize a very stable specific capacity of 485 mAh g-1 at a current density of 3.74 mA cm-2 after 1000 cycles. Meanwhile, a composite sulfur cathode with a high sulfur loading of 14.5 mg cm-2 in a lithium-sulfur pouch cell provides good flexibility and delivers a high capacity of 600 mAh g-1 at a current density of 0.72 mA cm-2 for 78 cycles.

Melt rheology of tadpole-shaped polystyrenes with different ring sizes

In this study, linear melt rheology of a single-tail tadpole-shaped polystyrene, ST-30/80, having ring and linear sizes of MR ∼ 30 kg mol-1 and ML ∼ 80 kg mol-1, respectively, was examined, and the effect of the ring size on rheological properties of tadpole polymers was discussed by comparing with the data of the previously reported tadpole samples having MR ∼ 60 kg mol-1. ST-30/80 exhibits an entanglement plateau and shows a clearly slower terminal relaxation than that of its component ring and linear polymers. When the zero-shear viscosity η0 for ST-30/80 is plotted against the molecular weight of a linear tail chain, the data point lies on the single curve of η0 for 4- and 6-arm star polymers and the single-tail tadpoles with MR ∼ 60 kg mol-1. These results suggest that the tadpole molecule in this study spontaneously forms a characteristic entanglement network, i.e., the intermolecular ring-linear threading, in the same manner as the previous tadpole samples, even though the size of the ring part is just slightly larger than the entanglement molecular weight (i.e., MR ∼ 1.8Me).

Dynamic Heterogeneity in Ring-Linear Polymer Blends

We present results from a direct statistical analysis of long molecular dynamics (MD) trajectories for the orientational relaxation of individual ring molecules in blends with equivalent linear chains. Our analysis reveals a very broad distribution of ring relaxation times whose width increases with increasing ring/linear molecular length and increasing concentration of the blend in linear chains. Dynamic heterogeneity is also observed in the pure ring melts but to a lesser extent. The enhanced degree of dynamic heterogeneity in the blends arises from the substantial increase in the intrinsic timescales of a large subpopulation of ring molecules due to their involvement in strong threading events with a certain population of the linear chains present in the blend. Our analysis suggests that the relaxation dynamics of the rings are controlled by the different states of their threading by linear chains. Unthreaded or singly-threaded rings exhibit terminal relaxation very similar to that in their own melt, but multiply-threaded rings relax much slower due to the long lifetimes of the corresponding topological interactions. By further analyzing the MD data for ring molecule terminal relaxation in terms of the sum of simple exponential functions we have been able to quantify the characteristic relaxation times of the corresponding mechanisms contributing to ring relaxation both in their pure melts and in the blends, and their relative importance. The extra contribution due to ring-linear threadings in the blends becomes immediately apparent through such an analysis.

Unexpected entanglement dynamics in semidilute blends of supercoiled and ring DNA

Blends of polymers of different topologies, such as ring and supercoiled, naturally occur in biology and often exhibit emergent viscoelastic properties coveted in industry. However, due to their complexity, along with the difficulty of producing polymers of different topologies, the dynamics of topological polymer blends remains poorly understood. We address this void by using both passive and active microrheology to characterize the linear and nonlinear rheological properties of blends of relaxed circular and supercoiled DNA. We characterize the dynamics as we vary the concentration from below the overlap concentration c* to above (0.5c* to 2c*). Surprisingly, despite working at the dilute-semidilute crossover, entanglement dynamics, such as elastic plateaus and multiple relaxation modes, emerge. Finally, blends exhibit an unexpected sustained elastic response to nonlinear strains not previously observed even in well-entangled linear polymer solutions.

Topology-dependent anomalous dynamics of ring and linear DNA are sensitive to cytoskeleton crosslinking

Cytoskeletal crowding plays a key role in the diffusion of DNA molecules through the cell, acting as a barrier to effective intracellular transport and conformational stability required for processes such as transfection, viral infection, and gene therapy. Here, we elucidate the transport properties and conformational dynamics of linear and ring DNA molecules diffusing through entangled and crosslinked composite networks of actin and microtubules. We couple single-molecule conformational tracking with differential dynamic microscopy to reveal that ring and linear DNA exhibit unexpectedly distinct transport properties that are influenced differently by cytoskeleton crosslinking. Ring DNA coils are swollen and undergo heterogeneous and biphasic subdiffusion that is hindered by crosslinking. Conversely, crosslinking actually facilitates the single-mode subdiffusion that compacted linear chains exhibit. Our collective results demonstrate that transient threading by cytoskeleton filaments plays a key role in the dynamics of ring DNA, whereas the mobility of the cytoskeleton dictates transport of linear DNA.

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.

Entangled polymer dynamics beyond reptation

Macroscopic properties of polymers arise from microscopic entanglement of polymer chains. Entangled polymer dynamics have been described theoretically by time- and space-averaged relaxation modes of single chains occurring at different time and length scales. However, theoretical and experimental studies along this framework provide oversimplified picture of spatiotemporally heterogeneous polymer dynamics. Characterization of entangled polymer dynamics beyond this paradigm requires a method that enables to capture motion and relaxation occurring in real space at different length and time scales. Here we develop new single-molecule characterization platform by combining super-resolution fluorescence imaging and recently developed single-molecule tracking method, cumulative-area tracking, which enables to quantify the chain motion in the length and time scale of nanometres to micrometres and milliseconds to minutes. Using linear and cyclic dsDNA molecules as model systems, our new method reveals chain-position-dependent motion of the entangled linear chains, which is beyond the scope of current theoretical framework.

Shaping membrane vesicles by adsorption of a semiflexible polymer

The adsorption of polymers onto fluid membranes is a problem of fundamental interest in biology and soft materials, in part because the flexibility of membranes can lead to nontrivial coupling between polymer and membrane configurations. Here, we use Monte Carlo computer simulations to study the adsorption of a semiflexible polymer onto a fluid membrane vesicle. Polymer adsorption can significantly impact both the vesicle and polymer shapes, and we identify distinct classes of configurations that emerge as a function of polymer persistence length, membrane bending rigidity, adsorption strength, and vesicle size. Large-scale deformations of the vesicle include invaginations of the membrane that internalize the polymer in a membrane bud. The buds range from disk-like shapes surrounding a collapsed polymer to tubular deformations enveloping rod-like polymers. For small vesicles, polymer adsorption also induces dumbbell-like vesicle shapes with a narrow membrane constriction circled by the polymer. Vesicles with sufficiently small or large bending rigidities adopt configurations similar to those without the polymer present. We further characterize statistical properties of the membrane and polymer configurations and identify distinct classes of polymer configurations that emerge within membrane buds. Analysis of idealized polymer-membrane configurations provides additional insight into transitions between bud shapes.

Ring Polymers: Threadings, Knot Electrophoresis and Topological Glasses

Elucidating the physics of a concentrated suspension of ring polymers, or of an ensemble of ring polymers in a complex environment, is an important outstanding question in polymer physics. Many of the characteristic features of these systems arise due to topological interactions between polymers, or between the polymers and the environment, and it is often challenging to describe this quantitatively. Here we review recent research which suggests that a key role is played by inter-ring threadings (or penetrations), which become more abundant as the ring size increases. As we discuss, the physical consequences of such threadings are far-reaching: for instance, they lead to a topologically-driven glassy behaviour of ring polymer melts under pinning perturbations, while they can also account for the shape of experimentally observed patterns in two-dimensional gel electrophoresis of DNA knots.

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.

Single-molecule imaging reveals topological isomer-dependent diffusion by 4-armed star and dicyclic 8-shaped polymers

The diffusion modes of the fluorophore incorporated 4-armed star and dicyclic 8-shaped polymers were investigated at the molecular level by means of single-molecule fluorescence imaging.

Single-molecule study on polymer diffusion in a melt state: effect of chain topology

We report a new methodology for studying diffusion of individual polymer chains in a melt state, with special emphasis on the effect of chain topology. A perylene diimide fluorophore was incorporated into the linear and cyclic poly(THF)s, and real-time diffusion behavior of individual chains in a melt of linear poly(THF) was measured by means of a single-molecule fluorescence imaging technique. The combination of mean squared displacement (MSD) and cumulative distribution function (CDF) analysis demonstrated the broad distribution of diffusion coefficient of both the linear and cyclic polymer chains in the melt state. This indicates the presence of spatiotemporal heterogeneity of the polymer diffusion which occurs at much larger time and length scales than those expected from the current polymer physics theory. We further demonstrated that the cyclic chains showed marginally slower diffusion in comparison with the linear counterparts, to suggest the effective suppression of the translocation through the threading-entanglement with the linear matrix chains. This coincides with the higher activation energy for the diffusion of the cyclic chains than of the linear chains. These results suggest that the single-molecule imaging technique provides a powerful tool to analyze complicated polymer dynamics and contributes to the molecular level understanding of the chain interaction.

Density functional theory for inhomogeneous ring polymeric fluids

The modeling of ring polymers remains a challenge in classical density functional theory (DFT) due to the difficulty in solving the direct bond connectivity of the ring architecture without free ends. By considering the feature that all of the segments in a ring are equivalent, we give an algorithm to solve the integral of direct bond connectivity for ideal ring polymers, and therefore propose a DFT for inhomogeneous ring polymers, where the excess free energy functional is extended from an equation of state (EOS). This EOS exhibits better agreement than other EOSs for the compressibility factors, compared to Monte Carlo data. Importantly, the DFT satisfactorily reproduces the data of the configurational-bias Monte Carlo (CBMC) simulations for ring polymers. The local density profiles from the DFT show that the bead density of inhomogeneous ring fluids is independent of ring size, which is also confirmed by the CBMC simulations. Interestingly, the behavior of solvation force for ring polymers is quite similar to that of the polymers with infinite chain length.

Rheology of ring polymer melts: from linear contaminants to ring-linear blends

Ring polymers remain a challenge to our understanding of polymer dynamics. Experiments are difficult to interpret because of the uncertainty in the purity and dispersity of the sample. Using both equilibrium and nonequilibrium molecular dynamics simulations we have investigated the structure, dynamics, and rheology of perfectly controlled ring-linear polymer blends of chains of up to about 14 entanglements per chain, comparable to experimental systems. Linear contaminants increase the zero-shear viscosity of a ring polymer melt by about 10% around one-fifth of their overlap concentration. For equal concentrations of linear and ring polymers, the blend viscosity is about twice that of the pure linear melt. The diffusion coefficient of the rings decreases dramatically, while the linear polymers are mostly unaffected. Our results are supported by a primitive path analysis.