Dynamics of poly[n]catenane melts

Shape and size tunability of sheets of interlocked ring copolymers

Mechanically bonded membranes of interlocked ring polymers are a significant generalization of conventional elastic sheets, where connectivity is provided by covalent bonding, and represent a promising class of topological meta-materials. In this context, two open questions regard the large-scale reverberations of the heterogeneous composition of the rings and the inequivalent modes of interlocking neighboring rings. We address these questions with Langevin dynamics simulations of chainmails with honeycomb-lattice connectivity, where the rings are block copolymers with two segments of different rigidity. We considered various combinations of the relative lengths of the two segments and the patterns of the over- and under-passes linking neighboring rings. We find that varying ring composition and linking patterns have independent and complementary effects. While the former sets the overall size of the chainmail, the latter defines the shape, enabling the selection of starkly different conformation types. Notably, one of the considered linking patterns favors saddle-shaped membranes, providing a first example of spontaneous negative Gaussian curvature in mechanically bonded sheets. The results help establish the extent to which mechanically bonded membranes can differ from conventional elastic ones, particularly for the achievable shape and size tunability.

Polycatenanes Formed of Self-Assembled Metal-Organic Cages

Poly-[n]-catenanes (PCs) self-assembled of three-dimensional (3D) metal organic cages (MOCs) (hereafter referred to as PCs-MOCs) are a relatively new class of mechanically interlocked molecules (MIMs) that combine the properties of MOCs and polymers. The synthesis of PCs-MOCs is challenging because of the difficulties associated with interlocking MOCs, the occurrence of multiple weak supramolecular electrostatic interactions between cages, and the importance of solvent templating effects. The high density of mechanical bonds interlocking the MOCs endows the MOCs with mechanical and physical properties such as enhanced stability, responsive dynamic behavior and low solubility, which can unlock new functional properties. In this Minireview, we highlight the benefit of interlocking MOCs in the formation of PCs-MOCs structures as well as the synthetic approaches exploited in their preparation, from thermodynamic to kinetic methods, both in the solution and solid-states. Examples of PCs-MOCs self-assembled from various types of nanosized cages (i.e., tetrahedral, trigonal prismatic, octahedral and icosahedral) are described in this article, providing an overview of the research carried out in this area. The focus is on the structure-property relationship with examples of functional applications such as electron conductivity, X-ray attenuation, gas adsorption and molecular sensing. We believe that the structural and functional aspects of the reviewed PCs-MOCs will attract chemists in this research field with great potential as new functional materials in nanotechnological disciplines such as gas adsorption, sensing and photophysical properties such as X-ray attenuation or electron conductivity.

Unified understanding of the impact of semiflexibility, concentration, and molecular weight on macromolecular-scale ring diffusion

Conformationally fluctuating, globally compact macromolecules such as polymeric rings, single-chain nanoparticles, microgels, and many-arm stars display complex dynamic behaviors due to their rich topological structure and intermolecular organization. Synthetic rings are hybrid objects with conformations that display both ideal random walk and compact globular features, which can serve as models of genomic DNA. To date, emphasis has been placed on the effect of ring molecular weight on their unusual behaviors. Here, we combine simulations and a microscopic force-level theory to build a unified understanding for how key aspects of ring dynamics depend on different tunable molecular properties including backbone rigidity, monomer concentration, degree of traditional entanglement, and molecular weight. Our large-scale molecular dynamics simulations of ring melts with very different backbone stiffnesses reveal unanticipated behaviors which agree well with our generalized theory. This includes a universal master curve for center-of-mass diffusion constants as a function of molecular weight scaled by a chemistry and thermodynamic state-dependent critical molecular weight that generalizes the concept of an entanglement cross-over for linear chains. The key physics is how backbone rigidity and monomer concentration induced changes of the entanglement length, interring packing, degree of interpenetration, and liquid compressibility slow down space-time dynamic-force correlations on macromolecular scales. A power law decay of the center-of-mass diffusion constant with inverse molecular weight squared is the first consequence, followed by an ultraslow activated hopping transport regime. Our results set the stage to address slow dynamics and kinetic arrest in different families of compact synthetic and biological polymeric systems.

Unraveling the Influence of Topology and Spatial Confinement on Equilibrium and Relaxation Properties of Interlocked Ring Polymers

We use Langevin dynamics simulations to study linked ring polymers in channel confinement. We address the in- and out-of-equilibrium behavior of the systems for varying degrees of confinement and increasing topological and geometrical complexity of the interlocking. The main findings are three. First, metric observables of different link topologies collapse onto the same master curve when plotted against the crossing number, revealing a universal response to confinement. Second, the relaxation process from initially stretched states is faster for more complex links. We ascribe these properties to the interplay of several effects, including the dependence of topological friction on the link complexity. Finally, we show that transient forms of geometrical entanglement purposely added to the initial stressed state can leave distinctive signatures in force-spectroscopy curves. The insight provided by the findings could be leveraged in single-molecule nanochannel experiments to identify geometric entanglement within topologically linked rings.

Effects of Linking Topology on the Shear Response of Connected Ring Polymers: Catenanes and Bonded Rings Flow Differently

We perform computer simulations of mechanically linked (poly[2]catenanes, PC) and chemically bonded (bonded rings, BR) pairs of self-avoiding ring polymers in steady shear. We find that BRs develop a novel motif, termed gradient tumbling, rotating around the gradient axis. For the PCs the rings are stretched and display another new pattern, termed slip tumbling. The dynamics of BRs is continuous and oscillatory, whereas that of PCs is intermittent between slip-tumbling attempts. Our findings demonstrate the interplay between topology and hydrodynamics in dilute solutions of connected polymers.

Mixing Linear Polymers with Rings and Catenanes: Bulk and Interfacial Behavior

We derive and parameterize effective interaction potentials between a multitude of different types of ring polymers and linear chains, varying the bending rigidity and solvent quality for the former species. We further develop and apply a density functional treatment for mixtures of both disconnected (chain-ring) and connected (chain-polycatenane) mixtures of the same, drawing coexistence binodals and exploring the ensuing response functions as well as the interface and wetting behavior of the mixtures. We show that worsening of the solvent quality for the rings brings about a stronger propensity for macroscopic phase separation in the linear-polycatenane mixtures, which is predominantly of the demixing type between phases of similar overall particle density. We formulate a simple criterion based on the effective interactions, allowing us to determine whether any specific linear-ring mixture will undergo a demixing phase separation.

Spatial Organization of Slit-Confined Melts of Ring Polymers with Nonconserved Topology: A Lattice Monte Carlo Study

We present Monte Carlo computer simulations for melts of semiflexible randomly knotted and randomly concatenated ring polymers on the fcc lattice and in slit confinement. Through systematic variation of the slit width at fixed melt density, we explore the influence of confinement on single-chain conformations and interchain interactions. We demonstrate that confinement makes chains globally larger and more elongated while enhancing both contacts and knottedness propensities. As for multichain properties, we show that ring-ring contacts decrease with the confinement, yet neighboring rings overlap more as confinement grows. These aspects are accompanied by a marked decrease in the links formed between pairs of neighboring rings. In connection with the quantitative relation between links and entanglements in polymer melts recently established by us [Ubertini M. A.; Rosa A.Macromolecules2023, 56, 3354-3362], we propose that confinement can be used to set polymer networks that act softer under mechanical stress and suggest a viable experimental setup to validate our results.

Topological Entanglement of Linear Catenanes: Knots and Threadings

We used molecular dynamics simulations to investigate the self-entanglements of the collapsed linear catenanes. We found two different types of topologically complex states. First, we observed numerous long-lived knotting events of the catenane backbone. However, comparison with conventional polymers reveals that knots are suppressed in catenanes. Next, we observed topologically complex states with no analogue in polymers, where a concatenated ring was threaded by other near or distal rings sliding through it. Differently from knots, these threaded states can disentangle by becoming fully tightened. A detailed thermodynamic and microscopic analysis is employed to rationalize the persistence of threaded states, which can survive significant internal reorganizations of the entire catenane. We finally discuss the broader implications of these previously unreported types of entanglements for other systems, such as noncollapsed and interacting catenanes.

Unexpected Slow Relaxation Dynamics in Pure Ring Polymers Arise from Intermolecular Interactions

Ring polymers have fascinated scientists for decades, but experimental progress has been challenging due to the presence of linear chain contaminants that fundamentally alter dynamics. In this work, we report the unexpected slow stress relaxation behavior of concentrated ring polymers that arises due to ring-ring interactions and ring packing structure. Topologically pure, high molecular weight ring polymers are prepared without linear chain contaminants using cyclic poly(phthalaldehyde) (cPPA), a metastable polymer chemistry that rapidly depolymerizes from free ends at ambient temperatures. Linear viscoelastic measurements of highly concentrated cPPA show slow, non-power-law stress relaxation dynamics despite the lack of linear chain contaminants. Experiments are complemented by molecular dynamics (MD) simulations of unprecedentedly high molecular weight rings, which clearly show non-power-law stress relaxation in good agreement with experiments. MD simulations reveal substantial ring-ring interpenetrations upon increasing ring molecular weight or local backbone stiffness, despite the global collapsed nature of single ring conformation. A recently proposed microscopic theory for unconcatenated rings provides a qualitative physical mechanism associated with the emergence of strong inter-ring caging which slows down center-of-mass diffusion and long wavelength intramolecular relaxation modes originating from ring-ring interpenetrations, governed by the onset variable N/ND, where the crossover degree of polymerization ND is qualitatively predicted by theory. Our work overcomes challenges in achieving ring polymer purity and by characterizing dynamics for high molecular weight ring polymers. Overall, these results provide a new understanding of ring polymer physics.

Structure and Thermodynamics of Linear, Ring, and Catenane Polymers in Solutions and at Liquid-Liquid Interfaces

In recent decades, advances in the syntheses of mechanically interlocked macromolecules, such as catenanes, have led to much greater interest in the applications of these complexes, from molecular motors and actuators to nanoscale computational memory and nanoswitches. Much remains to be understood, however, regarding how catenated ring compounds behave as a result of the effects of different solvents as well as the effects of solvent/solvent interfaces. In this work, we have investigated, using molecular dynamics simulations, the effects of solvation of poly(ethylene oxide) chains of different topologies─linear, ring, and [2]catenane─in two solvents both considered favorable toward PEO (water, toluene) and at the water/toluene interface. Compared to ring and [2]catenane molecules, the linear PEO chain showed the largest increase in size at the water/toluene interface compared to bulk water or bulk toluene. Perhaps surprisingly, observations indicate that the tendency of all three topologies to extend at the water/toluene interface may have more to do with screening the interaction between the two solvents than with optimizing specific solvent-polymer contacts.

Kinetic trapping of 2,4,6-tris(4-pyridyl)benzene and ZnI2 into M12L8 poly-[n]-catenanes using solution and solid-state processes

Here, we show that in a supramolecular system with more than 20 building blocks forming large icosahedral M₁₂L₈ metal-organic cages (MOCs), using the instant synthesis method, it is possible to kinetically trap and control the formation of interlocking M₁₂L₈ nanocages, giving rare M₁₂L₈ TPB-ZnI₂ poly-[n]-catenane. The catenanes are obtained in a one-pot reaction, selectively as amorphous (a1) or crystalline states, as demonstrated by powder X-ray diffraction (powder XRD), thermogravimetric (TG) analysis and ¹H NMR. The 300 K M₁₂L₈ poly-[n]-catenane single crystal X-ray diffraction (SC-XRD) structure including nitrobenzene (1) indicates strong guest binding with the large M₁₂L₈ cage (i.e., internal volume ca. 2600 ų), allowing its structural resolution. Conversely, slow self-assembly (5 days) leads to a mixture of the M₁₂L₈ poly-[n]-catenane and a new TPB-ZnI₂ (2) coordination polymer (i.e., thermodynamic product), as revealed by SC-XRD. The neat grinding solid-state synthesis also yields amorphous M₁₂L₈ poly-[n]-catenane (a1'), but not coordination polymers, selectively in 15 min. The dynamic behavior of the M₁₂L₈ poly-[n]-catenanes demonstrated by the amorphous-to-crystalline transformation upon the uptake of ortho-, meta- and para-xylenes shows the potential of M₁₂L₈ poly-[n]-catenanes as functional materials in molecular separation. Finally, combining SC-XRD of 1 and DFT calculations specific for the solid-state, the role of the guests in the stability of the 1D chains of M₁₂L₈ nanocages is reported. Energy interactions such as interaction energies (E), lattice energies (E*), host-guest energies (E_host-guest) and guest-guest energies (E_guest-guest) were analysed considering the X-ray structure with and without the nitrobenzene guest. Not only the synthetic control achieved in the synthesis of the M₁₂L₈ MOCs but also their dynamic behavior either in the crystalline or amorphous phase are sufficient to raise scientific interest in areas ranging from fundamental to applied sides of chemistry and material sciences.

The effect of thread-like monomer structure on the synthesis of poly[n]catenanes from metallosupramolecular polymers

The main-chain poly[n]catenane consists of a series of interlocked rings that resemble a macroscopic chain-link structure. Recently, the synthesis of such intriguing polymers was reported via a metallosupramolecular polymer (MSP) template that consists of alternating units of macrocyclic and linear thread-like monomers. Ring closure of the thread components has been shown to yield a mixture of cyclic, linear, and branched poly[n]catenanes. Reported herein are studies aimed at accessing new poly[n]catenanes via this approach and exploring the effect the thread-like monomer structure has on the poly[n]catenane synthesis. Specifically, the effect of the size of the aromatic linker and alkenyl chains of the thread-like monomer is investigated. Three new poly[n]catenanes (with different ring sizes) were prepared using the MSP approach and the results show that tailoring the structure of the thread-like monomer can allow the selective synthesis of branched poly[n]catenanes.

Circular Polycatenanes: Supramolecular Structures with Topologically Tunable Properties

Polycatenanes, macrochains of topologically interlocked rings with unique physical properties have recently gained considerable interest in supramolecular chemistry, biology, and soft matter. Most of the work has been, so far, focused on linear chains and on their variety of conformational properties compared to standard polymers. Here we go beyond the linear case and show that, by circularizing such macrochains, one can exploit the topology of the local interlockings to store twist in the system, significantly altering its metric and local properties. Moreover, by properly defining the twist (Tw) and writhe (Wr) of these macrorings we show the validity of a relation equivalent to the Călugăreanu-White-Fuller theorem Tw+Wr=const, originally proved for ribbonlike structures such as double stranded DNA. Our results suggest that circular polycatenanes with storable and tunable twist can form a new category of highly designable multiscale structures with potential applications in supramolecular chemistry and material science.

Random Knotting in Fractal Ring Polymers

Many ring polymer systems of physical and biological interest exhibit both pronounced topological effects and nontrivial self-similarity, but the relationship between these two phenomena has not yet been clearly established. Here, we use theory and simulation to formulate such a connection by studying a fundamental topological property—the random knotting probability—for ring polymers with varying fractal dimension, d_f. Using straightforward scaling arguments, we generalize a classic mathematical result, showing that the probability of a trivial knot decays exponentially with chain size, N, for all fractal dimensions: P₀(N) ∝ exp(−N/N₀). However, no such simple considerations can account for the dependence of the knotting length, N₀, on d_f, necessitating a more involved analytical calculation. This analysis reveals a complicated double-exponential dependence, which is well supported by numerical data. By contrast, functional forms typical of simple scaling theories fail to adequately describe the observations. These findings are equally valid for two-dimensional ring polymer systems, where “knotting” is defined as the intersection of any two segments.

Polycatenanes: synthesis, characterization, and physical understanding

Chemical composition and architecture are two key factors that control the physical and material properties of polymers. Some of the more unusual and intriguing polymer architectures are the polycatenanes, which are a class of polymers that contain mechanically interlocked rings. Since the development of high yielding synthetic routes to catenanes, there has been an interest in accessing their polymeric counterparts, primarily on account of the unique conformations and degrees of freedom offered by non-bonded interlocked rings. This has lead to the synthesis of a wide variety of polycatenane architectures and to studies aimed at developing structure-property relationships of these interesting materials. In this review, we provide an overview of the field of polycatenanes, exploring synthesis, architecture, properties, simulation, and modelling, with a specific focus on some of the more recent developments.

Metastable doubly threaded [3]rotaxanes with a large macrocycle

Ring size is a critically important parameter in many interlocked molecules as it directly impacts many of the unique molecular motions that they exhibit. Reported herein are studies using one of the largest macrocycles reported to date to synthesize doubly threaded [3]rotaxanes. A large ditopic 46 atom macrocycle containing two 2,6-bis(N-alkyl-benzimidazolyl)pyridine ligands has been used to synthesize several metastable doubly threaded [3]rotaxanes in high yield (65-75% isolated) via metal templating. Macrocycle and linear thread components were synthesized and self-assembled upon addition of iron(ii) ions to form the doubly threaded pseudo[3]rotaxanes that could be subsequently stoppered using azide-alkyne cycloaddition chemistry. Following demetallation with base, these doubly threaded [3]rotaxanes were fully characterized utilizing a variety of NMR spectroscopy, mass spectrometry, size-exclusion chromatography, and all-atom simulation techniques. Critical to the success of accessing a metastable [3]rotaxane with such a large macrocycle was the nature of the stopper group employed. By varying the size of the stopper group it was possible to access metastable [3]rotaxanes with stabilities in deuterated chloroform ranging from a half-life of <1 minute to ca. 6 months at room temperature potentially opening the door to interlocked materials with controllable degradation rates.

Sliding Dynamics of Ring Chains on Two Asymmetric/Symmetric Chains in a Simple Slide-Ring Gel

The sliding dynamics along two asymmetric/symmetric axial chains of ring chains linked by a linear chainis investigated using molecular dynamics (MD) simulations. A novel sub-diffusion behavior is observed for ring chains sliding along eithera fixed rod-like chain or fluctuating axial chain on asymmetric/symmetric axial chainsat the intermediate time range due to their strongly interplay between two ring chains. However, two ring chains slide in the normal diffusion at along time range because their sliding dynamics can be regarded as an overall motion of two ring chains. For ring chains sliding on two symmetric/asymmetricaxial chains, the diffusion coefficient D of ring chains relies on the bending energy of axial chains (Kb) as well as the distance of two axial chains (d). There exists a maximum diffusion coefficient Dmax at d = d* in which ring chains slide at the fastest velocity due to the maximum conformational entropy for the linking chain between two ring chainsat d = d*. Ring chain slide on fixed rod-like axial chainsfaster in the symmetric axial chain case than that in the asymmetric axial chain case. However, ring chains slide on fluctuatingaxial chainsslower in the symmetric axial chain case than that in the asymmetric axial chain case. This investigation can provide insights into the effects of the linked chain conformation on the sliding dynamics of ring chains in a slide-ring gel.

Flatness and intrinsic curvature of linked-ring membranes

Recent experiments have elucidated the physical properties of kinetoplasts, which are chain-mail-like structures found in the mitochondria of trypanosome parasites formed from catenated DNA rings. Inspired by these studies, we use Monte Carlo simulations to examine the behavior of two-dimensional networks ("membranes") of linked rings. For simplicity, we consider only identical rings that are circular and rigid and that form networks with a regular linking structure. We find that the scaling of the eigenvalues of the shape tensor with membrane size are consistent with the behavior of the flat phase observed in self-avoiding covalent membranes. Increasing ring thickness tends to swell the membrane. Remarkably, unlike covalent membranes, the linked-ring membranes tend to form concave structures with an intrinsic curvature of entropic origin associated with local excluded-volume interactions. The degree of concavity increases with increasing ring thickness and is also affected by the type of linking network. The relevance of the properties of linked-ring model membranes to those observed in kinetoplasts is discussed.

Effects of chain length on Rouse modes and non-Gaussianity in linear and ring polymer melts

The dynamics of ring polymer melts are studied via molecular dynamics simulations of the Kremer-Grest bead-spring model. Rouse mode analysis is performed in comparison with linear polymers by changing the chain length. Rouse-like behavior is observed in ring polymers by quantifying the chain length dependence of the Rouse relaxation time, whereas a crossover from Rouse to reptation behavior is observed in linear polymers. Furthermore, the non-Gaussian parameters of the monomer bead displacement and chain center-of-mass displacement are analyzed. It is found that the non-Gaussianity of ring polymers is remarkably suppressed with slight growth for the center-of-mass dynamics at long chain length, which is in contrast to the growth in linear polymers for both the monomer bead and center-of-mass dynamics.

Double Asymptotic Structures of Topologically Interlocked Molecules

The mean square size of topologically interlocked molecules (TIMs) is presented as a linear combination of contributions from the backbone and subcomponents. Using scaling analyses and extensive molecular dynamics simulations of polycatenanes as a typical example of TIMs, we show that the effective exponent ν(m) for the size dependence of the backbone on the monomer number of subcomponent m is asymptotic to a value ν (∼0.588 in good solvents) with a correction of m^-0.47, which is the same as for the covalently linked polymer. However, the effective exponent for the size dependence of subcomponents on m is asymptotic to the same value ν but with a new correction of m^-1.0. The different corrections to the scaling on the backbone and subcomponent structure induce a surprising double asymptotic behavior for the architecture of the TIMs. The scaling model that takes into account the double asymptotic behavior is in good quantitative agreement with the simulation result that the effective exponent for the size dependence of TIMs on m increases with the subcomponent number n. The full scaling functional form of the size dependence on m and n for polycatenanes in a good solvent is well described by a simple sum of two limiting behaviors with different corrections.

Effect of metallosupramolecular polymer concentration on the synthesis of poly[n]catenanes

Poly[n]catenanes are a class of polymers that are composed entirely of interlocked rings. One synthetic route to these polymers involves the formation of a metallosupramolecular polymer (MSP) that consists of alternating units of macrocyclic and linear thread components. Ring closure of the thread components has been shown to yield a mixture of cyclic, linear, and branched poly[n]catenanes. Reported herein are investigations into this synthetic methodology, with a focus on a more detailed understanding of the crude product distribution and how the concentration of the MSP during the ring closing reaction impacts the resulting poly[n]catenanes. In addition to a better understanding of the molecular products obtained in these reactions, the results show that the concentration of the reaction can be used to tune the size and type of poly[n]catenanes accessed. At low concentrations the interlocked product distribution is limited to primarily oligomeric and small cyclic catenanes . However, the same reaction at increased concentration can yield branched poly[n]catenanes with an ca. 21 kg mol-1, with evidence of structures containing as many as 640 interlocked rings (1000 kg mol-1).

Sliding dynamics of multi-rings on a semiflexible polymer in poly[n]catenanes

The sliding dynamics of one- or multi-ring structures along a semiflexible cyclic polymer in radial poly[n]catenanes is investigated using molecular dynamics simulations. The fixed and fluctuating (non-fixed) semiflexible central cyclic polymers are considered, respectively. With increasing bending energy of the central cyclic polymer, for the fixed case, the diffusion coefficient increases monotonically due to the reduction of the tortuous sliding path, while for the fluctuating case, the diffusion coefficient decreases. This indicates that the contribution of the polymer fluctuation is suppressed by a further increase in the stiffness of the central cyclic chain. Compared with the one ring case, the mean-square displacement of the multiple rings exhibits a unique sub-diffusive behavior at intermediate time scales due to the repulsion between two neighboring rings. In addition, for the multi-ring system, the whole set of rings exhibit relatively slower diffusion, but faster local dynamics of threading rings and rotational diffusion of the central cyclic polymer arise. These results may help us to understand the diffusion motion of rings in radial poly[n]catenanes from a fundamental point of view and control the sliding dynamics in molecular designs.

Hydrodynamic interactions in topologically linked ring polymers

Despite decades of interdisciplinary research on topologically linked ring polymers, their dynamics remain largely unstudied. These systems represent a major scientific challenge as they are often subject to both topological and hydrodynamic interactions (HI), which render dynamical solutions either mathematically intractable or computationally prohibitive. Here we circumvent these limitations by preaveraging the HI of linked rings. We show that the symmetry of ring polymers leads to a hydrodynamic decoupling of ring dynamics. This decoupling is valid even for nonideal polymers and nonequilibrium conditions. Physically, our findings suggest that the effects of topology and HI are nearly independent and do not act cooperatively to influence polymer dynamics. We use this result to develop highly efficient Brownian dynamics algorithms that offer enormous performance improvements over conventional methods and apply these algorithms to simulate catenated ring polymers at equilibrium, confirming the independence of topological effects and HI. The methods developed here can be used to study and simulate large systems of linked rings with HI, including kinetoplast DNA, Olympic gels, and poly[n]catenanes.