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: P 0(N) ∝ exp(-N/N 0). However, no such simple considerations can account for the dependence of the knotting length, N 0, 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.

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.

Correction: Polycatenanes: synthesis, characterization, and physical understanding.

Correction for 'Polycatenanes: synthesis, characterization, and physical understanding' by Guancen Liu et al., Chem. Soc. Rev., 2022, https://doi.org/10.1039/d2cs00256f.

Nonequilibrium statistical thermodynamics of multicomponent interfaces.

Nonequilibrium interfacial thermodynamics has important implications for crucial biological, physical, and industrial-scale transport processes. Here, we discuss a theory of local equilibrium for multiphase multicomponent interfaces that builds upon the "sharp" interface concept first introduced by Gibbs, allowing for a description of nonequilibrium interfacial processes such as those arising in evaporation, condensation, adsorption, etc. By requiring that the thermodynamics be insensitive to the precise location of the dividing surface, one can identify conditions for local equilibrium and develop methods for measuring the values of intensive variables at the interface. We then use extensive, high-precision nonequilibrium molecular dynamics (NEMD) simulations to verify the theory and establish the validity of the local equilibrium hypothesis. In particular, we demonstrate that equilibrium equations of state are also valid out of equilibrium, and can be used to determine interfacial temperature and chemical potential(s) that are consistent with nonequilibrium generalizations of the Clapeyron and Gibbs adsorption equations. We also show, for example, that, far from equilibrium, temperature or chemical potential differences need not be uniform across an interface and may instead exhibit pronounced discontinuities. However, even in these circumstances, we demonstrate that the local equilibrium hypothesis and its implications remain valid. These results provide a thermodynamic foundation and computational tools for studying or revisiting a wide variety of interfacial transport phenomena.

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.

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.

Dynamics of poly[n]catenane melts.

Inspired by advances in the chemical synthesis of interlocking polymer architectures, extensive molecular dynamics simulations have been conducted to study the dynamical properties of poly[n]catenanes-polymers composed entirely of interlocking rings-in the melt state. Both the degree of polymerization (number of links) and the number of beads per ring are systematically varied, and the results are compared to linear and ring polymers. A simple Rouse-like model is presented, and its analytical solution suggests a decomposition of the dynamics into "ring-like" and "linear-like" regimes at short and long times, respectively. In agreement with this picture, multiple sub-diffusive regimes are observed in the monomer mean-squared-displacements even though interchain entanglement is not prevalent in the system. However, the Rouse-type model does not account for the topological effects of the mechanical bonds, which significantly alter the dynamics at intermediate length scales both within the rings and at the chain segment scales. The stress relaxation in the system is extremely rapid and may be conveniently separated into ring-like and linear-like contributions, again in agreement with the Rouse picture. However, the viscosity has a non-monotonic dependence on the ring size for long chains, which disagrees strongly with theoretical predictions. This unexpected observation cannot be explained in terms of chain disentanglement and is inconsistent with other measures of polymer relaxation. Possible mechanisms for this behavior are proposed and implications for materials design are discussed.

Crossover from Rouse to Reptation Dynamics in Salt-Free Polyelectrolyte Complex Coacervates.

Considerable interest in the dynamics and rheology of polyelectrolyte complex coacervates has been motivated by their industrial application as viscosity modifiers. A central question is the extent to which classical Rouse and reptation models can be applied to systems where electrostatic interactions play a critical role on the thermodynamics. By relying on molecular simulations, we present a direct analysis of the crossover from Rouse to reptation dynamics in salt-free complex coacervates as a function of chain length. This crossover shifts to shorter chain lengths as electrostatic interactions become stronger, which corresponds to the formation of denser coacervates. To distinguish the roles of Coulomb interactions and density, we compare the dynamics of coacervates to those of neutral, semidilute solutions at the same density. Both systems exhibit a universal dynamical behavior in the connectivity-dominated (subdiffusion and normal diffusion) regimes, but the monomer relaxation time in coacervates is much longer and increases with increasing Bjerrum length. This is similar to the cage effect observed in glass-forming polymers, but the local dynamical slowdown is caused here by strong Coulomb attractions (ion pairing) between oppositely charged monomers. Our findings provide a microscopic framework for the quantitative understanding of coacervate dynamics and rheology.

Topological Effects in Isolated Poly[n]catenanes: Molecular Dynamics Simulations and Rouse Mode Analysis.

Poly[n]catenanes are mechanically interlocked polymers consisting of interlocking ring molecules. Over the years, researchers have speculated that the permanent topological interactions within the poly[n]catenane backbone could lead to unique dynamical behaviors. To investigate these unusual polymers, molecular dynamics simulations of isolated poly[n]catenanes have been conducted, along with a Rouse mode analysis. Owing to the mechanical bonds within the molecule, the dynamics of poly[n]catenanes at short length scales are significantly slowed and the distribution of relaxation times is broadened; these same behaviors have been observed in melts of linear polymers and are associated with entanglement. Despite these entanglement-like effects, at large length scales poly[n]catenanes do not relax much slower than isolated linear polymers and are less strongly impacted by increased segmental stiffness.

Poly[n]catenanes: Synthesis of molecular interlocked chains.

As the macromolecular version of mechanically interlocked molecules, mechanically interlocked polymers are promising candidates for the creation of sophisticated molecular machines and smart soft materials. Poly[n]catenanes, where the molecular chains consist solely of interlocked macrocycles, contain one of the highest concentrations of topological bonds. We report, herein, a synthetic approach toward this distinctive polymer architecture in high yield (~75%) via efficient ring closing of rationally designed metallosupramolecular polymers. Light-scattering, mass spectrometric, and nuclear magnetic resonance characterization of fractionated samples support assignment of the high-molar mass product (number-average molar mass ~21.4 kilograms per mole) to a mixture of linear poly[7-26]catenanes, branched poly[13-130]catenanes, and cyclic poly[4-7]catenanes. Increased hydrodynamic radius (in solution) and glass transition temperature (in bulk materials) were observed upon metallation with Zn2.