HPLC Characterization of Cyclization Reaction Product Obtained by End-to-End Ring Closure Reaction of a Telechelic Polystyrene

Metal-Organic Frameworks for Practical Separation of Cyclic and Linear Polymers

The purification step in the manufacturing of cyclic polymers is difficult as complete fractionation to eliminate linear impurities requires considerable effort. Here, we report a new polymer separation methodology that uses metal-organic frameworks (MOFs) to discriminate between linear and cyclic polyethylene glycols (PEGs) via selective polymer insertion into the MOF nanopores. Preparation of a MOF-packed column allowed analytical and preparative chromatographic separation of these topologically distinct pairs. In addition, gram-scale PEGs with only cyclic structures were successfully obtained from a crude reaction mixture by using MOF as an adsorbent.

Universal Synthetic Strategy for the Construction of Topological Polystyrenesulfonates: The Importance of Linkage Stability during Sulfonation

Polystyrenesulfonate (PSS), as one of the most important categories of polyelectrolytes, has received increasing attention due to its great potential in the applications of energy- and biomedical-related fields. However, most of the previous studies only focused on linear PSS and its derivatives, but little attention was paid to nonlinear topological PSSs. So far, the synthesis of nonlinear PSSs with well-defined structures is still a challenging task, and the main obstacle lies in the stability issue of functional chemical linkages during the sulfonation process of polystyrene (PS) precursors, such as the carbon-oxygen-containing linkages. Herein, by rationally designing the chemical structure of the functional linkage, we introduce a versatile and efficient strategy for the preparation of topological PSSs. Specifically, by embedding firm triazole linkages (without carbon-oxygen linkages) into the backbone structure of cyclic and hyperbranched PS precursors, the backbone and functional linkages are found to present excellent chemical stability under certain sulfonation conditions, which eventually lead to the successful preparation of cyclic and hyperbranched PSSs. By using two sets of PSS samples with varied molar masses, the scaling relations between the number of repeating units and the sedimentation coefficient are established for both linear and cyclic PSSs. We believe that our proposed synthetic strategy is universal and could be extended to the synthesis of other types of topological PSSs.

Synthesis of a cyclic-brush polymer with a high grafting density using activated ester chemistry via the “grafting onto” approach

A concept for preparing cyclic-brush polymers with high grafting densities combining a ring-closure strategy and activated ester chemistry as a post-modification method via the grafting onto approach.

Topology-Sensitive Microfluidic Filter for Polymers of Varying Stiffness

The separation of polymers based on their size, rigidity, and topology is an essential but also highly challenging task for nanoscience and engineering. Using hybrid molecular dynamics simulations that correctly take into account hydrodynamics, we have designed microfluidic channels for separating linear from ring polymers in dilute solutions. We establish that the transport velocity of the polymers is independent of their topology and rigidity when the channel walls are smooth and repulsive. However, when the walls are decorated with attractive spots arranged on lines parallel to the flow, ring polymers exhibit an order of magnitude higher transport velocity compared to linear chains. The spots induce a homeotropic-like reorientation of ring polymers close to walls leading to a tank treading motion along them, whereas linear chains are immobilized upon adsorption. This mechanism becomes more enhanced with increasing polymer rigidity. The presented technique holds thus promise for reliably separating nanoparticles based on their topology.

Scaling behavior of knotted random polygons and self-avoiding polygons: Topological swelling with enhanced exponent

We show that the average size of self-avoiding polygons (SAPs) with a fixed knot is much larger than that of no topological constraint if the excluded volume is small and the number of segments is large. We call it topological swelling. We argue an "enhancement" of the scaling exponent for random polygons with a fixed knot. We study them systematically through SAP consisting of hard cylindrical segments with various different values of the radius of segments. Here we mean by the average size the mean-square radius of gyration. Furthermore, we show numerically that the topological balance length of a composite knot is given by the sum of those of all constituent prime knots. Here we define the topological balance length of a knot by such a number of segments that topological entropic repulsions are balanced with the knot complexity in the average size. The additivity suggests the local knot picture.

Knotting probability of self-avoiding polygons under a topological constraint

We define the knotting probability of a knot K by the probability for a random polygon or self-avoiding polygon (SAP) of N segments having the knot type K. We show fundamental and generic properties of the knotting probability particularly its dependence on the excluded volume. We investigate them for the SAP consisting of hard cylindrical segments of unit length and radius r_ex. For various prime and composite knots, we numerically show that a compact formula describes the knotting probabilities for the cylindrical SAP as a function of segment number N and radius r_ex. It connects the small-N to the large-N behavior and even to lattice knots in the case of large values of radius. As the excluded volume increases, the maximum of the knotting probability decreases for prime knots except for the trefoil knot. If it is large, the trefoil knot and its descendants are dominant among the nontrivial knots in the SAP. From the factorization property of the knotting probability, we derive a sum rule among the estimates of a fitting parameter for all prime knots, which suggests the local knot picture and the dominance of the trefoil knot in the case of large excluded volumes. Here we remark that the cylindrical SAP gives a model of circular DNA which is negatively charged and semiflexible, where radius r_ex corresponds to the screening length.

Statistical and Dynamical Properties of Topological Polymers with Graphs and Ring Polymers with Knots

We review recent theoretical studies on the statistical and dynamical properties of polymers with nontrivial structures in chemical connectivity and those of polymers with a nontrivial topology, such as knotted ring polymers in solution. We call polymers with nontrivial structures in chemical connectivity expressed by graphs "topological polymers". Graphs with no loop have only trivial topology, while graphs with loops such as multiple-rings may have nontrivial topology of spatial graphs as embeddings in three dimensions, e.g., knots or links in some loops. We thus call also such polymers with nontrivial topology "topological polymers", for simplicity. For various polymers with different structures in chemical connectivity, we numerically evaluate the mean-square radius of gyration and the hydrodynamic radius systematically through simulation. We evaluate the ratio of the gyration radius to the hydrodynamic radius, which we expect to be universal from the viewpoint of the renormalization group. Furthermore, we show that the short-distance intrachain correlation is much enhanced for real topological polymers (the Kremer⁻Grest model) expressed with complex graphs. We then address topological properties of ring polymers in solution. We define the knotting probability of a knot K by the probability that a given random polygon or self-avoiding polygon of N vertices has the knot K. We show a formula for expressing it as a function of the number of segments N, which gives good fitted curves to the data of the knotting probability versus N. We show numerically that the average size of self-avoiding polygons with a fixed knot can be much larger than that of no topological constraint if the excluded volume is small. We call it "topological swelling".

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.

Characteristic length of the knotting probability revisited

We present a self-avoiding polygon (SAP) model for circular DNA in which the radius of impermeable cylindrical segments corresponds to the screening length of double-stranded DNA surrounded by counter ions. For the model we evaluate the probability for a generated SAP with N segments having a given knot K through simulation. We call it the knotting probability of a knot K with N segments for the SAP model. We show that when N is large the most significant factor in the knotting probability is given by the exponentially decaying part exp(-N/NK), where the estimates of parameter NK are consistent with the same value for all the different knots we investigated. We thus call it the characteristic length of the knotting probability. We give formulae expressing the characteristic length as a function of the cylindrical radius rex, i.e. the screening length of double-stranded DNA.

Kinetic analysis for macrocyclizations involving anionic template at the transition state

Several kinetic models for the macrocyclization of a C₂ pseudopeptide with a dihalide through a S_N2 reaction have been developed. These models not only focus on the kinetic analysis of the main macrocyclization reaction, but also consider the competitive oligomerization/polymerization processes yielding undesired oligomeric/polymeric byproducts. The effect of anions has also been included in the kinetic models, as they can act as catalytic templates in the transition state reducing and stabilizing the transition state. The corresponding differential equation systems for each kinetic model can be solved numerically. Through a comprehensive analysis of these results, it is possible to obtain a better understanding of the different parameters that are involved in the macrocyclization reaction mechanism and to develop strategies for the optimization of the desired processes.

Statistics and geometrical picture of ring polymer melts and solutions

We present a detailed account of a recently proposed phenomenological theory for noncatenated ring polymer melts [Sakaue, Phys. Rev. Lett. 106, 167802 (2011)]. A basic assumption lies in the implementation of the noncatenation constraint via the effective excluded-volume effect, from which a geometrical picture of melts emerges. The result captures many of the salient features observed so far, including (i) the overall spatial size of rings, (ii) the coordinate number, i.e., the number of rings surrounding a given ring, (iii) the topological length scale as a function of the molecular weight, and (iv) the effect of the chain stiffness and concentration. We also suggest a geometrical interpretation of the topological length scale, which may shed some light on the entanglement concept in polymeric systems.

Looped star polymers show conformational transition from spherical to flat toroidal shapes

Inspired by the topological organization of the circular Escherichia coli chromosome, which is compacted by separate domains, we study a polymer architecture consisting of a central ring to which either looped or linear side chains are grafted. A shape change from a spherical to a toroidal organization takes place as soon as the inner ring becomes large enough for the attached arms to fit within its circumference. Building up a torus, the system flattens, depending on the effective bending rigidity of the chain induced by entropic repulsion of the attached loops and, to a lesser extent, linear arms. Our results suggest that the natural formation of a toroidal structure with a decreased amount of writhe induced by a specific underlying topology could be one driving force, among others, that nature exploits to ensure proper packaging of the genetic material within a rod-shaped, bacterial envelope.

Zwitterionic polymerization: a kinetic strategy for the controlled synthesis of cyclic polylactide

The zwitterionic ring-opening polymerization of lactide initiated by N-heterocyclic carbenes generates cyclic polylactides with well-defined molecular weights between M(n) = 5000 and 30,000 g/mol with narrow polydispersities (M(w)/M(n) < or = 1.31). These zwitterionic polymerizations are extremely rapid (k(p) = 48.7 M(-1) s(-1)), but also exhibit exceptional control of molecular weight and molecular weight distribution. The unusual kinetic features of these zwitterionic polymerizations are illuminated with kinetic and mechanistic investigations, which implicate a mechanism that involves a slow initiation step (second order in [M]), a propagation step (first order in [M]) that is much faster than initiation (k(i) = 0.274 M(-2) s(-1)), cyclization (k(c) = 0.0575 s(-1)), and depropagation (k(d) = 0.208 s(-1)). Numerical and stochastic simulations of the kinetic data provide a kinetic rationale for the evolution of molecular weight with monomer conversion: the molecular weights increase with increasing monomer conversion, exhibit a nonzero intercept near 0% monomer conversion, and are relatively insensitive to the initial monomer-to-initiator ratio. The observed narrow molecular weight distributions are due to a high rate of propagation relative to cyclization and chain transfer. Kinetic simulations define the kinetic criteria under which the active zwitterions remain in solution; these simulations were substantiated by chain-extension experiments, which provide experimental evidence for chain extension of the zwitterions and reinitiation by the N-heterocyclic carbenes liberated upon macrocyclization. The kinetic model rationalizes some of the unique features of zwitterionic ring-opening polymerization and provides a useful mechanistic framework to optimize these polymerizations as a strategy to generate well-defined cyclic polyesters.

Universality in the diffusion of knots

We have evaluated a universal ratio between diffusion constants of the ring polymer with a given knot K and a linear polymer with the same molecular weight in solution through the Brownian dynamics under hydrodynamic interaction. The ratio is found to be constant with respect to the number of monomers, N , and hence the estimate at some N should be valid practically over a wide range of N for various polymer models. Interestingly, the ratio is determined by the average crossing number (N{AC}) of an ideal conformation of knotted curve K , i.e., that of the ideal knot. The N{AC} of ideal knots should therefore be fundamental in the dynamics of knots.