The second virial coefficients of highly-purified ring polystyrenes in cyclohexane

Solution characterization of a hyperbranched polysaccharide carbamate derivative and specific phase separation behavior due to chain branching

A hyperbranched polymer consisting of rigid helical part chains was prepared as highly branched cyclic dextrin tris(phenylcarbamate) (HTPC) with the weight-average molar masses of 880 and 590 kg mol-1. Small-angle X-ray scattering (SAXS) measurement and viscometry were performed on the samples in good and poor solvents to determine the dimensional and hydrodynamic properties in solution. The HTPC molecule has a much more compact conformation than the linear chain with the same molar mass as expected for the hyperbranched architecture. While the corresponding linear polymer is soluble in methyl acetate (MEA) over a wide temperature range, HTPC is only soluble in the solvent at low temperatures. A typical LCST-type phase diagram was observed for the HTPC-MEA system, indicating that the interactions between the polymer segment of HTPC and the MEA molecules are substantially different from those of the linear chains. This is most likely due to the bending helical chains near the branching point of HTPC having different interactions with solvent molecules.

Comparison of Cyclic and Linear Poly(lactide)s Using Small-Angle Neutron Scattering

Small-angle neutron scattering (SANS) experiments were conducted on cyclic and linear polymers of racemic and l-lactides (PLA) with the goal of comparing chain configurations, scaling, and effective polymer-solvent interactions of the two topologies in acetone-d ₆ and THF-d ₈. There are limited reports of SANS results on cyclic polymers due to the lack of substantial development in the field until recently. Now that pure, well-defined cyclic polymers are accessible, unanswered questions about their rheology and physical conformations can be better investigated. Previously reported SANS experiments have used cyclic and linear polystyrene samples; therefore, our work allowed for direct comparison using a contrasting (structurally and sterically) polymer. We compared SANS results of cyclic and linear PLA samples with various microstructures and molecular weights at two different temperatures, allowing for comparison with a wide range of variables. The results followed the trends of previous experiments, but much greater differences in the effective polymer-solvent interaction parameters between cyclic and linear forms of PLA were observed, implying that the small form factor and hydrogen bonding in PLA allowed for much more compact conformations in the cyclic form only. Also, the polymer microstructure was found to influence polymer-solvent interaction parameters substantially. These results illustrate how the difference in polymer-solvent interactions between cyclic and linear polymers can vary greatly depending on the polymer in question and the potential of neutron scattering as a tool for identification and characterization of the cyclic topology.

Molecular dynamics study of the swelling and osmotic properties of compact nanogel particles

Owing to their great importance in materials science and other fields, we investigate the solution and osmotic properties of uncharged compact nanogel particles over a wide range of solvent quality and particle concentration by molecular dynamics (MD) simulations. We characterize the osmotic pressure by estimating the second and third virial coefficients, and by extension, we identify the θ-point where the second virial coefficient vanishes. Calculations of the structure factor indicate that these particles are similar to macrogels in that the particle-like scattering profile disappears at moderate concentrations. We also find that improving the solvent quality enhances the spatial segmental uniformity, while significant heterogeneous structure arises near the θ-point. Well below the θ-point where the second osmotic virial coefficient vanishes, these heterogeneous structures become less prevalent as the particles tend to collapse. We also investigate the degree of swelling and structure of compact nanogel particles with a variable excluded volume interaction and gel particle concentration. The osmotic modulus and the scaling exponents in good and θ-point conditions of these gels are characteristic of interacting randomly branched polymers, i.e., "lattice animals".

On the Interfacial Behavior of Catenated Poly(l-lactide) at the Air-Water Interface

Interfacial properties of polymeric materials are significantly influenced by their architectural structures and spatial features, while such a study of topologically interesting macromolecules is rarely reported. In this work, we reported, for the first time, the interfacial behavior of catenated poly(l-lactide) (C-PLA) at the air-water interface and compared it with its linear analogue (L-PLA). The isotherms of surface pressure-area per repeating unit showed significant interfacial behavioral differences between the two polymers with different topologies. Isobaric creep experiments and compression-expansion cycles also showed that C-PLA demonstrated higher stability at the air-water interface. Interestingly, when the films at different surface pressures were transferred via the Langmuir-Blodgett method, successive atomic force microscopy imaging displayed distinct nanomorphologies, in which the surface of C-PLA exhibited nanofibrous structures, while that of the L-PLA revealed a smoother topology with less fiber-like structures.

Local Effects of Ring Topology Observed in Polymer Conformation and Dynamics by Neutron Scattering-A Review

The physical properties of polymers depend on a range of both structural and chemical parameters, and in particular, on molecular topology. Apparently simple changes such as joining chains at a point to form stars or simply joining the two ends to form a ring can profoundly alter molecular conformation and dynamics, and hence properties. Cyclic polymers, as they do not have free ends, represent the simplest model system where reptation is completely suppressed. As a consequence, there exists a considerable literature and several reviews focused on high molecular weight cyclics where long range dynamics described by the reptation model comes into play. However, this is only one area of interest. Consideration of the conformation and dynamics of rings and chains, and of their mixtures, over molecular weights ranging from tens of repeat units up to and beyond the onset of entanglements and in both solution and melts has provided a rich literature for theory and simulation. Experimental work, particularly neutron scattering, has been limited by the difficulty of synthesizing well-characterized ring samples, and deuterated analogues. Here in the context of the broader literature we review investigations of local conformation and dynamics of linear and cyclic polymers, concentrating on poly(dimethyl siloxane) (PDMS) and covering a wide range of generally less high molar masses. Experimental data from small angle neutron scattering (SANS) and quasi-elastic neutron scattering (QENS), including Neutron Spin Echo (NSE), are compared to theory and computational predictions.

Orientational relaxation of ring polymers in dilute solutions

The segmental relaxation dynamics of ring polymers in dilute solutions is investigated via optimized Rouse-Zimm theory. To the best of our knowledge, this is the first study that characterizes the orientational relaxation dynamics of ring polymers in dilute solutions. The orientational time autocorrelation functions are governed by two major processes that span a broad range of timescales: (i) local segmental motion at short times, independent of the ring size, and (ii) overall motion of the ring at long times that depends on the limiting ring size. Smaller rings relax faster than larger rings and their respective linear analogues. The hydrodynamic interactions decrease the higher relaxation rates corresponding to the local relaxation modes and increase the smaller relaxation rates which correspond to the collective relaxation modes. The spectral density is independent of frequency in the low frequency regime while it decreases with increasing frequency. Regardless of the ring size, the spin-lattice relaxation rate exhibits a single characteristic maximum as a function of frequency that shifts to a lower value with increasing strength of hydrodynamic interactions.

Dimensions of catenated ring polymers in dilute solution studied by Monte-Carlo simulation

Interaction between two simple ring chains catenated in a molecule was estimated by a Metropolis Monte Carlo simulation, and the result was compared with a model. We employed catenated ring chains in this study; they were composed of two simple ring chains, and the topology was kept as 2 1 2 . The temperature dependence of the distance between two ring chains in a molecule was discussed using Flory's scaling exponent, ν, in R _g ∝ N ^ν , where R _g is the radius of gyration of a simple ring chain catenated in a molecule. In the simulation, the topology of the component rings and their links were kept because chain crossing was prohibited. The excluded volume of chains was screened by the attractive force between polymer segments, and the strength of the attractive force depends on temperature, T. At the θ temperature for trivial ring polymers, where the condition ν = 1/2 holds, their trajectories can be described statistically as a closed-random walk, i.e., a closed-phantom chain model. The temperature at which interaction between trivial ring polymers, i.e., inter-molecular interaction, is repulsive; trivial ring polymer molecules show the excluded volume generated with keeping their own topology, 0₁. A catenated molecule is composed of two simple rings, and so forth a component ring can be affected by the existence of the counterpart rings. Under that temperature, the mean-square distance between two rings in a catenated molecule, ⟨L ²⟩, was obtained and compared with that of the simple model composed of two circles in three-dimensions, where interaction between circles is set as zero. It has been found that the simulated ⟨L ²⟩ values were constantly larger than those of the model owing to the excluded volume of rings in a molecule.

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".

Influence of the Solvent Quality on Ring Polymer Dimensions

We present a systematic investigation of well-characterized, experimentally pure polystyrene (PS) rings with molar mass of 161 000 g/mol in dilute solutions. We measure the ring form factor at θ- and good-solvent conditions as well as in a polymeric solvent (linear PS of roughly comparable molar mass) by means of small-angle neutron scattering (SANS). Additional dynamic light scattering (DLS) measurements support the SANS data and help elucidate the role of solvent quality and solution preparation. The results indicate the increase of ring dimensions as the solvent quality improves. Furthermore, the experimental form factors in both θ-solvent and linear matrix behave as ideal rings and are fully superimposable. The nearly Gaussian conformations of rings in a melt of linear chains provide evidence of threading of linear chains through rings. The latter result has implications for the dynamics of ring-linear polymer mixtures.

Interactions between ring polymers in dilute solution studied by Monte Carlo simulation

The second virial coefficient, A2, for trivial-ring polymers in dilute condition was estimated from a Metropolis Monte Carlo (MC) simulation, and the temperature dependence of A2 has been discussed with their Flory's scaling exponent, ν, in Rg ∝ N(ν), where Rg is radius of gyration of a polymer molecule. A limited but not too small number of polymer molecules were employed in the simulation, and the A2 values at various temperatures were calculated from the molecular density fluctuation in the solution. In the simulation, the topology of ring polymers was kept, since chain crossing was prohibited. The excluded volume effects can be screened by the attractive force between segments, which depends on the temperature, Tα, defined in the Metropolis MC method. Linear and trivial-ring polymers have the ν value of 1/2 at Tα = 10.605 and 10.504. At Tα = 10.504, the excluded volume effects are screened by the attractive force generated between segments in a ring polymer, but the A2 value for ring polymers is positive. Thus, the temperature at A2 = 0 for a ring polymer is lower than that at ν = 1/2, and this fact can be explained with the following two reasons. (a) Rg value for a ring polymer is much smaller than that for a linear polymer at the same temperature and molecular weight, where interpenetration of a ring polymer chain into neighboring chains is apparently less than a linear chain. (b) The conformation of trivial rings can be statistically described as a closed random walk at ν = 1/2, but their topologies are kept, being produced topological constraints, which strongly relate not only to the long-distance interaction between segments in a molecule but also the inter-molecular interaction.

Multi-blob coarse graining for ring polymer solutions

We present a multi-scale molecular modeling of concentrated solutions of unknotted and non-concatenated ring polymers under good solvent conditions. The approach is based on a multi-blob representation of each ring polymer, which is capable of overcoming the shortcomings of single-blob approaches that lose their validity at concentrations exceeding the overlap density of the solution [A. Narros, A. J. Moreno, and C. N. Likos, Soft Matter, 2010, 6, 2435]. By means of a first principles coarse-graining strategy based on analytically determined effective pair potentials between the blobs, computed at zero density, we quantitatively reproduce the single molecule and solution properties of a system with well-defined topological constraints. Detailed comparisons with the underlying, monomer-resolved model demonstrate the validity of our approach, which employs fully transferable pair potentials between connected and unconnected blobs. We demonstrate that the pair structure between the centers of mass of the rings is accurately reproduced by the multi-blob approach, thus opening the way for simulation of arbitrarily long polymers. Finally, we show the importance of the topological constraint of non-concatenation on the structure of the concentrated solution and in particular on the size of the correlation hole and the shrinkage of the rings as melt concentrations are approached.

Effects of Knots on Ring Polymers in Solvents of Varying Quality

We employ extensive computer simulations to investigate the conformations and the interactions of ring polymers under conditions of worsening solvent quality, in comparison with those for linear polymers. We determine the dependence of the Θ-temperature on knotedness by considering ring polymers of different topologies. We establish a clear decrease of the former upon changing the topology of the polymer from linear to an unknotted ring and a further decrease of the same upon introducing trefoil- or 5-fold knots but we find no difference in the Θ-point between the two knotted molecules. Our results are based on two independent methods: one considering the scaling of the gyration radius with molecular weight and one based on the dependence of the effective interaction on solvent quality. In addition, we calculate several shape-parameters of the polymers to characterize linear, unknotted, and knotted topologies in good solvents and in the proximity of the Θ-point. The shape parameters of the knotted molecules show an interesting crossover at a degree of polymerization that depends on the degree of knottedness of the molecule.

Universal properties of knotted polymer rings

By performing Monte Carlo sampling of N-steps self-avoiding polygons embedded on different Bravais lattices we explore the robustness of universality in the entropic, metric, and geometrical properties of knotted polymer rings. In particular, by simulating polygons with N up to 10(5) we furnish a sharp estimate of the asymptotic values of the knot probability ratios and show their independence on the lattice type. This universal feature was previously suggested, although with different estimates of the asymptotic values. In addition, we show that the scaling behavior of the mean-squared radius of gyration of polygons depends on their knot type only through its correction to scaling. Finally, as a measure of the geometrical self-entanglement of the self-avoiding polygons we consider the standard deviation of the writhe distribution and estimate its power-law behavior in the large N limit. The estimates of the power exponent do depend neither on the lattice nor on the knot type, strongly supporting an extension of the universality property to some features of the geometrical entanglement.