Rotaxane Formation of Multicyclic Polydimethylsiloxane in a Silicone Network: A Step toward Constructing "Macro-Rotaxanes" from High-Molecular-Weight Axle and Wheel Components.

Rotaxanes consisting of a high-molecular-weight axle and wheel components (macro-rotaxanes) have high structural freedom, and are attractive for soft-material applications. However, their synthesis remains underexplored. Here, we investigated macro-rotaxane formation by the topological trapping of multicyclic polydimethylsiloxanes (mc-PDMSs) in silicone networks.

Miscibility and exchange chemical potential of ring polymers in symmetric ring-ring blends.

Generally, differences of polymer topologies may affect polymer miscibility even with the same repeated units. In this study, the topological effect of ring polymers on miscibility was investigated by comparing symmetric ring-ring and linear-linear polymer blends. To elucidate the topological effect of ring polymers on mixing free energy, the exchange chemical potential of binary blends was numerically evaluated as a function of composition by performing semi-grand canonical Monte Carlo and molecular dynamics simulations of a bead-spring model.

Efficient compressed database of equilibrated configurations of ring-linear polymer blends for MD simulations.

To effectively archive configuration data during molecular dynamics (MD) simulations of polymer systems, we present an efficient compression method with good numerical accuracy that preserves the topology of ring-linear polymer blends. To compress the fraction of floating-point data, we used the Jointed Hierarchical Precision Compression Number - Data Format (JHPCN-DF) method to apply zero padding for the tailing fraction bits, which did not affect the numerical accuracy, then compressed the data with Huffman coding. We also provided a dataset of well-equilibrated configurations of MD simulations for ring-linear polymer blends with various lengths of linear and ring polymers, including ring complexes composed of multiple rings such as polycatenane.

Programmed folding into spiro-multicyclic polymer topologies from linear and star-shaped chains.

The development of precise folding techniques for synthetic polymer chains that replicate the unique structures and functions of biopolymers has long been a key challenge. In particular, spiro-type (i.e.

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.

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.

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.

Statistical and hydrodynamic properties of topological polymers for various graphs showing enhanced short-range correlation.

For various polymers with different structures in chemical connectivity expressed by graphs, we numerically evaluate the mean-square radius of gyration and the hydrodynamic radius systematically through simulation. We call polymers with nontrivial structures in chemical connectivity and those of nontrivial topology of spatial graphs as embeddings in three dimensions topological polymers. We evaluate the two quantities both for ideal and real chain models and show that the ratios of the quantities among different structures in chemical connectivity do not depend on the existence of excluded volume if the topological polymers have only up to trivalent vertices, as far as the polymers investigated.

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.

Statistical and hydrodynamic properties of double-ring polymers with a fixed linking number between twin rings.

For a double-ring polymer in solution we evaluate the mean-square radius of gyration and the diffusion coefficient through simulation of off-lattice self-avoiding double polygons consisting of cylindrical segments with radius rex of unit length. Here, a self-avoiding double polygon consists of twin self-avoiding polygons which are connected by a cylindrical segment. We show numerically that several statistical and dynamical properties of double-ring polymers in solution depend on the linking number of the constituent twin ring polymers.

Exact relaxation dynamics of a localized many-body state in the 1D Bose gas.

Through an exact method, we numerically solve the time evolution of the density profile for an initially localized state in the one-dimensional bosons with repulsive short-range interactions. We show that a localized state with a density notch is constructed by superposing one-hole excitations. The initial density profile overlaps the plot of the squared amplitude of a dark soliton in the weak coupling regime.

Dimension of ring polymers in bulk studied by Monte-Carlo simulation and self-consistent theory.

We studied equilibrium conformations of ring polymers in melt over the wide range of segment number N of up to 4096 with Monte-Carlo simulation and obtained N dependence of radius of gyration of chains R(g). The simulation model used is bond fluctuation model (BFM), where polymer segments bear excluded volume; however, the excluded volume effect vanishes at N-->infinity, and linear polymer can be regarded as an ideal chain. Simulation for ring polymers in melt was performed, and the nu value in the relationship R(g) proportional to N(nu) is decreased gradually with increasing N, and finally it reaches the limiting value, 1/3, in the range of N>or=1536, i.

Level statistics of a pseudo-Hermitian Dicke model.

A non-Hermitian operator that is related to its adjoint through a similarity transformation is defined as a pseudo-Hermitian operator. We study the level statistics of a pseudo-Hermitian Dicke Hamiltonian that undergoes quantum phase transition (QPT). We find that the level-spacing distribution of this Hamiltonian near the integrable limit is close to Poisson distribution, while it is Wigner distribution for the ranges of the parameters for which the Hamiltonian is nonintegrable.

Quantum phase transition in a pseudo-Hermitian Dicke model.

We show that a Dicke-type non-Hermitian Hamiltonian admits entirely real spectra by mapping it to the "dressed Dicke model" through a similarity transformation. We find a positive-definite metric in the Hilbert space of the non-Hermitian Hamiltonian so that the time evolution is unitary and allows a consistent quantum description. We then show that this non-Hermitian Hamiltonian describing nondissipative quantum processes undergoes quantum phase transition.

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.

Scattering functions of knotted ring polymers.

We discuss the scattering function of a Gaussian random polygon with N nodes under a given topological constraint through simulation. We evaluate the form factor PK(q) of a Gaussian polygon of N = 200 having a fixed knot K for some different knots such as the trivial, trefoil, and figure-eight knots. Here the Gaussian polygons with different knots K have distinct values of the mean-square radius of gyration, R2(G,K).

Geometrical complexity of conformations of ring polymers under topological constraints.

One measure of geometrical complexity of a spatial curve is the average of the number of crossings appearing in its planar projection. The mean number of crossings averaged over some directions have been numerically evaluated for N-noded ring polymers with a fixed knot type. When N is large, the average crossing number of ring polymers under the topological constraint is smaller than that of no topological constraint.

Average size of random polygons with fixed knot topology.

We have evaluated by numerical simulation the average size R(K) of random polygons of fixed knot topology K=,3(1),3(1) musical sharp 4(1), and we have confirmed the scaling law R(2)(K) approximately N(2nu(K)) for the number N of polygonal nodes in a wide range; N=100-2200. The best fit gives 2nu(K) approximately 1.11-1.

Knot complexity and the probability of random knotting.

The probability of a random polygon (or a ring polymer) having a knot type K should depend on the complexity of the knot K. Through computer simulation using knot invariants, we show that the knotting probability decreases exponentially with respect to knot complexity. Here we assume that some aspects of knot complexity are expressed by the minimal crossing number C and the "rope length" of K, which is defined by the smallest length of rope with unit diameter that can be tied to make the knot K.

Finite-size and asymptotic behaviors of the gyration radius of knotted cylindrical self-avoiding polygons.

Several nontrivial properties are shown for the mean-square radius of gyration R2(K) of ring polymers with a fixed knot type K. Through computer simulation, we discuss both finite size and asymptotic behaviors of the gyration radius under the topological constraint for self-avoiding polygons consisting of N cylindrical segments with radius r. We find that the average size of ring polymers with the knot K can be much larger than that of no topological constraint.