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In this study, we examined the viscoelastic properties of a series of comb-shaped ring (RC) polystyrene samples with different branch chain lengths, i.e., the molecular weights of the ring backbone Mbb (≃ 4Me where Me is the entanglement molecular weight) and branch chains Mbr (≃ Me, 2Me, and 4Me). Even for the RC sample with the shortest branch chains, a plateau region was observed for the dynamic modulus G*(ω) in the middle angular frequency ω region, suggesting that intermolecular branch chain entanglement occurred. In the ω region between the plateau and terminal regions, G*(ω) was observed with a weaker ω dependence than the terminal relaxation. This behavior was more pronounced for RC samples with shorter branch chains and for the corresponding linear comb (LC) samples than for the RC ones. The molecular weight dependence of the zero-shear viscosity η0 and the steady-state recoverable compliance Jeo of the RC and LC samples was evaluated, and the effects of different backbone molecular structures (i.e., ring or linear) on the terminal relaxation behavior was discussed. Moreover, the G*(ω) data were analyzed with two models: the comb-Rouse model, in which the structures of the RC/LC molecules are taken into account by graph theory, and the Milner-McLeish model for entangled star-shaped polymers. The former model qualitatively described the terminal relaxation behavior of G*(ω) at low ω but failed to reproduce the plateau in the middle ω range. Conversely, the latter model described the entanglement plateau in the middle ω range, but the difference in the terminal relaxation regime for the RC/LC samples seen in the data and the comb-Rouse model disappeared.

The multichain slip-spring (MCSS) model is one of the coarse-grained models of polymers developed in the niche between bead-spring models and tube type descriptions. In this model, polymers are represented by Rouse chains connected by virtual springs that temporally connect the chains, hop along the chain, and are constructed and annihilated at the chain ends. Earlier studies have shown that MCSS simulations can nicely reproduce entangled and unentangled polymer dynamics. However, the model parameters have been chosen arbitrarily, and their effects have not been reported. In this study, for the first time, we systematically investigated the effects of model parameters: fugacity of virtual springs, its intensity, and the Rouse bead density. We validated the employed simulation code by confirming that the statistics of the system follow the theoretical setup. Namely, the virtual spring density is correctly controlled, and polymer chains exhibit ideal chain statistics irrespective of the chosen parameter values. For diffusion and linear viscoelasticity, simulation results obtained for different parameters can be superposed with each other by conversion factors for the bead number per chain and units of length, time, and modulus. These conversion factors follow scaling laws concerning the number of Rouse segments between two consecutive anchoring points of virtual springs along the polymer chain. Besides, diffusion and viscoelasticity excellently agree with literature data for the standard bead-spring simulation. These results imply that the coarse-graining level for the MCSS model can be arbitrarily chosen and controlled by model parameters.

Using Brownian dynamics simulations, we studied the mechanical properties of tetra-gel-type networks with varying arm length contrast of prepolymers to modulate the number of double linkages.

Rod-shaped particles embedded in certain matrices have been reported to exhibit an increase in their center of mass diffusivity upon increasing the matrix density. This increase has been considered to be caused by a kinetic constraint in analogy with tube models. We investigate a mobile rodlike particle in a sea of immobile point obstacles using a kinetic Monte Carlo scheme equipped with a Markovian process, that generates gaslike collision statistics, so that such kinetic constraints do essentially not exist. Even in such a system, provided the particle's aspect ratio exceeds a threshold value of about 24, the unusual increase in the rod diffusivity emerges. This result implies that the kinetic constraint is not a necessary condition for the increase in the diffusivity., Physical Review E, 107 (4), ISSN:1539-3755, ISSN:1063-651X, ISSN:1095-3787, ISSN:1550-2376

Diblock copolymers consisting of flexible polystyrene and semiflexible poly(n-hexyl isocyanate), P(S-b-HIC)s, were prepared by anionic polymerization, followed by size-exclusion chromatography (SEC) fractionation. Two P(S-b-HIC) samples having PHIC volume fractions (ΦPHIC) of 0.18 and 0.31, sufficiently high molecular weights (Mn,PS and Mn,PHIC are both larger than 10 kg/mol), and narrow molecular weight distributions (Mw/Mn

We synthesized model polyethylene (PE) samples with nearly monodispersed molecular weight distribution and different fractions of short-chain branches to investigate the influences of the branch structure on the rheological properties. The activation energy of the flow process showed linear relations with the weight fractions of comonomer and the short-chain branch. These results suggest that the friction of the chain motion is dominated by the amount of the short-chain branch. The zero-shear viscosity showed a power law to Mw with an exponent of about 3.3 for all linear and branched PEs. The branched PEs showed large entanglement molecular weights compared with the linear PEs because the introduction of the short-chain branches results in the increase of the average molecular weight per backbone unit. Moreover, the introduction of the short-chain branches has no influence on the strain-hardening under an extensional elongation process, indicating that the short-chain branches only affect the friction.

This study examined viscoeslastic properties of two flexible dumbbell-shaped polystyrene (PS) samples, D-30/120/30 and D-30/240/30, which possess the same ring size (MR ∼ 30 kg/mol) and different lengths of central linear chains (ML ∼ 120 and 240 kg/mol) in bulk and solution. In bulk, both dumbbell PS samples exhibited an extremely long entanglement plateau in dynamic oscillatory measurements, and their terminal relaxation behavior was not observed in our experimental window. In stress relaxation measurements, dumbbell samples in bulk exhibited considerably slower terminal relaxation than high-molecular-weight linear PS samples (M ∼ 10^6 g/mol), and a clear difference between two dumbbell PS samples was observed in the long time regime, i.e., terminal relaxation of D-30/240/30 is much slower than that of D-30/120/30. These results suggest that (i) the dumbbell polymer forms a characteristic “network” where two rings on both ends in a molecule spontaneously thread the ring part of other dumbbell chains and (ii) this type of interchain interaction resulting in network formation becomes more dominant than usual entanglements similar to linear chains. The network of dumbbell chains starts to relax their stress when the intermolecular threading is released, and this release process tends to occur more frequently for the dumbbell with a shorter central chain, D-30/120/30, than that with a longer one, D-30/240/30. In dioctyl phthalate (DOP) solutions of D-30/240/30, where DOP is known as a θ-solvent for PS at 22 °C, in a relatively high concentration regime (i.e., 20–30 wt %), the solutions exhibited an entanglement plateau and higher viscosity than the corresponding linear PS solutions, suggesting that the characteristic entanglement originating from intermolecular threading of dumbbell chains is still dominant in the solution. On the contrary, in a lower-concentration regime (≤8 wt %), the D-30/240/30 solutions exhibited similar viscosities to the linear PS ones, wherein the dumbbell molecules behave like unentangled/isolated chains.

The non-linear viscoelastic response under interrupted shear flows is one of the interesting characteristics of entangled polymers. In particular, the stress overshoot in the resumed shear has been discussed concerning the recovery of the entanglement network in some studies. In this study, we performed multichain slip-link simulations to observe the molecular structure of an entangled polymer melt. After confirming the reasonable reproducibility of our simulation with the literature data, we analyzed the molecular characteristics following the decoupling approximation. We reasonably found that the segment orientation dominates the stress overshoot even under the resumed shear with minor contributions from the segment stretch and entanglement density. We defined the mitigation function for the recovery of the stress overshoot as a function of the rest time and compared it with the relaxation of the molecular quantities after the initial shear. As a result, we have found that the mitigation of the stress overshoot coincides with the relaxation of entanglement density., ファイル公開：2021/07/28

Diffusivity in some soft matter and biological systems changes with time, called the fluctuating diffusivity. In this work, we propose a novel origin for fluctuating diffusivity based on stochastic simulations of binary gas mixtures. In this system, the fraction of one component is significantly small, and the mass of the minor component molecule is different from that of the major component. The minor component exhibits fluctuating diffusivity when its mass is sufficiently smaller than that of the major component. We elucidate that this fluctuating diffusivity is caused by the time scale separation between the relaxation of the velocity direction and the speed of the minor component molecule., Comment: 20 pages, 12 figures

We derive the stress tensor of a rigid dumbbell by using the virtual work method. In the virtual work method, we virtually apply a small deformation to the system, and relate the change of the energy to the work done by the stress tensor. A rigid dumbbell consists of two particles connected by a rigid bond of which length is constant (the rigid constraint). The energy of the rigid dumbbell consists only on the kinetic energy. Also, only the deformations which do not violate the rigid constraint are allowed. Thus we need the dynamic equations which is consistent with the rigid constraint to apply the virtual deformation. We rewrite the dynamic equations for the underdamped SLLOD-type dynamic equations into the forms which are consistent with the rigid constraint. Then we apply the virtual deformation to a rigid dumbbell based on the obtained dynamic equations. We derive the stress tensor for the rigid dumbbell model from the change of the kinetic energy. Finally, we take the overdamped limit and derive the stress tensor and the dynamic equation for the overdamped rigid dumbbell. We show that the Green-Kubo type linear response formula can be reproduced by combining the stress tensor and the dynamic equation at the overdamped limit., Comment: 14 pages,to appear in Nihon Reoroji Gakkaishi (J. Soc. Rheol. Jpn.)

In this study, linear melt rheology of a single-tail tadpole-shaped polystyrene, ST-30/80, having ring and linear sizes of MR ∼ 30 kg mol^−1 and ML ∼ 80 kg mol^−1, respectively, was examined, and the effect of the ring size on rheological properties of tadpole polymers was discussed by comparing with the data of the previously reported tadpole samples having MR ∼ 60 kg mol^−1. ST-30/80 exhibits an entanglement plateau and shows a clearly slower terminal relaxation than that of its component ring and linear polymers. When the zero-shear viscosity η0 for ST-30/80 is plotted against the molecular weight of a linear tail chain, the data point lies on the single curve of η0 for 4- and 6-arm star polymers and the single-tail tadpoles with MR ∼ 60 kg mol^−1. These results suggest that the tadpole molecule in this study spontaneously forms a characteristic entanglement network, i.e., the intermolecular ring–linear threading, in the same manner as the previous tadpole samples, even though the size of the ring part is just slightly larger than the entanglement molecular weight (i.e., MR ∼ 1.8Me).

It has been established that the elongational rheology of polymers depends on their chemistry. However, the analysis of experimental data has been reported for only a few polymers. In this study, we analyzed the elongational viscosity of poly (propylene carbonate) (PPC) melts in terms of monomeric friction via primitive chain network simulations. By incorporating a small polydispersity of materials, the linear viscoelastic response was semi-quantitatively reproduced. Owing to this agreement, we determined units of time and modulus to carry out elongational simulations. The simulation with constant monomeric friction overestimated elongational viscosity, whereas it nicely captured the experimental data if friction decreased with increasing segment orientation. To see the effect of chemistry, we also conducted the simulation for a polystyrene (PS) melt, which has a similar entanglement number per chain and a polydispersity index. The results imply that PPC and PS behave similarly in terms of the reduction of friction under fast deformations.

Conformations of highly purified ring polystyrene, R-70, with the molar mass of 70 kg/mol, in a good solvent and in linear polymer homologue matrices were examined by small-angle neutron scattering (SANS) measurements. The radii of gyration Rg of R-70 were estimated by the Guinier’s approximation from the SANS profiles obtained, and the polymer volume fraction Φ dependence of Rg2 was discussed. In deuterated toluene as a good solvent, R-70 exhibits the Rg2 ∼ Φ–0.29±0.01 dependence at high Φ above the overlap volume fraction, Φ0* (i.e., 1 < Φ/Φ0* < 20). This exponent −0.29 shows stronger Φ dependence than that for semidilute solutions of linear polymers, −0.25, predicted from the scaling theory, suggesting that the ring expands more sensitively than linear chains when Φ decreases in semidilute regime. In contrast, the Φ dependence of Rg2 of R-70 is evidently weaker than that of the recent simulation for ring polymer solutions (Rg2 ∼ Φ−0.59) by Reigh et al. This difference is thought to originate from the d...

Highly purified hydrogenous ring poly(4-trimethylsilylstyrene) (h-PT) and deuterated ring polyisoprene (d-PI) samples as well as their linear counterparts were prepared, and the miscibility of three kinds of polymer blends, i.e., linear–linear, ring–linear, and ring–ring, denoted as L–L, R–L, and R–R, respectively, with all 50/50 vol % was evaluated by small angle neutron scattering (SANS) measurements. PT and PI are known as a miscible polymer pair with lower critical solution temperature (LCST) type phase diagram. At low-q regime of the scattering profiles, R–R blend exhibits much higher scattering intensity than L–L and R–L, while the latter two show similar profiles. Moreover, from Zimm’s analysis, the spinodal temperature of R–R was estimated to be about 100 °C lower than the other two blends, L–L and R–L. These results suggest that the miscibility of R–R is considerably lower than the other two blends, which is a clear manifestation of the topological effect on the phase behavior of the present blend.

We prepared five pairs of hydrogenous and deuterated ring polystyrene samples over a wide range of molecular weights (10 kg/mol ≤ Mw ≤ 400 kg/mol) and investigated their chain conformations in bulk by small-angle neutron scattering (SANS) measurements. From the SANS profiles obtained, we estimated the radii of gyration Rg of the ring polymers by the Guinier approximation. Rg can be related to the degree of polymerization N as Rg ∼ N0.47. This scaling exponent ν = 0.47 ± 0.01 is evidently smaller than that for the Gaussian chains (ν = 0.50) but higher than previous experimental reports (ν = 0.42–0.43). Then our data were compared with various simulation and experimental data by introducing the entanglement degree of polymerization Ne for linear polymers as a normalized parameter. Rg of three smaller rings, i.e., R-10, R-30, and R-70, where the numbers denote molecular weights in kg/mol unit, are in good agreement with simulation results, while two larger rings, R-100 and R-400, exhibit higher Rg values tha...

In this paper we study the (in)approximability of two distance-based relaxed variants of the maximum clique problem (Max Clique), named Max d-Clique and Max d-Club: A d-clique in a graph $$G = (V, E)$$ is a subset $$S\subseteq V$$ of vertices such that for every pair of vertices $$u, v\in S$$ , the distance between u and v is at most d in G. A d-club in a graph $$G = (V, E)$$ is a subset $$S'\subseteq V$$ of vertices that induces a subgraph of G of diameter at most d. Given a graph G with n vertices, the goal of Max d-Clique (Max d-Club, resp.) is to find a d-clique (d-club, resp.) of maximum cardinality in G. Since Max 1-Clique and Max 1-Club are identical to Max Clique, the inapproximabilty for Max Clique shown by Zuckerman in 2007 is transferred to them. Namely, Max 1-Clique and Max 1-Club cannot be efficiently approximated within a factor of $$n^{1-\varepsilon }$$ for any $$\varepsilon > 0$$ unless $$\mathcal{P} = \mathcal{NP}$$ . Also, in 2002, Marin $$\breve{\mathrm{c}}$$ ek and Mohar showed that it is $$\mathcal{NP}$$ -hard to approximate Max d-Club to within a factor of $$n^{1/3-\varepsilon }$$ for any $$\varepsilon >0$$ and any fixed $$d\ge 2$$ . In this paper, we strengthen the hardness result; we prove that, for any $$\varepsilon > 0$$ and any fixed $$d\ge 2$$ , it is $$\mathcal{NP}$$ -hard to approximate Max d-Club to within a factor of $$n^{1/2-\varepsilon }$$ . Then, we design a polynomial-time algorithm which achieves an optimal approximation ratio of $$O(n^{1/2})$$ for any integer $$d\ge 2$$ . By using the similar ideas, we show the $$O(n^{1/2})$$ -approximation algorithm for Max d-Clique for any $$d\ge 2$$ . This is the best possible in polynomial time unless $$\mathcal{P} = \mathcal{NP}$$ , as we can prove the $$\varOmega (n^{1/2-\varepsilon })$$ -inapproximability.

For high-molecular-weight (M) ring polymers with low contamination of linear chains, recent viscoelastic tests revealed broad terminal relaxation associated with no clear entanglement plateau. This relaxation behavior is qualitatively similar to that deduced from molecular models (double-folded lattice-animal model and the fractal loopy globule model) for entangled ring polymers, but quantitatively important differences are also noted: For example, the full terminal relaxation of those polymers is slower than the model prediction. This study re-examined the viscoelastic data of entangled high-M ring polystyrene (PS) samples (coded as R-240; M = 244×103) specifically for two points: the purity of the ring samples after the viscoelastic tests and the molecular origin of the stress. For the first point, the R-240 samples contaminated with linear chains at low but different levels were prepared by tuning either the purification efficiency or the retention time of the sample at high temperature (T) before/during the viscoelastic test. The fraction w L of the linear contaminant, determined after the viscoelastic measurement, was ranging from 0.7 to 4.9%, and the extrapolation of the modulus data to w L = 0 gave the data for the ideally pure ring melt. This pure ring melt exhibited broad terminal relaxation that started faster but completed slower compared to the model prediction, indicating that the ring relaxation is not well described by the current model(s) even in the absence of linear contaminant. For the second point, dynamic birefringence measurements were conducted for the R-240 samples with w L = 4.6 and 1.0%. These samples obeyed the stress-optical rule, and their stress-optical coefficient was indistinguishable from that for linear PS samples, revealing that the stress of the ring PS chains reflects the orientational anisotropy of the chains (as is the case also for linear chains). The relaxation behavior of pure ring PS melt is discussed on the basis of these findings, with the focus being placed on the ring-ring threading not considered in the models.

A dumbbell-shaped polystyrene (PS) sample, where rings are attached on both ends of a linear chain, was anionically synthesized by a coupling reaction between highly-purified ring PS (R-30; Mw = 33.7 kg/mol) with two diphenylethylene (DPE) type vinyl groups and bifunctional living linear PS (L-80; Mw = 84.0 kg/mol). The crude coupling product obtained was confirmed to include a targeted dumbbell polymer in addition to unreacted L-80 and R-30, and a single-tail tadpole polymer as byproducts. To isolate the dumbbell polymer, successive size-exclusion chromatography (SEC) and interaction chromatography (IC) fractionations were performed. It was found out that the dumbbell chain (D-30/80/30; Mw = 147 kg/mol) with sufficiently high purity (99.9%) was separated from the tadpole polymer with a single-tail in IC mode just due to a small difference in their chain architectures, i.e., the presence of the small rings on either one or both chain ends of a linear chain. This is the first report on successful preparation of a dumbbell-shaped polymer with high molecular weight and high purity, and the present product must be suitable for investigation of accurate physical properties.

A series of comb-shaped ring polystyrene samples, RC, composed of a common ring backbone and multiple linear branches with three different lengths were successfully prepared. First, a highly purified ring backbone was synthesized by anionic polymerizations followed by multistep size exclusion chromatography (SEC) and interaction chromatography (IC) fractionations. Second, this backbone was reacted with an excess molar amount of anionic living linear branch chains. Through the coupling reactions, the targeted RC samples were successfully obtained with high grafting efficiency without causing any side reactions. Their chain dimensions in THF were investigated by SEC-MALS measurements, and they were evaluated by estimating the shrinking factor g1/2 (= Rg,comb/Rg,linear). All RC samples prepared in this study exhibited smaller Rg and g1/2 values than those of the corresponding linear combs, LC. Moreover, combs with short branches exhibited an evidently larger g1/2 than the Gaussian chains due to the intramolecular excluded volume effects, being originated from the local high segment density in the vicinity of the branch points. In contrast, g1/2 for combs having long branches agreed reasonably well with the values based on the Gaussian approximation. This result suggests that the intramolecular excluded volume effect is weakened with increasing branch chain length, and consequently the comb behaves like a star polymer. Thus, characteristic chain dimensions of comb-shaped ring samples have been clarified, which strongly depend on the branch chain length and the backbone architecture.

A series of highly purified tadpole-shaped polystyrene (PS) samples were prepared by anionic polymerizations and multistep HPLC fractionations, and their linear melt rheology was investigated. Three tadpole samples were composed of a common ring chain (R-60; Mw = 59.8 kg/mol) and three different lengths of linear chains (L-30, L-70, and L-120; Mw = 27.1, 68.9, and 122 kg/mol, respectively), all of which were longer than the entanglement molecular weight (Me = 18.0 kg/mol for PS). All tadpole samples revealed remarkably slower terminal relaxation than the constituent ring or linear chains and also than the ring/linear blends. These results are evidently owing to the newly generated characteristic entanglements such as the intermolecular ring–linear penetrations. Two rheological parameters, i.e., the zero-shear viscosity, η0, and the steady-state recoverable compliance, Je, were estimated, and their molecular weight dependence was discussed. Tadpoles used in this study exhibited a drastic viscosity enhancem...

The melt rheology of highly-purified ring polystyrenes in a wide range of molecular weights (10K ≤ Mw ≤ 240K g/mol) was investigated. All the rings revealed no obvious rubbery plateau and faster terminal relaxation compared to the linear counterparts. The rheological data obtained were compared with some theoretical models such as the Rouse ring model and the lattice-animal model. Moreover, two rheological parameters, zero-shear viscosities η0 and the steady-state recoverable compliances Je, were estimated, and their Mw dependence was discussed. From these data analysis, it was found that relaxation mechanisms of ring chains can be divided into three categories depending on their Mw as follows: (i) Smaller rings (Mw ≤ 20K) exhibit faster terminal relaxation than the predicted Rouse rings. This behavior is related to the difference of the local chain conformation from linear chains. (ii) Rings with the moderate molecular weight (40K ≤ Mw ≤ 90K) exhibit dynamic moduli similar to the Rouse ring prediction. (...

Tadpole-shaped polystyrenes (PS) were successfully synthesized anionically by coupling reaction between ring polystyrene with two functional groups and living linear polystyrenes. Crude coupling product was confirmed to include unreacted linear tail polymer and tadpole polymer having one tail (single-tail tadpole polymer) and two tails (twin-tail tadpole polymer) by SEC analysis. In order to isolate the tadpole polymers, interaction chromatography (IC) was conducted to remove the linear polymer from the crude product; subsequently, two kinds of tadpole polymers were carefully separated by SEC fractionation. It has been found by SEC-MALS that the absolute molecular weights of the two fractionated products agree well with the sums of the component ring and linear chains, which prove two tadpole polymers, i.e., single-tail and twin-tail, were obtained as designed. It was confirmed that the two tadpole polymers have high molecular weight (58k for ring and 67k for a tail), narrow molecular weight distribution ...

A d-clique in a graph $$G = V, E$$G=V,E is a subset $$S\subseteq V$$S⊆V of vertices such that for pairs of vertices $$u, v\in S$$u,v∈S, the distance between u and v is at most d in G. A d-club in a graph $$G = V, E$$G=V,E is a subset $$S'\subseteq V$$S'⊆V of vertices that induces a subgraph of G of diameter at most d. Given a graph G with n vertices, the goal of Max d-Clique Max d-Club, resp. is to find a d-clique d-club, resp. of maximum cardinality in G. Max 1-Clique and Max 1-Club cannot be efficiently approximated within a factor of $$n^{1-\varepsilon }$$n1-e for any $$\varepsilon > 0$$e>0 unless $$\mathcal{P} = \mathcal{NP}$$P=NP since they are identical to Max Cliquei¾?[14, 21]. Also, it is knowni¾?[3] that it is $$\mathcal{NP}$$NP-hard to approximate Max d-Club to within a factor of $$n^{1/2 - \varepsilon }$$n1/2-e for any fixed $$d\ge 2$$di¾?2 and for any $$\varepsilon > 0$$e>0. As for approximability of Max d-Club, there exists a polynomial-time algorithm which achieves an optimal approximation ratio of $$On^{1/2}$$On1/2 for any even $$d\ge 2$$di¾?2i¾?[3]. For any odd $$d\ge 3$$di¾?3, however, there still remains a gap between the $$On^{2/3}$$On2/3-approximability and the $$\varOmega n^{1/2-\varepsilon }$$Ωn1/2-e-inapproximability for Max d-Clubi¾?[3]. In this paper, we first strengthen the approximability result for Max d-Club; we design a polynomial-time algorithm which achieves an optimal approximation ratio of $$On^{1/2}$$On1/2 for Max d-Club for any odd $$d\ge 3$$di¾?3. Then, by using the similar ideas, we show the $$On^{1/2}$$On1/2-approximation algorithm for Max d-Clique on general graphs for any $$d\ge 2$$di¾?2. This is the best possible in polynomial time unless $$\mathcal{P} = \mathcal{NP}$$P=NP, as we can prove the $$\varOmega n^{1/2-\varepsilon }$$Ωn1/2-e-inapproximability. Furthermore, we study the tractability of Max d-Clique and Max d-Club on subclasses of graphs.