Some remarks on the dynamics of polymer melts

Considering one long chain (N monomeric units) in a homodisperse melt of chemically identical, but shorter, “solvent” chains (P monomers per chain), we propose some tentative scaling laws for the self-diffusion constant D(N) and the relaxation time T(N) of the solute chain. We also discuss the viscosity increment δ_η due to a small volume fraction Φ of the long chains. We find three regimes of behavior, depending on N and P, and on the distance between entanglement points (assumed smaller than N and P): (A) reptation of the N chain; (B) Stokes–Einstein regime; the solute moves like a usual polymer coil in a viscous fluid of P chains; (C) mixed regime, where D(N) is controlled by reptation, while δ_η is of type B. Contrary to our earlier belief, we find no significant regime where the process of “tube renewal” could be dominant.     

Polymer chains in a soft nanotube: a Monte Carlo study

We study the equilibrium properties of flexible polymer chains confined in a soft tube by means of extensive Monte Carlo simulations. The tube wall is that of a single sheet six-coordinated self-avoiding tethered membrane. Our study assumes that there is no adsorption of the chain on the wall. By varying the length N of the polymer and the tube diameter D we examine the variation of the polymer gyration radius Rg and diffusion coefficient Ddiff in soft and rigid tubes of identical diameter and compare them to scaling theory predictions. We find that the swollen region of the soft tube surrounding the chain exhibits a cigarlike cylindrical shape for sufficiently narrow tubes with D相似文献   

Orientational relaxation times of worm-like chains

A semi-empirical formula for orientational relaxation times of worm-like chains in dilute solutions is proposed where τ_rod = τ₀N³ is the relaxation time of a rigid rod composed of N segments, and x = 2a/L is the chain rigidity, i.e. the ratio of the double persistence length to the chain contour length, L. The formula, which can be used in the entire range of molecular rigidities and chain lengths, has been tested against segment relaxation times for semi-rigid chains calculated from the optimized Rouse-Zimm model.     

Molecular geometry and chain entanglement: parameters for the tube model

The tube diameter in the reptation model is the distance between a given chain segment and its nearest segment in adjacent chains. This dimention is thus related to the cross-sectional area of polymer chains and the nearest approach among chains, without effects of thermal fluctuation and steric repulsion. Prior calculated tube diameters are much larger, about 5 times, than the actual chain cross-sectional areas. This is ascribed to the local freedom required for mutual rearrangement among neighboring chain segments. This tube diameter concept seems to us to infer a relationship to the corresponding entanglement spacing. Indeed, we report here that the critical molecular weight, M_c, for the onset of entanglements is found to be M_c = 28 A/(〈R²〉₀/M), where A is the chain cross-sectional area and 〈R²〉₀ the mean-square end-to-end distance of a freely jointed chain of molecular weight M. The new, computed relationship between the critical number of backbone atoms for entanglement and the chain cross-sectional area of polymers, N_c = A^0,44, is concordant with the cross-sectional area of polymer chains being the parameter controlling the critical entanglement number of backbone atoms of flexible polymers.     

Predictions of a recently proposed non-markovian model for polymer melts and concentrated solutions

Several predictions for a recently proposed mesoscopic model for polymer melts and concentrated solutions is presented. It is a single Kramers chain model in which elementary motions of the Orwoll-Stockmayer type are allowed. However, for this model, the bead jumps are no longer given by a Markovian probability, but rather are described by a fractal “waiting-time” distribution function, with a single adjustable parameter β, which describes the long-time behavior of the distribution: ∼ 1/t^1+β. We find that the model predicts D ∼ 1/N² and η₀ ∼ N^3.4 for β ≈︁ 1.4, where N is the degree of polymerization. The generalized model predicts that the relaxation spectrum has a plateau regime whose height is independent of N, but whose width is strongly N dependent, in agreement with experiment. The model also predicts that rings will diffuse somewhat more slowly than linear chains of the same molecular weight (about 80% as fast), with the same scaling dependence on N as linear chains, also in agreement with preliminary data.     

Dielectric Relaxation of Type‐A Polymers in Melts and Solutions

This article describes the dielectric relaxation behavior of flexible polymer chains having the so‐called type‐A dipoles parallel along the chain backbone. This behavior reflects the global chain motion. Viscoelastically well known features of this motion, such as the power‐law relationship between the relaxation time and molecular weight of entangled linear chains (τ₁ ∝ M^3.5), are also observed dielectrically. More importantly, the dielectric behavior of linear chains having once‐inverted type‐A dipoles enables us to find some detailed dynamic features such as changes in the eigenfunctions f_p of a local correlation function with the chain concentration in solutions. These changes are discussed in relation to motional coupling of concentrated chains. The dielectric properties detect the orientational correlation of two submolecules in the chain at two separate times, while the viscoelastic properties reflect the isochronal orientational anisotropy of individual submolecules. Thus the chain motion is differently averaged in the dielectric and viscoelastic properties, and comparison of these properties enables us to find novel dynamic features. Specifically, this comparison reveals the validity of the tube dilation molecular picture for entangled linear chains and weakening of the short‐time coherence of the submolecule motion due to the constraint release mechanism. Moreover, the dielectric method enables us to investigate the chain dynamics under strong flow and/or in a molecularly narrow space. In particular, the retarded dielectric relaxation found for homopolymers and block copolymers in such narrow spaces (in the microdomains for the latter) indicates important effects of the spatial and thermodynamic constraints on the global chain motion. All the above results in turn demonstrate the importance of the dielectric method in investigations of the polymer dynamics.     

A Monte Carlo study of a tethered polymer chain in a uniform field

We study the non‐uniform stretching and relaxation of a long flexible end‐anchored polymer chain of N monomers (32 ≤ N ≤ 1 024) in a uniform field B by means of an off‐lattice bead‐spring Monte Carlo model. Our simulational results for the case of a Rouse‐like polymer in the good solvent regime confirm the existence of “trumpet”‐ and “flower”‐type chain conformations, predicted recently by scaling analysis based on the notion of Pincus tensile blobs. The observed elongation of the chain and the critical fields, separating three different regimes of chain deformation, are found to obey the predicted scaling behavior. The segment density distribution matches that of a DNA molecule pulled from one end at constant velocity in a good solvent. As expected, the relaxation of the stretch to coil transition of the polymer of length N is determined by the typical Rouse time τ ∝ N^2ν+1.     

Unusual adsorption behavior of polymers in layered structures

We studied the conformations of polymer chains embedded in regular lamellar matrices such as microphase-separated block copolymer melts or lipid/water systems. In such quasi-one-dimensional systems, the host plays the role of a nearly-periodic external potential. A probe chain can get trapped with the localization length L at isolated defects, when its degree of polymerization N exceeds a cross-over value N_L. Interestingly, the radius of gyration, R_g, of the chain displays a minimum as a function of the defect density, ρ, the minimum being estimated by ρ_cL ≃ N_L/N. Using Monte Carlo simulation, we underline the non-monotonic behavior of R_g and calculate Green's function for various defect densities. On the same vein, the mean square displacement of the monomer units at large times shows a pronounced minimum as a function of the defect density, ρ.     

Cyano‐Bridged MnIII?MIII Single‐Chain Magnets with MIII=CoIII,FeIII, MnIII,and CrIII

A series of isostructural cyano‐bridged Mn^III(h.s.)–M^III(l.s.) alternating chains, [Mn^III(5‐TMAMsalen)M^III(CN)₆] ? 4H₂O (5‐TMAMsalen^2?=N,N′‐ethylenebis(5‐trimethylammoniomethylsalicylideneiminate), Mn^III(h.s.)=high‐spin Mn^III, M^III(l.s.)=low‐spin Co^III, Mn? Co ; Fe^III, Mn? Fe ; Mn^III, Mn? Mn ; Cr^III, Mn? Cr ) was synthesized by assembling [Mn^III(5‐TMAMsalen)]³⁺ and [M^III(CN)₆]^3?. The chains present in the four compounds, which crystallize in the monoclinic space group C2/c, are composed of an [‐Mn^III‐NC‐M^III‐CN‐] repeating motif, for which the ‐NC‐M^III‐CN‐ motif is provided by the [M^III(CN)₆]^3? moiety adopting a trans bridging mode between [Mn^III(5‐TMAMsalen)]³⁺ cations. The Mn^III and M^III ions occupy special crystallographic positions: a C₂ axis and an inversion center, respectively, forming a highly symmetrical chain with only one kind of cyano bridge. The Jahn–Teller axis of the Mn^III(h.s.) ion is perpendicular to the N₂O₂ plane formed by the 5‐TMAMsalen tetradentate ligand. These Jahn–Teller axes are all perfectly aligned along the unique chain direction without a bending angle, although the chains are corrugated with an Mn‐N_axis‐C angle of about 144°. In the crystal structures, the chains are well separated with the nearest inter‐chain M???M distance being relatively large at 9 Å due to steric hindrance of the bulky trimethylammoniomethyl groups of the 5‐TMAMsalen ligand. The magnetic properties of these compounds have been thoroughly studied. Mn? Fe and Mn? Mn display intra‐chain ferromagnetic interactions, whereas Mn? Cr is characterized by an antiferromagnetic exchange that induces a ferrimagnetic spin arrangement along the chain. Detailed analyses of both static and dynamic magnetic properties have demonstrated without ambiguity the single‐chain magnet (SCM) behavior of these three systems, whereas Mn? Co is merely paramagnetic with S_Mn=2 and D/k_B=?5.3 K (D being a zero‐field splitting parameter). At low temperatures, the Mn? M compounds with M=Fe, Mn, and Cr display remarkably large M versus H hysteresis loops for applied magnetic fields along the easy magnetic direction that corresponds to the chain direction. The temperature dependence of the associated relaxation time for this series of compounds systematically exhibits a crossover between two Arrhenius laws corresponding to infinite‐chain and finite‐chain regimes for the SCM behavior. These isostructural hetero‐spin SCMs offer a unique series of alternating [‐Mn‐NC‐M‐CN‐] chains, enabling physicists to test theoretical SCM models between the Ising and Heisenberg limits.     

A novel one‐dimensional double‐chain‐like ZnII coordination polymer: poly[bis(1‐benzyl‐1H‐imidazole‐κN3)tris(μ‐cyanido‐κ2C:N)(cyanido‐κC)disilver(I)zinc(II)]

One of most interesting systems of coordination polymers constructed from the first‐row transition metals is the porous Zn^II coordination polymer system, but the numbers of such polymers containing N‐donor linkers are still limited. The title double‐chain‐like Zn^II coordination polymer, [Ag₂Zn(CN)₄(C₁₀H₁₀N₂)₂]_n, presents a one‐dimensional linear coordination polymer structure in which Zn^II ions are linked by bridging anionic dicyanidoargentate(I) units along the crystallographic b axis and each Zn^II ion is additionally coordinated by a terminal dicyanidoargentate(I) unit and two terminal 1‐benzyl‐1H‐imidazole (BZI) ligands, giving a five‐coordinated Zn^II ion. Interestingly, there are strong intermolecular Ag^I…Ag^I interactions between terminal and bridging dicyanidoargentate(I) units and C—H…π interactions between the phenyl rings of BZI ligands of adjacent one‐dimensional linear chains, providing a one‐dimensional linear double‐chain‐like structure. The supramolecular three‐dimensional framework is stabilized by C—H…π interactions between the phenyl rings of BZI ligands and by Ag^I…Ag^I interactions between adjacent double chains. The photoluminescence properties have been studied.     

Chain behavior of an amphoteric cellulose derivative: O-carboxymethyl-O-2-hydroxy-3-(trimethylammonio) propylcellulose

The ¹³C NMR spin-lattice relaxation times (T₁) of anhydroglucose units vary with the number of substituents, and the T₁ values of unsubstituted anhydroglucose units of O-carboxymethylcellulose are longer than those of amylose. Those results indicate that in water, the rotational motions of anhydroglucose units of cellulose derivative are quite important local motions contributing to the ¹³C NMR spin-lattice relaxation, and within a cellulose chain, anhydroglucose units rotate with different degrees of freedom depending on their environment. Moreover, the ¹³C NMR spin-lattice relaxation data indicate that the mobilities of ionic substituents are dependent on substitution positions as well as their ionic interaction. © 1995 John Wiley & Sons, Inc.     

Using the reversible addition–fragmentation chain transfer process to synthesize core‐crosslinked micelles

Poly(2‐hydroxyethyl acrylate)–poly(n‐butyl acrylate) block copolymers were synthesized with the reversible addition–fragmentation chain transfer (RAFT) process. The block copolymers were synthesized successfully with either poly(2‐hydroxyethyl acrylate) or poly(n‐butyl acrylate) macro‐RAFT agents. The resulting block copolymers had narrow molecular weight distributions (polydispersity index = 1.3–1.4). Copolymer self‐aggregation in water yielded micelles, with the hydrodynamic diameter (D_h) values of the aggregates dependent on the length of both blocks according to D_h ～ N_BA^1.17N_HEA^0.57, where N_BA is the number of repeating units of n‐butyl acrylate and N_HEA is the number of repeating units of 2‐hydroxyethyl acrylate. The micelles were subsequently stabilized via chain extension of the block copolymer with a crosslinking agent. The successful chain extension in a micellar system was confirmed by an increase in the molecular weight, which was detected with membrane osmometry. The crosslinked particles showed noticeably different aggregation behavior in diverse solvent systems. The uncrosslinked micelles formed by the block copolymer (N_HEA = 260, N_BA = 75) displayed a definite critical micelle concentration at 5.4 × 10^?4 g L^?1 in aqueous solutions. However, upon crosslinking, the critical micelle concentration transition became obscure. © 2006Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 2177–2194, 2006     

Chiral Supramolecular Nanotubes of Single‐Chain Magnets

We report a single‐chain magnet (SCM) made of a terbium(III) building block and a nitronyl‐nitroxide radical (NIT) functionalized with an aliphatic chain. This substitution is targeted to induce a long‐range distortion of the polymeric chain and accordingly it gives rise to chains that are curled with almost 20 nm helical pitch. They self‐organize as a chiral tubular superstructure made of 11 chains wound around each other. The supramolecular tubes have a 4.5 nm internal diameter. Overall, this forms a porous chiral network with almost 44 % porosity. Ab initio calculations highlight that each Tb^III ion possesses high magnetic anisotropy. Indeed, notwithstanding the supramolecular arrangement each chain behaves as a SCM. Magnetic relaxation with both finite and infinite‐size regimes is observed and confirms the validity of the Ising approximation. This is associated with quite strong coercive field and magnetic remanence (H_c=2400 Oe M_R=2.09 μ_B at 0.5 K) for this class of compounds.     

Three‐Dimensional Antiferromagnetic Order of Single‐Chain Magnets: A New Approach to Design Molecule‐Based Magnets

Two one‐dimensional compounds composed of a 1:1 ratio of Mn^III salen‐type complex and Ni^II oximato moiety with different counter anions, PF₆^? and BPh₄^?, were synthesized: [Mn(3,5‐Cl₂saltmen)Ni(pao)₂(phen)]PF₆ ( 1 ) and [Mn(5‐Clsaltmen)Ni(pao)₂(phen)]BPh₄ ( 2 ), where 3,5‐Cl₂saltmen^2?=N,N′‐(1,1,2,2‐tetramethylethylene)bis(3,5‐dichlorosalicylideneiminate); 5‐Clsaltmen^2?=N,N′‐(1,1,2,2‐tetramethylethylene)bis(5‐chlorosalicylideneiminate); pao^?=pyridine‐2‐aldoximate; and phen=1,10‐phenanthroline. Single‐crystal X‐ray diffraction study was carried out for both compounds. In 1 and 2 , the chain topology is very similar forming an alternating linear chain with a [‐Mn^III‐ON‐Ni^II‐NO‐] repeating motif (where ‐ON‐ is the oximate bridge). The use of a bulky counteranion, such as BPh₄^?, located between the chains in 2 rather than PF₆^? in 1 , successfully led to the magnetic isolation of the chains in 2 . This minimization of the interchain interactions allows the study of the intrinsic magnetic properties of the chains present in 1 and 2 . While 1 and 2 possess, as expected, very similar paramagnetic properties above 15 K, their ground state is antiferromagnetic below 9.4 K and paramagnetic down to 1.8 K, respectively. Nevertheless, both compounds exhibit a magnet‐type behavior at temperatures below 6 K. While for 2 , the observed magnetism is well explained by a Single‐Chain Magnet (SCM) behavior, the magnet properties for 1 are induced by the presence in the material of SCM building units that order antiferromagnetically. By controlling both intra‐ and interchain magnetic interactions in this new [Mn^IIINi^II] SCM system, a remarkable AF phase with a magnet‐type behavior has been stabilized in relation with the intrinsic SCM properties of the chains present in 1 . This result suggests that the simultaneous enhancement of both intrachain (J) and interchain (J′) magnetic interactions (with keeping J ? J′), independently of the presence of AF phase might be an efficient route to design high temperature SCM‐based magnets.     

Chains of (4,4′‐bipyridyl)copper(I) bridged by dimolybdate units

The crystal structure of the title compound, poly­[bis‐[copper(I)‐μ‐(4,4′‐bipyridyl)‐N:N′]‐μ‐dimolybdato‐O:O′],[Cu₂(C₁₀H₈N₂)₂{Mo₂O₇}]_n, consists of {Mo₂O₇}^2? units (with the central O atom lying on twofold symmetry axes) and [Cu(4,4′‐bipy)]_nⁿ⁺ chains (bipy = bipyridyl); the chains are generated by a c‐glide‐plane operation. The {Mo₂O₇}^2? units are covalently bridged to two [Cu(4,4′‐bipy)]_nⁿ⁺ chains, forming a complex with a bridged double‐chain structure. The Cu—O and Cu—N distances are 2.191 (3) and 1.933 (3) Å, respectively.     

Computer simulation of brownian motion of a polyelectrolyte molecule (charged bead-spring model)

A Langevin equation of motion for a charged bead-spring statistical chain is written in difference form and the relaxation and equilibrium behavior of the chain is studied by computer simulation. Results are presented for the behavior of end-to-end length h, principal axes of the polymer ellipsoid L₁, L₂, L₃, and chain contour length c in terms of their averages, root mean square values, root mean square fluctuations, orientations, and relaxation strengths and times. The simulation was made with various sets of parameters, bead number N, charge on the bead q, and radius of ion atmosphere around the bead k^?1. It is found that 〈h²〉^1/2 and 〈L₁²〉^1/2 increase more strongly with increasing q and decreasing κ than 〈L₂²〉^1/2, 〈L₃²〉^1/2, and 〈c₁²〉^1/2, indicating that the chain is expanded in three dimensions and at the same time is extended along the end-to-end direction. The relaxation time τ_rot of rotation of the end-to-end vector, which is proportional to N² at q = 0, increases with increasing q and tends to be proportional to N³ for an extended chain, while the relaxation time τ_conf of the magnitude of h is almost independent of q and is always proportional to N². It is concluded that the extended chain possesses a well-defined end-to-end axis and the chain rotates as a whole with a relaxation time τ_rot which is much longer than τ_conf. The complex viscosity of the chain is calculated from the Fourier transform of the time–correlation function of momentum flux and is found to have a frequency spectrum similar to that observed for aqueous solutions of poly(acrylic acid). The dominant mode appearing in the low-frequency range is evidenced to arise from the rotation of the extended chain.     

Molecular Weight Development in Free‐Radical Polymerization with Polyfunctional Chain‐Transfer Agents, 1. Equal Reactivity Model

The analytic expression for the weight‐average molecular weight development in free‐radical polymerization that involves a polyfunctional chain‐transfer agent is proposed. Free‐radical polymerization is kinetically controlled; therefore, the probability of chain connection with a polyfunctional chain‐transfer agent as well as the primary chain‐length distribution changes during the course of polymerization. We consider the primary chains formed at different times as different types of chains, and the heterochain branching model is used to obtain the weight‐average chain length at a given conversion level in a matrix formula, described as P_w = W { D _w + ( I + T ) SP ( I – TSP )^–1 D_f }. Because the primary chains are formed consecutively, the number of chain types N is extrapolated to infinity, but such extrapolation can be conducted with the calculated values for only three different N values. The criterion for the onset of gelation is simply described as a point at which the largest eigenvalue of the product of matrixes, TSP reaches unity, i. e., det  ( I – TSP ) = 0. The present model can readily be extended for the star‐shaped polyfunctional initiators, and the relationships between the model parameters and kinetic rate expression for such reaction systems are also shown.     

Chiral Supramolecular Nanotubes of Single-Chain Magnets

We report a single-chain magnet (SCM) made of a terbium(III) building block and a nitronyl-nitroxide radical (NIT) functionalized with an aliphatic chain. This substitution is targeted to induce a long-range distortion of the polymeric chain and accordingly it gives rise to chains that are curled with almost 20 nm helical pitch. They self-organize as a chiral tubular superstructure made of 11 chains wound around each other. The supramolecular tubes have a 4.5 nm internal diameter. Overall, this forms a porous chiral network with almost 44 % porosity. Ab initio calculations highlight that each Tb^III ion possesses high magnetic anisotropy. Indeed, notwithstanding the supramolecular arrangement each chain behaves as a SCM. Magnetic relaxation with both finite and infinite-size regimes is observed and confirms the validity of the Ising approximation. This is associated with quite strong coercive field and magnetic remanence (H_c=2400 Oe M_R=2.09 μ_B at 0.5 K) for this class of compounds.     

Osmotic pressure,atomic pressure and the virial equation of state of polymer solutions: Monte Carlo simulations of a bead-spring model

A recently introduced coarse-grained model of polymer chains is studied analyzing various contributions to the pressure as obtained from the virial theorem as a function of chain length N, temperature T and density ϕ. The off-lattice model of the polymer chains has anharmonic springs between the beads, but of finite extensibility, and the Morse-type interaction between beads is repulsive at very short distances and attractive at intermediate distances. Solvent molecules are not explicitly included. It is found that the covalent forces along the chain (modelled by the spring potentials) contribute a negative term to the pressure, irrespective of temperature, which vanishes linearly in ϕ as ϕ → 0. In contrast, both contributions to the pressure due to intrachain nonbonded forces and due to forces between different chains change sign from high temperatures (T ≫ θ, θ the theta-temperature) where they are positive, to low temperature where both parts of the pressure become negative. It is shown that the total pressure has the expected behavior with temperature near the θ-temperature, i.e., Δp ≡ p_tot − k_B · T_p ∼ (T − θ). We study also the concentration and chainlength dependence of the various contributions to the pressure in the good solvent regime and interpret them with scaling predictions.