Dynamics of star branched polymers in a matrix of linear chains — a Monte Carlo study

A simple cubic lattice model of the melt of 3-arm star-branched polymers of various length dissolved in a matrix of long linear chains (n₁ = 800 beads) is studied using a dynamic Monte Carlo method. The total polymer volume fraction is equal to 0,5, while the volume fraction of the star polymers is about ten times smaller. The static and dynamic properties of these systems are compared with the corresponding model systems of isolated star-branched polymers and with the melt of linear chains. It has been found that the number of dynamic entanglements for the star polymers with arm length up to 400 segments is too small for the onset of the arm retraction mechanism of polymer relaxation. In this regime dynamics of star-branched polymers is close to the dynamics of linear polymers at corresponding concentration and with equivalent chain length. The entanglement length for star polymers appears to be somewhat larger compared with linear chains.     

Molecular tube models of branched polymer melts

We apply the tube model to a specific problem in polymer melt dynamics - the rheology of star polymers as an additive to a monodisperse linear matrix. We find that the tube dilation picture of constraint release may be applied to the relaxation of the star fraction. There are four qualitatively different cases depending on the relative concentrations and relaxation times of the two fractions. Terminal relaxation times and relaxation spectra are calculated for each case and compared with available experimental data. The existence of a modified Rouse relaxation such that G(t)∼t^−3/2 is argued.     

Rheology of Polydisperse Star Polymer Melts: Extension of the Parameter‐Free Tube Model of Milner and McLeish to Arbitrary Arm‐Length Polydispersity

Summary: This paper considers the extension of the parameter‐free tube model of Milner and McLeish for stress relaxation in melts of monodisperse star polymers to star polymers whose arms have a continuous molecular weight distribution such as the Flory distribution in the case of star‐nylons and star‐polyesters. Exact expressions are derived for the relaxation spectrum and the relaxation modulus for star polymers having an arbitrary continuous arm‐length distribution. For a Flory distribution a comparison is made with results of dynamic measurements on a melt of 8‐arm poly(ε‐caprolactone) (PCL) stars. An excellent quantitative agreement over a large frequency range is found, however, only if one treats, in contrast with the original parameter‐free tube model approach, the entanglement molecular weight that determines the relaxation spectrum as a fitting parameter independent of the entanglement molecular weight of the linear PCL. This discrepancy is not in anyway related to the polydispersity in arm‐length, but a consequence of the thermorheological complexity of the PCL stars. A similar discrepancy has been observed for hydrogenated polybutadiene stars, as described by Levine and Milner.

Measured and calculated storage moduli G′ as a function of frequency ω.     

Computer simulation of three‐arm star polymers

We show that Shaffer's version of the bond fluctuation model can be used to simulate three‐arm star polymers. We report a simulation study of both single stars and melts of star polymers with arm lengths up to 90 monomer units (approximately twice the entanglement crossover length for linear chains). Center‐of‐mass self‐diffusion of single stars is Rouse‐like (D ˜ N^–1). Due to a limited range of molecular weights we cannot distinguish between a power‐law and an exponential dependence of the star‐melt self‐diffusion coefficient on arm length.     

Full‐chain dynamics of entangled linear and star polymers

The full‐chain dynamics and the linear viscoelastic properties of monodisperse, entangled linear and star polymers are simulated consistently via an equilibrium stochastic algorithm, based on a recently proposed full‐chain reptation theory 1 that is able to treat self‐consistently mechanisms of chain reptation, chain‐length fluctuations, and constraint release. In particular, it is the first time that the full‐chain simulation for star polymers is performed without subjecting to the great simplifications usually made. To facilitate the study on linear viscoelasticity, we employ a constraint release mechanism that resembles the idea of tube dilation, in contrast to the one used earlier in simulating flows, where constraint release was performed in a fashion similar to double reptation. Predictions of the simulation are compared qualitatively and quantitatively with experiments, and excellent agreement is found for all investigated properties, which include the scaling laws for the zero‐shear‐rate viscosity and the steady‐state compliance as well as the stress relaxation and dynamic moduli, for both polymer systems. The simulation for linear polymers indicates that the full‐chain reptation theory considered is able to predict very well the rheology of monodisperse linear polymers under both linear viscoelastic and flow conditions. The simulation for star polymers, on the other hand, strongly implies that double reptation alone is insufficient, and other unexplored mechanisms that may further enhance stress relaxation of the tube segments near the star center seem crucial, in explaining the linear viscoelasticity of star polymers. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 248–261, 2000     

Synthesis of high‐order multiblock core cross‐linked star polymers

Core cross‐linked star (CCS) polymers with radiating arms composed of high‐order multiblock copolymers have been synthesized in a one‐pot system via iterative copper‐mediated radical polymerization. The employed “arm‐first” technique ensures the multiblock sequence of the macroinitiator is carried through to the star structure with no arm defects. The versatility of this approach is demonstrated by the synthesis of three distinct star polymers with differing arm compositions, two with an alternating ABABAB block sequence and one with six different block units (i.e. ABCDEF). Owing to the star architecture, CCS polymers in which the arm composition consists of alternating hydrophilic–hydrophobic (ABABAB) segments undergo supramolecular self‐assembly in selective solvents, whereas linear polymers with the same block sequence did not yield self‐assembled structures, as evidenced by DLS analysis. The combination of microstructural and topological control in CCS polymers offers exciting possibilities for the development of tailor‐made nanoparticles with spatially defined regions of functionality. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2016 , 54, 135–143     

Star PolyMOCs with Diverse Structures,Dynamics, and Functions by Three‐Component Assembly

We report star polymer metal–organic cage (polyMOC) materials whose structures, mechanical properties, functionalities, and dynamics can all be precisely tailored through a simple three‐component assembly strategy. The star polyMOC network is composed of tetra‐arm star polymers functionalized with ligands on the chain ends, small molecule ligands, and palladium ions; polyMOCs are formed via metal–ligand coordination and thermal annealing. The ratio of small molecule ligands to polymer‐bound ligands determines the connectivity of the MOC junctions and the network structure. The use of large M₁₂L₂₄ MOCs enables great flexibility in tuning this ratio, which provides access to a rich spectrum of material properties including tunable moduli and relaxation dynamics.     

Star PolyMOCs with Diverse Structures,Dynamics, and Functions by Three‐Component Assembly

We report star polymer metal–organic cage (polyMOC) materials whose structures, mechanical properties, functionalities, and dynamics can all be precisely tailored through a simple three‐component assembly strategy. The star polyMOC network is composed of tetra‐arm star polymers functionalized with ligands on the chain ends, small molecule ligands, and palladium ions; polyMOCs are formed via metal–ligand coordination and thermal annealing. The ratio of small molecule ligands to polymer‐bound ligands determines the connectivity of the MOC junctions and the network structure. The use of large M₁₂L₂₄ MOCs enables great flexibility in tuning this ratio, which provides access to a rich spectrum of material properties including tunable moduli and relaxation dynamics.     

Efficient water‐soluble drag reducing star polymers with improved mechanical stability

We describe here the first example of the synthesis of 4‐arm star poly(acrylic acid) for use as a water‐soluble drag reducing agent, by applying Cu(0)‐mediated polymerization technique. High molecular weight 4‐arm star poly(tert‐butyl acrylate) (M_n = 3.0–9.0 × 10⁵ g mol^?1) was first synthesized using 4,4′‐oxybis(3,3‐bis(2‐bromopropionate)butane as an initiator and a simple Cu(0)/TREN catalyst system. Then, 4‐arm star poly(tert‐butyl acrylate) were subjected to hydrolysis using trifluoroacetic acid resulting in water‐soluble 4‐arm star poly(acrylic acid). Drag reduction test rig analysis showed 4‐arm star poly(acrylic acid) to be effective as a drag reducing agent with drag reduction of 24.3%. Moreover, 4‐arm star poly(acrylic acid) exhibited superior mechanical stability when compared with a linear poly(acrylic acid) and commercially available drag reducing polymers; Praestol and poly(ethylene oxide). The linear poly(acrylic acid), Praestol, and poly(ethylene oxide) all showed a large decrease in drag reduction of 8–12% when cycled 30 times through the drag reduction test rig while, in contrast, 4‐arm star poly(acrylic acid) demonstrated much higher mechanical stability. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2016 , 54, 335–344     

Linear viscoelasticity of poly(acrylonitrile-co-itaconic acid)/1-butyl-3-methylimidazolium chloride extended from dilute to concentrated solutions

The solution rheology of poly(acrylonitrile-co-itaconic acid) (poly(AN-co-IA)) in 1-butyl-3-methylimidazolium chloride ([BMIM]Cl) spanning dilute, semidilute unentangled and entangled regimes were investigated. The exponents in the specific viscosity η_sp ～ overlap parameter c[η] power law were 1, 2 and 4.7 for dilute, semidilute unentangled and entangled regimes, respectively, which were found to be consistent with the scaling prediction for neutral linear polymers in θ-solvent. For dilute solutions (lower than 0.9 wt.%), the linear viscoelastic responses were observed to be in good agreement with the Zimm model (Flory exponent ν = 0.5). While for semidilute unentangled solutions (between 0.9 and 8 wt.%), results obtained had been found to be consistent with Rouse model. Considering Flory exponent ν = 0.5 and the concentration dependences of the specific viscosity and relaxation time, it had been evaluated that poly(AN-co-IA) in [BMIM]Cl behaves as a neutral polymer in θ-solvent. It had also been suggested that according to the unusual deviation of Cox-Merz rule, poly(AN-co-IA)/[BMIM]Cl solutions are typical neutral polymeric liquids for the concentrated solutions but have shown a more complicated behavior at high deformation rates.     

Synthesis of well‐defined 7‐arm and 21‐arm poly(N‐isopropylacrylamide) star polymers with β‐cyclodextrin cores via click chemistry and their thermal phase transition behavior in aqueous solution

The syntheses of well‐defined 7‐arm and 21‐arm poly(N‐isopropylacrylamide) (PNIPAM) star polymers possessing β‐cyclodextrin (β‐CD) cores were achieved via the combination of atom transfer radical polymerization (ATRP) and click reactions. Heptakis(6‐deoxy‐6‐azido)‐β‐cyclodextrin and heptakis[2,3,6‐tri‐O‐(2‐azidopropionyl)]‐β‐cyclodextrin, β‐CD‐(N₃)₇ and β‐CD‐(N₃)₂₁, precursors were prepared and thoroughly characterized by nuclear magnetic resonance and matrix‐assisted laser desorption/ionization time‐of‐flight mass spectrometry. A series of alkynyl terminally functionalized PNIPAM (alkyne‐PNIPAM) linear precursors with varying degrees of polymerization (DP) were synthesized via atom transfer radical polymerization (ATRP) of N‐isopropylacrylamide using propargyl 2‐chloropropionate as the initiator. The subsequent click reactions of alkyne‐PNIPAM with β‐CD‐(N₃)₇ and β‐CD‐(N₃)₂₁ led to the facile preparation of well‐defined 7‐arm and 21‐arm star polymers, namely β‐CD‐(PNIPAM)₇ and β‐CD‐(PNIPAM)₂₁. The thermal phase transition behavior of 7‐arm and 21‐arm star polymers with varying molecular weights were examined by temperature‐dependent turbidity and micro‐differential scanning calorimetry, and the results were compared to those of linear PNIPAM precursors. The anchoring of PNIPAM chain terminal to β‐CD cores and high local chain density for star polymers contributed to their considerably lower critical phase separation temperatures (T_c) and enthalpy changes during phase transition as compared with that of linear precursors. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 404–419, 2009     

Arm‐first synthesis of star polymers with polywedge arms using ring‐opening metathesis polymerization and bifunctional crosslinkers

This work presents a two‐step, one‐pot process to make star polymers with polywedge arms. In a one‐pot reaction, after the polywedge arms are synthesized, crosslinker species are added to the reaction, rapidly forming star polymers. Crosslinker species with different degrees of conformational freedom were designed and synthesized and their capacity to generate star polymers was evaluated. Mass conversions up to 92% and stars with up to 17 arms were synthesized with the most rigid crosslinker. The effects of arm molecular weight and molar ratio of crosslinker to arm on mass conversion and arms per star were explored further. Finally, the size‐molecular weight scaling relationship for polywedges with linear and star architectures was compared, corroborating theoretical results regarding star polymers with arms much larger than their core. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2018 , 56, 732–740     

Significance of cross correlations in the stress relaxation of polymer melts

According to linear response theory, all relaxation functions in the linear regime can be obtained using time correlation functions calculated under equilibrium. In this paper, we demonstrate that the cross correlations make a significant contribution to the partial stress relaxation functions in polymer melts. We present two illustrations in the context of polymer rheology using (1) Brownian dynamics simulations of a single chain model for entangled polymers, the slip-spring model, and (2) molecular dynamics simulations of a multichain model. Using the single chain model, we analyze the contribution of the confining potential to the stress relaxation and the plateau modulus. Although the idea is illustrated with a particular model, it applies to any single chain model that uses a potential to confine the motion of the chains. This leads us to question some of the assumptions behind the tube theory, especially the meaning of the entanglement molecular weight obtained from the plateau modulus. To shed some light on this issue, we study the contribution of the nonbonded excluded-volume interactions to the stress relaxation using the multichain model. The proportionality of the bonded/nonbonded contributions to the total stress relaxation (after a density dependent "colloidal" relaxation time) provides some insight into the success of the tube theory in spite of using questionable assumptions. The proportionality indicates that the shape of the relaxation spectrum can indeed be reproduced using the tube theory and the problem is reduced to that of finding the correct prefactor.     

Primitive chain network simulations for asymmetric star polymers

For branched polymers, the curvilinear motion of the branch point along the backbone is a significant relaxation source but details of this motion have not been well understood. This study conducts multi-chain sliplink simulations to examine effects of the spatial fluctuation and curvilinear hopping of the branch point on the viscoelastic relaxation. The simulation is based on the primitive chain network model that allows the spatial fluctuations of sliplink and branch point and the chain sliding along the backbone according to the subchain tension, chemical potential gradients, drag force against medium, and random force. The sliplinks are created and∕or disrupted through the motion of chain ends. The curvilinear hopping of the branch point along the backbone is allowed to occur when all sliplinks on a branched arm are lost. The simulations considering the fluctuation and the hopping of the branch point described well the viscoelastic data for symmetric and asymmetric star polymers with a parameter set common to the linear polymer. On the other hand, the simulations without the branch point motion predicted unreasonably slow relaxation for asymmetric star polymers. For asymmetric star polymers, further tests with and without the branch point hopping revealed that the hopping is much less important compared to the branch point fluctuation when the lengths of the short and long backbone arms are not very different and the waiting time for the branch point hopping (time for removal of all sliplinks on the short arm) is larger than the backbone relaxation time. Although this waiting time changes with the hopping condition, the above results suggest a significance of the branch point fluctuation in the actual relaxation of branch polymers.     

Linear rheology of water-soluble reversible neodymium(III) coordination polymers

The rheology of reversible coordination polymer networks in aqueous solution is studied. The polymers are formed by neodymium(III) ions and bifunctional ligands, consisting of two pyridine-2,6-dicarboxylate groups connected at the 4-positions by an ethylene oxide spacer. Neodymium(III) ions can bind three of these terdendate ligand groups. At high concentrations, the polymer networks yield viscoelastic materials, which can be described with the Maxwell model. The scaling of the elastic modulus, relaxation time, and zero-shear viscosity with concentration are in good agreement with the predictions of Cates' model that describes the dynamics of linear equilibrium polymers. This indicates that the networks have only few cross-links and can be described as linear equilibrium polymers. The gels are also thermo-reversible. At high temperatures, fast relaxation was found, resulting in liquidlike behavior. Upon cooling, the viscoelastic properties returned immediately. From the temperature dependence of the relaxation time, an activation energy of 49 kJ/mol was determined for the breaking and reptation of the polymers.     

Synthesis and Rheological Investigation ofModel Symmetric 3-Arm Star Polyethylene简

A variety of linear and 3-arm star polyethylene （PE） model polymers covering a wide range of molecular weight are synthesized by the living polymerization of butadiene and the subsequent hydrogenation. Several rheological parameters of these model linear and 3-arm star PE samples are analyzed for detecting the long chain branching （LCB） structure. It is found that the analyses based on zero shear viscosity, vGP plot and flow activation energy are very sensitive to the 3-arm star PEs. The information on the presence of LCB can be obtained with these methods even for low molecular weight samples, which can not be determined by GPC-MALLS. However the information about the LCB structure can not be obtained by the rheological methods alone.     

Synthesis and characterization of poly(ethylene oxide) star polymers possessing a tertiary amino group at each arm end by organized polymerization using macromonomers

Functional poly(ethylene oxide) star polymers possessing a tertiary amino group at each arm end were prepared by free-radical copolymerization of poly(ethylene oxide) macromonomers with divinylbenzene (DVB) in water or ethanol. The poly(ethylene oxide) arm was prepared by anionic polymerization using 2-[2-(N,N-dimethylamino)ethoxy]ethanol potassium alkoxide as the initiator. The star polymers had narrow molecular weight distribution. The arm number was controlled by varying the feed ratio [DVB]/[M], the initial concentration of macromonomer [M], and solvent media. The branching factor g' in methanol ([eta]S/[eta]L are the intrinsic viscosities of the star and linear molecules, respectively) exhibited a power-law dependence on the arm number, f, with a negative exponent. This means that the dimensions of a star were in agreement with the Daoud-Cotton scaling model.     

Synthesis and characterization of 4‐arm star side‐chain liquid crystalline polymers containing azobenzene with different terminal substituents via ATRP

4‐Arm star side‐chain liquid crystalline (LC) polymers containing azobenzene with different terminal substituents were synthesized by atom transfer radical polymerization (ATRP). Tetrafunctional initiator prepared by the esterification between pentaerythritol and 2‐bromoisobutyryl bromide was utilized to initiate the polymerization of 6‐[4‐(4‐methoxyphenylazo)phenoxy]hexyl methacrylate (MMAzo) and 6‐[4‐(4‐ethoxyphenylazo)phenoxy]hexyl methacrylate (EMAzo), respectively. The 4‐arm star side‐chain LC polymer with p‐methoxyazobenzene moieties exhibits a smectic and a nematic phase, while that with p‐ethoxyazobenzene moieties shows only a nematic phase, which derives of different terminal substituents. The star polymers have similar LC behavior to the corresponding linear homopolymers, whereas transition temperatures decrease slightly. Both star polymers show photoresponsive isomerization under the irradiation with UV–vis light. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 3342–3348, 2007     

Arm-first method as a simple and general method for synthesis of miktoarm star copolymers

Miktoarm star copolymers containing two or more arm species were synthesized by atom transfer radical polymerization using a simple and general "arm-first" method, that is, one-pot cross-linking a mixture of different linear macroinitiator (MI) species by a divinyl cross-linker, such as divinylbenzene. Using linear MIs with a high degree of bromine chain-end functionality, including polyacrylate, polystyrene, polymethacrylate and poly(ethylene oxide), resulted in high-yield star polymers (>90%). Characterized by liquid adsorption chromatography techniques, which separated star polymers on the basis of the chemical composition of arms, the obtained star product was proved to be miktoarm star copolymers containing two or more arm species in one molecule, instead of mixture of different homoarm star polymers. Within our investigation, the molar ratios of the arms in the miktoarm star copolymers were always in agreement with the composition of the initial MI mixture, indicating the powerful capacity of this arm-first method for synthesis of miktoarm star copolymers with potentially any molar ratios and species of the arms. By using a mixture containing five types of linear MIs with different chemical compositions, miktoarm star copolymers containing five kinds of arms were synthesized for the first time, which significantly expanded the methodologies for synthesis of miktoarm star copolymers by living polymerization techniques.