Integral equation theory study on the phase separation in star polymer nanocomposite melts

The polymer reference interaction site model theory is used to investigate phase separation in star polymer nanocomposite melts. Two kinds of spinodal curves were obtained: classic fluid phase boundary for relatively low nanoparticle-monomer attraction strength and network phase boundary for relatively high nanoparticle-monomer attraction strength. The network phase boundaries are much more sensitive with nanoparticle-monomer attraction strength than the fluid phase boundaries. The interference among the arm number, arm length, and nanoparticle-monomer attraction strength was systematically investigated. When the arm lengths are short, the network phase boundary shows a marked shift toward less miscibility with increasing arm number. When the arm lengths are long enough, the network phase boundaries show opposite trends. There exists a crossover arm number value for star polymer nanocomposite melts, below which the network phase separation is consistent with that of chain polymer nanocomposite melts. However, the network phase separation shows qualitatively different behaviors when the arm number is larger than this value.     

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.     

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.     

Motion of star‐branched chains in polymer networks: a Monte Carlo study

A simple model of branched polymers in confined space is developed. Star‐branched polymer molecules are built on a simple cubic lattice with excluded volume and no attractive interactions (good solvent conditions). A single star molecule is trapped in a network of linear polymer chains of restricted mobility. The simulations are carried out using the classical Metropolis algorithm. Static and dynamic properties of the star‐branched polymer are determined using various networks. The dependence of the longest relaxation time and the self‐diffusion coefficient on chain length and network properties are discussed and the proper scaling laws formulated. The possible mechanism of motion is discussed. The differences between the motion of star‐branched polymers in such a network are compared with the cases of a dense matrix of linear chains and regular rod‐like obstacles.     

Conformations of (AB)n star copolymers in dilute solution

Molecular simulation of amphiphilic (AB)_n star diblock copolymers in dilute solution shows that the properties of the polymer are significantly affected by which of the ends of the diblock arm is attached to the center of the star. In dilute aqueous solution, stars having the solvophilic end of the diblock attached to the center of the star can undergo a dramatic conformational change in which the outer solvophobic blocks aggregate into one or more compact solvophobic globules. This aggregation transition is accompanied by a significant change in the size of the polymer as measured by the radius of gyration.     

Models for the dynamics of monodisperse polymer melts based on lateral chain motions

A recently developed model for the dynamics of monodisperse polymer melts of linear chains is briefly reviewed. Within the simplifications inherent in the model, it is found that the obstacles to the motion of a given chain, which are imposed by neighboring chains, do not suppress the lateral chain motion. The model associates a length scale with each obstacle, and compares it with the length scale for chain motion. If the obstacle length is greater than the length scale for chain motion, the obstacle is deemed impassable. The cooperative motion of the mutually impassable obstacles is considered, and this gives rise to predictions that are in excellent agreement with experimental observations. If the model were modified to include the additional complexities of real polymer systems, various features of the model might change. The implications of a number of possible modifications in the model are explored. Specifically, the impact of varying the behavior of the function which determines the fraction of obstacles that are impassable is examined in detail. In addition, in the original model it is assumed that chain memory is relaxed due to the slowing of lateral chain motion by the obstacles imposed by neighboring chains. The effect of the opposite assumption of essentially no memory relaxation is also studied. Finally, the influence of limiting the extent of the correlations between the motions of various chain segments because of finite chain length is also considered. It is found that these features have effects that can largely cancel each other. As a result, a range of lateral motion models, which are consistent with the known phenomenology of these systems, are possible.     

Integral equation theory study on the structure and effective interactions in star polymer nanocomposite melts

The polymer reference interaction site model theory is used to investigate the radial distribution function, potential of mean force, depletion force, and second virial coefficient in star polymer nanocomposite melts. The contact aggregation of nanoparticles for relatively weak nanoparticle-monomer attraction and the bridging aggregation of nanoparticles for very large nanoparticle-monomer attraction are observed. The star architecture can well suppress the organization states of direct contact and bridging structure for the moderate nanoparticle-monomer attraction, and promote the bridging-type organization for relatively large nanoparticle-monomer attraction. At constant particle volume fraction, the arm length quantitatively affects the organization states of star polymer nanocomposite melt, and larger repulsive barriers are existent to prevent the contact aggregation of larger nanoparticles. These observations provide useful information for the development of new nanocomposite materials.     

Nano‐scaled intrachain ordering and dynamics in two‐dimensional orientational polymer domains1)

Statistical and local relaxation properties of two‐dimensional finite polymer systems (domains) are considered. The domains consist of a large number of semirigid chains with the finite contour length at free, half‐free and fixed boundary conditions for chain ends. The intermolecular orientational order at short distances between chains in the thick domains is similar to the order in infinite two‐dimensional systems. The correlations of orientation between sufficiently distant elements of different chains decay by the exponential law, but the effective constant of interchain interactions in the domain is proportional to the molecular weight of the chain. At the given intra‐and interchain interactions an elongtation of the chains leads to a local ordering of chains in the domain (at free boundary conditions) or, on the contrary, to the decreasing of the parameter of short‐range orientational order (at fixed and half‐free boundary conditions). Independently of type of boundary conditions the parameter of large‐range orientational order tends to zero with increasing of the chain contour length. Dynamical equations and relaxation spectrums for times of local motions are obtained. From time correlation functions of local relaxation the times of nano‐scaled mobility of chains were calculated in depending on the bending rigidity of chains, the parameter of interchain interactions, and the contour length of chains. At the given intra‐and interchain interactions an elongtation of chains forming the domain leads to to the slowing‐down of local mobility of chains in the domain. The comparison with experimental date obtained by dielectric relaxation and polarized luminescence methods on investigation of nano‐scaled mobility in the dilute melts of comb‐shaped polymers has been carried out.     

Span‐Length Distributions of Star‐Branched Polymers

Many relations between the physical, rheological or mechanical properties of linear polymers and their molar mass are well known. For disperse polymers, parameters that express these relations are typically related to (a combination of) the moments of the molar‐mass distribution. Properties of branched, nonlinear polymers have been far more difficult to describe in the form of general relations. Monodisperse star polymers or regular stars, with a distinct number of arms and equal arm length, are the simplest member of the family of branched polymers and have served as model compounds in many studies. For these regular stars, the relation between zero shear viscosity and arm or span length has been determined. To establish equivalent relations for disperse star‐branched polymers, it is important to assess the span‐length distribution and its moments; these parameters can be calculated when the distribution of the molar mass of the arms of a star‐branched polymer is known, for instance, for a known polymerization mechanism.

Span‐length probability functions of star‐branched polycondensates with x_n = 100: f = 1 (•), f = 2 (○), f = 3 (▪), f = 5 (□), and f = 10 (+).     

高分子/棒状纳米粒子复合物的分子动力学模拟

采用非平衡态分子动力学模拟研究了剪切场下棒状纳米粒子对高分子基体的结构、 动力学和流变性质的影响. 通过比较多种体积分数(0.8%~10%)的纳米复合物及纯熔体的模拟结果发现, 随着纳米粒子的增加, 高分子链的扩散和松弛逐渐受到抑制, 而链尺寸几乎保持不变. 从Weissenberg number(W_i)角度看, 在剪切流场下, 高分子链的结构性质(如归一化的均方回转半径、 回转张量和取向抑制参数)几乎与纳米粒子的体积分数无关, 而高分子链的Tumbling运动受到抑制. 研究还发现, 纳米复合物与纯熔体的剪切黏度曲线趋势基本一致, 即W_i=1将曲线分为平台区和剪切变稀区. 纳米棒的加入仅定量地改变了流体的剪切黏度.     

Monte Carlo simulation studies of the interfaces between polymeric and other solids as models for fiber-matrix interactions in advanced composite materials

As a coarse-grained model for dense amorphous polymer systems interacting with solid walls (i.e., the fiber surface in a composite), the bond fluctuation model of flexible polymer chains confined between two repulsive surfaces is studied by extensive Monte Carlo simulations. Choosing a potential for the length of an effective bond that favors rather long bonds, the full temperature region from ordinary polymer melts down to the glass transition is accessible. It is shown that in the supercooled state near the glass transition an “interphase” forms near the walls, where the structure of the melt is influenced by the surface. This “interphase” already shows up in static properties, but also has an effect on monomer mobilities and the corresponding relaxation behavior of the polymer matrix. The thickness of the interphase is extracted from monomer density oscillations near the walls and is found to be strongly temperature dependent. It is ultimately larger than the gyration radius of the polymer chains. Effects of shear deformation on this model are simulated by choosing asymmetric jump rates near the moving wall (large jump rate in the direction of motion, and a small rate against it). It is studied how this dynamic perturbation propagates into the bulk of the polymer matrix.     

在“高聚物的结构与性能”课程中讲透高聚物的特点

通过对高分子链的柔性、聚合物独有的熵弹性、显著的粘弹性、特有的描述链段运动的WLF方程,可能实现的大尺寸取向和小尺寸解取向、银纹、单链凝聚态、折叠链片晶和伸直链晶体、分子量的多分散性、高分子溶液特性和高聚物熔体的弹性行为等的讨论,希望能突出“高聚物的结构与性能”课程中高聚物的特点。     

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.     

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.     

Advanced Computational Strategies for Modelling the Evolution of Full Molecular Weight Distributions Formed During Multiarmed (Star) Polymerisations

Summary: A novel computational strategy is described for the simulation of star polymerisations, allowing for the computation of full molecular weight distributions (MWDs). Whilst the strategy is applicable to a broad range of techniques for the synthesis of star polymers, the focus of the current study is the simulation of MWDs arising from a reversible addition fragmentation chain transfer (RAFT), R‐group approach star polymerisation. In this synthetic methodology, the arms of the star grow from a central, polyfunctional moiety, which is formed initially as the refragmenting R‐group of a polyfunctional RAFT agent. This synthetic methodology produces polymers with complex MWDs and the current simulation strategy is able to account for the features of such complex MWDs. The strategy involves a kinetic model which describes the reactions of a single arm of a star, the kinetics of which are implemented and simulated using the PREDICI® program package. The MWDs resulting from this simulation of single arms are then processed with an algorithm we describe, to generate a full MWD of stars. The algorithm is applicable to stars with an arbitrary number of arms. The kinetic model and subsequent algorithmic processing techniques are described in detail. A simulation has been parameterised using rate coefficients and densities for a 2,2′‐azoisobutyronitrile (AIBN) initiated, bulk polymerisation of styrene at 60 °C. A number of kinetic parameters have been varied over large ranges. Conversion normalised simulations were performed, leading to information regarding star arm length, polydispersity index (PDI) and the fraction of living arms. These screening processes provided a rigorous test for the kinetic model and also insight into the conditions, which lead to optimal star formation. Finally, full MWDs are simulated for several RAFT agent/initiator ratios as well as for stars with a varying number of arms.

Full MWDs from a star with 1, 2, 4, 6 and 8 arms.     

Theory of nanoparticle diffusion in unentangled and entangled polymer melts

We propose a statistical dynamical theory for the violation of the hydrodynamic Stokes-Einstein (SE) diffusion law for a spherical nanoparticle in entangled and unentangled polymer melts based on a combination of mode coupling, Brownian motion, and polymer physics ideas. The non-hydrodynamic friction coefficient is related to microscopic equilibrium structure and the length-scale-dependent polymer melt collective density fluctuation relaxation time. When local packing correlations are neglected, analytic scaling laws (with numerical prefactors) in various regimes are derived for the non-hydrodynamic diffusivity as a function of particle size, polymer radius-of-gyration, tube diameter, degree of entanglement, melt density, and temperature. Entanglement effects are the origin of large SE violations (orders of magnitude mobility enhancement) which smoothly increase as the ratio of particle radius to tube diameter decreases. Various crossover conditions for the recovery of the SE law are derived, which are qualitatively distinct for unentangled and entangled melts. The dynamical influence of packing correlations due to both repulsive and interfacial attractive forces is investigated. A central finding is that melt packing fraction, temperature, and interfacial attraction strength all influence the SE violation in qualitatively different directions depending on whether the polymers are entangled or not. Entangled systems exhibit seemingly anomalous trends as a function of these variables as a consequence of the non-diffusive nature of collective density fluctuation relaxation and the different response of polymer-particle structural correlations to adsorption on the mesoscopic entanglement length scale. The theory is in surprisingly good agreement with recent melt experiments, and new parametric studies are suggested.