Monte Carlo simulations of star‐branched polymers in a network of obstacles

A simple model of branched polymers in space confined between two parallel plates is developed. Star‐branched polymer molecules are built on a simple cubic lattice with excluded volume and no attractive interactions. A single star molecule is immersed in a network of irregularly dispersed linear rod‐like obstacles. The classical Monte Carlo Metropolis sampling algorithm is employed in the simulation. The aim of this study is to determine the effects of changes in dynamic properties of the star‐branched polymer as functions of the length of the molecule and the concentration of obstacles. Also the mechanism of motion of the polymer is discussed.     

Monte Carlo simulation studies of the size and shape of linear and star‐branched polymers embedded in the tetrahedral lattice

In recent years the author investigated the size and shape of linear and star‐branched model chains as a function of functionality (number of arms) F and chain length (total number of beads) N for several lattice and off‐lattice random walk models as well as for self‐avoiding random walks (embedded in the tetrahedral lattice) subjected to various thermodynamic conditions. Not only mean values – characteristic quantities averaged over a large number of configurations of given length, functionality and thermodynamic conditions – have been computed, but also distributions of shape parameters, and the correlation operative between size and shape has been explored. The present feature article in principle summarizes these results, however, the data given in part are recomputed for still larger sample sizes and chain‐lengths as in the underlying papers. In addition to stars with F = 4, 8 and 12 arms, so far unpublished investigations on stars with F = 3, F = 6 and F = 10 arms are presented and discussed.     

Tube dilation process in star‐branched cis‐polyisoprenes

In current tube models for entanglement, the tube representing the topological constraint is considered to move with time. This tube motion results in the constraint release (CR) as well as the dynamic tube dilation (DTD), and an importance of DTD has been argued for entangled star chains. Under these backgrounds, this article examines the validity of the DTD molecular picture for the star chains. For monodisperse star chains having noninverted type‐A (parallel) dipoles in respective arms, the normalized viscoelastic and dielectric relaxation functions μ(t) and Φ(t) were found to obey a relationship μ(t) ≅ [Φ(t)]² if the tube actually dilates in the time scale of the star relaxation. For 6‐arm star cis‐polyisoprene (PI) chains (having those type‐A dipoles), dielectric and viscoelastic measurements were conducted to test this DTD relationship. Both viscoelastic and dielectric properties exhibited characteristic behavior expected from DTD models (assuming the arm retraction in the dilating tube), the exponential increase of the relaxation time and broadening of the relaxation mode distribution with increasing arm molecular weight M_a. However, in the range of M_a examined, M_a ≤ 8M_e (M_e = entanglement spacing), the above DTD relationship was not valid for a dominant part of the slow relaxation (and the models failed in this sense). Thus, for star chains at least in this range of M_a, the simple DTD picture assuming very rapid CR motion (rapid equilibration in the dilated tube) did not explain the slow relaxation behavior of star chains. This result in turn suggested the importance of the CR motion in this behavior. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 1024–1036, 2000     

Molecular mass distributions of star‐branched polycondensates

Of late much attention has been paid to star‐branched polymers, being a good reference model for branched polymers, in general. Usually, monodisperse or narrow disperse polymers are analysed. Knowledge of molecular mass distributions is a key factor in the analysis and study of these systems. Star‐branched polycondensates can be synthesised by reaction of a difunctional ( AB ) monomer with a compound RA _f . Weight and number molecular mass distributions of star‐branched polycondensates have been studied in relation to the initial molar ratio between R and the AB monomer (α), the average molecular mass per arm (β) and the number of arms (f ). Simple probability density functions can be derived at, if molecules are split into – R (without core R ) and + R (with core R ). This enables further the representation in molecular mass units next to the representation in monomer units. Proper choice of α, β and f can either give narrower or broader distributions compared to the most probable distribution, which is the theoretical distribution for the pure AB polymer (α = 0). The resulting polymer might either have a uni‐modal or a bi‐modal molecular mass distribution.     

Precise synthesis of well‐defined dendrimer‐like star‐branched polymers by iterative methodology based on living anionic polymerization

Dendrimer‐like star‐branched polymers recently developed as a new class of hyperbranched polymers, which resemble well‐known dendrimers in branched architecture, but comprise polymer chains between junctions, are reviewed in this highlight article. In particular, we focus on the precise synthesis of various dendrimer‐like star‐branched polymers and block copolymers by the recently developed methodology based on iterative divergent approach using living anionic polymers and 1,1‐bis(3‐tert‐butyldimethylsilyloxymethylphenyl)ethylene. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 6659–6687, 2006     

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.     

Pair distribution function and related properties of star‐branched chains

Pair configurations of linear and star‐branched chains with F = 4, 8 and 12 arms embedded in the tetrahedral lattice were investigated. Pair data were determined by exact enumeration of all possible pair configurations. When the separation between two (linear) chains reached zero (r → 0) the pair distribution function g (r) read ≈ 0.15 for athermal and ≈ 0.6 for theta conditions in full accordance with former work. For star‐branched chains, g (r) approached a value zero at small separations for both thermodynamic conditions and the range of g (r) = 0 increased with an increase of the number of arms. As a consequence, the characteristic maximum of g (r) for theta conditions was the more pronounced the larger the number of arms. For stars, the extent to which mean squared dimensions and shape parameters depend on intermolecular distance was similar to that of linear chains, at least in the region of intermediate and large intermolecular separations. Transformation of the data into a concentration dependence revealed that with an increase in concentration, the dimensions decreased in the case of athermal solvents while they increased for θ‐solvents regardless of the functionality given.     

Long‐chain branched ROMP polymers

This article describes the construction of branched ROMP‐polymer architectures via polycondensation of AB_n‐type macromonomers. For this convergent strategy, a polymer was synthesized that carries several hydroxyl‐groups along the polymer chain and one carboxylic acid group at the chain end. An esterification reaction between these functional groups yielded long‐chain branched polymers. The polymers were analyzed by NMR and SEC to monitor the condensation reaction. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2009     

Silicon surface modification with trialkoxysilyl‐functionalized star‐shaped polymers

The synthesis of trimethoxysilane end‐capped linear polystyrene (PS) and star‐branched PS and subsequent silicon (Si) surface modification with linear and star polymers are described. Trimethoxysilane terminated PS was synthesized using sec‐butyl lithium initiated anionic polymerization of styrene and subsequent end‐capping of the living anions with p‐chloromethylphenyl trimethoxysilane (CMPTMS). ¹H and ²⁹Si NMR spectroscopy confirmed the successful end‐capping of polystyryllithium with the trimethoxysilane functional group. The effect of a molar excess of end‐capper on the efficiency of functionalization was also investigated, and the required excess increased for higher molar mass oligomers. Acid catalyzed hydrolysis and condensation of the trimethoxysilane end‐groups resulted in star‐branched PS, and NMR spectroscopy and SEC analysis were used to characterize the star polymers. This is the first report of core‐functionalized star‐shaped polymers as surface modifiers and the first comparative study showing differences in surface topography between star and linear polymer modified surfaces. Surface‐sensitive techniques such as ellipsometry, contact angle goniometry, and AFM were used to confirm the attachment of star PS, as well as to compare the characteristics of the star and linear PS modified Si surfaces. The polymer film properties were referenced to polymer dimensions in dilute solution, which revealed that linear PS chains were in the intermediate brush regime and the star‐branched PS produced a surface with covalently attached chains in the mushroom regime. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 43: 3655–3666, 2005     

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.     

A Monte Carlo simulation study of branched polymers

Monte Carlo simulations are presented for the static properties of highly branched polymer molecules. The molecules consist of a semiflexible backbone of hard-sphere monomers with semiflexible side chains, also composed of hard-sphere monomers, attached to either every backbone bead or every other backbone bead. The conformational properties and structure factor of this model are investigated as a function of the stiffness of the backbone and side chains. The average conformations of the side chains are similar to self-avoiding random walks. The simulations show that there is a stiffening of the backbone as degree of crowding is increased, for example, if the branch spacing is decreased or side chain length is increased. The persistence length of the backbone is relatively insensitive to the stiffness of the side chains over the range investigated. The simulations reproduce most of the qualitative features of the structure factor observed in experiment, although the magnitude of the stiffening of the backbone is smaller than in experiment.     

Computer simulation of stress relaxation

Molecular dynamics simulations of stress relaxation have been performed for models of metals and polymers. The method which employs coupling between the simulation cell and an external thermal bath as well as an applied stress has been used. Two-dimensional models of the materials are defined with interactions described by the Lennard-Jones (Mie 6–12) and harmonic potentials. A special method is employed to generate chains in dense polymeric systems. Simulated stress relaxation curves are similar for metals and polymers, while there exist essential differences in the stress-strain behavior of the models. During the relaxation, trajectories of the particles in different materials display a common feature: there exist domains in which movement of the particles is highly correlated. This observation supports the cooperative theory of stress relaxation developed by one of us. The results of the simulations do not significantly depend on the number of particles in the system nor on other simulation details.     

Dielectric relaxation studies of some linear crosslinked and branched polymers

Dielectric loss measurements are reported for polystyrene, crosslinked polystyrene, polyacrylamide, branched polyacrylamide, and poly(methyl methacrylate) at 1 and 10 kHz f over the temperature range ?85 to +100°C. Crosslinking and branching have a pronounced effect on the dielectric relaxation spectra of polymers. The methods of preparation of these polymers and their viscosity molecular weight data are also reported.     

Micro‐rheology and relaxation phenomena in Tg vicinity

The present paper describes possible forms of relaxation phenomena in T_g vicinity. The individual types of relaxation in amorphous matrix are described in the areas of Vogel's temperature, T_g and crossover temperature. Special emphasis is laid on the difference between Flory's relaxation mechanism and the approach by Di Marzio, Adams and Gibbs. These authors used the common formula of Flory, however, their interpretations of the so‐called “configurational phenomena” reflect completely different realities. The theoretical justification of heat capacity as a complex number (recently introduced by Hutchinson, Monserrat and Schawe in T_g vicinity) is provided in detail and compared with the existing experimental evidence quoted by the authors of these contributions as well as with data published in literature. The relation between complex heat capacity and relaxation of internal stress in amorphous matrixes or the effects of ageing are discussed.     

Coarse‐grained models and collective phenomena in membranes: Computer simulation of membrane fusion

We discuss the role coarse‐grained models play in investigating collective phenomena in bilayer membranes and place them in the context of alternative approaches. By reducing the degrees of freedom and applying simple effective potentials, coarse‐grained models can address the large time scales and length scales of collective phenomena in membranes. Although the mapping from a coarse‐grained model onto chemically realistic models is a challenge, such models provide a direct view on the phenomena that occur on the length scales of a few tens of nanometers. Their relevance is exemplified by the study of fusion of model membranes. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1441–1450, 2003