Lattice models for the collapse of branched polymers

Randomly branched polymers in dilute solution in a good solvent can be modelled by lattice animals (i.e. connected clusters). Introducing attractive monomer—monomer interactions between nearest-neighbour sites causes the branched polymer to become more compact and a collapse transition, analogous to that in linear polymers, is expected to occur at low temperature. This model, and alternative lattice models for the collapse transition of branched polymers, will be described. In each of the models, the collapse is driven by some kind of near-neighbour fugacity, cycle fugacity or perimeter fugacity. In two and three dimensions and on several lattices, analytical results for the free energy and numerical results, using exact enumeration data, for the free energy and for the specific heat, will be presented with estimates of the cross-over exponent and the collapse temperature T_c.     

Properties of branched confined polymers

A model of star-branched polymer chains confined in a slit formed by two parallel surfaces was studied. The chains were embedded to a simple cubic lattice and consisted of f=3 branches of equal length. The macromolecules had the excluded volume and the confining surfaces were impenetrable for polymer segments. No attractive interactions between polymer segments and then between polymer segments and the surfaces were assumed and therefore the system was a thermal. Monte Carlo simulations were carried out employing the sampling algorithm based on chain's local changes of conformation. Lateral diffusion of star-branched chains was studied. Dynamic properties of star-branched chains between the walls with impenetrable rod-like obstacles were also studied and compared to the previous case. The density profiles of polymer segments on the slit were determined. The analysis of contacts between the polymer chain and the surfaces was also carried out.     

Dilute solution properties of branched polymers

For calculating the ratio of the intrinsic viscosities of branched and linear polymers of the same molecular weight, [η]_B/[η]_L, a new theory taking into account the excluded volume effect is presented. By using the modified Flory equation, the excluded volume effect of branched polymers is predicted with the aid of the first-order perturbation theory. The linear expansion factor α_s is converted to the hydrodynamic expansion factor α_η by using the Kurata-Yamakawa theory. Our calculated results, i.e., [η]_B/[η]_L and 〈s²〉_B/〈s²〉_L, agree well with experiment for various type branched polymers, i.e., randomly branched and comb-shaped polymers of poly(vinyl acetate).     

Intrinsic viscosity of polydisperse branched polymers

The relationships between molecular weight distribution and structure in polymerizations with long-chain branching were reviewed and extended. Results were applied to an experimental examination of intrinsic viscosity in polydisperse, trifunctionally branched systems. Several samples of poly(vinyl acetate) were prepared by bulk polymerization under conditions of very low radical concentration. The relative rate constants for monomer transfer, polymer transfer, and terminal double-bond polymerization were established from the variation of M _n and M _w with the extent of conversion. Average branching densities were then calculated for each sample and ranged as high as 1.5 branch points/molecule. Intrinsic viscosities [η]_B were measured in three systems: a theta-solvent, a good solvent, and one that was intermediate in solvent interaction. These results were compared with calculated viscosities, [η]_L, which would have been observed if all the molecules had been linear. The values of [η]_B/[η]_L were substantially the same in all three solvents. The variation of this ratio with branching density was compared with the theory of Zimm and Kilb as adapted to polydisperse systems. Discrepancies were noted, and the adequacy of present model distribution functions for branched polymers was questioned.     

Flow properties of branched polydisperse polymers

Viscosity and normal-stress difference as functions of shear rate were examined for poly(vinyl acetate) samples with various degrees of random branching in diethyl phthalate solution. At moderate concentration (c = 0.17 g/ml) the viscosities were depressed by branching, in fair accord with the Bueche theory. However, at higher concentrations (c = 0.35 g/ml) the data showed a progressive trend in the direction of viscosity enhancement by branching, a characteristic which has been observed by other workers in undiluted branched polymers. Accompanying viscosity enhancement was an increase in the temperature coefficient of viscosity, an increase in recoverable compliance (estimated from steady-state normal-stress data) and an early onset in the shear rate dependence of viscosity.     

Synthesis of new branched urethane-triazole polymers

An AB₂ monomer obtained via the reaction of hexamethylene diisocyanate with 1,3-diazidopropan-2-ol and 2-propyn-1-ol contains one triple bond (A) and two azide groups (B₂). Hyperbranched urethane-triazole polymers were synthesized through stepwise polymerization of this monomer via 1,3-dipolar cycloaddition. Owing to the presence of impurities of compounds with four azide groups (the products of the reaction of hexamethylene diisocyanate with two molecules of diazidopropanol) and two propyne groups (the product of the reaction of hexamethylene diisocyanate with two molecules of 2-propyne-1-ol) in the aforementioned monomer, uretane-triazole “polymers” are not classical huperbranched polymers, they are highly branched polymers that are soluble in aprotic polar solvents and have weight-average molecular masses of 5–260 and values of the exponent in the Kuhn-Mark-Houwink equation of 0.23–0.36.     

Intrinsic viscosity in branched free-radical polymers

The intrinsic viscosity ratio [η]_B/[η]_L was calculated as a function of average branching density for trifunctionally branched, free-radical polymers. Calculations were made for the g^1/2, g^3/2, and h³ rules, using realistic distributions of molecular weights and branches. Experimental data on branched poly(vinyl acetate) lay between the curves obtained from the g^1/2 and h³ relations.     

Amphiphilic branched polymers as antimicrobial agents

Cationic amphiphilic polymers were prepared from PEI and functional ethylene carbonates bearing cationic, hydrophobic or amphiphilic groups. The polymers are designed to exhibit antimicrobial properties. In a one-step addition, different functional ethylene carbonates were added to react with the primary amine groups of PEI. The water soluble polymers were studied regarding their ability to form soluble aggregates. Their hydrodynamic radii, their inhibition potential against proliferation of E. coli and their hemolytic potential were determined. A structure-property relationship was established by analyzing the antimicrobial activity as a function of the ratio of alkyl to cationic groups, length of the alkyl chains, and molecular weight of the PEI.     

Molecular weight distribution in branched polymers

The distribution of molecular weights in realistic free-radical polymerizations with branching was analyzed theoretically. Series solutions were obtained for the fraction of molecules with r repeating units and the number of branch points contained in molecules with r repeating units. Branching by transfer processes was found to increase the proportion of both high and low molecular weight components in the system. The apportioning of branch points among r-mer molecules was shown to be somewhat narrower than a Poisson distribution. The major difference between the calculated distributions and previous model distributions for branched systems was the absence of discontinuities in the moments at all levels of branching.     

Size exclusion chromatography of branched polymers: Star and comb polymers

Monte Carlo simulations were conducted to estimate the elution curve of size exclusion chromatography (SEC). The present simulation can be applied to various types of branched polymers, as long as the kinetic mechanism of nonlinear polymer formation is given. We considered two types of detector systems, (1) a detector that measures the polymer concentration in the elution volume to determine the calibrated molecular weights, such as by using the differential refractive index detector (RI), and (2) a detector that determines the weight‐average molecular weight of polymers within the elution volume directly, such as a light scattering photometer (LS). For polydisperse star polymers, both detector systems tend to give a reasonable estimate of the true molecular weight distribution (MWD). On the other hand, for comb‐branched polymers, the RI detector underestimates the molecular weight of branched polymers significantly. The LS detector system improves the measured MWD, but still is not exact. The present simulation technique promises to establish various types of complicated reaction mechanisms for nonlinear polymer formation by using the SEC data quantitatively. In addition, the present technique could be used to reinvestigate a large amount of SEC data obtained up to the present to estimate the true MWD.     

Shear damping function measurements for branched polymers

We present results for step‐strain experiments and the resulting damping functions of polyethylene blends of different structures, including solutions of linear, star and comb polymers. Remarkably, an entangled melt of combs exhibits a damping function close to that for entangled linear chains. Diluting the combs with faster‐relaxing material leads to a more nearly constant damping function. We find similar behavior for blends of commercial low density polyethylene LDPE. Our results suggest a simple picture: on timescales relevant to typical damping‐function experiments, the rheologically active portions of our PE combs as well as commercial LDPE are essentially chain backbones. When strongly entangled, these exhibit the Doi‐Edwards damping function; when diluted, the damping function tends toward the result for unentangled chains described by the Rouse model – namely, no damping. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 3117–3136, 2007     

Porous coordination polymers with zeolite topologies constructed from 4-connected building units

Two novel 3D coordination polymers, Cd(CTC)(H2O).(H2PIP)(0.5)(H2O) (1) with zeolite ABW topology and Cd(CTC).(HIPA) (2) with zeolite BCT topology, have been synthesized by constructing inorganic and organic 4-connected building units and using the organic bases as templates, and the frameworks of and not only expand the original structures of zeolites ABW and BCT, but also exhibit significant advantages over them in terms of thermal stability, ion exchange and adsorption.     

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     

General description of the structure of branched polymers

A multidimensional distribution function is defined to describe the branching structure of branched homopolymers such as starch and polyacrylates. Averages of this function give distributions which can be measured using, for example, the number and weight distributions as a function of hydrodynamic volume from size‐exclusion chromatography and field‐flow fractionation, and two‐dimensional separation methods. This provides means to plot data to obtain physically meaningful quantities, and to test mechanistic postulates for the (bio)synthesis, of branched polymers. A simple enzyme‐kinetic model for a reduced form of this multidimensional distribution for starch biosynthesis is derived and solved. One application is to derive number distributions for the molecular weight distribution of debranched glycogen. Fitting this to experiment gives estimates of this ratio for two forms of glycogen. We propose that number distributions from size separation for starch (which, it is pointed out, are obtained directly from in‐line viscometric detection) have a simple and meaningful form when plotted as ln(number distribution) against V_h^p, where V_h is hydrodynamic volume, and p a parameter of order unity determined from multiple‐detection size separation measurements. The new function is also used to propose a two‐dimensional experiment which can yield an unambiguous measurement of the amylose: amylopectin ratio in starch. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 3914–3930, 2009     

A series of three-dimensional lanthanide coordination polymers with rutile and unprecedented rutile-related topologies

The complexes of formulas Ln(pydc)(Hpydc) (Ln = Sm (1), Eu (2), Gd (3); H2pydc = pyridine-2,5-dicarboxylic acid) and Ln(pydc)(bc)(H2O) (Ln = Sm (4), Gd (5); Hbc = benzenecarboxylic acid) have been synthesized under hydrothermal conditions and characterized by elemental analysis, IR, TG analysis, and single-crystal X-ray diffraction. Compounds 1-3 are isomorphous and crystallize in the orthorhombic system, space group Pbcn. Their final three-dimensional racemic frameworks can be considered as being constructed by helix-linked scalelike sheets. Compounds 4 and 5 are isostructural and crystallize in the monoclinic system, space group P2(1)/c. pydc ligands bridge dinuclear lanthanide centers to form the three-dimensional frameworks featuring hexagonal channels along the a-axis that are occupied by one-end-coordinated bc ligands. From the topological point of view, the five three-dimensional nets are binodal with six- and three-connected nodes, the former of which exhibit a rutile-related (4.6(2))(2)(4(2).6(9).8(4)) topology that is unprecedented within coordination frames, and the latter two species display a distorted rutile (4.6(2))(2)(4(2).6(10).8(3)) topology. Furthermore, the luminescent properties of 2 were studied.     

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.     

Prediction of molecular weight distributions in branched polymers

A theoretical treatment of long branching in radical polymerization in a continuous stirred tank reactor is presented. The treatment takes into account radical termination by disproportionation and/or combination, transfer to polymer and/or additive of low molecular weight, residence time, and injection of polymer. In the absence of added polymer, the molecular weight distribution depends upon four parameters which can be expressed as P?_n and ratios of the rate of polymerization to the rate at which radicals (a) leave the reactor, (b) combine, and (c) transfer to polymer. The kinetic equations were converted into a differential equation which was solved numerically to give polymer and radical moments. An analytical solution is presented for the case where combination is absent. P?_w/P?_n is predicted to increase smoothly in a markedly exponential manner with increasing polymer transfer, combination, P?_n, and mean residence time. At no stage do any of the moments become infinite unless the residence time is infinite. For polymers of wide distribution, the second and higher radical moments can exceed the corresponding polymer moments so that most of the large molecules leave the reactor as radicals. Beasley's and Nicolas's treatments are shown to be limiting cases at low polymer transfer.     

Separation of complex branched polymers by size-exclusion chromatography probed with multiple detection

Size-exclusion chromatography (SEC) separates polymers by hydrodynamic volume (the universal calibration principle). Molecular weights can be determined using viscometry (relying on universal calibration) and light scattering (independent of universal calibration). In the case of complex branched polyacrylates with tetrahydrofuran as eluent, universal calibration is valid, although the separation in term of molecular weight is incomplete: a given elution slice contains a range of molecular weights, described in terms of a 'local polydispersity'. The local polydispersity index decreases when the number of branches per chain increases and complete separation is reached for highly branched chains.     

Lattice models of DNA electrophoresis

This article presents an overview of lattice polymer models used to study DNA electrophoresis. Three commonly used models--the bond fluctuation model, the cage model, and the repton model--are discussed, as well their extension to include electric fields, and simulation results obtained. Physical properties that are shared amongst these models are identified, and differences between the models are discussed.