Molecular Weight Development in Free‐Radical Polymerization with Polyfunctional Chain‐Transfer Agents, 2. Substitution Effects

The analytic expression for the weight‐average molecular weight development proposed in the first part of this series^[1] is extended to account for the substitution effects of the functional groups in a polyfunctional chain‐transfer agent. The obtained functional form of the matrix formula, P_w = W { D _w + ( I + T ) SPE ( I – TSPE )^–1 D _f } is essentially the same as for equal reactivity model, but the NxN square matrixes, S and P in the equal reactivity model is now expanded to NxfN and fNxfN matrixes, respectively, to account for the different reactivities of f functional groups. The number of chain types, N is extrapolated to infinity to obtain the P_w‐value for free‐radical polymerization. Practically, such extrapolation can be conducted with calculated P_w‐values for only three different N‐values. Illustrative calculations show that the obtained results agree with those from the Monte Carlo method. The present approach provides newer insight into the branched structure formation during complex polymerization reactions.     

Synthesis of Branched Polymer Architectures from Molecular Weight and Branching Distributions for Radical Polymerisation with Long‐Chain Branching,Accounting for Topology‐Controlled Random Scission

Branched polymers like LDPE are known to possess a wide range of architectures. In this paper a modelling approach is developed, describing the relation between architectures, chemistry and reactor conditions with the general objective of improving characterisation and controlling visco‐elastic properties. More specifically, the particular scission kinetics of branched molecules as strongly contrasting with linear scission is described. A new method to synthesise branched architectures is developed as an alternative to full Monte Carlo (MC) sampling. It employs MC sampling for coupling of primary polymers only. Graph theory is used as an efficient storage method containing all topological information of individual molecules. The algorithm synthesises molecules for any given combination of chain length (n) and number of branches (N). The explicit and detailed knowledge of branched architectures allows finding the correct topological scission kinetics. Distributions of fragment lengths and numbers of branches on fragments after scission are obtained, showing a preference for short and long fragments. Approximate functions describing this have been implemented in another model, predicting molecular weight (MWD) and degree of branching (DBD) distributions using a Galerkin finite element method. Topological scission is seen to give MWD broadening and a higher branching density for long chains. Distributions of longest end‐to‐end distances could be computed for all architectural alternatives for given n, N. In conclusion, it is demonstrated that this method yields distributions of architectures consistent with MWD/DBD for radical polymerisation with long‐chain branching and random scission.     

Free volume,glass transition,and degree of branching in metallocene‐based propylene/α‐olefin copolymers: Positron lifetime,density, and differential scanning calorimetric studies

Positron annihilation lifetime spectroscopy (PALS), density, and differential scanning calorimetric (DSC) measurements were used to study systematically the variation of the glass‐transition temperature (T_g) and the size v and number density N_h of local free volumes in n‐alkyl‐branched polypropylenes. The samples were metallocene‐catalyzed propylene copolymers with different α‐olefins (from C₄ to C₁₆) and a different α‐olefin content (between 0 and 20 mol %). From the total specific volume and crystallinity the specific volume of the amorphous phase V_a was estimated and used to calculate the fractional free (hole) volume h and value of N_h. The variations of T_g, v, V_a, h, and N_h were related to the degree (number and length) of branching. T_g decreases and v increases linearly with the number and length of n‐alkyl branches. This behavior was attributed to an increased segmental mobility caused by branching. Both values, T_g and v, follow linear master curves as a function of the degree of branching (DB) if this is defined as the number of all side‐chain carbons with respect to a total of 1000 (main‐chain and side‐chain) carbons. Only propylene/1‐butene copolymers deviated from these relations. A linear relation between v and T_g was also found. The number density of holes was estimated to be N_h = 0.49(±0.07) nm^?3 and N_h′ = 0.58(±0.11) × 10²¹ g^?1, respectively. It shows a slight variation with the DB, which is also seen in the behavior of the specific volume V_a. This variation was explained by the appearance of sterical hindrances resulting from short‐chain branches that may prevent an efficient packing of the chains. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 434–453, 2002; DOI 10.1002/polb.10108     

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.     

Polyethylene Made with Combinations of Single‐Site‐Type Catalysts: Monte Carlo Simulation of Long‐Chain Branch Formation

The formation of long‐chain branches (LCBs) during ethylene polymerization with a combination of catalysts was studied by Monte Carlo simulation. The model describes polymerization with a non‐branching catalyst that produces linear macromonomers, and a branching catalyst that produces linear and branched macromonomers. The LCBs are formed when the branching catalyst incorporates a macromonomer. The discussion is based on the three types of chain topology obtained during the synthesis: linear, comb‐branched, or hyperbranched. Simulation results show how the chain length distribution and the number of LCBs change according to the ratio between the two catalysts present in the reactor. The ratio hyperbranched/comb‐branched is defined to evaluate the system composition and the contribution of each catalyst.     

Dimensions of branched polymers formed in simultaneous long‐chain branching and random scission

In free‐radical olefin polymerizations, the polymer‐transfer reactions could lead to chain scission as well as the formation of long‐chain branches. The Monte Carlo simulation for free‐radical polymerization that involves simultaneous long‐chain branching and random scission is used to investigate detailed branched structure. The relationship between the mean‐square radius of gyration 〈s²〉 and degree of polymerization P as well as that between the branching density and P is the same for both with and without random scission reactions—at least for smaller frequencies of scission reactions. The 〈s²〉 values were larger than those calculated from the Zimm–Stockmayer (Z‐S) equation in which random distribution of branch points is assumed, and therefore, the Z‐S equation may not be applied for low‐density polyethylenes. The elution curves of size exclusion chromatography were also simulated. The molecular weight distribution (MWD) calibrated relative to standard linear polymers is much narrower than the true MWD, and high molecular weight tails are clearly underestimated. A simplified method to estimate the true MWD from the calibrated MWD data is proposed. The MWD obtained with a light scattering photometer in which the absolute weight‐average molecular weight of polymers at each retention volume is determined directly is considered a reasonable estimate of the true MWD. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2960–2968, 2001     

Ultrahigh Branching of Main‐Chain‐Functionalized Polyethylenes by Inverted Insertion Selectivity

Branched polyolefin microstructures resulting from so‐called “chain walking” are a fascinating feature of late transition metal catalysts; however, to date it has not been demonstrated how desirable branched polyolefin microstructures can be generated thereby. We demonstrate how highly branched polyethylenes with methyl branches (220 Me/1000 C) exclusively and very high molecular weights (ca. 10⁶ g mol^?1), reaching the branch density and microstructure of commercial ethylene–propylene elastomers, can be generated from ethylene alone. At the same time, polar groups on the main chain can be generated by in‐chain incorporation of methyl acrylate. Key to this strategy is a novel rigid environment in an α‐diimine Pd^II catalyst with a steric constraint that allows for excessive chain walking and branching, but restricts branch formation to methyl branches, hinders chain transfer to afford a living polymerization, and inverts the regioselectivity of acrylate insertion to a 1,2‐mode.     

Highly branched polyethylene graft copolymers prepared by means of migratory insertion polymerization combined with TEMPO‐mediated controlled radical polymerization

New families of highly branched polyethylenes containing alkyl short chain branches as well as polar and non‐polar long‐chain branches were prepared by combining migratory insertion copolymerization with controlled radical graft copolymerization. Key intermediate was a novel alkoxyamine‐functionalized 1‐alkene which was copolymerized with ethylene using a palladium catalyst. The resulting highly branched polyethylene with alkoxyamine‐functionalized short chain branches was used as macroinitiator to initiate controlled radical graft copolymerization of styrene and styrene/acrylonitrile. Novel polyethylene graft copolymers with molecular masses of M_w >100 000 g/mol and narrow polydispersities were obtained. Transmission electron microscopic studies (TEM) and the presence of two glass transition temperatures at –67 and +100°C indicated microphase separation.     

Cationic para‐phenyl‐substituted α‐diimine nickel catalyzed ethylene and 4‐methyl‐1‐pentene (co)polymerizations via living/controlled chain‐walking

Upon activation with diethylaluminium chloride (Et₂AlCl), a series of phenyl‐substituted α‐diimine nickel precatalysts conducted 4‐methyl‐1‐pentene (4MP) and ethylene (E) (co)polymerizations via controlled chain‐walking to generate branched amorphous polymers with high molecular weight and narrow molecular weight distribution (M_w/M_n < 1.6). The obtained poly(4MP)s were amorphous elastomers with glass transition temperature (T_g) of ?10 ~ ?24 °C, which are higher than that of E‐4MP copolymer ( ? 63.0 °C). At room temperature (25 °C), 4MP polymerization proceeds in a living manner. The microstructures of the produced poly(4MP)s indicated the 2,1‐ and 1,2‐insertion followed by chain‐walking, the latter being predominant. The NMR analyses of the polymers showed that the obtained poly(4MP) possessed methyl, isobutyl, 2,4‐dimethylpentyl and 2‐methylhexyl groups, while the isobutyl and 2,4‐dimethylalkyl branches derived from 4MP were observed in the E‐4MP copolymer. The branch structures and the insertion‐type of monomer were depended on the polymerization temperature, and the content of methyl branch increased with an increase in the polymerization temperature.     

Arborescent polypeptides from γ‐benzyl l‐glutamic acid

The synthesis of arborescent polymers with poly(γ‐benzyl L‐glutamate) (PBG) side chains was achieved through successive grafting reactions. The linear PBG building blocks were produced by the ring‐opening polymerization of γ‐benzyl L‐glutamic acid N‐carboxyanhydride initiated with n‐hexylamine. The polymerization conditions were optimized to minimize the loss of amino chain termini in the reaction. Acidolysis of a fraction of the benzyl groups on a linear PBG substrate and coupling with linear PBG using a carbodiimide/hydroxybenzotriazole promoter system yielded a comb‐branched or generation zero (G0) arborescent PBG. Further partial deprotection and grafting cycles led to arborescent PBG of generations G1 to G3. The solvent used in the coupling reaction had a dramatic influence on the yield of graft polymers of generations G1 and above, dimethylsulfoxide being preferable to N,N‐dimethylformamide. This grafting onto scheme yielded well‐defined (M_w/M_n ≤ 1.06), high molecular weight arborescent PBG in a few reaction cycles, with number‐average molecular weights and branching functionalities reaching over 10⁶ and 290, respectively, for the G3 polymer. α‐Helix to coiled conformation transitions were observed from N,N‐dimethylformamide to dimethyl sulfoxide solutions, even for the highly branched polymers. © 2013 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2013 , 51, 5270–5279     

Synthesis of branched polyethylenes by the tandem catalysis of silica‐supported linked cyclopentadienyl‐amido titanium catalysts and a homogeneous dibromo nickel catalyst having a pyridylimine ligand

The synthesis of branched polyethylenes by ethylene polymerization with new tandem catalyst systems consisting of methylaluminoxane‐preactivated linked cyclopentadienyl‐amido titanium catalysts [Ti(η⁵:η¹‐C₅Me₄SiMe₂NR)Cl₂ (R = Me or ^tBu)] supported on pyridylethylsilane‐modified silica (PySTiNMe and PySTiN^tBu) and homogeneous dibromo nickel catalyst having a pyridyl‐2,6‐diisopropylphenylimine ligand (PyminNiBr₂) in the presence of modified methylaluminoxane was investigated. Ethylene polymerization with only PyminNiBr₂ yielded a mixture of 1‐ and 2‐olefin oligomers with methyl branches [weight‐average molecular weight (M_w) ～ 460)] with a ratio of about 1:7. By the combination of this nickel catalyst with PySTiN^tBu, polyethylenes with long‐chain branches (M_w = 15,000–50,000) were produced. No incorporation of 2‐olefin oligomers was observed in the ¹³C NMR spectra. Unexpectedly, the combination of the nickel catalyst with PySTiNMe produced lower molecular weight polyethylenes with only methyl branches. The molecular weight distributions of branched polyethylenes obtained with both PySTiNMe and PySTiN^tBu combined with the nickel catalyst were broad (weight‐average molecular weight/number‐average molecular weight < 9). Bimodal gel permeation chromatography (GPC) curves were clearly observed in the PySTiNMe system, whereas GPC curves with small shoulders in low molecular weight areas were observed for PySTiN^tBu. The synthesis of branched polyethylenes with tandem catalyst systems of corresponding homogeneous titanium catalysts and the nickel catalyst was also investigated for comparison. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 528–544, 2003     

Branched Azobenzene Side‐Chain Liquid‐Crystalline Copolymers Obtained by Self‐Condensing ATR Copolymerization

Summary: The one step synthesis of a series of branched azobenzene side‐chain liquid‐crystalline copolymers by the self‐condensing vinyl copolymerization (SCVCP) of a methyl acrylic AB* inimer, 2‐(2‐bromoisobutyryloxy)ethyl methacrylate (BIEM), with the monomer 6‐(4‐methoxy‐azobenzene‐4′‐oxy)hexyl methacrylate (M), by atom transfer radical polymerization (ATRP) in the presence of CuBr/N,N,N′,N′,N″‐pentamethyldiethylenetriamine as a catalyst system, and in chlorobenzene solvent, is reported. The degree of branching (DB), and the molecular weights and polydispersities of the resultant polymers were determined by NMR spectroscopy and size exclusion chromatography, respectively. The phase behaviors of the branched copolymers were characterized by differential scanning calorimetry (DSC) and thermal polarized optical microscopy (POM). The degree of branching of the branched copolymers could be controlled by the comonomer ratio in the feed and influenced their liquid‐crystal properties. Liquid‐crystal properties were not exhibited when the comonomer ratio was low. Comonomer ratios greater than 8 gave polymers with a lower number of branches, which exhibited both a smectic and a nematic phase.

A polarized optical micrograph of the smectic phase texture of a polymer synthesized here with a higher comonomer feed ratio (magnification × 400).     

Hamiltonian Paths and Self‐Avoiding Walks of Lesser Length on Various Surfaces: A Monte Carlo Estimate

Given some lattice, the number Z_HP of Hamiltonian paths and also the number Z_N of N‐step shorter self‐avoiding walks on the surface of cylinders, cones, tori, and spheres has been Monte Carlo estimated. The procedure is an extension of the technique used in a previous paper for plane squares and rectangles, which is based on the Rosenbluth‐Rosenbluth chain‐generation procedure. Starting from a rectangle having m lines and n columns, and thus m×n lattice sites, one may obtain cylindrical, conical, toroidal and spherical surfaces through continuous deformations, which respect the topology. Then a correspondence is established between a plane figure of the ‘polar’ coordinates kind and the topology of the above surfaces. Using this topological equivalence, and thus operating exclusively on the plane ‘polar’ figure, Monte Carlo simulations show that for given m and n, Z_HP and Z_N increase when going from the plane rectangle to the cylinder and then to the cone and the torus. The number Z_NC of N‐step cycles (closed configurations) has also been Monte Carlo estimated. The Monte Carlo results for the surfaces studied here have been condensed in fifth degree polynomials in Φ, where Φ is the fraction of available lattice sites on the surface which are occupied by the N‐step self‐avoiding walk. The variation of the ratio Z_NC /Z_N with m and n has been estimated for cylindrical and conical surfaces. Finally, an effective coordination number q_eff has been introduced for finite surfaces, and its variation with Φ studied.     

Methyl branching in polyethylene from homopolymerization of ethylene with N‐arylcyano‐β‐diketiminate methallyl nickel‐B(C6F5)3

N‐Arylcyano‐β‐diketiminate methallyl nickel complexes activated with B(C₆F₅)₃ were used in the polymerization of ethylene. The microstructure analysis of obtained polyethylene (PE) was done by differential scanning calorimetry and ¹³C nuclear magnetic resonance (NMR). The branched polymer structures produced by these catalysts were attributed to one step isomerization mechanism of the catalyst along the polymer chain. The ortho or para position of the cyano group with co‐ordinated B(C₆F₅)₃ in both methallyl nickel catalysts influenced the polymer molecular weight, branching, and consequently melting and crystallization temperatures. NMR spectroscopic studies showed predominantly the formation of methyl branches in the obtained PE. Catalysts under study gave linear low‐density PEs with good crystallinities at temperatures of reaction between 50 °C and 70 °C at moderate pressures (12.3 atm). A propylene–ethylene copolymer produced by the metallocene catalyst had the same concentration of branches as the PE synthesized from methallyl nickel/B(C₆F₅)₃. Comparing the two polyolefins with the same degree of branching, it was observed that the polymer obtained with the nickel catalyst proved to be twice more crystalline and had greater T_m. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 452–458     

Novel Polymer Clusters with a Uniform Chain Density

The thermal decomposition of polystyrene clusters prepared by using 4‐methacryloyloxy‐2,2,6,6‐tetramethylpiperidine‐N‐oxyl as a branching agent was studied. The weight‐average molar mass (M̄_w) and the radius of gyration (R_g) of the resultant clusters can be scaled as M̄_w ∝︁ R_g^{3.0 ± 0.1}, revealing that these clusters have a uniform chain density. Moreover, the dynamic properties of such clusters converge unexpectedly faster than the static properties as the molar mass decreases.     

Synthesis of ultra‐small branched star poly(ε‐caprolactone)s and their high end group concentration effects on crystallization

We successfully synthesize the three‐ and six‐branched star poly(ε‐caprolactone)s with extremely small branched segments (USB‐SPCLs) using a facile pseudo‐one‐pot process in a pilot scale and investigate the effect of ultra‐small branches on their crystallization behaviors. The number of branched segments and the individual branched segment lengths for USB‐SPCLs are precisely controlled via manipulating monomer‐to‐core ratio, adjusting monomer‐to‐polymer conversion, end‐capping the terminal hydroxyl groups, and vacuum purification, which results in USB‐SPCLs having the branched segments below five degree of polymerization with a high yield exceeding 93%. The molecular weights obtained from ¹H NMR spectroscopy are consistent with that obtained from MALDI‐TOF‐MS and the molecular weight distributions are narrow with M_w/M_n ≤ 1.2, indicating that USB‐SPCLs have mono‐dispersed branches. USB‐SPCLs have low melting temperatures and broad double‐melting peaks attributed to their extremely small branches, and the crystallization behaviors for USB‐SPCLs depend on the end group concentration. On the other hand, the glass transitions for USB‐SPCLs depend on the total molecular weights, regardless of the number and length of branched segments. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 1134–1142     

Heterochain model for simultaneous long‐chain branching and crosslinking. III. Multicomponent polymerization

The matrix formula developed in the context of heterochain theory, M?_w = M?_wp + WF ( I ? M )^?1 S , was applied to describe the molecular weight development during free‐radical multicomponent polymerization. All of the required probabilistic parameters are expressed in terms of the kinetic‐rate constants and the various concentrations associated with them. In free‐radical polymerization, the number of heterochain types, N, needs to be extrapolated to infinity, and such extrapolation is conducted with only three different N values. This matrix formula can be used as a benchmark test if other approximate approaches can give reasonable estimates of the weight‐average molecular weights. The moment equations with the average pseudo‐kinetic‐rate constants for branching and crosslinking reactions may provide poor estimates when the copolymer composition drift during polymerization is very significant. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 2801–2812, 2004     

Monte carlo study of star-branched polymers on the tetrahedral lattice. II. Statistical thermodynamics of single macromolecules

The thermodynamical properties of the star-branched polymers on the tetrahedral lattice are studied taking into account nearest-neighbor interactions. The excess free energy and energy and heat capacities are computed for wide ranges of chain lengths, reduced potential ?/kT, and number of branches. A significant influence of the degree of branching on long-range interactions in the polymer random coil is observed. The possibilities of phase transitions in both linear and branched systems are discussed on the basis of the Monte Carlo data.     

Alkylaluminum activator effects on polyethylene branching using a N,N′‐nickel precatalyst appended with bulky 4,4′‐dimethoxybenzhydryl groups

Ten unsymmetrical N,N'‐bis (imino) acenaphthene‐nickel (II) halide complexes, [1‐[2,6‐{(4‐MeOC₆H₄)₂CH}₂–4‐MeC₆H₂N]‐2‐(ArN)C₂C₁₀H₆]NiX₂, each appended with one N‐2,6‐bis(4,4'‐dimethoxybenzhydryl)‐4‐methylphenyl group, have been synthesized and characterized. The molecular structures of Ni1 , Ni3 , Ni5 and Ni6 highlight the variation in steric protection afforded by the inequivalent N‐aryl groups; a distorted tetrahedral geometry is conferred about each nickel center. On activation with diethylaluminum chloride (Et₂AlCl) or methylaluminoxane (MAO), all complexes showed high activity at 30°C for the polymerization of ethylene with the least bulky bromide precatalysts ( Ni1 and Ni4 ), generally the most productive, forming polyethylenes with narrow dispersities [M_w/M_n: < 3.4 (Et₂AlCl), < 4.1 (MAO)] and various levels of branching. Significantly, this level of branching can be influenced by the type of co‐catalyst employed, with Et₂AlCl having a predilection towards polymers displaying significantly higher branching contents than with MAO [T_m: 33.0–82.5°C (Et₂AlCl) vs. 117.9–119.4°C (MAO)]. On the other hand, the molecular weights of the materials obtained with each co‐catalyst were high and, in some cases, entering the ultra‐high molecular weight range [M_w range: 6.8–12.2 × 10⁵ g mol^?1 (Et₂AlCl), 7.2–10.9 × 10⁵ g mol^?1 (MAO)]. Furthermore, good tensile strength (ε_b up to 553.5%) and elastic recovery (up to 84%) have been displayed by selected more branched polymers highlighting their elastomeric properties.     

Randomly branched bisphenol A polycarbonates. I. Molecular weight distribution modeling,interfacial synthesis,and characterization

Randomly branched bisphenol A polycarbonates (PCs) were prepared by interfacial polymerization methods to explore the limits of gel‐free compositions available by the adjustment of various composition and process variables. A molecular weight distribution (MWD) model was devised to predict the MWD, G, and weight‐average molecular weight per arm (M_w /arm) values based on the composition variables. The amounts of the monomer, branching agent, and chain terminator must be adjusted such that the weight‐average functionality of the phenolic monomers (F_OH ) was less than 2 to preclude gel formation in both the long‐ and short‐chain branched (SCB) PCs. Several series of SCB and long‐chain branched PCs were prepared, and those lacking gels showed molecular weights measured by gel permeation chromatography–UV and gel permeation chromatography–LS consistent with model calculations. In SCB PCs, the minimum M_w /arm that could be realized without gel formation depended on both composition (molecular weight, terminator type) and process (terminator addition point, coupling catalyst) variables. The minimum M_w /arm achieved in the low molecular weight series studied ranged from ∼3300 to ∼1000. The use of long chain alkyl phenol terminators gave branched PCs with lower glass‐transition temperatures but a higher gel‐free minimum M_w /arm. SCB PCs where M_w /arm was less than ∼M_c spontaneously cracked after compression molding, a result attributed to their lack of polymer chain entanglements. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 560–570, 2000