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‐Radical Polymerization with Long‐Chain Branching and Scission in a Continuous Stirred‐Tank Reactor

Free‐radical polymerization that involves the polymer transfer reactions leading to both long‐chain branching and scission, as in the cases of high‐pressure olefin polymerization, is considered. In CSTR, the residence time distribution is broad and the primary polymer chain, whose residence time is large, is subjected to polymer transfer reaction for a longer time, leading to a larger number of branching and scission points. The distributions of both branching and scission density are much broader in a CSTR than in a batch, or equivalently, a PFR. The radius of gyration for larger sized polymers formed in a CSTR tends to be much smaller than that for randomly branched polymers.

     

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     

Molecular Weight Development during Simultaneous Chain Scission,Long‐Chain Branching and Crosslinking, 1

A general matrix formula is proposed for the weight‐average molecular weights of the polymer systems formed through simultaneous scission, branching and crosslinking of N types of chains, assuming the chain connection statistics are Markovian. For the polymerization systems in which chains are generated consecutively, such as for free‐radical polymerization, the present theory can be applied by increasing the number of chain types N to infinity, by considering the chains formed at different times as different types of chains. The gel point determination reduces to the eigenvalue problem and the present theory extends the classical gelation theory to non‐random, history‐dependent reaction systems. From the mathematical point of view, this theory is capable of describing complex molecular build‐up processes through end‐linking, T‐ and H‐shaped chain connections, irrespective of reaction/reactor types used.

Schematic representation of the 0^th generation segment and the connection to the 1^st generation segments.     

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.     

Prediction of the Bivariate Molecular Weight‐Long Chain Branching Distribution in High‐Pressure Low‐Density Polyethylene Autoclaves

In the present study a population balance approach is described to follow the time evolution of bivariate molecular weight‐long chain branching (MW‐LCB) distributions in high pressure low density polyethylene autoclaves. The model formulation is based on a sectional grid method, the so‐called fixed pivot technique (FPT). According to this method, the ‘live’ and ‘dead’ polymer chain populations are assigned to a selected number of discrete points. Then, the resulting dynamic discrete‐continuous molar species equations for ‘live’ and ‘dead’ polymer chains are solved at the specified grid points. It is shown that a very good agreement exists between theoretical results and experimental data which proves the capability of the FPT method in calculating the joint MW‐LCB distribution for branched polymers.

     

Branching degree and rheological response correlation in peroxide‐modified linear low‐density polyethylene

Correlations between rheological behavior and degree of long chain branching (LCB) of linear low‐density polyethylene (LLDPE) upon a peroxide (dicumyl peroxide [DCP]) modification process under various conditions are discussed in this paper. The gel content analysis revealed negligible insoluble crosslinked fraction implying that incorporation of DCP to LLDPE predominately leads to branching rather than crosslinking. The slight changes in average molecular weight and molecular weight distribution induced by peroxide modification under various conditions revealed that formation of low‐molecular‐weight fractions due to chain scission is also negligible. The changes in terminal, trans, and pendant double bonds concentration of the modified samples with different amounts of peroxide were well depicted by Fourier transform infrared spectroscopy. Considering insignificant changes in molecular weight and molecular weight distribution during peroxide modification, the deviation observed in zero‐shear‐rate viscosity (η₀) values of the modified LLDPE with that of power‐law equation related to the linear PEs could be reliably attributed to the presence of LCB in the peroxide modified samples. Increasing the DCP content at roughly constant molar mass led to increasing of η₀ values as a result of increased degree of LCB. The increase in η₀ values was ascribed to prolonged relaxation times of the polymer molecules due to the retarded reptation motion‐driven relaxation mechanism. Copyright © 2014 John Wiley & Sons, Ltd.     

Rheological Characterization of Computationally Synthesized Reactor Populations of Hyperbranched Macromolecules: Bivariate Seniority‐Priority Distribution of ldPE

Summary: Hyperbranched molecules like low‐density polyethylene (ldPE) adopt a huge variety of molecular architectures. Previous work has shown that it is possible to computationally synthesize these architectures and to characterize them according to radius of gyration. Here, a method is presented and applied on ldPE to characterize populations using rheological quantities in terms of comb‐shaped and Cayley tree structures. Interbranch segments are assigned seniority and priority values that quantify their behavior in relaxation and elastic deformation processes. New general‐purpose algorithms have been developed to derive the full bivariate seniority/priority distribution using a representation from the graph theory of branched architectures. This paper describes the computation of bivariate chain length/degree of branching distributions (CLD/DBD) using a Galerkin finite element method for two scission mechanisms: linear and topological scission. The DBD is calculated using pseudo‐distributions. Random scission is treated with fragment length and branch point redistribution functions as obtained from scission statistics of branched molecules, preferentially yielding short and long fragments. Reactor populations of ldPE architectures are then obtained using computational synthesis. The seniority and priority distributions calculated indeed prove to be an adequate characterization method. They show good comparison, although not a complete overlap, with size characterization using a variant of the radius of gyration. It was possible to calculate a full bivariate seniority/priority fraction distribution, but due to the limited sample size its surface was not smooth. Subsequent work has shown the consequences for the prediction of rheological properties.

Seniority/priority values for segments of molecules for one chain length/number of branch points combination.     

Evaluation of the Chain Length Distribution in Free‐Radical Polymerization, 1. Bulk Polymerization

A method for the direct computation of the chain length distribution in a bulk polymerization is developed, based on the discretization procedure introduced by Kumar and Ramkrishna (Chem. Eng. Sci. 1996 , 51, 1311) in the context of particle size distribution. The overall distribution of chain lengths is partitioned into a finite number of classes which are supposed to be concentrated at some appropriate pivotal chain lengths. Several of the involved reactions lead to the formation of chain whose length differs from the pivotal values. Rules have been introduced in order to share chains between two contiguous classes, which have been designed so as to preserve two well‐defined properties of the distribution, such as, for example, two of its moments. The method has been applied to a polymerization system including propagation, bimolecular terminations and two different chain branching mechanisms: chain transfer to polymer and crosslinking. In addition, complex systems such as one with chain length‐dependent kinetic constants or a two‐dimensional distribution of chain length and number of branches have been considered.     

Main‐Chain Scission of a Charged Fifth‐Generation Dendronized Polymer

During divergent synthesis of the next higher‐generation dendronized polymer (DP), the fifth‐generation DP, PG5, with a number‐average degree of polymerization, (i.e., number of monomeric units) P_n, of ca. 500 underwent main‐chain scission. This happened in the step when its peripheral Boc groups were removed by the treatment with trifluoroacetic acid (TFA), and thus a heavily charged polyelectrolyte formed as an intermediate. Atomic Force Mircoscopy (AFM) analysis of the product after drop‐casting onto mica showed a large majority of short deprotected PG5 chains with P_n of ca. 40, as well as some smaller features that by MALDI‐TOF mass spectrometry and ¹H‐NMR spectroscopy were assigned to the hypothetical monomer, deprotected MG5. This behavior is compared to a recently reported main‐chain scission of a closely related PG5 which, however, resulted in significantly longer fragments. While this difference cannot yet be fully explained, questions are formulated which will guide future research.     

Comparing Long‐Chain Branching Mechanisms for Ethylene Polymerization with Metallocenes and Other Single‐Site Catalysts: What Simulated Microstructures Can Teach Us

Three different long‐chain branch (LCB) formation mechanisms for ethylene polymerization with metallocenes in solution polymerization semi‐batch and continuous stirred‐tank reactors are modeled to predict the microstructure of the resulting polymer. The three mechanisms are terminal branching, C–H bond activation, and intramolecular random incorporation. Selected polymerization parameters are varied to observe how each mechanism affects polymer microstructure. Increasing the ethylene concentration during semi‐batch polymerization reduces the LCB frequency of polymers made with the terminal branching and intramolecular mechanisms, but has no effect on those made with the C–H bond activation mechanism, which disagrees with most previous data published in the literature. The intramolecular mechanism predicts that LCB frequencies hardly depend on polymerization time or ethylene conversion, which also disagrees with the published experimental data for these systems. For continuous polymerization reactors, experimental data relating polydispersity to LCB frequency can be well described with the terminal branching mechanism, but both C–H bond activation and intramolecular models fail to describe this experimental relationship. Therefore, detailed simulations confirm that the terminal branching mechanism is indeed the most likely mechanism for LCB formation when ethylene is polymerized with single‐site coordination catalysts such as metallocenes in solution polymerization reactors.     

Ethylene/1‐octene copolymerization studies with in situ supported metallocene catalysts: Effect of polymerization parameters on the catalyst activity and polymer microstructure

Ethylene and 1‐octene copolymerizations were carried out with an in situ supported rac‐[dimethylsilylbis(methylbenzoindenyl)] zirconium dichloride catalyst. In a previous study, it was found that some in situ supported metallocenes produced polyethylene/α‐olefin copolymers with broad and bimodal short chain branching distributions and narrow molecular weight distributions. The ability to produce polyolefins with multimodal microstructural distributions in a single metallocene and a single reactor is attractive for producing polymers with balanced properties with simpler reactor technology. In this study, a factorial experimental design was carried out to examine the effects of the polymerization temperature and ethylene pressure, the presence of hydrogen and an alkylaluminum activator, and the level of the comonomer in the feed on the catalyst activity, short chain branching distribution, and molecular weight distribution of the polymer. The temperature had the most remarkable effect on the polymer microstructure. At high 1‐octene levels, the short chain branching distribution of the copolymer broadened significantly with decreasing temperature. Several factor interactions, including the hydrogen and alkylaluminum concentrations, were also observed, demonstrating the sensitivity of the catalyst to the polymerization conditions. For this catalyst system, the responses to the polymerization conditions are not easily predicted from typical polymerization mechanisms, and several two‐factor interactions seem to play an important role. Given the multiple‐site nature of the catalyst, it has been shown that predicting the polymerization activity and the resulting microstructure of the polymer is a challenging task. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 40: 4426–4451, 2002     

Mathematical Modelling and Computer Simulation of Linear Polymer Degradation: Simple Scissions

Summary: Degradation of a polymer in a reactor by the degrading agent(s) follows a distinct pattern, primarily influenced by structural integrity and reactor environment. This distinct pattern is recorded in the changes in the evolved molecular weight distribution (MWD) or polymer chain length distribution (PCLD) curve characteristics from the initial intact state. Modern size exclusion chromatography (SEC) is the best laboratory‐based method that can clearly provide these plots in the form of chromatogram; however, detailed molecular information is not available. The nature of molecular destruction can be well‐characterised if the distinct MWD shift patterns can be simulated to fingerprint the different chain scission dynamics. This is investigated by our current research using the power of computer simulation techniques to gain insight into the polymer ageing processes. One such technique for studying simple decay processes is presented here, and the results are compared with experimental findings. The concept of a binary tree scission model is introduced to show chain rupture as a sequence of probabilistic events and as a non‐linear function of time. Two new mathematical algorithms, an iterative Monte Carlo structured probability scheme and a semi‐iterative algebraic exact statistical formulation method, are investigated to implement this model and simulate the evolution of resultant temporal MW distribution. The latter, an innovative approach to mathematical modelling, has the potential to generate a statistically perfect instant MWD decay curve. A statistical comparison of the product yield is presented from the data obtained using a wide variety of simulated scission regimes to determine the sources of variability.

Simulated MWD lateral shift for percent cut scission model showing deviation from the initial MWD (red) over degradation time zones T_j(0 ≥ j ≤ 9) with bimodal and curve broadening effect due to accumulation of varied percent cut range 5–30%.     

Effect of the Side‐Chain‐Distribution Density on the Single‐Conjugated‐Polymer‐Chain Conformation

The spatial arrangement of the side chains of conjugated polymer backbones has critical effects on the morphology and electronic and photophysical properties of the corresponding bulk films. The effect of the side‐chain‐distribution density on the conformation at the isolated single‐polymer‐chain level was investigated with regiorandom (rra‐) poly(3‐hexylthiophene) (P3HT) and poly(3‐hexyl‐2,5‐thienylene vinylene) (P3HTV). Although pure P3HTV films are known to have low fluorescence quantum efficiencies, we observed a considerable increase in fluorescence intensity by dispersing P3HTV in poly(methyl methacrylate) (PMMA), which enabled a single‐molecule spectroscopy investigation. With single‐molecule fluorescence excitation polarization spectroscopy, we found that rra‐P3HTV single molecules form highly ordered conformations. In contrast, rra‐P3HT single molecules, display a wide variety of different conformations from isotropic to highly ordered, were observed. The experimental results are supported by extensive molecular dynamics simulations, which reveal that the reduced side‐chain‐distribution density, that is, the spaced‐out side‐chain substitution pattern, in rra‐P3HTV favors more ordered conformations compared to rra‐P3HT. Our results demonstrate that the distribution of side chains strongly affects the polymer‐chain conformation, even at the single‐molecule level, an aspect that has important implications when interpreting the macroscopic interchain packing structure exhibited by bulk polymer films.     

Size‐exclusion chromatography coupled to multiangle light scattering detection of long‐chain branching in polyethylene made with phillips catalyst

Size‐exclusion chromatography coupled to multiangle light scattering (SEC‐MALS) has been used to detect long‐chain branching (LCB) in polyethylene (PE) from Cr/silica catalysts for the first time. The observed LCB response to several catalyst and reactor variables mostly confirms earlier conclusions drawn from rheological measurements. However, SEC‐MALS has also shed additional light on a few previously unanswered questions. Above all, SEC‐MALS shows the placement of branching within the MW distribution, which was not previously known, and which may explain some of the unique molding behavior of Cr‐derived PE. This new SEC‐MALS data also provide insight into the mechanism of LCB formation, which is discussed. Like earlier studies based on rheology, this new study demonstrates that the commonly accepted view of macromer incorporation may be overly simplistic. © 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2012     

Structural and free‐volume analysis for alkyl‐substituted palladium‐catalyzed poly(norbornene): A combined experimental and Monte Carlo investigation

Increasing the length of an alkyl side chain on a rigid polynorbornene (PNB) backbone is shown to decrease the glass‐transition temperature of the resultant polymer, decrease the density of the bulk polymer, decrease the number and average free‐volume element size present, and decrease the permeability of gases through the polymer. Methyl‐, butyl‐, and hexyl‐substituted PNBs were investigated. Experimental results were compared with predictions based on molecular modeling. By using models that provided good agreement between the experimental and simulated wide‐angle diffraction patterns, the distributions of free‐volume elements were predicted. These predictions clearly indicate that the number of large free‐volume elements decreases as the length of the side chain increases, suggesting that the flexible aliphatic side chains can be largely accommodated within the free volume between the rigid PNB backbones. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 215–233, 2006     

A Conditional Monte Carlo Method to Determine the Architectures of Metallocene Catalyzed Polyethylene

Two polymer molecules of the same length (n) and the same number of branch points (N) can have different properties, since they may possess distinct architectures. In this paper we present a conditional Monte Carlo algorithm for the virtual synthesis of metallocene‐catalyzed polyethylene (PE) in a continuous stirred tank reactor (CSTR). The condition for the Monte Carlo method consists of a fixed chain length distribution (CLD) and a degree of branching distribution (DBD). These distributions are calculated with a Galerkin finite element method. The synthesis method is a recursive algorithm that subsequently creates insertions of sub‐structures containing numbers of branch points according to a certain probability density function. This provides an adjacency matrix describing the connectivity between the branch points, while separately a vector containing the length of segments between branch points and terminal segments is generated. Characterization of the architectures proceeds by rheological features, seniorities and priorities, and molecular properties like the radius of gyration. Comparing the radii of gyration of metallocene polyethylene and low density PE (ldPE) shows the former to possess a more comb‐like structure on average. This is confirmed by the rheological characterization. The found bivariate seniority/priority distribution agrees well to the results of an analytical study of the same chemical system.

Constituting elements of the algorithm.     

Simultaneous long‐chain branching and random scission: I. Monte Carlo simulation

In free‐radical olefin polymerizations, the polymer transfer reactions could lead to chain scission as well as forming long‐chain branches. For the random scission of branched polymers, it is virtually impossible to apply usual differential population balance equations because the number of possible scission points is dependent on the complex molecular architecture. On the other hand, the present problem can be solved on the basis of the probability theory by considering the history of each primary polymer molecule in a straightforward manner. The random sampling technique is used to solve this problem and a Monte Carlo simulation method is proposed. In this simulation method, one can observe the structure of each polymer molecule formed in this complex reaction system, and virtually any structural information can be obtained. In the illustrative calculations, the full molecular weight distribution development, the gel point determination, and examples of two‐ and three‐dimensional polymer structure are shown. © 2001 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 391–403, 2001     

Hydrolytic degradation of poly(rac‐lactide) and poly[(rac‐lactide)‐co‐glycolide] at the air–water interface

The understanding of the simultaneous transport and chain‐scission phenomena involved in the hydrolysis of bulk‐degrading polymers requires the experimental separation of chain cleavage and water diffusion. The hydrolytic chain cleavage of poly(rac‐lactide) rac‐(PLA) and poly[(rac‐lactide)‐co‐glycolide] (PLGA) is analysed on the basis of monolayer degradation experiments combined with an improved data reduction procedure. Different, partly contradictory models of the hydrolytic degradation and erosion mechanism of PLA and PLGA, namely random chain scission and chain‐end scission, are discussed in the literature. The instantaneous linear area reduction observed for the polymer Langmuir films indicates a chain‐end scission mechanism. As monolayers of end‐capped and non‐end‐capped polymers degrade with exactly the same rate, the observed differences in the degradation kinetics of bulk samples do clearly result from differences in the water penetration into these polymers. A pronounced ‘auto‐inhibition’ effect is observed for the polymers degraded at initially high pH of the aqueous subphase in the absence of buffers. Copyright © 2007 John Wiley & Sons, Ltd.