Molecular‐Weight Multimodality of Multiple Flory Distributions

It is well‐known that the final end‐use properties of polymer resins depend on the shape of the molecular‐weight distribution (MWD) very strongly. Particularly, polymer resins with bimodal MWDs are required for certain special applications, as they may simultaneously present enhanced mechanical and flow properties. A theoretical framework for the characterization of bimodality (or multimodality) of MWDs of polymers produced through linear polymerizations at steady‐state or quasi‐steady‐state conditions is developed and presented here. Conditions for the development of bimodality in generalized NS‐Schulz–Flory distributions are characterized for different forms of presentation of the MWDs. It is shown that the bimodal character of the MWD depends on the particular form used to represent it, which can then be used to generate an index of bimodality of the MWD. The theoretical results are finally used to compute the index of bimodality of actual polymer materials obtained at plant site.     

Addition‐Fragmentation Chain Transfer to Polymer in the Free Radical Ring‐Opening Polymerization of an Eight‐membered Cyclic Allylic Sulfide Monomer

Summary: A detailed investigation of chain transfer to polymer during free radical ring‐opening polymerization of the eight‐membered disulfide monomer 2‐methyl‐7‐methylene‐1,5‐dithiacyclooctane (MDTO) is presented. It has been shown that extensive chain transfer to polymer occurs involving both poly(MDTO) radicals and cyanoisopropyl radicals. Significant decreases in molecular weight were observed when cyanoisopropyl radicals were generated in the presence of poly(MDTO) in the absence of monomer. The molecular weight distribution (MWD) obtained from polymerization of MDTO in the presence of pre‐added poly(MDTO) was markedly different from that obtained without pre‐added polymer. A kinetic model was constructed in an attempt to quantitatively describe the chain transfer to polymer process based on the addition fragmentation chain transfer mechanism. It was found however that the simulated MWDs were considerably broader than the experimental MWDs, which were similar to the Schulz‐Flory distribution.

Mechanism for chain transfer to polymer.     

Evolution of molecular weight distributions in the catalytic chain transfer polymerization of methyl methacrylate up to high monomer conversions

The evolution of molecular weight distributions (MWDs) with monomer conversion in the catalytic chain transfer (CCT) polymerization of methyl methacrylate at 60 °C is investigated by simulation (via the program package PREDICI®) and experiment. A Co(III)‐based complex is used as the precursor for the CCT agent, which is formed in situ by initiator‐derived (2,2′‐azobisisobutyronitrile) radicals to yield the catalytically active Co(II) species. The small shifts seen in the MWD toward lower molecular weights with increasing monomer conversion are shown to be of the same order of magnitude as the associated changes in the MWD in non‐CCT controlled free‐radical polymerization, indicating that no significant change in the MWD with monomer conversion is associated with the CCT process. These results are compared to the evolution of MWDs in conventional chain transfer polymerizations with thiols as transfer agents. A clear shift toward higher molecular weights is seen with increasing monomer conversion, indicating disparate rates of thiol and monomer consumption. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 3303–3312, 2000     

Modeling molecular weight distribution in emulsion polymerization reactions with transfer to polymer

A mathematical model was developed for the computation of the dynamic evolution of molecular weight distributions (MWDs) during nonlinear emulsion polymerization reactions. To allow the direct computation of the whole MWD, an adaptive orthogonal collocation technique was applied. The model was validated with experimental methyl methacrylate/butylacrylate (BuA) semicontinuous and vinyl acrylate (VA)/Veova10 continuous emulsion polymerization results. Both systems considered introduce significant chain‐transfer reactions to polymer chains as a result of the presence of BuA and VA, respectively. The model developed was able to represent quite properly the kinetics and MWD of polymer samples during emulsion polymerizations. © 2001 John Wiley & Sons, Inc. J Polym Sci Part A: Polym Chem 39: 3513–3528, 2001     

Exploring Reaction Kinetics of a Multi‐Site Ziegler‐Natta Catalyst Using Deconvolution of Molecular Weight Distributions for Ethylene‐Hexene Copolymers

Industrial ethylene‐hexene copolymer samples produced using a supported Ti‐based Ziegler‐Natta catalyst were deconvoluted into five Flory molecular weight distributions (MWDs). Relationships between reactor operating conditions and deconvolution parameters confirmed that temperature and hydrogen and hexene concentrations influenced the MWD. The two sites that produced low‐molecular‐weight polymer responded similarly to changes in reactor operating conditions, as did the three sites that produce high‐molecular‐weight polymer. Increasing hexene concentration resulted in relatively more polymer being produced at the two low‐molecular‐weight sites and less at the high‐molecular‐weight sites. The information obtained will be useful for making simplifying assumptions during kinetic model development.

     

Organic nanoparticles as fragmentable support for Ziegler–Natta catalysts

A fragmentable support material for Ziegler–Natta catalysts is presented based on micrometer‐sized aggregates of polystyrene nanoparticles. Hydroxyl anchoring groups are introduced by copolymerization of hydroxymethylstyrene in emulsion process to immobilize the catalysts. The catalytic activity in ethylene slurry polymerizations is found to be directly correlated to the hydroxyl group content of the supports. Furthermore, the fragmentation behavior of dye‐labeled support aggregates into the initial nanoparticles is demonstrated using laser scanning confocal fluorescence microscopy as a nondestructive method. These supported catalysts fulfill two important design criteria, high fragmentability and high catalyst loading, and produce high‐density polyethylene with medium molecular weight distributions (MWDs = 3–4). These values lie between those obtained using single‐site metallocene‐based (narrow MWD < 3) or inorganic supported multi‐site Ziegler–Natta‐based (broad MWD = 4–12) polymerizations without the need of blending. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 15–22     

Dynamic Optimization of Molecular Weight Distribution Using Orthogonal Collocation on Finite Elements and Fixed Pivot Methods: An Experimental and Theoretical Investigation

In the present study, two numerical methods, namely the orthogonal collocation on finite elements and the fixed pivot technique, are employed to calculate the MWD in an MMA free‐radical batch suspension polymerization reactor operating up to very high conversions (e.g., ≥95%). The theoretical MWD predictions are directly compared with experimentally measured MWDs, obtained from a pilot‐scale batch MMA suspension polymerization reactor. It is shown that there is a very good agreement between model predictions and experimental measurements on both monomer conversion and MWD. Subsequently, two different time‐optimal temperature trajectories are calculated to obtain a polymer having either a narrow or a bimodal MWD in minimum batch time. The calculated time optimal trajectories are then applied, as set point temperature changes, to a pilot plant batch polymerization reactor. It is shown that the measured MWDs are in very good agreement with the off‐line calculated optimal MWDs.

     

Thermosensitive diblock copolymers with designed molecular weight distribution: Synthesis by continuous living cationic polymerization and micellization behavior

The synthesis of diblock copolymers with designed molecular weight distributions (MWDs) was successfully demonstrated in a continuous living cationic polymerization system using simple equipment. The control of MWDs was achieved by gradually feeding a polymerization reaction mixture into a terminating agent. As thermosensitive diblock copolymers, poly(vinyl ethers) containing a thermosensitive segment with oxyethylene side chains and a hydrophilic segment were prepared. The polymerization was carried out in a gas‐tight microsyringe, and the polymerization mixture was added continuously into methanol during the second‐stage polymerization. The self‐association behavior of the resulting diblock copolymers was evaluated by dynamic light scattering in water. MWD‐designed polymers with thermosensitive segments that varied continuously in length and hydrophilic segments of nearly uniform lengths formed micelles with a broad size distribution. Conversely, polymers with nearly uniform thermosensitive segments and hydrophilic segments of different lengths formed micelles with a narrow size distribution, as observed with conventional narrow MWD diblock copolymers. Thus, the MWD of the thermosensitive segment proved a decisive factor in achieving fine control of self‐association. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 2212–2221, 2008     

Molecular Weight Distributions in Coupled Polymers Using Z‐Transform

A study of end‐to‐end coupling reactions of bifunctional linear prepolymers of the Schulz–Flory type is presented by use of Z‐transform. The application of Z‐transform allows one to derive exact expressions for the change of molecular weight distributions in dependence of the progress of the coupling reaction. After coupling the distribution is also of Schulz–Flory type, however, it depends in a relatively complicated manner on the extent of reaction of the prepolymer and on the extent of the coupling reaction. Further expressions are derived for various dependencies including the influence of the extent of coupling on the average molecular weights, chain lengths, and polymolecularity. By implementation of the Mark–Houwink–Sakurada equation, a correlation between intrinsic viscosity and the progress of coupling reaction is shown.     

Simulation as a Tool for Feasibility Studies about PIP‐SEC Experiments

Advanced homo‐ and copolymerization models have been used to perform a feasibility study on the potential of pulse‐initiated polymerization (PIP) experiments for ethene (co)polymerizations. An application of PIP experiments directly to the ethene homo‐polymerization appears not as a very promising strategy to derive the homo‐propagation rate coefficient k_p of ethene. This failure can be attributed to the special characteristics of high temperature size exclusion chromatographs, being required to determine the molecular weight distribution (MWD) of polyethylene. PI copolymerizations appear as an interesting alternative to provide access to the homo‐propagation rate coefficient of ethene. Most advantageous in this strategy is the fact that even a simple convergence contemplation (using a variation in monomer composition) yields the ethene homo‐propagation rate coefficient k_p. Simply aiming at this coefficient, there is no necessity of knowing the detailed kinetic parameters of the copolymerization. In a further part, the extended kinetic information being available about branching processes in ethene polymerizations was used to test for the potential influence of a slower propagation rate of secondary macroradicals on the PIP structure in MWDs. Even at the significant level of branching present in ethene homopolymerizations still a PIP structure inside the MWD remains observable, assuming retardation up to an extend of almost two orders of magnitude. In order to perform these studies a kinetic model was designed explicitly accounting for the formation of secondary macroradicals by transfer. The kinetic information about branching being available in literature was adopted toward this scheme.     

Branched structure formation in free radical polymerization of vinyl acetate

The branched structure formation during free radical polymerization of vinyl acetate is investigated in detail by application of the computer simulations on the basis of the Monte Carlo sampling technique. Simulations are made for the whole molecular weight distribution (MWD), the MWDs for polymer molecules containing 0, 1, 2, 3, etc., branch points, the branching density as functions of both size and the number of branch points, the spatial distribution of the branched chains, etc. It was found that the effect of polyradicals on the formed MWD could be neglected for batch polymerizations of the present reaction system. A large number of relatively small branch chains are formed due to both chain transfer to polymer (CTP) and the terminal double-bond polymerization (TDBP). The radius of gyration at a Θ state is found to agree satisfactorily with the Zimm-Stockmayer equation for random branching in spite of the heterogeneous branched structure formed in the polymerization. The present investigation reveals important characteristics of the complex molecular structure formation during free radical polymerization that involves both CTP and TDBP. © 1996 John Wiley & Sons, Inc.     

“Living” Radical Polymerization in Tubular Reactors, 2 – Process Optimization for Tailor‐Made Molecular Weight Distributions

“Living” radical polymerization is a relatively new polymerization process that can be used to prepare resins with controlled structures. In this work, a mathematical model developed previously to describe nitroxide‐mediated “living” radical polymerizations performed in tubular reactors is used for the optimization of the process and obtainment of tailor‐made MWDs. Operating conditions and design variables are determined with the help of optimization procedures in order to produce polymers with specified MWDs. It is shown that bimodal and trimodal MWDs, with given peak locations, can be obtained through proper manipulation of the operating conditions. This indicates that the technique discussed in this work is suitable for detailed design of the MWD of the final polymer.

     

Copolymerization of Ethylene with 1,9‐Decadiene: Part II—Prediction of Molecular Weight Distributions

In the present work, adaptive orthogonal collocation and a Monte Carlo method are used to compute the molecular weight distributions (MWD) of ethylene/1,9‐decadiene copolymers produced with a constrained geometry catalyst. Predictions from each model are compared to each other and to the experimental MWDs, allowing for the evaluation of relative strengths and weaknesses of each mathematical modeling method. Comparisons with experimental results indicate that the rate of macromonomer incorporation in the growing polymer chains decays with the macromonomer radius of gyration. In all cases, the proposed models are able to fit appropriately the available experimental MWDs.     

Determination of the Catalytic Sites for Ziegler‐Natta Homo‐Polymerization from GPC Data

Deconvolution of the MWD of a polymer produced by multi‐site catalysts into independent Flory modes is the first step in modeling the polymerization process. A new deconvolution procedure for GPC data is developed that does not require an a priori assumption concerning the nature of the discrete distribution and can be used with a continuous distribution. The MWD measured via GPC is a linear function of the individual catalytic sites, but it is numerically ill‐conditioned, preventing direct inversion of the GPC data. Tikhonov regularization has been developed to uniquely invert the MWD. Applying the regularizing method to a polyethylene produced via a Ziegler‐Natta catalyst, seven discrete sites were found, and the kinetic constant ratios were determined for each of these sites.

     

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.     

Copolymerization of ethylene and α‐olefins with combined metallocene catalysts. I. A formal criterion for molecular weight bimodality

Metallocene and other single‐site catalysts can be combined to produce polyolefins with broadened distributions of molecular weight, chemical composition, and long‐chain branching. These resins are finding increasing applications because of their enhanced properties compared to ones made with conventional Ziegler–Natta catalysts. Resins with bimodal molecular weight distributions (MWDs) have especially attractive mechanical and rheological properties. Although the use of these resins is expected to increase, there are very few studies available to quantify MWD bimodality or to decide a priori which combinations of metallocene catalysts will lead to the formation of polyolefins with bimodal MWDs. In this article, a necessary condition for the production of polymer with bimodal MWD using two single‐site‐type catalysts is derived. Additionally, a bimodality index is defined to quantify MWD bimodality. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 1408–1416, 2000     

Copolymerization of ethylene and α‐olefins with combined metallocene catalysts. II. Mathematical modeling of polymerization with single metallocene catalysts

The relation between the polymerization conditions and the distributions of molecular weight (MWD) and chemical composition (CCD) of poly(ethylene‐co‐1‐hexene) made with single supported metallocene catalysts was investigated. Understanding the behavior of each metallocene under different polymerization conditions is necessary for designing combined metallocene catalysts to produce tailor‐made polyolefins. In this article, a simple mathematical model based on experimental results is developed and combined with the bimodality criterion developed in Part I of this series to predict polymerization conditions and metallocene combinations that will produce polymers with desired MWDs and CCDs. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 1417–1426, 2000     

Design strategies for controlling the molecular weight and rate using reversible addition–fragmentation chain transfer mediated living radical polymerization

Living radical polymerization has allowed complex polymer architectures to be synthesized in bulk, solution, and water. The most versatile of these techniques is reversible addition–fragmentation chain transfer (RAFT), which allows a wide range of functional and nonfunctional polymers to be made with predictable molecular weight distributions (MWDs), ranging from very narrow to quite broad. The great complexity of the RAFT mechanism and how the kinetic parameters affect the rate of polymerization and MWD are not obvious. Therefore, the aim of this article is to provide useful insights into the important kinetic parameters that control the rate of polymerization and the evolution of the MWD with conversion. We discuss how a change in the chain‐transfer constant can affect the evolution of the MWD. It is shown how we can, in principle, use only one RAFT agent to obtain a polymer with any MWD. Retardation and inhibition are discussed in terms of (1) the leaving R group reactivity and (2) the intermediate radical termination model versus the slow fragmentation model. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 43: 3189–3204, 2005