Deconvolution of Molecular Weight Distributions Using Dynamic Flory‐Schulz Distributions

Summary: The deconvolution of molecular weight distributions (MWDs) may be useful for obtaining information about the polymerization kinetics and properties of catalytic systems. However, deconvolution techniques are normally based on steady‐state assumptions and very little has been reported about the use of non‐stationary approaches for the deconvolution of MWDs. In spite of this, polymerization reactions are often performed in batch or semi‐batch modes. For this reason, dynamic solutions are proposed here for simple kinetic models and are then used for deconvolution of actual MWD data. Deconvolution results obtained with dynamic models are compared to deconvolution results obtained with the standard stationary Flory‐Schulz distributions. For coordination polymerizations, results show that dynamic MWD models are able to describe experimental data with fewer catalytic sites, which indicates that the proper interpretation of the reaction dynamics may be of fundamental importance for kinetic characterization. On the other hand, reaction dynamics induced by modification of chain transfer agent concentration seem to play a minor role in the shape of the MWD in free‐radical polymerizations.

This Figure illustrates that MWDs obtained at unsteady conditions should not be deconvoluted with standard steady‐state Flory‐Schulz distributions.     

Kinetics of redox polymerizations of acrylic acid in inverse dispersion and in aqueous solution

Redox polymerizations of acrylic acid in inverse dispersion and in aqueous solution (with surfactant) were conducted by using sodium metabisulphite/potassium bromate initiators. The monomer conversions were determined by using high‐performance liquid chromatography, and the polymer particles in the final lattices were examined using a scanning electron microscope with freeze‐fracture equipment. Experimental rate expressions implied that complex reactions are involved in the redox polymerizations. A chemical reaction scheme was proposed, and kinetic models were developed for the redox polymerization in aqueous solution. Comparison between the experimental rate expressions and the kinetic models suggested a combination of bimolecular and monomolecular termination modes, a chain transfer function of the surfactant, and an oxidizing role of the oxygen molecules. The differences in the experimental rate expressions between the redox polymerization in inverse dispersion and that in aqueous solution are in good agreement with the kinetic model predictions. © 1999 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 37: 313–324, 1999     

A survey of kinetic and mechanistic similarities in living polymerization systems

In the light of recent discoveries in the field of living polymerizations it seems inevitable to reconsider our views on these polymerization systems. This paper surveys the kinetic and mechanistic similarities in living polymerizations, and analyses and compares chain transfer dominated nonliving polymerizations and living systems to conclude on the nature of propagating species, shelflife and livingness. Some recently raised specific problems are also summarized and discussed. It has been found that most of the living polymerizations known to date, such as living anionic, cationic ring opening, group transfer, carbocationic, ring opening metathesis, Ziegler-Natta, free radical and immortal polymerizations, exhibit the characteristics of quasiliving polymerization, i.e., an equilibrium exists between propagating (active) and inactive (dormant) species. On the basis of this finding and a comparison between mechanistic and kinetic models of quasiliving and ideal living polymerizations, it is suggested that the former is the general phenomenon, and ideal living polymerization is a subclass of quasiliving polymerizations.     

Model simulation of cationic polymerization by an iterative technique

Using a simple, instructive model of living polymerizations, we develop an economic technique for solving a complete set of kinetic equations for active polymers P_n^* with chain length n. The method consists of two steps: (i) solution of the global kinetic equations for the global species P^* which comprises all individual polymers P_n^* together with monomer M and other non-polymeric species, followed by (ii) iterative solution of the original kinetic equations, successively for the individual polymers P₁^*, P₂^*, P₃^*, …, P_n^*. The method is applied successfully to a rather complex, realistic model of cationic polymerizations. In general, our iterative scheme serves to reduce numerical storage requirements, in comparison with traditional direct integrations for all individual kinetic equations. Therefore it may allow simulations of very demanding polymerizations, including very high degrees of polymerization, which cannot be evaluated by traditional techniques.     

Thermal polymerization of styrene in the presence of some ferrocene derivatives. II. Low concentrations of vinylferrocene and ethylferrocene

Styrene has been polymerized thermally at 60°C in the presence of low concentrations of vinylferrocene and in the presence and absence of 2,2′-azobisisobutyronitrile (AIBN). The polymerizations were studied in bulk and also in benzene solution. The thermal polymerization of styrene in the presence of ethylferrocene, but without added AIBN or solvent, was also examined. The bulk polymerizations exhibited high initial rates of polymerization followed by a decrease in rate. Initial rates of polymerization for bulk polymerizations in the absence of AIBN have been interpreted by means of a kinetic scheme involving propagation with styrene participating in a specific interaction with the ferrocene derivative and some kinetic parameters associated with this scheme have been evaluated. The decrease in the rate of polymerization is due to the formation of a retarder. The benzene solution polymerizations fitted a simple kinetic scheme and the transfer constant for vinylferrocene with respect to polystyryl radicals C_s, has been evaluated as 1.98 × 10^?3.     

Determination of initial reaction rates using wilkinson's relation

Uncertainty sometimes exists in determining initial reaction rates from experimental data. A method, originally proposed by Wilkinson [9] for estimating orders and rate constants for simple batch nth order reactions, has been generalized to complex kinetic systems. This method yields very accurate initial rates for all systems and extends the conversion range of experimental investigation of initial rates well beyond the “zero-order” region. Accurate initial rates are required in analytical methods used for screening alternate reaction mechanisms.     

Analysis of nonlinear reactions in modulated molecular beam surface experiments

An approximate method of analyzing nonlinear reaction models in modulated molecular beam surface kinetic studies is developed. The exact method for treating nonlinear surface mechanisms is tedious and almost always requires computer analysis. The proposed approximate method is a simple extension of the Fourier expansion technique valid for linear surface reactions; it quickly provides analytical expressions for the phase lag and amplitude of the reaction product for any type of nonlinear surface mechanism, which greatly facilitates comparison of theory and experiment. The approximate and exact methods are compared for a number of prototypical adsorption–desorption reactions which include coverage-dependent adsorption and desorption kinetics of order greater than unity. Except for certain extreme forms of coverage-dependent adsorption, the approximate method provides a good representation of the exact solution. The errors increase as the nonlinearities become stronger. Fortunately, when the discrepancy between the two methods is substantial, the reaction product signal is so highly demodulated that reliable experimental data usually cannot be obtained in these regions anyway.     

Communication between calorimetric experiment and continuous data evaluation during the incomplete reaction run

The kinetic control of exothermic reactions plays an important role in chemical safety technology. Predictions of thermal explosions by runaway, e.g. redox reactions, poly reactions, and decompositions, are required in case of simple as well as of complex reactions. Activation parameters of chemical reactions are often stressed by systematic errors which are caused by failures in accuracy of measurement and calibration. The influence of systematic errors for some selected reactions has been investigated by the software package TA-kin. The coupling between continuous data evaluation software and the precision calorimeter ACTRON 5.0, including a safety scenario equipment, was established to test the validity of calculated runaway predictions under practical conditions. Measured data are applied for kinetic evaluations by nonlinear optimization methods in real time. In this way, the experimental investigations of reaction systems become possible beyond the point of no return without any danger for the laboratory.     

Safety and Quality Aspects of Acrylic Monomers

Summary : Acrylic monomers are important intermediates for the chemical industry. Especially acrylic acid (AA) is the basis for various reactions, such as polymerizations and esterifications and is, therefore, responsible for high product diversity. Spontaneous polymerization is a safety problem during the transportation and storage of acrylic monomers. In the production process, polymerization leads to blockages in the apparatus. For the prevention of these issues, special stabilizer systems are used such as hydroquinone monomethyl ether (MeHQ)/oxygen and phenothiazine (PTZ). The reactions of these stabilizer systems are not well understood at the moment. Therefore a lot of expertise and experience are necessary to guarantee safe handling. In this paper some methods for the investigation of stability related reaction kinetics are presented. A better comprehension of the mechanism of the polymerization inhibition is generated by the kinetic simulation with these data.     

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.     

Polymerization in catalyst particles: Calculation of molecular weight distribution

Factors which affect and broaden molecular weight distributions are investigated for polymerizations in catalyst particles which are accumulating polymer, such as Ziegler polymerizations of olefins. Expressions for the Q value, or ratio of weight-average to number-average molecular weight, are derived for simple, but important linear models. These expressions are of the form Q = FSX for a kinetic scheme in which polymer chains have an infinite growing lifetime, and Q = 2FGSX for a kinetic scheme in which termination by hydrogen is dominant and each site produces many chains. The factors F, G, S, and X are equal to or greater than unity: F depends on particle geometry and the nature of diffusion and reaction; G depends on environmental history of the particle; S depends on the distribution of site activities; and X depends on the distribution of particle sizes. Effects of reactor type are also studied quantitatively.     

Statistical derivation of kinetic molecular weight development equations in nonlinear free-radical polymerization

We propose a statistical method to derive the differential equations that describe the weight-average molecular weight development during nonlinear free-radical polymerizations, by using the random sampling technique. We consider two types of nonlinear free-radical polymerization schemes, free-radical polymerization with chain transfer to polymer and free-radical crosslinking (co)polymerization. The obtained equations fully agree with those obtained through the kinetic approach using the method of moments. The physical meaning of each term in the differential equations as well as the implication of the functional form is discussed from the point of view of the statistical derivation.     

Modeling Kinetics and Structural Properties in High‐Pressure Fluid‐Phase Polymerization

The present contribution provides an overview of actual applications in modeling free‐radical polymerizations. Topics of interest are the simulation of pulsed laser polymerization experiments with subsequent analysis of the formed product by size exclusion chromatography (PLP‐SEC), single pulse laser experiments, experimental techniques for determining rate coefficients of elementary reactions that control polymer properties, and technical applications. Aspects being investigated are model validation and testing predictive potential in polymerization models using well‐defined experiments as well as developing and testing experimental strategies for deriving rate coefficients of elementary reactions that exist (especially when dealing with copolymerizations) within a network of complex coupled reactions. In any of these fields remarkable success in modeling can be achieved. This demonstrates the great potential that can grow from combining modern mathematical methods, computational power and detailed kinetic insights into the mechanism of polymerization. It is the wide scope of applications, e. g. ranging from modeling kinetics to the investigation of termination processes being dependent on the chain‐length of the macroradical (as an example of pure fundamental research) to modeling of technical reactors, that provides attractiveness and defines challenges. Especially, the success in transforming results directly from laboratory experiments into technical applications justifies laborious efforts in determining highly precise rate coefficients and proves the concept breaking down a complex process into elementary subparts. A necessary boundary condition for this is keeping in mind the demands along the whole scope of applications and avoiding simplifications that are only applicable for part of them. Although at a first glance this may appear to hinder fast progress in one discipline, it is the essential requirement for final success.     

Location of the Catalytic Site in Supported Atom Transfer Radical Polymerization

Summary: A supported and highly recyclable catalyst complex, CuBr/HMTETA physically adsorbed to silica gel, was used for the ATRP of MMA to elucidate the nature of the catalytic site. In some polymerizations, the reaction solutions were filtered and compared with their unfiltered references for catalytic activity. The filtered systems had high catalyst activity indicating the presence of active catalyst sites in solution. These sites are the primary catalytic contributors.

Conversion versus time and first‐order kinetic plot for the control and filtered polymerizations.     

On the Statistical Calibration of Physical Models

We introduce a novel statistical calibration framework for physical models, relying on probabilistic embedding of model discrepancy error within the model. For clarity of illustration, we take the measurement errors out of consideration, calibrating a chemical model of interest with respect to a more detailed model, considered as “truth” for the present purpose. We employ Bayesian statistical methods for such model‐to‐model calibration and demonstrate their capabilities on simple synthetic models, leading to a well‐defined parameter estimation problem that employs approximate Bayesian computation. The method is then demonstrated on two case studies for calibration of kinetic rate parameters for methane air chemistry, where ignition time information from a detailed elementary‐step kinetic model is used to estimate rate coefficients of a simple chemical mechanism. We show that the calibrated model predictions fit the data and that uncertainty in these predictions is consistent in a mean‐square sense with the discrepancy from the detailed model data.     

Vinyl polymerization by a manganese(III) complex

Bis(diethanolamine) manganate(III) was prepared. The polymerization of acrylamide and methacrylamide initiated by this complex in aqueous solution at pH 0.9 was studied at 45°. The rate of polymerization was followed by bromometry, the rate of complex disappearance spectrophotometrically and the molecular weights of the polymers were determined viscometrically. The rate of polymerization was found to be proportional to [Monomer]^1.0. The order with respect to initiator was found to be 0.5 for acrylamide and 0.3 for methacrylamide. The apparent overall activation energies for the polymerizations are ?87 kJ mol^?1 and ?59 kJ mol^?1 for acrylamide and methacrylamide respectively. A kinetic reaction scheme is proposed on the basis of the experimental data; kinetic parameters have been evaluated.     

Mechanism and kinetics of dithiobenzoate‐mediated RAFT polymerization. I. The current situation

Investigations into the kinetics and mechanism of dithiobenzoate‐mediated Reversible Addition–Fragmentation Chain Transfer (RAFT) polymerizations, which exhibit nonideal kinetic behavior, such as induction periods and rate retardation, are comprehensively reviewed. The appreciable uncertainty in the rate coefficients associated with the RAFT equilibrium is discussed and methods for obtaining RAFT‐specific rate coefficients are detailed. In addition, mechanistic studies are presented, which target the elucidation of the fundamental cause of rate retarding effects. The experimental and theoretical data existing in the literature are critically evaluated and apparent discrepancies between the results of different studies into the kinetics of RAFT polymerizations are discussed. Finally, recommendations for further work are given. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 5809–5831, 2006