Pseudo-kinetic Rate Constant Method Research Articles

Dynamic modeling of crosslinking and gelation in continuous ethylene–propylene–diene polymerization reactors using the pseudo–kinetic constant approach

A dynamic model for continuous ethylene–propylene–diene terpolymerization reactors in which crosslinking and gel formation are attributable to reactions between the pendant double bonds of diene units has been developed. The model is applicable to other types of crosslinking reactions such as those due to aging, polymer blending, and vulcanization. The polymer properties at the gel point and in the post-gel region are computed using the numerical fractionation method. Direct application of this method to the prediction of terpolymer properties in the gel or post-gel region can lead to severe numerical problems, due to large differences in order of magnitude of various moments across the generations. These problems are overcome by applying a pseudo-kinetic rate constant method, i.e., by constructing a moment model for a pseudo-homopolymer that approximates the behavior of the actual terpolymer under the long chain and quasi-steady state assumptions. The pseudo-homopolymer model is then used as the basis for application of the numerical fractionation method. We show that the proposed dynamic model is capable of predicting realistic polydispersities and molecular weight distributions even near the gel point with as few as 11 generations, and in the post-gel region with as few as five generations. The largest steady-state polydispersities of the soluble polymer are obtained when the crosslinking rate just exceeds the critical value for gelation. The steady-state polydispersity decreases exponentially in the post-gel region at higher values of the rate constant, while the sol fraction decreases in a more linear fashion. The overall molecular weight distribution (MWD) of the sol is constructed assuming a Schulz two parameter distribution for each generation. For the industrial case of a small number of crosslinks, the first two generations contribute the most to the MWD, which is unimodal. The tail of the MWD is longest near the initial gelation time; the tail is shortened in the post-gel region as higher generations are consumed.

Free-radical copolymerization of methyl methacrylate with styrene in the presence of 2-mercaptoethanol II. Influence of methyl methacrylate/styrene ratio

The free-radical copolymerization of methyl methacrylate (MMA) with styrene (St) in the presence of 2-mercaptoethanol (ME) was investigated in order to obtain ω-hydroxy oligomers with random copolymer-type chains of various compositions and molecular weights. Polymerizations at three different MMA/St molar ratios were carried out, while keeping constant the ME/monomer ratio. Monomer mixtures richer in MMA than in St were employed in order to attempt preparing lower polydispersity oligomers with monomodal molecular weight distribution (MWD). The molecular weights of the resulting oligomers increased with both conversion and MMA fraction in the feed, while polydispersities increased with conversion and decreased with MMA concentration in the initial monomer mixture. For the lower MMA fractions in the monomer feed, bimodal MWDs resulted beyond a certain conversion due to the faster relative consumption of ME than of monomer. Based on the pseudo-kinetic rate constant method, apparent chain transfer constants corresponding to the three different compositions of the monomer feed were estimated. The values obtained were in good agreement with the evolution of molecular weights and polydispersities with conversion and MMA fraction in the monomer feed. The co-oligomers prepared displayed functionalities around unity, making them suitable for the synthesis of macromonomers.

Molecular weight distributions in terpolymerization of α‐methylstyrene/styrene/acrylic acid in continuous stirred tank reactor

Free-radical terpolymerization of α-methylstyrene/styrene/acrylic acid (AMS/ST/AA) in a continuous stirred tank reactor (CSTR) was studied theoretically and experimentally. A series of reactions were conducted to investigate the effect of the reaction temperature, residence time, solvent contents, initiator concentration, chain-transfer agent level, and different monomer compositions on the molecular weight distribution (MWD). The mathematical model based on “the instantaneous property method (IPM)” and the pseudokinetic rate constant method was introduced and modified to predict the MWD of the bulk terpolymer produced in a CSTR. The effects of process variables were included in the mathematical model. Extensive comparisons were made between the theoretical results and experimental values. The rate of the free-radical terpolymerization of AMS/ST/AA was comparatively slow and it was found that the concentration of α-methylstyrene had a strong influence on the molecular weight and polydispersity of the MWD. This presumably reflected the existence of a normal bimolecular termination rate and a slow propagation rate due to steric hindrances at the doubly substituted vinyl carbon atom or transfer of the allylic hydrogen from the α-methyl group. Good agreement was obtained between calculated MWD and the experimental one by assuming that the disproportionation termination of free radicals was the dominant chain-stoppage mechanism. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 79: 479–487, 2001

Molecular weight distributions in terpolymerization of ?-methylstyrene/styrene/acrylic acid in continuous stirred tank reactor

Free-radical terpolymerization of α-methylstyrene/styrene/acrylic acid (AMS/ST/AA) in a continuous stirred tank reactor (CSTR) was studied theoretically and experimentally. A series of reactions were conducted to investigate the effect of the reaction temperature, residence time, solvent contents, initiator concentration, chain-transfer agent level, and different monomer compositions on the molecular weight distribution (MWD). The mathematical model based on “the instantaneous property method (IPM)” and the pseudokinetic rate constant method was introduced and modified to predict the MWD of the bulk terpolymer produced in a CSTR. The effects of process variables were included in the mathematical model. Extensive comparisons were made between the theoretical results and experimental values. The rate of the free-radical terpolymerization of AMS/ST/AA was comparatively slow and it was found that the concentration of α-methylstyrene had a strong influence on the molecular weight and polydispersity of the MWD. This presumably reflected the existence of a normal bimolecular termination rate and a slow propagation rate due to steric hindrances at the doubly substituted vinyl carbon atom or transfer of the allylic hydrogen from the α-methyl group. Good agreement was obtained between calculated MWD and the experimental one by assuming that the disproportionation termination of free radicals was the dominant chain-stoppage mechanism. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 79: 479–487, 2001

Kinetic parameter estimation for the MMA/MA copolymerization system

The mathematical model for a batch methyl methacrylate (MMA) and methyl acrylate (MA) copolymerization system was simplified by applying the pseudo-kinetic rate constant method and used for the estimation of parameters for various reaction rate constants. For this, the polymerization experiment was conducted under a variety of reaction conditions and the Levenberg-Marquardt method was employed. The state variables selected for the parameter estimation were the solid content and the number and weight average molecular weights. The validity of the estimated parameters was corroborated by conducting experiments under different reaction conditions and the newly estimated parameter values were found to be superior to the ones reported in the literature. Thus the kinetic parameters estimated in this study may be used to predict the properties of polymer product obtained by solution MMA/MA copolymerization. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 78: 2554–2564, 2000

Free-radical copolymerization of methyl methacrylate with styrene in the presence of 2-mercaptoethanol

Abstract The free-radical copolymerization of methyl methacrylate (MMA) with styrene (St) in the presence of 2-mercaptoethanol (ME) was investigated in order to obtain ω -hydroxy oligomers with random copolymer-type main chains and which may be further used to prepare macromonomers. Weak retardation was noticed during polymerization. It was demonstrated that the pseudo-kinetic rate constant method can be successfully applied to this copolymerization, irrespective of the initial monomer ratio. On this basis, the apparent transfer constant of ME was determined under azeotropic copolymerization conditions, and it was found to be intermediate between the corresponding values for MMA and St homopolymerizations. M n and M w /M n of the oligomers prepared increased with conversion. As the copolymerization proceeded, MWD turned from monomodal to slightly bimodal at a conversion depending on the ME/monomer ratio. The functionalities of the oligomers prepared were around unity, making them suitable for macromonomer synthesis.

Application and validation of the pseudo-kinetic rate constant method to high pressure LDPE tubular reactors

The present study is concerned with the application of the pseudo-kinetic rate constant method to high pressure LDPE tubular reactors. The utilization of multiple monomers is of profound importance to the polymer manufacturing industry since it leads to the production of an unlimited number of polymer types with a wide range of properties. However, by increasing the number of comonomers the kinetic rate expressions describing the appearance / disappearance of the various molar species become fairly complex. In order to simplify the kinetic treatment of the resulting expressions, the pseudo-kinetic rate constant method can be employed, in combination with the method of moments, to model the multicomponent free-radical polymerization of ethylene in a high pressure tubular reactor.

Modeling and control of continuous stirred tank reactor for thermal copolymerization

A mathematical model is developed for solution copolymerization in a continuous stirred tank reactor. For the thermal copolymerization of styrene and acrylonitrile (SAN), the kinetic rate expression for thermal initiation is derived by applying the pseudo-steady-state hypothesis to the intermediates, and the kinetic parameters are estimated by experimental investigation. The moment equations of living and dead polymer concentrations are derived by applying the pseudokinetic rate constantmethod. The model is used to calculate the conversion, the copolymer composition, the weight-average molecular weight, and the polydispersity. It is demonstrated that this model can predict the industrial data very well under various operating conditions. The dynamic analysis of the reaction system enables us to determine the polymer properties against the changes in the operation parameters. It is noticed that the monomer conversion is controlled to some extent by the reaction temperature and the feed monomer fraction. The monomer conversion control of a solution copolymerization reactor is treated with different control algorithms. The fuzzy/proportional–integral–derivative controller shows satisfactory performances for both setpoint tracking and disturbance rejection and can be easily applied to continuous polymerization processes. © 1998 John Wiley & Sons, Inc. J Appl Polym Sci 67:921–931, 1998

Modeling of bulk copolymerization reactor using chain-length-dependent rate constants

A mathematical model is developed for a batch reactor in which binary free radical copolymerization occurs. The diffusion-controlled features of the propagation and termination reactions are taken into account by applying the free volume theory, whereas the chain-length-dependent termination rate constant is formulated by using the continuous probability function. Application of the pseudokinetic rate constant method, as well as the terminal model, reduces the complex rate expressions for the copolymerization system to those for the corresponding homopolymerization system. In addition, the moment equations of the living and dead polymer concentrations, as well as the equation for copolymer composition, are derived to compute the average molecular weight and the copolymer composition. The model is proven adequate when applied to the copolymerization system of styrene and acrylonitrile with AIBN(2,2'-azobisisobutyronitrile) initiator. The results of model prediction clearly show that even the propagation reaction is limited by the diffusion of monomers at higher conversion and that the azeotropic fraction of styrene is about 0.6. It is noticed that as the monomer conversion increases, the molecular weight distribution tends to become broader because the weight-average molecular weight increases at a faster rate than the number-average molecular weight.

Modelling free‐radical copolymerization kinetics—evaluation of the pseudo‐kinetic rate constant method, 2. Molecular weight calculations for copolymers with long chain branching

AbstractThe full moment equations and equations using pseudo‐kinetic rate constants for binary copolymerization with chain transfer to polymer in the context of the terminal model have been developed and solved numerically for a batch reactor operating over a wide range of conditions. Calculated number‐ and weight‐average molecular weights (M̄n and M̄w) were compared with those found using the pseudo‐kinetic rate constant method (PKRCM). The results show that the weight‐average molecular weights calculated using PKRCM are in agreement with those found using the method of full moments for binary copolymerization when polymeric radical fractions φ1˙ and φ2˙ of type 1 and 2 (radical centers are on monomer types 1 and 2 for a binary copolymerization) are calculated accounting for chain transfer to small molecules and polymer reactions in addition to propagation reactions. Errors in calculating M̄w using PKRCM are not always negligible when polymer radical fractions are calculated neglecting chain transfer to small molecules and polymer. In this case, the relative error in M̄w by PKRCM increases with increase in monomer conversion, extent of copolymer compositional drift and chain transfer to polymer rates. The errors in calculating M̄w, however, vanish over the entire monomer conversion range for all polymerization conditions when chain transfer reactions are properly taken into account. It is theoretically proven that the pseudo‐kinetic rate constant for chain transfer to polymer is valid for copolymerizations. One can therefore conclude that the pseudo‐kinetic rate constant method is a valid method for molecular weight modelling for binary and multicomponent polymerizations.

Kinetics of free-radical copolymerization: the pseudo-kinetic rate constant method

The pseudo-kinetic rate constant method in which the kinetic treatment of a multicomponent polymerization reduces to that of a homopolymerization is a useful tool to analyse the kinetics of copolymerization. This method is applicable not only for the systems described by the terminal model for copolymerization but also for those described by the higher order Markov chain statistics such as the penultimate model. It has also been clarified that this method can practically be applicable even when the effect of the chain length dependent bimolecular termination reaction is significant.

Kinetics of the free-radical copolymerization of methyl methacrylate/ethylene glycol dimethacrylate: 2. Analysis of gelation and the pregel region

A kinetic model based on the pseudo-kinetic rate constant method is used to investigate the kinetics of the bulk free-radical copolymerization of methyl methacrylate (MMA) and the divinyl monomer ethylene glycol dimethacrylate (EGDMA) initiated with AIBN, with and without chain transfer agent (CBr 4) during the pregel region. Most of the kinetic parameters were estimated from experimental data. The agreement of predicted with measured onset of gelation is found to be satisfactory.

Kinetics of the free-radical copolymerization of methyl methacrylate/ethylene glycol dimethacrylate: 1. Experimental investigation

An experimental investigation was made of the kinetics of the bulk free-radical copolymerization of methyl methacrylate (MMA) and the divinyl monomer ethylene glycol dimethacrylate (EGDMA) initiated with AIBN at 70°C for levels of EGDMA of 0–25 wt% and chain transfer agent (CBr 4) levels of 0–1 wt%. The conversion of monomers to sol and crosslinked gel was measured by gravimetry using Soxhlet extraction. This provided measurements of monomer conversion at the gelation point as a function of EGDMA and CBr 4 levels and the weights of sol and gel as a function of time and monomer conversion. The experimental data are interpreted in terms of the pseudo-kinetic rate constant method in Part 2 of this series.