Radiation-induced catalytic polymerization. I. Theory

The kinetics of radiation polymerization on a solid catalyst is discussed, under the condition that only linear termination of the chain takes place. All the kinetic equations are balance equations of particles of each type adsorbed by unit mass of the catalyst, and this makes it possible to account for the effect on the kinetics of the time dependence of the magnitude of the part of its surface on which the reactions we are considering may take place. Integro-differential equations are used for calculating the molecular weight distribution of the resulting polymer; this ensures higher accuracy of the formulas obtained than when differential equations are used and makes it possible to eliminate a number of limitations generally involved in the transition to differential equations. An expression has been found for the molecular weight distribution of the polymer product which allows for the possibility of radiation-induced catalytic polymerization on the resulting adsorbed polymer. Expressions have been derived for the average molecular weight and yield (weight and molecular) of the polymer formed. Asymptotic formulas have been obtained (for large irradiation times) for all the above values. The conclusions that can be drawn concerning the mechanism of the process based on a comparison of the formulas obtained with kinetic curves plotted from experimental data are given. It is shown how such a comparison can be utilized for calculating the rate constants for polymerization and chain termination reactions.     

Kinetic study of the thermal degradation of high density polyethylene

The thermal degradation of high density polyethylene has been modelled by the random breakage of polymer bonds, using a set of population balance equations. A model was proposed in which the population balances were lumped into representative sizes so that the experimentally determined molecular weight distribution of the original polymer could be used as the initial condition. This model was then compared to two different cases of the unlumped population balance which assumed unimolecular initial distributions of 100 and 500 monomer units, respectively. The model that utilised the experimentally determined molecular weight distribution was found to best describe the experimental data. The model fits suggested a second mechanism in addition to random breakage at slow reaction rates.     

Simulation model for the molecular weight distribution in emulsion polymerization

A Monte Carlo simulation model for the kinetics of emulsion polymerization is proposed. In the present model, the formation of each polymer molecule is simulated by the use of only a couple of probability functions; therefore, the calculation can be handled well even on personal computers. It is straightforward to account for virtually any kinetic event, such as the desorption of oligomeric radicals and chain length dependence of kinetic parameters, and as a consequence very detailed information such as the full distributions of the dead polymer molecular weights and the macroradicals among various polymer particles can be obtained. When bimolecular terminations are the dominant chain stoppage mechanism, the instantaneous molecular weight distribution (produced in a very small time interval) becomes broader than that for homogeneous polymerizations due to a higher possibility that short and long polymer radicals react with each other if bimolecular reactions are fast enough. The increase in the polydispersity of the MWD is fairly large, especially when bimolecular termination by disproportionation is significant; however, the gel permeation chromatography (GPC) may not be a suitable analytical technique to detect such broadening since oligomeric peaks may not be observed in the elution curve. The present simulation method provides greater insight into the complicated phenomena of emulsion polymerizations. © 1995 John Wiley & Sons, Inc.     

Mathematical models of the initial stage of the thermal degradation of poly(vinyl chloride). I. The thermal degradation of the low molecular weight models of macromolecules of poly(vinyl chloride) having no abnormal fragments

Some mathematical models of the initial stage of the thermal degradation of the low molecular weight models for the normal units of PVC have been considered. The equations have been deduced. If the values of the average length of kinetic chain of dehydrochlorination of the polymer and the model compound are equal to each other, these equations may be used for the thermal degradation of idealized PVC.     

Molecular‐Weight‐Distribution Modelling of Radical Polymerization in Batch and Continuous Reactors with Transfer to Polymer Leading to Gel Formation

Full chain‐length distribution (CLD) modelling applying the Galerkin finite‐element method^[1] (FEM) to polymerization reactors featuring a certain degree of gel formation is confronted with extremely long computation times. The paper describes a new method to predict CLDs for systems where gel formation may occur. The new concept is to model a part of the CLD up to a cut‐off length L, while satisfying the full set of population balances. With transfer to polymer as the mechanism responsible for gelation, this gives rise to a closure problem, which has been solved by assuming the dead CLD beyond L to be represented by a part of a Flory distribution. The method could be proved to work by performing simulations and comparing cut‐off CLDs to full CLDs for non‐gelling systems and comparing results for different L for systems with gelation. The model is demonstrated for polymerization reactors, the batch reactor and the continuous stirred‐tank reactor (CSTR), with either disproportionation or recombination termination. Reliable results are obtained for systems with moderate gel formation. Comparing these results to those from moment models including balance equations up to the fourth moment, a number of interesting differences have been found.     

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.     

A discretization method for computing chain length distributions

The method of Kumar and Ramkrishna is a numerical technique to solve population balance equations (PBEs) by discretization while preserving two moments of the distribution. When the method is used to calculate chain length distributions in polymerization reactions, the polydispersity, which depends on the first three moments of the distribution, cannot be estimated correctly. This work presents a modification of the method that allows to preserve three moments and thus calculate the polydispersity correctly, independently of the number of grid points. The modified method is applied to a model of controlled radical polymerization via RAFT and compared with the original one.     

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     

The quenched instationary polymerization system, 1. Derivation of the radical and polymer chain length distributions for the case of δ‐pulse like initiation

A new instationary polymerization system is presented including, as an essential element, the complete deactivation of all active radicals by reaction with an inhibitor at a certain time after chain initiation. The complete kinetic scheme, the set of differential equations, as well as the analytical solutions are presented. A proof is presented that the reaction with the inhibitor during the quench period dominates any other possible reaction such as propagation and bimolecular termination. As a result, the radical spectrum present at the beginning of the quench period is converted (almost) completely unchanged and instantaneously into a polymer chain length distribution. The quenched radical spectrum appears as a single additional peak in the experimentally observable total chain length distribution. In the case of δ‐pulse initiation the analytical solutions of the differential equations reduce to a simple poisson distribution for the radical concentrations as a function of time. Theoretical expressions for the maximum and the points of inflections (low and high molecular weight side) were derived and their applicability for the direct determination of k_p was tested. All of them turned out to be equally well suited for this purpose.     

A Detailed Polymerization Model for Cerenol® Polyols

Summary: A mathematical model of the acid catalyzed 1,3-propanediol polymerization has been developed. Two catalysts investigated include sulfuric acid and superacid (tetrafluoroethane sulfonic acid or triflic acid). Based on a detailed reaction mechanism, population and mass balance equations have been derived for small molecules as well as for polymeric species of numerous chain distributions, which are distinguishable in terms of protonation state and end group functionality. Due to the interaction of the sulfuric acid catalyst with the polymer ends, a novel, dual index polymer chain distribution was derived and implemented. The model has been validated with various sets of experimental data obtained in a lab-scale reactor setup. Dynamic model outputs such as monomer concentration, molecular weight averages, unsaturated and sulfate end groups, water evaporation rate and sulfate middle groups have been compared with experimental data of sulfuric and super acid catalyzed polymerization runs. Very good agreement between model predictions and experimental data has been obtained for both catalyst systems over an extended range of conditions using the same set of model parameter values. It is worth noting that the model is also capable of predicting polymerization equilibrium.     

Bivariate chain length and long chain branching distribution for copolymerization of olefins and polyolefin chains containing terminal double-bonds

Polyolefins containing long chain branches can be synthesized using certain metallocene catalysts such as Dow Chemical's constrained geometry catalyst. These polyolefins combine the excellent mechanical properties of polymers with narrow molecular weight distribution with the easy processability of polymers containing long chain branches. A mathematical model for the chain length distribution for these novel polyolefins was derived from basic principles and an analytical solution for the chain length distributions of the populations containing different number of long chain branches per polymer molecule was obtained. The analytical solution agrees with the direct solution of the population balances and with a Monte-Carlo simulation model. It is also shown that this solution applies for copolymers using pseudo-kinetic rate constants and Stockmayer's bivariate distribution.     

Application of numerical methods for the calculation of size distribution functions of polymers produced by ionic polymerizations

The application of numerical methods for the calculation of the molecular weight distribution of living ionic polymerization, carried out in the presence of monofunctional and polyfunctional transfer agents, is described. The methods used include the numerical solution of a system of differential equations by the Runge-Kutta-Merson procedure and the statistical Monte-Carlo approach. When a monofunctional transfer agent is used, the Runge-Kutta-Merson procedure is quite useful for the calculation of the molecular weight distribution for various polymerization systems. When a polymerization is carried out in the presence of a polyfunctional transfer agent, the mechanism includes the coupling of polymer chains. Due to the complexity of the system, the Runge-Kutta-Merson procedure is hardly applicable and problems of this type should be solved by a Monte Carlo simulation procedure. Once a computer program has been written, both methods allow the chemist to calculate the molecular weight distribution of a polymer as a function of the different kinetic parameters of the polymerization.     

Kinetic Modeling of Non-Linear Polymerization

Recent developments of a method based upon population balances of generating functions of polymer chain length distributions (CLD) are presented. The calculation of the CLD and how to take into account chain length dependent reactivity are discussed. Prediction of polymer properties is also possible but only easily done for the average molecular radius of gyration; some results are presented for a radical polymerization including transfer to polymer and propagation with terminal double bonds.     

Kinetics of long-chain branching in emulsion polymerization

A kinetic model suitable to deal with the case of branched polymers produced in emulsion both in the case of chain transfer to polymer and propagation to terminal double bond is briefly presented and numerically solved through the method of the moments. Thanks to the “numerical fractionation” approach, the whole molecular weight distribution of the polymer is evaluated while accounting for the compartmentalized nature of the system. The results of some illustrative calculations concerning the effects upon the molecular properties of the final polymer of starved semibatch monomer feed policies, addition of a chain transfer agent and propagation to terminal double bond are discussed.     

Molecular weight determination of bulk polymer surfaces by static secondary ion mass spectrometry

The determination of molecular weights at surfaces of bulk polymer materials can be accomplished by static secondary ion mass spectrometry (SIMS) via fragments originating from repeat units and end groups. The intensity ratio of these fragments depends on the polymer chain length as seen for bisphenol-A-polycarbonate and perfluorinated polyethers (Krytox). A kinetic model of fragment ion formation explains the molecular weight dependent fragment intensities and links them to properties of the molecular weight distribution. In the most simple case one obtains the number average molecular weight <M_n> at the surface. This technique can be used for the determination of the molecular weight at bulk polymer surfaces such as a CD-ROM made from polycarbonate by injection molding.     

Change of polymer reactivity in the course of macromolecular reaction

For a macromolecular reaction in polymer solution or melt, the reactivity of the transforming unit is supposed to depend upon its microenvironment, including both the nearest neighbours of the same chain and units belonging to the other chains or low molecular components of the reaction mixture. The microenvironment changes with conversion. Hence, the reactivity of polymer functional groups should change during the reaction. Such a reaction belongs to Markovian processes with locally interacting components and time-dependent transient probabilities. To describe the reaction kinetics, it is necessary: (1) to analyse statistical properties of an individual transforming chain, (2) to derive the kinetic equations using relations based on these properties, (3) to determine the dependence of the transient probabilities upon time or upon degree of conversion. For a general reaction model, equations are derived to describe the kinetic curve, distribution of units and composition heterogeneity of the products. The calculations reveal the influence of an interchain interaction on the reaction rate and also on the distribution of units for reaction in a polymer melt.     

Theoretical depolymerization kinetics. IV. Effect of a volatile fraction on the degradation of an initial “most probable” polymer

The effect of volatilization of molecules larger than monomer has been introduced into the solution of the Simha, Wall, and Blatz kinetic equations for the degradation of a high polymer with an initial “most probable” distribution. Equations describing the rate of sample weight and average molecular weight change result. They differ from the previous “most probable” equations primarily in the presence of an additive term representing the random splitting near the chain ends due to bond scission or transfer attack. Equations are also obtained for the rate of formation of each volatile species and hence the product distribution. The effect of volatilization of larger fragments is discussed in detail for the special case of random scission initiation. The product distribution is discussed for two special cases.     

Statistical thermodynamics in the framework of the lattice fluid model, 1. Polydisperse polymers of special distribution

The Gibbs free energies and equations of state of polymers with special molar mass distributions, e.g., Flory distribution, uniform distribution and Schulz distribution, are derived based on a lattice fluid model. The influence of the polydispersity (or the chain length) on the close-packed mass density, the close-packed volume of a mer and the mer-mer interaction energy or the scaling temperature is discussed. The diagrams of the Gibbs free energies as a function of temperature and chain length are simulated with a computer. The results suggest that a polydisperse polymer is thermodynamically more stable than the corresponding monodisperse polymer and that the thermodynamical properties of a polydisperse polymer are identical with those of the corresponding monodisperse polymer when the average degree of polymerization is sufficiently high.