Kinetic models propagation

Torkelson and coworkers [274,275] have developed kinetic models to describe the formation of gels in free-radical pol5nnerization. They have incorporated diffusion limitations into the kinetic coefficient for radical termination and have compared their simulations to experimental results on methyl methacrylate polymerization. A basic kinetic model with initiation, propagation, and termination steps, including the diffusion hmitations, was found to describe the gelation effect, or time for gel formation, of several samples sets of experimental data. 

Monomer concentrations Ma a=, ...,m) in a reaction system have no time to alter during the period of formation of every macromolecule so that the propagation of any copolymer chain occurs under fixed external conditions. This permits one to calculate the statistical characteristics of the products of copolymerization under specified values Ma and then to average all these instantaneous characteristics with allowance for the drift of monomer concentrations during the synthesis. Such a two-stage procedure of calculation, where first statistical problems are solved before dealing with dynamic ones, is exclusively predetermined by the very specificity of free-radical copolymerization and does not depend on the kinetic model chosen. The latter gives the explicit dependencies of the instantaneous statistical characteristics on monomers concentrations and the rate constants of the elementary reactions. 

The kernel (26) and the absorbing probability (27) are controlled by the rate constants of the elementary reactions of chain propagation kap and monomer concentrations Ma(x) at the moment r. These latter are obtainable by solving the set of kinetic equations describing in terms of the ideal kinetic model the alteration with time of concentrations of monomers Ma and reactive centers Ra. 

This assumption is implicitly present not only in the traditional theory of the free-radical copolymerization [41,43,44], but in its subsequent extensions based on more complicated models than the ideal one. The best known are two types of such models. To the first of them the models belong wherein the reactivity of the active center of a macroradical is controlled not only by the type of its ultimate unit but also by the types of penultimate [45] and even penpenultimate [46] monomeric units. The kinetic models of the second type describe systems in which the formation of complexes occurs between the components of a reaction system that results in the alteration of their reactivity [47-50]. Essentially, all the refinements of the theory of radical copolymerization connected with the models mentioned above are used to reduce exclusively to a more sophisticated account of the kinetics and mechanism of a macroradical propagation, leaving out of consideration accompanying physical factors. The most important among them is the phenomenon of preferential sorption of monomers to the active center of a growing polymer chain. A quantitative theory taking into consideration this physical factor was advanced in paper [51]. 

In this paper, the kinetics and polymerization behavior of HEMA and DEGDMA initiated by a combination of DMPA (a conventional initiator) and TED (which produces DTC radicals) have been experimentally studied. Further, a free volume based kinetic model that incorporates diffusion limitations to propagation, termination by carbon-carbon radical combination and termination by carbon-DTC radical reaction has been developed to describe the polymerization behavior in these systems. In the model, all kinetic parameters except those for the carbon-DTC radical termination were experimentally determined. The agreement between the experiment and the model is very good. 

Equilibrium studies under anaerobic conditions confirmed that [Cu(HA)]+ is the major species in the Cu(II)-ascorbic acid system. However, the existence of minor polymeric, presumably dimeric, species could also be proven. This lends support to the above kinetic model. Provided that the catalytically active complex is the dimer produced in reaction (26), the chain reaction is initiated by the formation and subsequent decomposition of [Cu2(HA)2(02)]2+ into [CuA(02H)] and A -. The chain carrier is the semi-quinone radical which is consumed and regenerated in the propagation steps, Eqs. (29) and (30). The chain is terminated in Eq. (31). Applying the steady-state approximation to the concentrations of the radicals, yields a rate law which is fully consistent with the experimental observations ... 

To illustrate the conditions under which a system that includes chain propagating, chain branching, and chain terminating steps can generate an explosion, one chooses a simplified generalized kinetic model. The assumption is made that for the state condition just prior to explosion, the kinetic steady-state assumption with respect to the radical concentration is satisfactory. The generalized mechanism is written as follows ... 

In order to estimate kinetic constants for elementary processes in template polymerization two general approaches can be applied. The first is based on the generalized kinetic model for radical-initiated template polymerizations published by Tan and Alberda van Ekenstein. The second is based on the direct measurement of the polymerization rate in a non-stationary state by rotating sector procedure or by post-effect in photopolymerization. The first approach involves partial absorption of the monomer on the template. Polymerization proceeds according to zip mechanism (with propagation rate constant kp i) in the sequences filled with the monomer, and according to pick up mechanism (with rate constant kp n) at the sites in which monomer is outside the template and can be connected by the macroradical placed onto template. This mechanism can be illustrated by the following scheme ... 

The problem of predicting copolymer composition and sequence in the case of chain copolymerizations is determined by a set of differential equations that describe the rates at which both monomers, Ma and MB, enter the copolymer chain by attack of the growing active center. This requires a kinetic model of the copolymerization process. The simplest one is based on the assumption that the reactivity of a growing chain depends only on its active terminal unit. Therefore when the two monomers MA and MB are copolymerized, there are four possible propagation reactions (Table 2.17). 

A valid kinetic model of stage 3 emulsion polymerization must account for diffusion-controlled termination and propagation reactions. Marten and Hamielec (J) have proposed such a model based on a free-volune theory and have confirmed its validity for the bulk polymerization of methyl methacrylate (7). Herein is reported an evaluation of this model for the emulsion... 

The results definitely prove our hypotheses in the kinetic model for vinyl acetate emulsion polymerization (10), that vinyl radical, CH2=C-0Ac, is the major monomer radical formed and is a stable radical which reinitiates relatively slowly compared to the propagation step. 

At high particle velocity (above 50 m/s), the wear rate increases more than what would be expected from straight kinetic modeling (Arnold and Hutchings, 1990). This is believed to be from higher formation and propagation of microcracks. 

The core of the kinetic model describes the thermal oxidation at low temperature (typically at T < 200 C) at low conversion ([PH] = [PH]o = constant) and in oxygen excess of unfilled and unstabilized saturated hydrocarbons. It is derived from the closed-loop mechanistic scheme (CLM) of which the main characteristic is that radicals are formed by the thermal decomposition of its main propagation product the hydroperoxide group POOH 12). This closed-loop character explains the sharp auto-acceleration of oxidation at the end of an induction period (Figure 1). 

Yet Chien demostrated that it would be possible to obtain polyethylene with a Q value near the theoretical 2, with the homogeneous (CjH5)2TiCl2—A1(CH3)2C1 catalytic system, only if carefully controlled pseudosteady-state conditions are employed. In fact he showed mathematically that the relatively high experimental polydispersiiy (<,) tfom 2 to 5 in function of reaction time), is a natural consequence of a polymerization kinetic model based on non stationary first order initiation, chain propagation and bimolecular chain termination by recombination. 

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