Penultimate propagation kinetics

Fukuda and coworkers u 2 have recently derived a model equivalent to the Russo-Munari model but where the implicit penultimate model is used to describe the propagation kinetics. 

There are two cases to consider when predicting flie effect of solvent polarity on copolymerization propagation kinetics (1) the solvent polarity is dominated by an added solvent and polarity is thus independent of the comonomer feed ratio, or (2) the solvent polarity does depend on the comonomer feed ratio, as it would in a bulk copolymerization. In the first case, the effect on copolymerization kinetics is simple. The monomer reactivity ratios (and additional reactivity ratios, depending on which copolymerization model is appropriate for that system) would vary fi om solvent to solvent, but, for a given copolymerization system they would be constant as a function of the monomer feed ratios. Assuming of course that there were no additional types of solvent effect present, fliese copolymerization systems could be described by their appropriate base model (such as the terminal model or the explicit or implicit penultimate models), depending on the chemical structure of the monomers. 

While copolymer composition is well-described by the terminal model, the copolymer-averaged propagation rate coefficient (kp. Equation 3.45) for many common systems [10, 26, 27] is not. The measured kp values can be higher or lower than the terminal model predictions, with the deviation substantial in some cases. The implicit penultimate unit effect model, which accounts for the influence of the penultimate monomer-unit of the growing polymer radical on the propagation kinetics [26, 27], provides a good representation of this behavior ... 

Assuming terminal propagation kinetics, the best fit for was found to be much greater than unity such that kt was greater than either homo-termination value, a non-sensical result as is diffusion-controlled. When the deviation of propagation kinetics from the terminal model is taken into account, however, the estimates for kt become well-behaved and bounded by the homo-termination values [26]. Various penultimate models that account for the influence of polymer composition on segmental diffusion have been proposed to fit low-conversion kt data. Equation 3.49 emphasizes the role of the whole chain composition [26], while other formulations use both the terminal and penultimate units to represent the conformational characteristics of the last portion of the polymer chain [33, 34]. 

To summarize, we know firstly from simple model-testing studies spanning the last 20 years that for almost all systems tested, the terminal model can be fitted to (kp) or composition data for a copolymerization system, but not both simultaneously. Secondly, more recent experimental and theoretical studies have demonstrated that the assiunption of the implicit penultimate model— that the penultimate imit affects radical reactivity but not selectivity—cannot be justified. Therefore, on the basis of existing evidence, the explicit penultimate model should replace the terminal model as the basis of free-radical copolymerization propagation kinetics, and hence the failure of the terminal model kp) equation must be taken as a failure of the terminal model and hence of the terminal model composition equation. This means that the terminal model composition equation is not physically valid for the majority of systems to which it has been apphed. 

The implicit penultimate unit effect model, which accounts for the influence of the penultimate monomer unit of the growing polymer radical on the propagation kinetics [51], provides a good representation of this behavior [Eqs. (52)]. 

When solvent effects on the propagation step occnr in free-radical copolymerization reactions, they result not only in deviations from the expected overall propagation rate, but also in deviations from the ejqiected copolymer composition and microstracture. This may be trae even in bulk copolymerization, if either of the monomers exerts a direct effect or if strong cosolvency behavior causes preferential solvation. A number of models have been proposed to describe the effect of solvents on the composition, microstmcture and propagation rate of copolymerization. In deriving each of these models, an appropriate base model for copolymerization kinetics is selected (such as the terminal model or the implicit or explicit penultimate models), and a mechanism by which the solvent influences the propagation step is assumed. The main mechanisms by which the solvent (which may be one or both of the comonomers) can affect the propagation kinetics of free-radical copolymerization reactions are as follows ... 

The influence of penultimate units on the kinetics of copolymerization and the composition of copolymers was first considered in a formal way by Merz et al and Ham.8 They consider eight propagation reactions (Scheme 7.2). 

This is the simplest of the models where violation of the Flory principle is permitted. The assumption behind this model stipulates that the reactivity of a polymer radical is predetermined by the type of bothjts ultimate and penultimate units [23]. Here, the pairs of terminal units MaM act, along with monomers M, as kinetically independent elements, so that there are m3 constants of the rate of elementary reactions of chain propagation ka ]r The stochastic process of conventional movement along macromolecules formed at fixed x will be Markovian, provided that monomeric units are differentiated by the type of preceding unit. In this case the number of transient states Sa of the extended Markov chain is m2 in accordance with the number of pairs of monomeric units. No special problems presents writing down the elements of the matrix of the transitions Q of such a chain [ 1,10,34,39] and deriving by means of the mathematical apparatus of the Markov chains the expressions for the instantaneous statistical characteristics of copolymers. By way of illustration this matrix will be presented for the case of binary copolymerization ... 

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]. 

The former theory suggests that the reactivity of the polymeric radical is determined by the type of both ultimate and penultimate units. In this case the kinetic scheme of propagation reaction can be presented as follows [25] ... 

The kinetic copolymerization models, which are more complex than the terminal one, involve as a rule no less than four kinetic parameters. So one has no hope to estimate their values reliably enough from a single experimental plot of the copolymer composition vs monomer feed composition. However, when in certain systems some of the elementary propagation reactions are forbidden due to the specificity of the corresponding monomers and radicals, the less number of the kinetic parameters is required. For example, when the copolymerization of two monomers, one of which cannot homopolymerize, is known to follow the penultimate model, the copolymer composition is found to be dependent only on two such parameters. It was proposed [26, 271] to use this feature to estimate the reactivity ratios in analogous systems by means of the procedures similar to ones outlined in this section. 

Even though the discussion has been mainly on homopolymerization, the same polymerization mechanism steps are valid for copolymerization with coordination catalysts. In this case, for a given catalyst/cocatalyst system, propagation and transfer rates depend not only on the type of coordinating monomer, but also on the type of the last monomer attached to the living polymer chain. It is easy to understand why the last monomer in the chain will affect the behavior of the incoming monomer as the reacting monomer coordinates with the active site, it has to be inserted into the carbon-metal bond and will interact with the last (and, less likely, next-to-last or penultimate) monomer unit inserted into the chain. This is called the terminal model for copolymerization and is also commonly used to describe free-radical copolymerization. In the next section it will be seen that, with a proper transformation, not only the same mechanism, but also the same polymerization kinetic equations for homopolymerization can be used directly to describe copolymerization. 

Fukuda et al [22] have carried out an extensive study of the kinetics of the S/M copolymer system. They showed that, although copolymer compositions conform to the terminal model, rate constants for propagation are best represented by the penultimate model. There must therefore be some doubt over the assignments of Uebel and Dinan [18] since these were made on the premise of terminal model conformity. 

Quantum chemistry thus provides an invaluable tool for studying the mechanism and kinetics of free-radical polymerization, and should be seen as an important complement to experimental procedures. Already quantum chemical studies have made major contributions to our understanding of free-radical copolymerization kinetics, where they have provided direct evidence for the importance of penultimate imit effects (1,2). They have also helped in our understanding of substituent and chain-length effects on the frequency factors of propagation and transfer reactions (2-5). More recently, quantum chemical calculations have been used to provide an insight into the kinetics of the reversible addition fragmentation chain transfer (RAFT) polymerization process (6,7). For a more detailed introduction to quantum chemistry, the interested reader is referred to several excellent textbooks (8-16). 

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