Chain copolymerization terminal model

Under the condition that the reaction capability is only affected by the nature of the last monomer unit of the growing polymer chain end (terminal model, Bernoulli statistics), the copolymerization equation can be transformed according to Kelen and Tudos ... 

The analysis of copolymer composition in Section 4.6.4.1 has been carried out using the terminal model which assumes that radical reactivity is solely determined by the terminal unit on the free-radical chain. The terminal model has been successfully applied to represent the monomer and copolymer compositions for a wide variety of systems, mostly studied at ambient pressure. This model is, however, not capable of describing both copolymer composition and copolymerization kinetics with a single set of reactivity ratios [63]. 

The rate of copolymerization often shows a strong dependence on the monomer feed composition. Many theories have been developed to predict the rate of copolymerization based on the terminal model for chain propagation (Section 7.3.1.1), This usually requires an overall rate constant for termination in copolymerization that is substantially different from that observed in homopolymerization of any of the component monomers. 

Values of 0 required to fit the rate of copolymerization by the chemical control model were typically in the range 5-50 though values <1 are also known. In the case of S-MMA copolymerization, the model requires 0 to be in the range 5-14 depending on the monomer feed ratio. This "chemical control" model generally fell from favor wfith the recognition that chain diffusion should be the rate determining step in termination. 

More complex models for diffusion-controlled termination in copolymerization have appeared.1 tM7j Russo and Munari171 still assumed a terminal model for propagation but introduced a penultimate model to describe termination. There are ten termination reactions to consider (Scheme 7.1 1). The model was based on the hypothesis that the type of penultimate unit defined the segmental motion of the chain ends and their rate of diffusion. 

Harwood112 proposed that the solvent need not directly affect monomer reactivity, rather it may influence the way the polymer chain is solvated. Evidence for the proposal was the finding for certain copolymerizations, while the terminal model reactivity ratios appear solvent dependent, copolymers of the same overall composition had the same monomer sequence distribution. This was explained in... 

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). 

For example, under kinetic modeling of "living" anionic copolymerization in the framework of the terminal model, a macromolecule is associated with the realization of a certain stochastic process. Its states (a,r) are monomeric units, each being characterized along with chemical type a and also by some label r. This random quantity equals the moment when this monomeric unit entered in a polymer chain as a result of the addition of o-type monomer to the terminal active center. It has been... 

A generalization of the theory of the binary copolymerization for multicomponent systems in the case of the terminal model (2.8) is not difficult since the copolymer microstructure is still described by the Markov chain with states S corresponding to the monomer units Mj. The number m of their types determines the order of... 

You have no doubt been asking yourself Why are these dirty, rotten swine doing this to me Step back and think about what we ve said about copolymerization. There is something called the terminal models where the rate constants depend only on the nature of the terminal group, and the penultimate model, where there is a dependence on the character of the final two units in the chain (Figure 6-17). The use of conditional probabilities is the starting point for distinguishing between these experimentally, as we will see. 

We ve now defined all the stuff we want to define and we can now apply this to copolymers and copolymerization. However, some of the applications of probability theory won t be apparent until we discuss spectroscopy and the characterization of chain micro-structure. Here we will start with the terminal model, then we will see howcomposition and things like the parameters lA and % vary with conversion. First, remind yourself of the four possibilities for chain addition that occur in the copolymerization of A and B units when the rate of addition of a monomer depends only on the nature of the terminal group on a growing chain (Figure 6-20). 

A key facet of copolymerization is the possible disparity of reactivities of the monomers. Traditional procedure is to assume, at least as an approximation, that the reactivity of a growing propagating center depends only on the identity of its reactive end unit (i.e., the last monomer added), not on the composition and length of the rest of its chain [124-126] (first-order Markov or terminal model see also... 

Theoretical studies of the polymerization of two or more monomers have been conducted on various reaction systems since Dostal first proposed the concept of the terminal model in 1936. In the terminal model, it is assumed that the reactivity of the growing polymer chain is determined merely by the last added monomer unit (i.e., terminal unit), independently of the chain length and composition. For a two-component (Ml and M2) copolymerization, the terminal model leads to the following four propagation reaction equations ... 

To predict the course of a copolymerization we need to be able to express the composition of a copolymer in terms of the concentrations of the monomers in the reaction mixture and the relative reactivities of these monomers. In order to develop a simple model, it is necessary to assume that the chemical reactivity of a propagating chain (which may be free-radical in a radical chain copolymerization and carbocation or carboanion in an ionic chain copolymerization) is dependent only on the identity of the monomer unit at the growing end and independent of the chain composition preceding the last monomer unit [2-5]. This is referred to as the first-order Markov or terminal model of copolymerization. 

A question that continually arises when the topic of stable free radical copolymerization is discussed is what is the composition and microstructure of the copolymers Scheme 1 shows the four possible propagation reactions for a stable free radical copolymerization based on the terminal model. It is expected that if in the uncapped form, the nitroxide leaves the vicinity of the propagating chain end the copolymer microstructure should not be affected by the presence of nitroxide. Unsuccessful attempts by Sogah and Puts to influence the microstructure of polymers prepared by the SFRP process using chiral nitroxides suggest that the nitroxide does leave the vicinity of the propagating chain end (3). This is in agreement with Fukuda s results, which show that the microstructure of styrene-acrylonitrile (SAN) copolymers... 

As early as the 1940s, radical copolymerization models were already developed to describe specific features of the process. Initially, these models were relatively simple models where the reactivity of chain-ends was assumed to depend only on the nature of the terminal monomer unit in the growing chain (Mayo-Lewis model or terminal model (TM)). This model by definition leads to first-order Markov chains. 

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. 

The models considered earlier were developed for homopolymerization of olefins with single- and multiple-site catalysts. As has aheady been seen, several industrial polyolefins are, however, copolymers of ethylene, propylene and higher a-olefins. Because, for copolymerization, the kinetic rate constants depend on monomer and chain end type (in the terminal model), modeling these systems may seem daunting at first sight, but it will now be shown that, using the concept of pseudo-kinetic constants, the same equations derived for homopolymerization can be applied for copolymerization as well. 

FRP leads to the formation of statistical copolymers, where the arrangement of monomers within the chains is dictated purely by kinetic factors. The most common treatment of free-radical copolymerization kinetics assumes that radical reactivity depends only on the identity of the terminal unit on the growing chain. The assumption provides a good representation of polymer composition and sequence distribution, but not necessarily polymerization rate, as discussed later. This terminal model is widely used to model free-radical copolymerization according to the set of mechanisms in Scheme 3.11. 

It was reported by Barb in 1953 that solvents can affect the rates of copolymerization and the composition of the copolymer in copolymerizations of styrene with maleic anhydride [145]. Later, Klumperman also observed similar solvent effects [145]. This was reviewed by Coote and coworkers [145]. A number of complexation models were proposed to describe copolymerizations of styrene and maleic anhydride and styrene with acrylonitrile. There were explanations offered for deviation from the terminal model that assumes that radical reactivity only depends on the terminal unit of the growing chain. Thus, Harwood proposed the bootstrap model based upon the study of styrene copolymerized with MAA, acrylic acid, and acrylamide [146]. It was hypothesized that solvent does not modify the inherent reactivity of the growing radical, but affects the monomer partitioning such that the concentrations of the two monomers at the reactive site (and thus their ratio) differ from that in bulk. 

This model for the copolymerization described by Eq. (3.17) is called the terminal model. The model needs four propagation steps to describe the rate of copolymerization, implying that only the last unit in the growing chain determines the reactivity. However, it is known in literature (9) that for a system like, e.g., styrene-methylmethacrylate, the expression for the propagation rate [Eq. (3.17)] is not sufficient to give a good description of the polymerization. Not only the last unit, but also the one before the last unit may influence the reactivity. This eflect is known as the penultimate ejfect. 

The simplest copolymerization model is the terminal model, according to which the reactivity of the growing chain is determined by the last (terminal) monomer imit. In this case two reactivity ratios (ri and T2) are sufficient to describe the instantaneous composition of a binary copolymer ... 

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