Copolymerization equation terminal model

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 ... 

It has been argued that for a majority of copolymerizations, composition data can be adequately predicted by the terminal model copolymer composition equation (eqs. 5-9). However, in that composition data are not particularly good for model discrimination, any conclusion regarding the widespread applicability of the implicit penultimate model on this basis is premature. 

Thus, the terminal model for copolymerization gives us expressions for copolymer composition (Eqs. 6-12 and 6-15), propagation rate constant (Eq. 6-71), and polymerization rate (Eq. 6-70). The terminal model is tested by noting how well the various equations describe the experimental variation of F, kp, and Rp with comonomer feed composition. 

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 ... 

A number of copolymerizations involving macromonomer(s) have been studied and almost invariably treated according to the terminal model, Mayo-Lewis equation, or its simplified model [39]. The Mayo-Lewis equation relates the instantaneous compositions of the monomer mixture to the copolymer composition ... 

So far we have discussed reactivity ratios as if they are known quantities. And many of them are (you can find their values in the Polymer Handbook), thanks to sterling work by many polymer chemists over the years. But what if you re confronted with a situation where you don t have this information - how would you determine the reactivity ratios of a given pair of monomers Essentially, there are two sets of approaches, both of which depend upon using the copolymer equation in one form or another, hence, the assumption that the terminal model applies to the copolymerization we are considering. A form we will use as a starting point was... 

The terminal and penultimate models then correspond to first- and second-order Markovian statistics, respectively. But you don t actually have to know this, in the sense that we can just proceed using common sense. For example, the probability of finding the sequence ABABA in a system obeying first-order Markovian statistics (i.e., copolymerization where the terminal model applies) is given in Equations 6-31. 

All IR investigations of sequence distribution so far published rely on the terminal copolymerization model, which assumes that the kinetics of copolymerization are governed only by the probability that monomer units from the feed will be added to the last unit of the growing chain, and that there is only one active site present in the catalyst system, whether homogeneous or heterogeneous. As will be shown later (Section 3.4), this is only an approximation multiple active species are formed by many soluble Ziegler-Natta catalysts, so that the product of reactivity ratios determined from the normal copolymerization equation does not always exactly predict the actual sequence distribution in the copolymer. 

The terminal model for copolymerization can be naturally extended to multicomponent systems involving three or more monomers. Multicomponent copolymerizations And practical application in many commercial processes that involve three to five monomers to impart different properties to the final polymer (e.g., chemical resistance or a certain degree of crosslinking) [134]. There is a classical mathematical development for the terpolymerization or three-monomer case, the Alfrey-Goldfinger equation (Eq. 6.43)... 

Using terminal model kinetics as an example, the propagation equations for copolymerization are as follows ... 

In the second case, the effect of the solvent on copolymerization kinetics is much more complicated. Since the polarity of the reacting medium would vary as a function of the comonomer feed ratios, the monomer reactivity ratios would no longer be constant for a given copolymerization system. To model such behavior, it would be first necessary to select an appropriate base model for the copolymerization, depending on the chemical structure of the monomers. It would then be necessary to replace the constant reactivity ratios in this model by functions of the composition of the comonomer mixture. These functions would need to relate the reactivity ratios to the solvent polarity, and then the solvent polarity to the comonomer feed composition. The overall copolymerization kinetics would therefore be very complicated, and it is difficult to suggest a general kinetic model to describe these systems. However, it is obvious that such solvent effects would cause deviations fi om the behavior predicted by their appropriate base model and might therefore account for the deviation of some copolymerization systems from the terminal model composition equation. 

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. 

This equation follows from the kinetic analysis of copolymerization by Melville and co-workers (8) and Walling (9), who arrived at an expression for the overall rate of copolymerization assuming a terminal model for both propagation and termination. 

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 basic kinetic equations for chain addition copolymerization are given in Table I for three termination models geometric mean (GM), phi factor (PF) and penultimate effect (PE). It Is important to note the symmetry in form created by confining the effect of choice of termination model to a single factorable function H. 

A few years later, Tobolsky and Owen showed that the equilibrium copolymer composition is related to the propagation equilibrium constants in simple systems such as the copolymerization of sulfur with selenium (the terminal model of copolymerization without any effect of ultimate units, i.e., Kaa = Kba/ Kbb = Kab cf. propagation equations of Scheme 1). 

Scheme 5 Copolymerization equations for cyclotrisiloxanes according to terminal model.

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). 

Equations (7.58) and (7.68) yield the rate of copolymerization, and may be taken from previous studies of the chemical control model, or from an empirical correlation between this parameter and the r T2 product [27] which is based on the fact that cross-termination over homo-termination. Direct measurements of have been obtained [26] by measuring the absolute values of the rates of propagation and termination in pure monomers and in mixtures of various compositions. In the case of styrene-/7-metho)q styrene, = 1, indicating that no polar or other influences favor cross-termination. In most cases, however, cross-termination is... 

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