Terminal model of copolymerization

Using the terminal model of copolymerization, there are four possibilities for propagation [Eq. (22)]. 

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 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). 

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. 

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)... 

The theory presented in Section 1.6.1 is based upon the assumption that the reactivity of an active centre depends only upon the terminal repeat unit in which it is located. However, there are many examples where this assumption is not strictly valid because the nature of the penultimate repeat unit influences the reactivity of the terminal free radical. In such cases, eight propagation reactions need to be considered and four reactivity ratios are required to define copolymerization behaviour. Penultimate repeat unit effects are most obvious when attempting to predict sequence distributions ftom reactivity ratios, and often have a smaller effect on the prediction of overall copolymer composition. Given the inherent difficulties in determining accurate values of reactivity ratios, it is common practice to use the terminal model, especially in view of the fact tiiat it gives reasonable predictions of copolymer composition for most copolymerizations and is easy to implement. More detailed reviews should be consulted for accounts of the various theoretical models of copolymerization [3,5,6]. 

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 early kinetic models for copolymerization, Mayo s terminal mechanism (41) and Alfrey s penultimate model (42), did not adequately predict the behavior of SAN systems. Copolymerizations in DMF and toluene indicated that both penultimate and antepenultimate effects had to be considered (43,44). The resulting reactivity model is somewhat compHcated, since there are eight reactivity ratios to consider. 

The various copolymerization models that appear in the literature (terminal, penultimate, complex dissociation, complex participation, etc.) should not be considered as alternative descriptions. They are approximations made through necessity to reduce complexity. They should, at best, be considered as a subset of some overall scheme for copolymerization. Any unified theory, if such is possible, would have to take into account all of the factors mentioned above. The models used to describe copolymerization reaction mechanisms arc normally chosen to be the simplest possible model capable of explaining a given set of experimental data. They do not necessarily provide, nor are they meant to be, a complete description of the mechanism. Much of the impetus for model development and drive for understanding of the mechanism of copolymerization conies from the need to predict composition and rates. Developments in models have followed the development and application of analytical techniques that demonstrate the inadequacy of an earlier model. 

Tire simplest model for describing binary copolyinerization of two monomers, Ma and Mr, is the terminal model. The model has been applied to a vast number of systems and, in most cases, appears to give an adequate description of the overall copolymer composition at least for low conversions. The limitations of the terminal model generally only become obvious when attempting to describe the monomer sequence distribution or the polymerization kinetics. Even though the terminal model does not always provide an accurate description of the copolymerization process, it remains useful for making qualitative predictions, as a starting point for parameter estimation and it is simple to apply. 

Cases have been reported where the application of the penultimate model provides a significantly better fit to experimental composition or monomer sequence distribution data. In these copolymerizations raab "bab and/or C BA rBBA- These include many copolymerizations of AN, 4 26 B,"7 MAH28" 5 and VC.30 In these cases, there is no doubt that the penultimate model (or some scheme other than the terminal model) is required. These systems arc said to show an explicit penultimate effect. In binary copolynierizations where the explicit penultimate model applies there may be between zero and three azeotropic compositions depending on the values of the reactivity ratios.31... 

For many systems, the copolymer composition appears to be adequately described by the terminal model yet the polymerization kinetics demand application of the penultimate model. These systems where rAAB=rliAR and aha bba hut sAfsB are said to show an implicit penultimate effect. The most famous system of this class is MMA-S copolymerization (Section 7.3.1.2.3). 

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. 

Mechanisms for copolymerization involving complexes between the monomers were first proposed to explain the high degree of alternation observed in some copolymerizations. They have also been put forward, usually as alternatives to the penultimate model, to explain anomalous (not consistent with the terminal model) composition data in certain copolymerizations.65"74... 

Terpolymerizations or ternary copolymerizations, as the names suggest, are polymerizations involving three monomers. Most industrial copolymerizations involve three or more monomers. The statistics of terpolymerization were worked out by Alfrey and Goldfinger in 1944.111 If we assume terminal model kinetics, ternary copolymerization involves nine distinct propagation reactions (Scheme 7.9). 

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. 

In evaluating the kinetics of copolymerization according to the chemical control model, it is assumed that the termination rate constants k,AA and A,Br are known from studies on homopolymerization. The only unknown in the above expression is the rate constant for cross termination (AtAB)- The rate constant for this reaction in relation to klAA and kmB is given by the parameter . 

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... 

Under current treatment of statistical method a set of the states of the Markovian stochastic process describing the ensemble of macromolecules with labeled units can be not only discrete but also continuous. So, for instance, when the description of the products of living anionic copolymerization is performed within the framework of a terminal model the role of the label characterizing the state of a monomeric unit is played by the moment when this unit forms in the course of a macroradical growth [25]. 

Further examination of these reactivity and Q,e data allow for a clear understanding of the polymerization behavior and how it is modified by the phosphazene. Although most systems are best described by the terminal model, the reactivity patterns exhibited in the copolymerization of o-methylstyryl penta-fluorocyclophosphazene (6) with methylmethacrylate can only by quantitatively fit by a pennultimate model.14... 

The rate of copolymerization, unlike the copolymer composition, depends on the initiation and termination steps as well as on the propagation steps. In the usual case both monomers combine efficiently with the initiator radicals and the initiation rate is independent of the feed composition. Two different models, based on whether termination is diffusion-controlled, have been used to derive expressions for the rate of copolymerization. The chemical-controlled termination model assumed that termination proceeds with chemical control, that is, termination is not diffusion-controlled [Walling, 1949]. But this model is of only historical interest since it is well established that termination in radical polymerization is generally diffusion-controlled [Atherton and North, 1962 Barb, 1953 Braun and Czerwinski, 1987 North, 1963 O Driscoll et al., 1967 Prochazka and Kratochvil, 1983] (Sec. 3-10b). 

Figures 6-12 and 6-13 shows plots of copolymer composition and propagation rate constant, respectively, versus comonomer feed composition for styrene-diethyl fumarate copolymerization at 40°C with AIBN [Ma et al., 2001]. The system follows well the implicit penultimate model. The copolymer composition data follow the terminal model within experimental error, which is less than 2% in this system. The propagation rate constant shows a penultimate effect, and the results conform well to the implicit penultimate model with si = 0.055, S2 — 0.32.

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