Terminal model

If the terminal model applies to the copolymerization that interests us and we happen to know the rate constants (we ll discuss how to measure these later), then we can immediately describe some interesting limiting conditions. 

Finally, if the rate of adding either Mj or M2 to a radical is the same, and the rate of adding either or M2 to a M2 radical is equivalent, then the copolymers are truly random. More on this shortly  

The simplest model for describing binary copolymerization 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. 

The ratio of these equations provides an expression for the instantaneous copolymer composition (eq. 3). 

Other convenient fonns of the copolymer composition equation are eq. 8  

Grccnlcy s compilation. All values rounded to two significant figures. 

the terminal model allows the copolymer composition for a given monomer feed to be predicted from just two parameters the reactivity ratios aq and tba. Some values of terminal model reactivity ratios for common monomer pairs are given in Table 7.1. Values for other monomers can be found in data 

This allows elimination of the radical concentrations from the above equation and the copolymer composition equation (eq. 5),14-16 also known as the Mayo-Lewis equation, can now be derived. 

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Equation (7.32) shows that pjj is constant for a particular copolymer if the terminal model applies therefore the ratio NmjMi/Nmi also equals this constant. Equation (7.49) shows that Pj u is constant for a particular copolymer if the penultimate model applies therefore the ratio NmiMiMi/NmiMi equals this constant, but the ratio NmjMj/Nmj does not have the same value. 

The first quantitative model, which appeared in 1971, also accounted for possible charge-transfer complex formation (45). Deviation from the terminal model for bulk polymerization was shown to be due to antepenultimate effects (46). Mote recent work with numerical computation and C-nmr spectroscopy data on SAN sequence distributions indicates that the penultimate model is the most appropriate for bulk SAN copolymerization (47,48). A kinetic model for azeotropic SAN copolymerization in toluene has been developed that successfully predicts conversion, rate, and average molecular weight for conversions up to 50% (49). 

Table 7.1 Terminal Model Reactivity Ratios for Some Common Monomer Pairs9...

It is informative to consider some of the implications of the terminal model and, in particular, how the relative magnitudes of the reactivity ratios affect the copolymer composition (Figure 7.1) ... 

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). 

Triad information is more powerful, but typically is subject to more experimental error and signal assignments are often ambiguous (Section 7.3.3.12). Triad data for the MMA-S system are consistent with the terminal model and support the view that any penultimate unit effects on specificity are small.Mv lS... 

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... 

If chains are long such that the initiation and termination reactions have a negligible effect on the average sequence distribution, then according to the terminal model, PAA, the probability that a chain ending in monomer unit MA adds another unit MA, is given by eq. 22 8... 

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). 

It is also possible to process copolymer composition data to obtain reactivity ratios for higher order models (e.g. penultimate model or complex participation, etc.). However, composition data have low power in model discrimination (Sections 7.3.1.2 and 7.3.1.3). There has been much published on the subject of the design of experiments for reactivity ratio determination and model discrimination.49 "8 136 137 Attention must be paid to the information that is required the optimal design for obtaining terminal model reactivity ratios may not be ideal for model discrimination.49... 

Terminal model reactivity ratios may be estimated from the initial monomer feed composition and the dyad concentrations in low conversion polymers using the following relationships (eqs. 45, 46). 

Various methods for predicting reactivity ratios have been proposed. These schemes are largely empirical although some have offered a theoretical basis for their function. They typically do not allow for the possibility of variation in reactivity ratios with solvent and reaction conditions. They also presuppose a terminal model. Despite their limitations they are extremely useful for providing an initial guess in circumstances where other data is unavailable. 

More recent work has shown that the observed variation in propagation rate constants with composition is not sufficient to define the polymerization rates.5" 161,1152 There remains some dependence of the termination rate constant on the composition of the propagating chain. Thus, the chemical control (Section 7.4.1) and the various diffusion control models (Section 7.4.2) have seen new life and have been adapted by substituting the terminal model propagation rate constants (ApXv) with implicit penultimate model propagation rate constants (kpKY -Section 7.3.1.2.2). 

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 the classical diffusion control model it is assumed that propagation occurs according to the terminal model (Scheme 7.1). The rate of the termination step is limited only by the rates of diffusion of the polymer chains. This rate may be dependent on the overall polymer chain composition. However, it does not depend solely on the chain end.166,16... 

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 apparent terminal model reactivity ratios are then r => aK and c =rR, K It follows that rABVBf = rABrBA - const. The bootstrap effect does not require the terminal model and other models (penultimate, complex participation) in combination with the bootstrap effect have been explored.103,1 4215 Variants on the theory have also appeared where the local monomer concentration is a function of the monomer feed composition.11[Pg.431]

Usually, reactions 1 and 2 take place in the aqueous jiiase, yttiile all the other kinetic events can occur both in the aqueous and in the polymer phases. Note that Pj,n indicates the concentration of active polymer chains with nTronaner units and tenninal unit of type j (i. e. of monomer j) Hi is the concentration of monomer i and T is the concentration of the chain transfer agent. Reactions 4 and 5 are responsible for chain desorption from the polymer pjarticles reactions 6 and 7 describe bimolecular temination by conJoination and disproportionation, respiectively. All the kinetic constants are depsendent upon the last monomer unit in the chain, i. e. terminal model is assumed. 

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]. 

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