Terminal copolymerization model

In fact, all of the experimental parameters can be calculated in terms of the two conditional probabilities P(B/A ) and P(A/B ), or the correspondingly r, and x, which are the basic parameters of the terminal model copolymerization mechanism. The order parameter is given by... 

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). 

Unsymmetrical azo-compounds find application as initiators of polymerization in special circumstances, for example, as initiators of living radical polymerization [e.g. triphenylmethylazobenzene (30) (see 9.3.4)], as hydroxy radical sources [e.g. a-hydroperoxydiazene (31) (see 3.3.3,1)1, for enhanced solubility in organic solvents [e.g. f-butylazocyclohexanecarbonitrile (32)J, or as high temperature initiators [e.g. t-butylazoformamide (33)]. They have also been used as radical precursors in model studies of cross-termination in copolymerization (Section... 

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. 

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. 

Perhaps because of this complexity, few studies on determining kld/ktt, in cross termination in copolymerization have been reported and most of the available data come from model studies, it is also usually assumed, without specific justification, that penultimate unit effects are unimportant in determining which reactions occur and that values of k klt for the homotermination reactions are similar to those in the corresponding homopolymerizations. 

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... 

Table 9 Reactivity ratios determined for 2-oxazoline copolymerizations utilizing both the Mayo-Lewis terminal model (MLTD) and the extended Kelen-Tiidds (KT) method. Initial defines - 20% conversion and final defines >50% conversion...

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). 

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 implicit penultimate model was proposed for copolymerizations where the terminal model described the copolymer composition and monomer sequence distribution, but not the propagation rate and rate constant. There is no penultimate effect on the monomer reactivity ratios, which corresponds to... 

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.

In contrast to the kinetic approach, deviations from the terminal model have also been treated from a thermodynamic viewpoint [Kruger et al., 1987 Lowry, 1960 Palmer et al., 2000, 2001]. Altered copolymer compositions in certain copolymerizations are accounted for in this treatment in terms of the tendency of one of the monomers (M2) to depropagate. An essential difference between the kinetic and thermodynamic treatments is that the latter implies that the copolymer composition can vary with the concentrations of the monomers. If the concentration of monomer M2 falls below its equilibrium value [M]c at the particular reaction temperature, terminal M2 units will be prone to depropagate. The result would be a... 

Fig. 6-12 Plot of Fi versus/i for copolymerization of styrene (MJ and diethyl fumarate (M2). The solid line represents the terminal model with r — 0.22, r2 — 0.021. After Ma et al. [2001] (by permission of American Chemical Society, Washington, DC) an original plot, from which this figure was drawn, was kindly supplied by Dr. T. Fukuda.

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 ... 

Copolymer composition curves in the copolymerization of MMA (Mi) and TrMA ( 2) with BuLi at -78°C are shown in Figure 1. From these curves the monomer reactivity ratios were determined to be ri=6.28 and 2 = 0.13 in toluene and ri = 0.62 and 2 0.62 in THF. In THF both the monomers showed similar reactivity. The compositional and configurational analyses of the copolymers indicated that the copolymerization approximately follows the terminal model in this solvent. 

For the copolymerization of epoxides with cyclic anhydrides and curing of epoxy resins, Lewis bases such as tertiary amines are most frequently used as initiators. In this case, terminal epoxides react with cyclic anhydrides at equimolar ratios. The time dependence of the consumption of epoxide and anhydride is almost the same for curing 35-36> and for model copolymerizations 39,40,45). The reaction is specific 39,40) to at least 99 %. In contrast, the copolymerization with non-terminal epoxides does not exhibit this high specificity, probably because of steric hindrances. The copolymerization of vinylcyclohexene oxide or cyclohexene oxide is specific only to 75-80 % and internal epoxides such as alkylepoxy stearates react with anhydrides only to 60-65 %. On the other hand, in the reaction of epoxy resins with maleic anhydride the consumption of anhydride is faster 65the products are discoloured and the gel is formed at a low anhydride conversion 39). Fischer 39) assumes that the other resonance form of maleic anhydride is involved in the reaction according to Eq. (33). 

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 ... 

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... 

The majority of the above mentioned kinetic schemes were used for the description of the multicomponent copolymerization of three or more types m of monomers. The generalization of the scheme (2.1) for m = 3 was carried out by Alfred and Goldfinger [45] and for arbitrary m — by Walling and Briggs [46]. In this case the terminal model... 

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