Copolymerization models for

The sequence distributions expected for the different models have been described [Hill et al., 1982, 1983 Howell et al., 1970 Tirrell, 1986] (Sec. 6-5a). Sequence distributions obtained by 13C NMR are sometimes more useful than composition data for discriminating between different copolymerization models. For example, while composition data for the radical copolymerization of styrene-acrylonitrile are consistent with either the penultimate or complex participation model, sequence distributions show the penultimate model to give the best fit. 

The same procedure can be applied to different copolymerization models, for example, PUM, and different types of data sets, for example, triad distributions as a function of monomer feed composition. Both will lead to more complex mathematical... 

A number of examples may be found in the literature. The polymerization of ethylene with q -olefins is a particularly simple example as it obeys the ideal copolymerization model. For the polymerization of ethylene-1-butene produced by zirconocene/ methylaluminoxane catalysis [23], the triad concentrations measured by NMR are shown in Table 7.12. 

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 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). 

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. 

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. 

The reaction between the PMMA and PS model radicals (4 and 5, generated from the unsymmetrical azo-compound 3) has been studied as a model for crosstermination in MMA-S copolymerization (Scheme 7.13).178,179 The value for tcross reaction was 0.56. In disproportionation, transfer of hydrogen from the PS model 5 to the PMMA radical 4 was ca 5.1 times more prevalent than transfer in the reverse direction (from 4 to 5). The value of kJklc(90°C) is between those of Atd/ tc(90oC) for the self-reaction of these radicals... 

Figure 5. Exact and approximate computer models for SAN copolymerization (2)...

Appendix 1 Toward a Model for Step-Growth Copolymerization w ith Phase Separation... 

A general method has been developed for the estimation of model parameters from experimental observations when the model relating the parameters and input variables to the output responses is a Monte Carlo simulation. The method provides point estimates as well as joint probability regions of the parameters. In comparison to methods based on analytical models, this approach can prove to be more flexible and gives the investigator a more quantitative insight into the effects of parameter values on the model. The parameter estimation technique has been applied to three examples in polymer science, all of which concern sequence distributions in polymer chains. The first is the estimation of binary reactivity ratios for the terminal or Mayo-Lewis copolymerization model from both composition and sequence distribution data. Next a procedure for discriminating between the penultimate and the terminal copolymerization models on the basis of sequence distribution data is described. Finally, the estimation of a parameter required to model the epimerization of isotactic polystyrene is discussed. 

Mayo-Lewis Binary Copolymeriration Model. In this exeimple we consider the Mayo-Lewis model for describing binary copolymerization. The procedure for estimating the kinetic parameters expressed as reactivity ratios from composition data is discussed in detail in our earlier paper (1 ). Here diad fractions, which are the relative numbers of MjMj, MiMj, M Mj and MjMj sequences as measured by NMR are used. NMR, while extremely useful, cannot distinguish between MiM and M Mi sequences and... 

Using copolymerization theory and well known phase equilibrium laws a mathematical model is reported for predicting conversions in an emulsion polymerization reactor. The model is demonstrated to accurately predict conversions from the head space vapor compositions during copolymerization reactions for two commercial products. However, it appears that for products with compositions lower than the azeotropic compositions the model becomes semi-empirical. 

Development of a reduced-order model for metallocene-catalyzed ethylene-norbornene copolymerization reaction... 

To our knowledge, this is the first time that an emulsion copolymerization model has been developed based on a population balance approach. The resulting differential equations are more involved and complex than those of the homopolymer case. Lack of experimental literature data for the specific system VCM/VAc made it impossible to directly check the model s predictive powers, however, successful simulation of extreme cases and reasonable trends obtained in the model s predictions are convincing enough about the validity and usefulness of the mathematical model per se. 

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 original mathematical treatment of the penultimate effect was presented by Merz and coworkers [Barb, 1953 Ham, 1964 Merz et al., 1946], Fukuda and coworkers developed a more extensive treatment, which distinguished between two penultimate models of copolymerization behavior—the explicit penultimate and implicit penultimate models [Coote and Davis, 1999, 2000 Davis, 2001 Fukuda et al., 1985, 1987, 1992, 2002 Ma et al., 2001], The explicit penultimate model for copolymerization involves the use of eight propagation reactions... 

The termination rate constants and molecular weights for the different copolymerization models have also been studied for purposes of discriminating between different copolymerization models [Buback and Kowollik, 1999 Landry et al., 1999]. 

Kawaguchi et al. [12] have modified Paine s model for the dispersion copolymerization of amphiphilic PEO macromonomers. The authors have modeled the variation of the particle size and its distribution with reaction conditions. For example, the expressions for the critical conversion (x Xthe particle radius (r), and the surface area (S) occupied by a PEO chain are as follows ... 

The formation of the donor-acceptor complexes Mt. .. M2 between the monomers M, and M2 is regarded as being an additional important factor responsible for the deviations of some certain systems from the classic copolymerization model. Also it should be noted that besides the single monomer entrance into the polymer chain a possibility of the monomer addition in pairs as a complex also exists. The corresponding kinetic scheme of the propagation reaction parallel with reactions (2.1) involves four additional ones [36] ... 

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