Copolymerization kinetic models

Regarding the question of alternative copolymerization kinetic models, as mentioned earlier, in the event of discriminating between competing models (e.g., terminal model kinetics vs penultimate model kinetics), a set of equidistant monomer feed compositions along the entire composition range can serve as an appropriate design of experiments. Once one has determined that an alternative model is operative, the same four questions noted for the terminal model above should be revisited. There are several examples of the estimation of penultimate unit kinetic parameters in the literature [125, 112]. 

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). 

Multi-State Models. In studies of copolymerization kinetics and polymer microstructure, the use of reaction probability models can provide a convenient framework whereby the experimental data can be organized and interpreted, and can also give insight on reaction mechanisms. (1.,2) The models, however, only apply to polymers containing one polymer component. For polymers with mixtures of different components, the one-state simple models cannot be used directly. Generally multi-state models(11) are needed, viz. 

Gel Permeation Chromatography (CPC) is often the source of molecular wei t averages used in polymerization kinetic modelling Q.,2). Kinetic models also r uire measurement of molecular weight distribution, conversion to polymer, composition of monomers in a copolymerization rea tion mixture, copolymer composition distribution, and sequence length distribution. The GPC chromatogram often reflects these properties (3,. ... 

Monomer concentrations Ma a=, ...,m) in a reaction system have no time to alter during the period of formation of every macromolecule so that the propagation of any copolymer chain occurs under fixed external conditions. This permits one to calculate the statistical characteristics of the products of copolymerization under specified values Ma and then to average all these instantaneous characteristics with allowance for the drift of monomer concentrations during the synthesis. Such a two-stage procedure of calculation, where first statistical problems are solved before dealing with dynamic ones, is exclusively predetermined by the very specificity of free-radical copolymerization and does not depend on the kinetic model chosen. The latter gives the explicit dependencies of the instantaneous statistical characteristics on monomers concentrations and the rate constants of the elementary reactions. 

AG Mikos, CG Takoudis, NA Peppas. Kinetic modeling of copolymerization/ cross-linking reactions. Macromolecules 19 2174-2182, 1986. 

This assumption is implicitly present not only in the traditional theory of the free-radical copolymerization [41,43,44], but in its subsequent extensions based on more complicated models than the ideal one. The best known are two types of such models. To the first of them the models belong wherein the reactivity of the active center of a macroradical is controlled not only by the type of its ultimate unit but also by the types of penultimate [45] and even penpenultimate [46] monomeric units. The kinetic models of the second type describe systems in which the formation of complexes occurs between the components of a reaction system that results in the alteration of their reactivity [47-50]. Essentially, all the refinements of the theory of radical copolymerization connected with the models mentioned above are used to reduce exclusively to a more sophisticated account of the kinetics and mechanism of a macroradical propagation, leaving out of consideration accompanying physical factors. The most important among them is the phenomenon of preferential sorption of monomers to the active center of a growing polymer chain. A quantitative theory taking into consideration this physical factor was advanced in paper [51]. 

Attempts to elucidate the polymerization or copolymerization kinetics of ethynyl and maleimide-functionalized monomers have been undertaken via vibrational spectroscopy (137). The thermal polymerization of A-(3-ethynyl-phenyl) maleimide (the structure is given in Fig. 48) was studied via IR and Raman spectroscopy. This model compound is interesting because it carries maleimide and ethynyl groups attached to the same aromatic ring. Kinetic studies indicate that both the acetylene and maleimide group react at the same rate, which strongly suggests the formation of a copolymer rather than a mixture of homopolymers. 

Fig. 13. Copolymerization DADMAC/acrylamide in aqueous solution. Comparison of experimental results with kinetic models (Data taken from [38])...

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). 

The kinetic model proposed by Ng and Manas-Zloczower (1989) for the copolymerization of unsaturated polyester (UP) with styrene (S) is one example of this type of hybrid model. Relevant species considered in the analysis are C=C double bonds of the UP (E) styrene (S) initiators (In 1,2,..., N) and an inhibitor (Z). The model is based on the following hypotheses ... 

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... 

In addition to the above mentioned dynamic problems of copolymerization theory this review naturally dwells on more traditional statistical problems of calculation of instantaneous composition, parameters of copolymer molecular structure and composition distribution. The manner of presentation of the material based on the formalism of Markov s chains theory allows one to calculate in the uniform way all the above mentioned copolymer characteristics for the different kinetic models by means of elementary arithmetical operations. In Sect. 3 which is devoted to these problems, one can also find a number of original results concerning the statistical description of the copolymers produced through the complex radical mechanism. 

The practical value of the quantitative theory of radical copolymerization depends to a great extent on the adequacy of the applied kinetic model to the real systems. Hence, in Sect. 6 we shall discuss the issues of model discrimination and also the problems of reliability and validity of the calculations of the model parameters with an account of the potentialities of the modern experimental techniques. 

Penultimate and similar kinetic models are used nowadays principally for the treatment of the experimental data, obtained from the copolymerization of certain monomers like fumaronitrile or maleic anhydride which are characterized by rather strong steric and polar effects. 

This important peculiarity, which allows one to determine the kinetic parameters of m-component copolymerization on the basis of the analysis of the experimental data obtained under the copolymerization of m(m — l)/2 monomer pairs vanishes if one uses other kinetic models instead of the terminal one. There are a number of models describing multicomponent systems which account for the influence of the penultimate unit [47], the formation of the binary [48] and triple [49] complexes and also for the depolymerization reactions [50], However, up to now, all such models have a limited range of application since the current experimental techniques do not allow one to determine correctly a great number of their kinetic parameters. 

Hence, within the framework of the traditional kinetic model (2.8) there is a mathematically rigorous solution of the problem of the calculations of the azeotropic composition x under the copolymerization of any number of monomer types knowing their reactivity ratios, i.e. the elements of matrix ay. However, since the values of au can be estimated from the experiment with certain errors Say, the calculated location of azeotrope x is also determined with an accuracy, the degree of which is characterized by vector 8x with components 8xj (k = 1,2,..., m) and modulus 8X ... 

The first important question we need to answer is how the monomer feed composition x(p) will be changed with conversion at various initial values 5° and the parameters of kinetic copolymerization model. When such a trajectory x(p) is known, on the base of the formulae (5.1), (5.3), and (5.7) one can find the main statistical copolymer characteristics at any number of its components within the framework of the chosen kinetic model. 

When the copolymerization is carried out under real conditions, each researcher is to answer a question which kinetic model is preferable for the proper description of the experimental data. One should also know the validity of the model under consideration, the numerical values of its parameters, and the expected accuracy of the calculated copolymer characteristics predicted within the framework of this model. Modern experimental methods for analyzing the copolymer composition... 

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