Copolymerization model

Carbon monoxide does not obey general copolymerization models. 

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. 

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. 

Penultimate Group Effects Copolymerization Model. This model represents an extension of the Mayo-Lewis model in which the next to last or penultimate group is assumed to affect the reaction rate. Under this assumption the eight reactions represented by the following equations are of importance ( ) ... 

In their paper Hill and coworkers discriminate between alternative copolymerization models by fitting the models to composition data and then predicting sequence distributions based on the fitted models. Measured and fitted sequence distributions are then compared. A better approach taken here is to fit the models to the sequence distribution data directly. 

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. 

The ability to determine which copolymerization model best describes the behavior of a particular comonomer pair depends on the quality of the experimental data. There are many reports in the literature where different workers conclude that a different model describes the same comonomer pair. This occurs when the accuracy and precision of the composition data are insufficient to easily discriminate between the different models or composition data are not obtained over a wide range of experimental conditions (feed composition, monomer concentration, temperature). There are comonomer pairs where the behavior is not sufficiently extreme in terms of depropagation or complex participation or penultimate effect such that even with the best composition data it may not be possible to conclude that only one model fits the composition data [Hill et al., 1985 Moad et al., 1989]. 

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 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]. 

This extremely simple model provided a reasonable fit of experimental DSC scans performed in the 3-7°C/min range. Better mechanistic models should consider separately the consumption of E and S double bonds in the frame of typical copolymerization models. 

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] ... 

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. 

Each trajectory x(p) is regarded to be a solution of the universal set of dynamic equations (5.2), the form of the right-hand parts of which is determined by the selected copolymerization model. 

All the mentioned types of the nontrivial dynamic behavior are excluded for the systems where the reactivity ratios ry can be described by the expressions of the well-known Alfrey-Price Q-e scheme [20], and as a result they are to follow the simplified terminal model (see Sect. 4.6). In these systems, due to the relations Bj(X)/Bj(x) = ajj/ajj which holds for all i and j, the functions 7e,-(2) according to relations (4.10) are the ratios of the homogeneous polynomials of degree 2. Besides, for the calculations of the coefficients ak of Eq. (5.11) one can use the simple formulae presented in terms of determinants Dj and D [6, p. 265]. The theoretical analysis [202] leads to the conclusion that in such systems even the limited cycles are not possible and all azeotropes are certainly unstable. Hence any trajectory H(p) and X(p) when p -> 1 inevitably approaches the SP corresponding to the homopolymer the number of which can be from 1 to m. The set of systems obtained due to the classification within the framework of the simplified model essentially impoverishes in comparison with the general case of the terminal copolymerization model since some types of systems cannot be principally realized under the restrictions which the Q-e scheme puts on the reactivity ratios r. ... 

A number of well-known procedures of r, r2 estimations within the framework of the terminal copolymerization model share the following general expression ... 

The kinetic copolymerization models, which are more complex than the terminal one, involve as a rule no less than four kinetic parameters. So one has no hope to estimate their values reliably enough from a single experimental plot of the copolymer composition vs monomer feed composition. However, when in certain systems some of the elementary propagation reactions are forbidden due to the specificity of the corresponding monomers and radicals, the less number of the kinetic parameters is required. For example, when the copolymerization of two monomers, one of which cannot homopolymerize, is known to follow the penultimate model, the copolymer composition is found to be dependent only on two such parameters. It was proposed [26, 271] to use this feature to estimate the reactivity ratios in analogous systems by means of the procedures similar to ones outlined in this section. 

Now let us make a short survey of quite different approaches to the problem of identifying the kinetic copolymerization models based on the investigations of the model reactions between the low-molecular compounds only. In this very promising direction it is hard to overestimate the paramount contribution made by Bevington, Tirrell et al. [286-295], whose publications could be divided into two groups. 

The thermodynamic parameters ASP° and AHP° may also be affected by the microstructure of the resulting polymer or copolymer. In particular, low values of AHP° lead to reversible propagation, which in turn results in significant deviation of the copolymer composition as described by the terminal copolymerization model discussed below. On the other hand, the microstructure of the polymer affects ASP°, with atactic polymers and more random copolymers having higher entropies than tactic polymers and more regular copolymers, respectively. 

Errors in variables methods are particularly suited for parameter estimation of copolymerization models not only because they provide a better estimation in general but also, because it is relatively easy to incorporate error structures due to the different techniques used in measuring copolymer properties (i.e. spectroscopy, chromatography, calorimetry etc.). The error structure for a variety of characterization techniques has already been identified and used in conjunction with EVM for the estimation of the reactivity ratios for styrene acrylonitrile copolymers (12). 

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