Prediction of Reactivity Ratios

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. 

The most popular methods are the Q-e (Section 7.3.4.1) and Patterns of Reactivity schemes (Section 7.3.4.2). Both methods may also be used to predict transfer constants (Section 6.2.1). For furtherdiscussionontheapplicationofthe.se and other methods to predict rate constants in radical reactions, see Section 2.3.7. 

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The method for the prediction of reactivity ratios in most widespread usage is the Q-e scheme.17 147 This scheme was devised in 1947 by Alfrey and Price148 who... 

PAM sec polyacrylamide PAM see polyacrylonitrile palladium complexes, as catalysts for ATRP 492 Patterns of Reactivity scheme 11,26, 31 for prediction of reactivity ratios 365-6 lor prediction of iransfer constants 287 PB see polybutadieue PE see polyethylene... 

Several attempts have been made to codify the relations between monomer reactivity ratios and structures. These approaches are essentially empirical but they are useful for predictions of reactivity ratios. 

Though molecular orbital calculations allow accurate predictions of reactivity ratios, many chemists rely upon the Price-Alfrey Q-e equations. These are based on (1) the polarity of the double bonds of the monomers or of the propagating chain ends, (2) mesomerism of the substituents with the double bonds or with the chain ends, and (3) the steric hindrance of the substituents. This relationship is expressed in the following equation ... 

The complexity of copolymerization makes theoretical prediction of reactivity ratios rather difficult. Nevertheless the semi-empirical Q-e scheme is often used for estimating reactivity ratios and also provides an approximate ranking of the reactivities and polarities of monomers. The basis of the scheme is that the rate constant kp for reaction of a polymeric... 

One final point should be made. The observation of significant solvent effects on kp in homopolymerization and on reactivity ratios in copolymerization (Section 8.3.1) calls into question the methods for reactivity ratio measurement which rely on evaluation of the polymer composition for various monomer feed ratios (Section 7.3.2). If solvent effects arc significant, it would seem to follow that reactivity ratios in bulk copolymerization should be a function of the feed composition.138 Moreover, since the reaction medium alters with conversion, the reactivity ratios may also vary with conversion. Thus the two most common sources of data used in reactivity ratio determination (i.e. low conversion composition measurements and composition conversion measurements) are potentially flawed. A corollary of this statement also provides one explanation for any failure of reactivity ratios to predict copolymer composition at high conversion. The effect of solvents on radical copolymerization remains an area in need of further research. 

As already indicated, values of reactivity ratios apply only to a given pair of monomers. There have been many attempts, especially for radical copolymerization, to derive parameters from the reactivity ratios, representing individual constants for each monomer which can be related to the structure of the monomers, and can be used to make predictions. 

Obviously the precision of this procedure is not very great, since the assumptions underlying the calculations of Q and e values can be regarded at best as semiquantitative. However, it has been shown that when the reactivity ratios are back-calculated from the Q,e values, quite good agreement is obtained with the experimental values, so that it is possible to make useful predictions of reactivity... 

The Subtle Role of Symmetry. An appeal to symmetry seems like one of the least controversial and most rigorous bases for making an argument about the behavior of a physical system. And indeed it is. However, it is possible for arguments that appear to be based solely on symmetry actually to depend on some additional ancillary assumptions that may be less obviously valid. An important example for the present discussion concerns prediction of product ratios from reactive intermediates. 

It proved possible to estimate the reactivity of a particular nitrogen in an azine ring, hence the ratio of isomeric quaternary salts which would be expected, and the approximate second-order rate constant for methylation of the azine. The method is, however, better suited to prediction of product ratios than rate coefficients, since steric effects largely cancel in the former. Examples of the predictive value of this approach will be included with the discussion of the individual azines. 

As long ago as 1960, Tarasov et al. [121] presented some examples of the concrete three-component systems for which the existence of the azeotropic composition had already been predicted theoretically. The list of such systems was widened substantially after publication of the important paper [125], where a set of the known tabulated values of 653 pairs of reactivity ratios for a computer search of the possible multicomponent azeotropes was employed. For this aim one should, at first, reveal all the completely characterized multicomponent systems for which the values of reactivity ratios of all monomer pairs are tabulated. This problem can be formalized by reducing it to the search on the graph with 653 lines of a... 

First let us demonstrate the possibilities of predicting transparency and heat resistance of (styrene + methylacrylate + heptyl acrylate) terpolymerization of the products prepared at complete conversion of monomers. The elements ry of the matrix of reactivity ratios ... 

An examination of reported reactivity ratios (Table 6) shows that the behaviour rj > 1, r2 1 or vice versa is a common feature of anionic copolymerization. Only in copolymerizations involving the monomers 1,1-diphenylethylene and stilbene, which cannot homopolymerize, do we find <1, r2 <1 [212—215], and hence the alternating tendency so characteristic of many free radical initiated copolymerizations. Normally one monomer is much more reactive to either type of active centre in the order acrylonitrile > methylmethacrylate > styrene > butadiene > isoprene. This is the order of electron affinities of the monomers as measured polarographically in polar solvents [216, 217]. In other words, the reactivity correlates well with the overall thermodynamic stability of the product. Variations of reactivity ratio occur with different solvents and counter-ions but the gross order is predictable. 

If two of the three monomers belong to the group described above and one is weakly conjugated, i.e. of the group of vinyl chloride, vinyl esters, olefins and the like, the product probabilities are approximately 0.006. It is evident that knowledge of the product probabilities permits to predict relative reactivity ratios for a wide variety of monomers. 

All IR investigations of sequence distribution so far published rely on the terminal copolymerization model, which assumes that the kinetics of copolymerization are governed only by the probability that monomer units from the feed will be added to the last unit of the growing chain, and that there is only one active site present in the catalyst system, whether homogeneous or heterogeneous. As will be shown later (Section 3.4), this is only an approximation multiple active species are formed by many soluble Ziegler-Natta catalysts, so that the product of reactivity ratios determined from the normal copolymerization equation does not always exactly predict the actual sequence distribution in the copolymer. 

As regards the methyl rocking region, Drushel et at. showed that at certain composition levels (ca. 22 wt %C3) maximum absorption occurs near 10.4 p and they assume that it could be produced by two contiguous C3 units with C2 units on each side. Such an attribution is quite reasonable not only could it offer an alternative interpretation of this complex band, but it could also extend the relationship between IR absorption and sequence distribution beyond the values of the copolymer composition covered by Eq. (2). Unfortunately the uncertainty regarding the absorptivities makes it impossible to reconcile the fractions of C3 units present in sequences of one, two, three or more members, as deduced from the bands at 10.67, 10.4 and 10.3 p and reported in Table 5 of Ref. (25), with those predicted for any value of the product of reactivity ratios. 

The theory presented in Section 1.6.1 is based upon the assumption that the reactivity of an active centre depends only upon the terminal repeat unit in which it is located. However, there are many examples where this assumption is not strictly valid because the nature of the penultimate repeat unit influences the reactivity of the terminal free radical. In such cases, eight propagation reactions need to be considered and four reactivity ratios are required to define copolymerization behaviour. Penultimate repeat unit effects are most obvious when attempting to predict sequence distributions ftom reactivity ratios, and often have a smaller effect on the prediction of overall copolymer composition. Given the inherent difficulties in determining accurate values of reactivity ratios, it is common practice to use the terminal model, especially in view of the fact tiiat it gives reasonable predictions of copolymer composition for most copolymerizations and is easy to implement. More detailed reviews should be consulted for accounts of the various theoretical models of copolymerization [3,5,6]. 

It was alieady pointed out that reactivity ratios are the cnidal parameters in describing, among others, copolymer composition versus monomer feed composition. Knowledge of these parameters is, therefore, of great importance if one wishes to control the synthesis of a copolymer. Already in the early days of copolymerization smdies, experimental determination of reactivity ratios as well as their theoretical prediction was investigated. In this chapter, some of the important aspects will be reviewed. 

Throughout the history of radical copolymerization, attempts have been made to predict the value of reactivity ratios. One of the earliest attempts was the so-called Q-e scheme. The basic approach in this method is that the general reactivity of a propagating radical M, is represented by B,-, whereas the general reactivity of a monomer Mj is represented by Qj. The polarity of monomer and radical are assumed to be identical and are represented by e,- and ej, respectively, in the example above. The rate constant for the addition of monomer M2 to radical M is then written as shown in eqn ]41] ... 

The dyad probabilities for copolymer were calculated as a function of reactivity ratios and monomer composition. A first-order Markov model was developed to predict the chain sequence distribution of SAN and AMS-AN copolymers. The six triad concentrations for SAN copolymer were calculated. Nin dyad and 27 triads for random terpolymers were calculated and tabulated in Tables 11.3, 11.5-11.7. 

The study of the mechanisms of copolymerization reactions has attracted the attention of polymer scientists over many years. Copolymerization reactions have been described in terms of various mathematical models, terminal, penultimate, complex, etc., which are characterized by a particular set of reactivity ratios. These mathematical models can be used to make predictions about the variation of the copolymer composition or the copolymer microstructure (for example, the triad fractions, number average sequence lengths, etc.) with comonomer feed composition or monomer conversion (1). 

A semi-empirical method has been devised to predict the reactivity ratios for a pair of monomers. It eliminates the need to determine the reactivity ratios experimentally for each monomer in a particular free-radical copolymerization reaction. The method is known as the Q-e scheme and each monomer is assigned a particular value of Q and e relative to styrene which is given arbitrary reference values of 0 = 1.0 and e = -0.8. The reactivity ratios are given for the two monomers by... 

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