Q and e scheme

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  

It is possible to calculate the Q and e values from n and n, or, conversely, r values can be obtained from the Q and e values. The relationship is as follows  

The 2 and e scheme is based on a semiempirical approach. Nevertheless, some attempts were made to develop theoretical interpretations. Thus Schwann and Price developed the following relationship  

A revised reactivity scheme was proposed by Jenkins, that he called U,V scheme [148], It is claimed to be more accurate and also capable of application to both copolymerizations and to transfer reactions. The scheme retains much of the format of the Q and e scheme. In this one, the intrinsic radical reactivity is quantified by reference to the rate of reaction of the radical with styrene monomer. 

The original approach for this scheme was based on copolymerizations of styrene with acrylonitrile and with other acrylic monomers [149]. In developing a more general approach, however, Jenkins concluded that while in principle, a general procedure involves fewer assumptions, in practice much of the utility is lost. He proposed that it is convenient in practice to employ from styrene copolymerizations. 

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The Q and e scheme has been the subject of considerable attention. It promised new possibilities of monomer classification, calculation of copolymerization parameters from known values of Q and and predictions about the behaviour of copolymerizing systems. However, to such ends the Q and e values should be known, and they are not directly measurable. In order to calculate them from copolymerization parameters, the easily copolymerized styrene has been selected as the reference monomer, with the assigned values Q = 1 and e = -0.8. Data on the Q and e factors of practically all copolymerizing monomers are now available (see Tables 3 and 3A) some kinetic significance is ascribed to monomer position in the Q — e plane (see Fig. 21). Monomers with a high Q value are expected to form poorly reactive radicals with a low tendency to add further monomer units monomers with widely differing e values usually copolymerize easily, etc. 

Explain the Q and e scheme and write the Price-Alfrey equation. 

The above equation represents a postulate on the same bases as does the Q and e scheme. This scheme contains an assumption that the intrinsic reactivity of a radical is measured by the value of and its polarity Ci (or tti). Thus,... 

The Price-Alfrey approach begins by defining three parameters-P, Q, and e-for each of the comonomers in a reaction system. We shall see presently that the parameter P is rapidly eliminated from the theory. As a result, the Price-Alfrey system is also called the Q-e scheme for copolymerization. 

Table 7.4 lists the Q and e values for an assortment of common monomers. The extremes in the column of e values in Table 7.4—which are listed in order-quantify the range of donor-acceptor properties which is used as the basis for ranking in Fig. 7.2. The Q values perform a similar ranking with respect to resonance effects. The eight different Q-e combinations in Table 7.4 allow the estimation of ri and values for 28 different copolymers. Of course, in these systems Q and e values were assigned to give the best fit to r values which had already been measured. As an illustration of the predictive values of the Q-e scheme, consider the following example ... 

When the axial dispersion terms are present, D > Q and E > Q, Equations (9.14) and (9.24) are second order. We will use reverse shooting and Runge-Kutta integration. The Runge-Kutta scheme (Appendix 2) applies only to first-order ODEs. To use it here. Equations (9.14) and (9.24) must be converted to an equivalent set of first-order ODEs. This can be done by defining two auxiliary variables ... 

Hence by assigning two parameters, a Q and an c, to each of a set of monomers, it should be possible according to this scheme to compute reactivity ratios ri and V2 for any pair. In consideration of the number of monomer pairs which may be selected from n monomers—about n /2—the advantages of such a scheme over copolymerization experiments on each pair are obvious. Price has assigned approximate values to Q and e for 31 monomers, based on copolymerization of 64 pairs. The latitude of uncertainty is unfortunately large assignment of more accurate values is hampered by lack of better experimental data. Approximate agreement between observed and predicted reactivity ratios is indicated, however. 

On this basis, values of Q and e can be calculated for each monomer, so long as two arbitrary reference values are assumed. For this purpose Price took the values for styrene as Q = 1.0 and e = -0.8. Q and e values can then be obtained for all monomers that are copolymerizable with styrene. These monomers in their turn can serve as reference compounds for further determinations with other monomers that do not copolymerize with styrene. One of the main advantages of the so-called Q,e scheme is that the data can be presented in the form of a diagram instead of very complex tables of reactivity ratios. 

The application of NMR spectroscopy data to estimate the reactivity ratios is regarded to be very promising [272]. The Q and e values of the Alfrey-Price scheme may be immediately calculated analyzing the shifts of the corresponding bands in carbon-NMR spectra Such data obtained for more than fifty pairs of monomers are tabulated in Ref. [273]. A quite different method based on the application of the trivial expressions ... 

Experimental reactivity ratios provide values for (QilQz) and ei — C ) for a given comonomer pair. In order to obtain numerical values of the four Q and e parameters from two reactivity ratios, two of the former are assigned arbitrary values. Originally, the scheme was anchored by selecting styrene as the reference monomer with 0 = 1.0 and e — —0.8. Later modifications have broadened the calculation base to include styrene copolymerization data of other well-researched monomers [26]. 

The value of KgJKg can be determined also from Q and e values (36). The basic relationship (11) of the Q—e scheme (76) yields... 

These are simply the equations of Alfrey and Price (1 j, which relate monomer reactivity ratios to Q and e values, and in which the reasonable values of 2 = and 2 = 1 re substituted, with the convention that the reference standard, ethylene, is monomer 2. In Equation 6 it is seen that the Qi value is simply a ratio of propagation rate constants unmodified by the presence of differences in e values, as is the case in the styrene-based scheme. This would seem to be a more desirable type of parameter to deal with, simply because its meaning is perfectly straightforward. 

Some data recently obtained on high pressure ethylene copolymerizations illustrate the quantitative aspects of an ethylene-based Q-e scheme (6). In Figures 3 and 4 copolymer composition curves for the ethylene-vinyl chloride and the ethylene-vinyl acetate copolymerizations are given. The monomer reactivity ratios for these two systems are tabulated in Table III along with Q values and e values for vinyl chloride and vinyl acetate calculated using ethylene as the standard (Q = 1.0 and g = 0). These Q and e values may be compared with those obtained using styrene as the standard. 

It appears that these ethylene-based Q and e values are capable of forming an internally consistent correlation scheme. It will be interesting to see whether this scheme is capable of yielding good results over the wide variety of monomers for which the styrene-based scheme has been so successful. 

All the above factors controlling monomer and radical reactivities contribute to the rate of polymerization, but in a manner which makes it difficult to distinguish the magnitude of each effect. Attempts to correlate copolymerization tendencies based on these factors are thus mainly of a semiempirical nature and can, at best, be treated as useful approximations rather than rigorous relations. However, a generally useful scheme was proposed by Alfrey and Price [23] to provide a quantitative description of the behavior of diferent monomers in radical polymerization, with the aid of two parameters, for each monomer rather than for a monomer pair. These parameters are denoted by Q and e and the method has been called the Q — e scheme. It allows calculation of monomer reactivity ratios r and T2 from properties of monomers irrespective of which pair is used. The scheme assumes that each radical or monomer can be classified according to its reactivity or resonance effect and its polarity so that the rate constant... 

In 1942, Alfrey and Price24 proposed the Q, e-scheme to account for the behavior of different monomers in free radical copolymerization. They attempted to describe the reactivity of monomer by means of Q and e values according to Eqs. (2.1)... 

Using the tabulated Q and e values for any two monomers, one can calculate the r and V2 values from Eqs. (7.33) and (7.34) for this monomer pair whether or not they were ever polymerized. The Q-e scheme is of the utmost utility, qualitatively, for predicting copolymerization behavior and for obtaining approximate estimates of r and V2 values. 

Thus, according to the Q-e scheme, by assigning values to Q and e, it should be possible to evaluate rj and rj for any monomer pair. A selected list of Q and e values is shown in Table 8.6. Negative values of e indicate electron-rich monomers, while positive e values indicate electron-poor monomers. 

The Q-e seheme is subject to criticisms. First, there seems to be no justification for assuming the same e values for the monomer and the radical derived ITom it. Second, the Q and e values for a particular monomer are not unique they vary with the monomer to which the monomer is paired (Table 8.7). In spite of these flaws, however, the Q-e scheme provides a semi-empirical basis for correlating the effect of structure on monomer reactivity. 

To determine the Q and e values of a monomer, the Q and e values of another monomer must be known. Styrene has been chosen as reference monomer for free radical copolymerizations, since it can be copolymerized with many other different monomers. Its values have been arbitrarily been given as 0 = 1 and e = —0.8. The Q and e values determined in this way are empirical values that often reproduce experimentally observed behavior quite satisfactorily. Large deviations are occasionally observed, however (see Table 22-4), especially for e values determined via the exponents, and so correspond to variations in r values that differ by large amounts. The Q-e scheme allows the copolymerization parameters of unknown monomer pairs to be estimated, and so allows their copolymerization capacity to be assessed. For this, the following guidelines apply (1) monomers with very different Q values cannot... 

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