Q — e scheme

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 

There have, however, been attempts to correlate Q-e values and hence reactivity ratios to, for example, c NMR chemical shifts 50 or the results of MO calculations 51153 and to provide a better theoretical basis for the parameters. Most recently, Zhan and Dixon153 applied density functional theory to demonstrate that Q values could be correlated to calculated values of the relative free energy for the radical monomer reaction (PA + Mn — PA ). The e values were correlated to values of the electronegativities of monomer and radical. 

The NMR method of predicting Q-e values appears attractive since spectra can be measured under the particular reaction conditions (solvent, temperature, pH). Thus, it may be possible to predict the dependence of the Q-e values and reactivity ratios on the reaction medium. 10 

S is taken as the reference monomer with =1.0 and e - -0.8. Values for other monomers are derived by regression analysis based on literature or measured reactivity ratios. The Q-e values for some common monomers as presented in the Polymer Ihwdbook are given in Table 7.7. The accuracy of Q-e parameters is limited by the quality of the reactivity ratio data and can also suffer from inappropriate statistical treatment employed in their derivation. A further problem is that the data analysis makes no allowance for the dependence of reactivity ratios on reaction conditions. Reactivity ratios can he dependent on solvent (Section 7.3.1.2), reaction temperature, pll, etc. It follows that values of e and perhaps Q for a given monomer should depend on the medium, the monomer ratio and the particular comonomer. This is especially true for monomers which contain ionizablc groups e.g. MAA, A A, vinyl pyridine) or arc capable of fomiing hydrogen bonds e.g. HEMA, HEA). 

There have, however, been attempts to correlate Q-e values and hence 

Various attempts have been made to place the radical-monomer reaction on a quantitative basis in terms of correlating structure with reactivity. Success in this area would give a better understanding of copolymerization behavior and allow the prediction of the monomer reactivity ratios for comonomer pairs that have not yet been copolymerized. A useful correlation is the Q-e scheme of Alfrey and Price [1947], who proposed that the rate constant for a radical-monomer reaction, for example, for the reaction of Mp radical with M2 monomer, be written as 

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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 ... 

Chain-Growth Gopolymerization Theory. The theory of chain-growth (eg, radical, anionic, etc) copolymerisation has received more attention than that of step-growth or other copolymerisations. In the case of chain-growth copolymerisation, growing polymer chains must choose between more than one monomer. Such a choice or relative reactivity has been quantitatively treated by the reactivity ratio (6,7) and the Q-e schemes (8). 

The Q-e Scheme. The magnitude of and T2 can frequentiy be correlated with stmctural effects, such as polar and resonance factors. For example, in the free-radical polymerization of vinyl acetate with styrene, both styrene and vinyl acetate radicals preferentially add styrene because of the formation of the resonance stabilized polystyrene radical. 

Various empirical schemes have also been proposed as predictive tools with respect to the outcome of radical addition reactions.9193 Two-parameter schemes, including the Q-e scheme (Section 7.3.4.1), Patterns of Reactivity (Section 7.3.4.2)... 

The complexity of the terpolymer composition equation (eq. 36) can be reduced to eq. 41 through the use of a modified steady slate assumption (eqs. 38-40), However, while these equations apply to component binary copolymerizations it is not clear that they should apply to terpolymerization even though they appear to work well. It can be noted that when applying the Q-e scheme a terpolymer equation of this form is implied. 

Bamford, Jenkins and coworkers131157 concluded that many of the limitations of the Q-e scheme stemmed from its empirical nature and proposed a new scheme containing a radical reactivity term, based on experimentally measured values of the rate constant for abstraction of benzylic hydrogen from toluene (Ay i), a polar term (the Hammett o value) and two constants a and J which are specific for a given monomer or substrate (eq. 57) 146... 

Numbers in parentheses refer to Price s values for e in the Q,e scheme. 

Mayo and Walling, who have given a penetrating critique of the Q,e scheme, point out that it represents in essence merely a transcription to equation form of the reactivity series of Table XX and the po-larity series of Table XXII. Regardless of the manner of interpretation adopted, it is apparent that monomer reactivity in copolymerization depends on two factors. One of these relates to the intrinsic characteristics of the monomer (and of the activated complex produced from it as well) as they tend to favor its addition to a radical. As we have seen, the capacity for resonance stabilization in the transition state is of foremost importance in determining the general level of monomer reactivity. The second factor has to do with the specificity... 

The Q-e scheme neglects steric factors, but it is a useful guide when data for r and Y2 are not available. Following is an approach that relates the reactivity ratios to the Alfrey-Price e values ... 

The patterns of reactivity scheme is a more advanced treatment of copolymerization behavior. It follows the general form of the Q-e scheme but does not assume that the same intrinsic reactivity or polarity factors apply both to a monomer and its corresponding radical [Bamford and Jenkins, 1965 Jenkins, 1999, 2000 Jenkins and Jenkins, 1999]. The monomer reactivity ratio for monomer 1 is expressed in terms of four parameters... 

Table 6-8 shows values of the various parameters needed to calculate monomer reactivity ratios from Eqs. 6-60 and 6-62 [Jenkins and Jenkins, 1999]. The monomers in Table 6-8 are lined up in order of their u values. The Patterns of Reactivity scheme, like the Q e. scheme, is an empirical scheme. Monomer reactivity ratios calculated by the patterns of reactivity scheme are generally closer to experimental values than those calculated by the Q e scheme, which supports the rationale of assigning different polarity values to a monomer and the radical derived from the monomer. 

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. 

Alfrey-Pnce Q-e scheme Alfven waves Algae... 

Considering now reactions (5 a) and (5 b) (p. 176), it was found that the addition of monomers to macroradicals produced by chain transfer depends directly on the reactivity and polarity of both the radical and the monomer (203) and that the Q—e scheme of Alfrey and Price can be applied to these graft copolymerizations by chain transfer (227). In this way some unsuccessful attempts for grafting were interpreted, e. g. vinyl acetate on polystyrene and methyl methacrylate on polyvinyl acetate and polyvinyl chloride. 

The effect of polarity on vinyl monomer copolymerization has long been recognized and is a major factor in the Q, e scheme and copolymerization theory. Mayo, Lewis, and Walling tabulated a number of vinyl monomers into an average activity series and an electron donor-acceptor series (62). The activity series showed the effect of substituents on the ease with which an ethylene derivative reacted with an average radical and on stabilizing the radical which was formed thereby. The electron donor-acceptor series indicated the ability of the substituents to serve as donors or acceptors in radical-monomer interactions. It is significant that in both series the dominant factor is the radical-monomer interaction. 

When the reactivity ratios ry can be expressed in terms of the parameters of the well-known Q-e scheme of Alfrey-Price [20,157], the condition (4.20) always holds [147, 150] and in the case of terpolymerization the general Eqs. (3.8) and (4.10) transform into the simplified equation [158]. It is rather curious that similar equations have been derived at the end of the 1940s [159] within the framework of the Alfrey-Price scheme, being investigated even for the general case of copolymerization of arbitrary number m of monomer types. 

The violation of the condition (4.20) for the particular system means that the latter can not be described by the Q-e scheme. However, in addition to this scheme, there are known some others, also applied for the description of the reactivity of the polymer radical in propagation reactions [160], One such scheme proposed by Bamford [161, 162] was successfully used by Jenkins [163] for an interpretation of the experimental values [150] of parameter H for a number of ternary systems. For many of these the values of H are noticeably different from unity, as it has to be according to the predictions of the Alfrey-Price scheme, but are in satisfactory agreement with the values calculated through the Bamford scheme [163, 160],... 

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. ... 

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