The Q-e Scheme

All these factors contribute to the rate of copolymraization, but in a manner that makes it difficult to distinguish the magnitude of each effect. 

Attempts to correlate copolymerization tendencies are thus mainly on a semi-empirical footing and must be treated as useful approximations rather than rigorous relations. A genraally useful scheme was proposed by Alfrey and Price, who denoted the reactivities or resonance effects of monomers by a quantity Q and radicals by F, whereas the polar properties were assigned a factor e, which is assumed to be the same for both a monomer and its radical. 

An expression for the rate constant of the cross-propagation reaction can then be derived as 

By choosing arbitrary reference values for styrene oi Q= 1.0 and e = -0.8, a table of relative values of Q and e for monomers can be compiled (see Table 5.2). 

On doing this, one finds that for substituents capable of conjugating with the double bond, Q 0.5, whereas for groups such as Cl, OR, and alkyl, (2 0.1, thereby reflecting the assumption that G is a measure of resonance stabilization. 

A useful scheme was proposed by Alfrey and Price (1947) to provide a quantitative description of the behavior of vinyl monomers in radical polymerization, in terms of two parameters for eac/t monomer rather than for a monomer pair. These parameters are denoted by Q and e and the method is known as the O - e scheme. An advantage of the method is that it allows calculation of monomer reactivity ratios ri and T2 from the same Q and e values of the monomers irrespective of which monomer 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 for a radical-monomer reaction, e.g., the reaction of Mi ° radical with M2 monomer, can be written as 

By assuming that the same e value applies to both a monomer and its radical (that is, e defines the polarities of Mi and Mi, while 62 defines the polarities of M2 and M2 ), one can write expressions for k, k22, and 21 analogous to Eq. (7.31). These can then be appropriately combined to yield the monomer reactivity ratios. Thus for k 1 one can write by analogy to Eq. (7.31), 

Thus ri and V2 can be calculated from Q and e values of monomers forming the pair. 

Equations (7.33) and (7.34) permit us to calculate the Q and e values for single monomers from the values of ri and r2, provided we have one monomer for which O and e have been arbitrarily decided. Price chose styrene as the standard monomer with the values 2=1 and e = -0.8. Table 7.2 gives a selection of Q and e values for some of most common monomers. As a general rule, monomers with electron-rich double bonds have more negative e values and those that form highly resonance-stabilized radical have higher 2 numbers. 

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 the effects of radical activity cancels, and r can be expressed analytically in terms of parameters independent of the paired interdependence of Ml and M2. (Note that in any given pair of monomers, the monomer cited first is considered as Mi and the other as M2.) An expression for T2 is similarly obtained, viz.. 

Source. Data from R. Z. Greenley, Q and e Values for Free Radical Copolymerizations of Vinyl Monomers and Telogens, pp 267-274 in Chap. II in Polymer Handbook (J. Brandnip and E. H. Immergut, eds.), 3rd ed., Wiley-Interscience, New York (1989). 

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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. 

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... 

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 ... 

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... 

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. 

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. 

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. ... 

Some systems actually behave as the Q - e scheme predicts while other copolymerizations deviate from the proposed pattern. On the whole, the scheme is regarded with some misgivings at present. It is quite clear that a complicated chemical reaction, comprising the mutual interaction of many kinds of radicals and molecules in various media, can hardly be described in its entirety... 

The Q-e scheme [2.5] assumes that each radical or monomer can be classified according to its general reactivity and its polarity. The general reactivity of radical M, is represented by 6,- while the corresponding factor for monomer M, is Qj. Polarity is denoted by e, and e, with the e values for a monomer and its resulting radical being assumed to be ec]ual. The rate constant for reaction (7-3) would then be expressed as l i2 = From analogous expressions... 

The relative ease of anionic polymerization can be correlated with the base strengths of the respective anions or crudely with the e values of the Q-e scheme (Section 7.11) as shown in Table 9-1. Polymerization of a given monomer can generally be initiated by a carbanion from any monomer higher in the list, but the... 

Foedyce, Chapin and Ham (P) have replaced reactivity ratios in multicomponent copolymer equations (7) and (10) with the values given by the Q—e scheme 10) ... 

Ham 18) and Mayo 19) have further observed that relationships (17) and (18) are consistent with the Q—e scheme (70). 

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... 

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