Q-e Parameters

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The Q-e Scheme
Price-Alfrey Equation

In 1947, Price and Alfrey1 developed a simple scheme to predict reactivity ratios of monomers participating in a free radical copolymerization. Although molecular orbital calculations allow for more accurate predictions of reactivity ratios, the Alfrey-Price Q-e scheme is still widely used. Its popularity is based on its simplicity, clarity, and general utility, as has been proven over many years.

In the case of two monomers, M1 and M2, four different propagation reactions can occur:

M1· + M1 → M1·

M1· + M2 → M2·

M2· + M2 → M2·

M2· + M1 → M1·

where M1· and M2· represent chain radicals with free radical bearing terminal units of type 1 and 2. Alfrey and Price1 suggested that the four different propagation rate constants, k11, k12, k22, and k21, of these reactions can be written as

k11 = P1 Q1 exp(-e12)

k12 = P1 Q2 exp(-e1 e2)

k22 = P2 Q2 exp(-e22)

k21 = P2 Q1 exp(-e2 e1)

where Pi and Qj are measures of the general reactivity of radical i and monomer j , respectively, while ei and ej are proportional to the residual charges in the respective reacting groups. Writing reactivity ratios r1 and r2 as

r1 = k11 / k12 = (Q1/Q2) exp[-e1(e1 - e2)]

r2 = k22 / k21 = (Q2/Q1) exp[-e2(e2 - e1)]

r1r2 = exp[-(e1 - e2)2]

eliminates P to give reactivity ratios r1 and r2 solely as function of Q and e, the Q-e scheme.

This simple scheme has been subject to much criticism on theoretical grounds. First, there seems to be no justification for assigning a single e value to both the monomer and the radical derived from it. Second, the Q and P values are not unique but depend on the type of monomer to which the radical is bonded to. And third, the effect of steric hindrance is not taken into account which can have a significant effect on reactivity.
Despite of all this criticism, the Q-e scheme has proven useful to describe the effect of structure on monomer reactivity at least on a semi-empirical basis.

The first requirement for the successful use of the Q-e scheme is the selection of a representative group of common monomers for which Q and e can be independently calculated. By definition, the Q-e scheme is based on styrene as the reference monomer (Q = 1.0 and e = -0.8) to which all other monomers relate. This is a major weakness, because if an alternative reference monomer is chosen or if different numerical values are assigned to the Q and e values of styrene, then not only all other Q and e values change but also their ranking.

References & Notes
  1. T. Alfrey, C. C. Price, J. Polym. Sci, Vol. 2, No. 1, 101 (1947)
  2. T. Alfrey, J. Polym. Sci, Vol. 1, No. 2, 83 (1946)
  3. F. M. Lewis, F. H. Majo, and m. F. IIulse, J . Am. Chem. Soc., 67, 1701 (1945)

  4. As Majo and Lewis, and Price have shown, the radical-monomer reaction is favored by dissimilarity in the polarization, that is, pairs of monomers will most readily copolymerize if an electron-rich (negative) double bond is present in one monomer and an electron poor (positive) double bond in the other.

  • Summary

    Q-e SCHEME

    allows for the prediction of relative rates of copolymerization. The two constants Q and e are characteristic of an individual monomer. It is assumed that each monomer and its corresponding radical has the same reactivity.

  • The constant Q describes the general monomer reactivity and is related to the possibilities for stabilization in a radical adduct. The constant e is proportional to the residual electrostatic charges in the respective reaction groups.

  • The product r1r2 tends to be smallest when one of the two monomers of the blend has strongly electropositive (electron-releasing) and the other strongly electronegative (electron-attracting) substituents.

    Thus, an opposite polarity of the monomers results in an alternate sequence in the polymer chain.

PRODUCT OF REACTIVITY
RATIOS OF MONOMERS
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