Copolymerization equation derivation

Cationic copolymerization can be treated in identical manner to anionic copolymerization with the mechanistic scheme for propagation obtained simply by replacing the negative signs in Eqs. (8.92) to (8.95) by positive signs. The copolymerization equation derived for free-radical initiation [Eqs. (7.11) and (7.17)] may also be applied to cationic polymerization to determine ratios, several of which are listed in Table 8.5. The situation is complicated, however, due to counterion effects (Ham, 1964 Kennedy and Marechal, 1982). Thus, unlike in free radical copolymerizatiqn, different initiators used in cationic polymerization can... 

Copolymerization equations for systems of more than two monomers have been derived, and several experimental studies of copolymerizations involving three monomers have been reported. Six reactivity ratios are required for treatment of the composition in a three-compo-... 

Although the derivation above involves the steady-state assumption, the copolymerization equation can also be obtained by a statistical approach without invoking steady-state conditions [Farina, 1990 Goldfinger and Kane, 1948 Melville et al., 1947 Tirrell, 1986 Vollmert, 1973]. We proceed to determine the number-average sequence lengths, h and h2, of monomers 1 and 2, respectively. h is the average number of Mj monomer units that follow each other consecutively in a sequence uninterrupted by M2 units but bounded on each end of the sequence by M2 units. h2 is the average number of M2 monomer units in a sequence uninterrupted by Mj units but bounded on each end by Mi units. 

The copolymerization equation is valid if all propagation steps are irreversible. If reversibility occurs, a more complex equation can be derived. If the equilibrium constants depend on the length of the monomer sequence (penultimate effect), further changes must be introduced into the equations. Where the polymerization is subjected to an equilibrium, a-methylstyrene was chosen as monomer. The polymerization was carried out by radical initiation. With methyl methacrylate as comonomer the equilibrium constants are found to be independent of the sequence length. Between 100° and 150°C the reversibilities of the homopolymerization step of methyl methacrylate and of the alternating steps are taken into account. With acrylonitrile as comonomer the dependence of equilibrium constants on the length of sequence must be considered. 

Considering the terpolymerization of CPT-SO2-AN system as a binary copolymerization of CPT-SO2 complex and free acrylonitrile, the copolymerization equation can be derived as follows, assuming a fast equilibrium. 

The copolymerization equation can also be derived from the mean sequence lengths, I, of Mj and M2 units in the chains. The number of M, monomer additions per single M2 addition will evidently be given by the corresponding rate ratio, and the sequence will be longer by one unit than the rate ratio... 

Copolymer composition can be expressed by a single parameter. The copolymerization equation (66) is simplified to a form derived by Wall [168] in the first attempt to characterize copolymerization quantitatively... 

From the copolymerization equation, a situation can easily be derived where the compositions of the monomer mixture and the generated polymer will be equal (F, = fx) and the copolymerization curve will intersect the diagonal of the square copolymerization diagram. This will occur, apart from the case of eqn. (71) when r,[Mj] + [M2])/(r2[M2] + [Mj]) = 1, i. e. when... 

Statistical Models. Due to the difficulties involved in calculating the composition distributions by purely deterministic techniques, statistical methods have been developed from which not only the CCD can be obtained but also the sequence length distribution. These methods view the chain growth as an stochastic process having possible states resulting from the kinetic mechanisms. Early work on this approach was reported by Merz, Alfrey and Goldfinger (4) who derived the copolymerization equation and the SLD for the ultimate effect case. Alfrey Bohrer and Mark (19) and Ham (9) formalized this approach. 

From these four basic kinetic equations, the Mayo-Lewis instantaneous copolymerization equation can be derived. Equation 7.31 (see also Chapter 6) ... 

To derive the copolymerization equation the following substitutions are appropriate ... 

When the limit for infinite long chains, i.e., ae = be 0 is formed, the traditional copolymerization equation is obtained. The derivation given here is more general. 

Eq. (11.14) is known as the copolymerization equation and was first derived by Mayo and Lewis on a kinetic basis. A derivation on a probabUity basis was later published. Feinman and Ross derived the linear equations. ... 

The models considered earlier were developed for homopolymerization of olefins with single- and multiple-site catalysts. As has aheady been seen, several industrial polyolefins are, however, copolymers of ethylene, propylene and higher a-olefins. Because, for copolymerization, the kinetic rate constants depend on monomer and chain end type (in the terminal model), modeling these systems may seem daunting at first sight, but it will now be shown that, using the concept of pseudo-kinetic constants, the same equations derived for homopolymerization can be applied for copolymerization as well. 

The important property of Equation 2.97 is that it is analogous to Equation 2.3 with the pseudo-kinetic constants replacing the actual polymerization kinetic constants. The same result would be obtained if it were decided to develop equations for any other species in the reactor. As a consequence, all the equations derived above for the homopolymerization model are applicable to copolymerization as well, provided that the polymerization kinetic constants are replaced with pseudo-kinetic constants. Equations in Table 2.10 can be used either for homo- and copolymerization This elegant approach can considerably reduce the time and effort spent on developing models for copolymerization. Even though it was demonstrated for binary copolymers, this approach is equally valid for higher copolymers. Table 2.12 summarizes the pseudo-kinetic constants associated with the equations shown in Table 2.10. 

The copolymerization equation (22-14) was originally derived with the equations in (22-1) on considering that Vi = —d[Mi ]/dt together with the condition i i2 = 1 21- This assumption certainly seems to be fulfilled satisfactorily for free radical polymerizations, but it is doubtful in the case of ionic copolymerizations. The preceding statistical derivation holds good without assuming that V12 = 21 thus shows at the same time that the copolymerization equation must also apply to ionic copolymerizations when conditions 1-5 are fulfilled. The copolymerization equation also provides no information about the kinetics of copolymerization. 

With the same assumptions as in Section 22.1, copolymerization equations can also be derived for the copolymerization of more than two monomers. Relationships of this kind are very important in industry, since with terpolymerization, as many as 160,000 three-monomer combinations can be achieved with just 100 different monomer types. As the number of monomers per system increases, however, the equations rapidly become difficult to handle. 

Basically, just as in the case of two monomers, a copolymerization equation for three monomers can be derived using the probability of formation of the sequence. Here, however, the older kinetic derivation has been chosen for illustration purposes. It must also be stressed that the introduction of the steady-state condition required in this derivation is not absolutely necessary, since the same results can be obtained with a statistical derivation without this assumption. 

The copolymerization equation is derived under the assumption that all of the propagation steps are irreversible. But if one or more of the propagation steps is reversible, the copolymerization equation must be correspondingly extended. 

The copolymerization equation was derived under the assumption that the probability for the addition of a monomer to the growing chain is determined only by the last monomeric unit. With highly polar monomeric units, however, the penultimate chain end may be expected to have an influence (penultimate effect). Consequently, instead of one equation for the rate, for example, of the process —MJ -h there are... 

Thus, it is not the assumptions under which the copolymerization equation was derived which are untenable in ionic copolymerizations. A simple correlation of the overall and the active concentrations, on the other hand, is no longer possible, so that for a long time it was impossible to draw up a scheme similar to the free radical Q-e scheme for ionic copolymerization. 

If the two equations are combined under the assumption that 1 12 = then this again gives the copolymerization equation (22-14). The reactivity ratios r = k i/k 2 and r = i2/k i that occur in this derivation, however, in contrast to those that are otherwise applicable, do not relate to the reactivities of the growing chain end. 

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