Binary Copolymer Equation

The composition of a copolymer produced by simultaneous polymerization of two monomers is usually different from the composition of the comonomer feed from which it is produced. This shows that different monomers have different tendencies to undergo copolymerization. These tendencies often have little or no resemblance to their behavior in homopolymerization. Some monomers are more reactive in copolymerization than indicated by their rates of homopolymerization, and some monomers are less reactive. Thus, vinyl acetate polymerizes about twenty times as fast as styrene in a free-radical reaction, but the product in free-radical polymerization of a mixture of vinyl acetate and styrene is found to be almost pure polystyrene with practically no content of vinyl acetate. By contrast, maleic anhydride, which has very little or no tendency to undergo homopolymerization with radical initiation, undergoes facile copolymerization with styrene forming one-to-one copolymers. 

The composition of a copolymer thus cannot be determined simply from a knowledge of the homopolymerization rates of the two monomers. The simple copolymer model described here, however, accounts for the behavior of many important systems and the entire process is amenable to statistical calculations which provide a good deal of useful information from few data. Thus, it is possible to calculate the distribution of sequences of each monomer in the macromolecule and the drift of copolymer composition with the extent of conversion of monomers to polymer. 

To predict the course of a copolymerization we need to be able to express the composition of a copolymer in terms of the concentrations of the monomers in the reaction mixture and the relative reactivities of these monomers. In order to develop a simple model, it is necessary to assume that the chemical reactivity of a propagating chain (which may be free-radical in a radical chain copolymerization and carbocation or carboanion in an ionic chain copolymerization) is dependent only on the identity of the monomer unit at the growing end and independent of the chain composition preceding the last monomer unit [2-5]. This is referred to as the first-order Markov or terminal model of copolymerization. 

Let us consider the case for the copolymerization of two monomers Ml and M2. Although copolymerization has been more extensively studied using radical initiation, and radical copolyraerization is also more important than ionic copolyraerization, we will consider here the general case without specification as to whether polymerization occurs by a free-radical or ionic mechanism. To generalize, an asterisk( ) will be used—instead of the con- 

In order to simplify the kinetic formulation of copolymerization it is assumed that a steady state mechanism applies, in which the concentration of each propagating chain type, that is, the concentration of each of M and M2, remains constant. This assumption requires that the rate of conversion of Ml to Mg must equal that of Mg to M, or in mathematical terms 

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In Refs. [173-176] it was suggested to use the weight composition distributions instead of the molar ones and the results of their numerical calculation for some systems were reported The authors of Ref. [177] carried out a thorough theoretical study of the composition distribution and derived an equation for it without the Skeist formula. They, as the authors of Ref. [178], proposed to use dispersion of the distribution (5.3) as a quantitative measure of the degree of the composition inhomogeneity of the binary copolymers and calculated its value for some systems. Elsewhere [179-185] for this purpose there were used other parameters of the composition distribution. In particular the discussion of the different theoretical aspects of the binary copolymerization is reported in a number of reviews by Soviet authors [186-189], By means of numerical calculations there were analyzed [190-192] the limits of the validity of the traditional assumption which allows to ignore the instantaneous component of composition distribution of the copolymers produced at high conversions. 

When the boundary azeotrope is located on m-simplex edge (12), corresponding to the binary copolymer of monomers Mt and M2 the roots of the characteristic equation (5.11) are equal to... 

Another important recent contribution is the provision of a good measurement of the precision of estimated reactivity ratios. The calculation of independent standard deviations for each reactivity ratio obtained by linear least squares fitting to linear forms of the differential copolymer equations is invalid, because the two reactivity ratios are not statistically independent. Information about the precision of reactivity ratios that are determined jointly is properly conveyed by specification of joint confidence limits within which the true values can be assumed to coexist. This is represented as a closed curve in a plot of r and r2- Standard statistical techniques for such computations are impossible or too cumbersome for application to binary copolymerization data in the usual absence of estimates of reliability of the values of monomer feed and copolymer composition data. Both the nonlinear least squares and the EVM calculations provide computer-assisted estimates of such joint confidence loops [15]. 

Since considerations of sequence distributions can be used to derive the simple copolymer equation, it is not surprising that measured values of triad distributions in binary copolymers [by H or C NMR analyses] can be inserted into the copolymer equation to calculate reactivity ratios [19]. 

The monomer reactivity ratios r and r2 can be determined from the experimental conversion-composition data of binary copolymerization using both the instantaneous and integrated binary copolymer composition equations, described previously. However, in the former case, it is essential to restrict the conversion to low values (ca. < 5%) in order to ensure that the feed composition remains essentially unchanged. Various methods have been used to obtain monomer reactivity ratios from the instantaneous copolymer composition data. Several procedures for extracting reactivity ratios from the differential copolymer equation [Eq. (7.11) or (7.17)] are mentioned in the following paragraphs. Two of the simpler methods involve plotting of r versus r2 or F versus f. ... 

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. 

Equation 2.98 is very easy to justify. Consider the following binary copolymer chain ... 

Substitution and rearrangement of this equation yields the polymer composition equation 3.35. Thus it is possible to estimate reactivity ratios for binary copolymers from sequence distributions measured by NMR. 

Stockmayer s bivariate distribution for Unear binary copolymers can be expressed by the simple equations,... 

Here xi is the interaction parameter of a binary copolymer with the pure solvent. The interaction parameters of the corresponding homopolymers in the same solvent are xia and xib- The interaction between the A and B units in the chain is given by Xab- The volume fractions and mole fractions of the comonomers in the copolymer molecule are va, ub, xa and jcb respectively. Equation (5.51) or (5.52) should hold for all types of copolymers. However, the entropy of mixing will depend on the copolymer sequence distribution.(3)... 

The statistics of binary copolymers k one of the most advanced parts of polymer kinetics. All tte necessary equations for the copolymer composition and monomer distribution in the simple four-reaction scheme (t,2 for the pmultimate effect (3,4) and the pen-penultimate effea (5) wer obtained by the kinetic method and by an approach based on the Markov chain tteory (6). Some of the recent papers in this field are very sophisticated (5,7, S) and the theoretical achievements often far exceed the experimental possibilities for testing them. 

Equation relating the instantaneous composition of a binary copolymer to the monomer reactivity ratios and the ratio of instantaneous monomer eoncentrations. 

The interaction parameter between the ethylene (E) units and octene (O) units for a binary blend of two random copolymers can be obtained from the copolymer equation ... 

There are many examples known where a random copolymer Al, comprised of monomers 1 and 2, is miscible with a homopolymer B, comprised of monomer 3, even though neither homopolymer 1 or 2 is miscible with homopolymer 3, as illustrated by Table 2. The binary interaction model offers a relatively simple explanation for the increased likelihood of random copolymers forming miscible blends with other polymers. The overall interaction parameter for such blends can be shown (eg, by simplifying eq. 8) to have the form of equation 9 (133—134). 

The values of K and (3(K > 0 and 0 < P < 1) were calculated for each monomer pair from the logarithmic plot of the ratio of the monomers in the monomer feed [MJ/[M2] to the comonomer units in the copolymer using a modified equation of binary copolymerization ... 

For a detailed analysis of monomer reactivity and of the sequence-distribution of mers in the copolymer, it is necessary to make some mechanistic assumptions. The usual assumptions are those of binary, copolymerization theory their limitations were discussed in Section III,2. There are a number of mathematical transformations of the equation used to calculate the reactivity ratios and r2 from the experimental results. One of the earliest and most widely used transformations, due to Fineman and Ross,114 converts equation (I) into a linear relationship between rx and r2. Kelen and Tudos115 have since developed a method in which the Fineman-Ross equation is used with redefined variables. By means of this new equation, data from a number of cationic, vinyl polymerizations have been evaluated, and the questionable nature of the data has been demonstrated in a number of them.116 (A critique of the significance of this analysis has appeared.117) Both of these methods depend on the use of the derivative form of,the copolymer-composition equation and are, therefore, appropriate only for low-conversion copolymerizations. The integrated... 

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