Copolymerization composition equation

Equation (10.8) is called the copolymerization composition equation. In a CSTR, the effluent concentration and the reactor concentration are the same. The reactor concentrations of monomers 1 and 2 are [MJ and [Mj], respectively. The polymer composition of monomer 1 repeat unit in the polymer F, can be seen to be... 

When the product of two reactivity ratios is equal to one, what can be said about the copolymerization composition equation ... 

Can the copolymerization composition equation yield multiple roots If so, what can be the possible physical significance of a second composition ... 

The parameters rj and T2 are the vehicles by which the nature of the reactants enter the copolymer composition equation. We shall call these radical reactivity ratios, although similarly defined ratios also describe copolymerizations that involve ionic intermediates. There are several important things to note about radical reactivity ratios ... 

The reactivity ratios of a copolymerization system are the fundamental parameters in terms of which the system is described. Since the copolymer composition equation relates the compositions of the product and the feedstock, it is clear that values of r can be evaluated from experimental data in which the corresponding compositions are measured. We shall consider this evaluation procedure in Sec. 7.7, where it will be found that this approach is not as free of ambiguity as might be desired. For now we shall simply assume that we know the desired r values for a system in fact, extensive tabulations of such values exist. An especially convenient source of this information is the Polymer Handbook (Ref. 4). Table 7.1 lists some typical r values at 60°C. 

It has been argued that for a majority of copolymerizations, composition data can be adequately predicted by the terminal model copolymer composition equation (eqs. 5-9). However, in that composition data are not particularly good for model discrimination, any conclusion regarding the widespread applicability of the implicit penultimate model on this basis is premature. 

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. 

Polymerization equilibria frequently observed in the polymerization of cyclic monomers may become important in copolymerization systems. The four propagation reactions assumed to be irreversible in the derivation of the Mayo-Lewis equation must be modified to include reversible processes. Lowry114,11S first derived a copolymer composition equation for the case in which some of the propagation reactions are reversible and it was applied to ring-opening copalymerization systems1 16, m. In the case of equilibrium copolymerization with complete reversibility, the following reactions must be considered. 

Equation 6-12 is known as the copolymerization equation or the copolymer composition equation. The copolymer composition, d M /d Mi, is the molar ratio of the two monomer units in the copolymer. monomer reactivity ratios. Each r as defined above in Eq. 6-11 is the ratio of the rate constant for a reactive propagating species adding tis own type of monomer to the rate constant for its additon of the other monomer. The tendency of two monomers to copolymerize is noted by r values between zero and unity. An r value greater than unity means that Mf preferentially adds M2 instead of M2, while an r value less than unity means that Mf preferentially adds M2. An r value of zero would mean that M2 is incapable of undergoing homopolymerization. 

For any specific type of initiation (i.e., radical, cationic, or anionic) the monomer reactivity ratios and therefore the copolymer composition equation are independent of many reaction parameters. Since termination and initiation rate constants are not involved, the copolymer composition is independent of differences in the rates of initiation and termination or of the absence or presence of inhibitors or chain-transfer agents. Under a wide range of conditions the copolymer composition is independent of the degree of polymerization. The only limitation on this generalization is that the copolymer be a high polymer. Further, the particular initiation system used in a radical copolymerization has no effect on copolymer composition. The same copolymer composition is obtained irrespective of whether initiation occurs by the thermal homolysis of initiators such as AIBN or peroxides, redox, photolysis, or radiolysis. Solvent effects on copolymer composition are found in some radical copolymerizations (Sec. 6-3a). Ionic copolymerizations usually show significant effects of solvent as well as counterion on copolymer composition (Sec. 6-4). 

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... 

An alternative rationale for the unusual RLi (hydrocarbon) copolymerization of butadiene and styrene has been presented by O Driscoll and Kuntz (71). Rather than invoking selective solvation, these workers stated that classical copolymerization kinetics is sufficient to explain this copolymerization. They adapted the copolymer-composition equation, originally derived from steady-state assumptions for free-radical copolymerizations, to the anionic copolymerization of butadiene and styrene. Equation (20) describes the relationship between the instantaneous copolymer composition c/[M,]/rf[M2] with the concentrations of the two monomers in the feed, M, and M2, and the reactivity ratios, rt, r2, of the monomers. The rx and r2 values are measures of the preference of the growing chain ends for like or unlike monomers. 

Equation (7.113b) gives the instantaneous copolymer composition in terms of the feed composition and the reactivity ratio. Figure 7.18 shows the copolymer composition for an ideal copolymerization (r,r2 = 1). In this case, the copolymer composition equation becomes ... 

When r, r2 values are rather close to unity, one can use for their estimation the so-called approximation method [225, 256-258]. Its idea is based on the fact that if the copolymerization is carried out at low concentrations of one of the monomers, the instantaneous composition of the copolymer depends only on one reactivity ratio. In this case the composition equation in both differential and integrated forms is fairly simple. 

Liquid sulfur-dicyclopentadiene (DCP) solutions at 140°C undergo bulk copolymerization where the melt viscosity and surface tension of the solutions increase with time. A general melt viscosity equation rj == tj0 exp(aXH), at constant temperature, has been developed, where tj is the viscosity at time t for an S -DCP feed composition of DCP mole fraction X and rj0 (in viscosity units), a (in time 1), and b (a dimensionless number, -f- ve for X < 0.5 and —ve for X > 0.5) are empirical constants. The structure of the sul-furated products has been analyzed by NMR. Sulfur non-crystallizable copolymeric compositions have been obtained as shown by thermal analysis (DSC). Dodecyl polysulfide is a viscosity suppressor and a plasticizer for the S8-DCP system. 

The occurrence of a homogeneous reaction system is also implicit i n the derivation of the copolymer composition equation. Some polymers, like poly(vinylidene chloride), are insoluble in their own monomer and are not highly swollen by monomer. In emulsion copolymerizations of such reactants the relative concentrations of the comonomers in the polymerizing particles will be influenced by the amounts that can be adsorbed on the surface or absorbed into the interior of these polymerization loci. 

In the study of anionic copolymerization it is possible to use two types of approach. The first method is the use of the classical copolymer composition equations developed for free radical polymerization. The second is unique to anionic polymerization and depends on the fact that for living systems it is possible to prepare an active polymer of one monomer and to study its reaction with the second monomer. The initial rate of disappearance of one type of active end, or the appearance of the other type (usually determined spectroscopically) or the rate of monomer consumption gives directly the reactivity of polymer-1 with monomer-2. It is in principle possible to compare the two methods to see if additional complications occur when both monomers are present together. 

Early investigations of ionic copolymerization [202] led to the conclusion that the product r, rj is approximately unity. This result will be produced exactly if fej j/fe, 2 = 21/ 22. expected if the competition between two monomers for reaction with an ionic centre is independent of the nature of that centre [202]. The copolymer composition equation then becomes... 

Equation (26) is the ideal copolymer composition equation suggested [203] early in the development of copolymerization theory but which had to be abandoned in favour of eqn. (23) as a general description of radical copolymerization. Only in this particular case are the rates of incorporation of each monomer proportional to their homopolymerization rates. It was shown that the reactivity of a series of monomers in stannic chloride initiated copolymerization followed the same order as their homopolymerization rates [202] and so eqn. (26) could be at least qualitatively correct for carbonium-ion polymerizations and possibly for reactions carried by carbanions. This, in fact, does not seem to be correct for anionic polymerizations since the reactivities of the ion-paired species at least, differ greatly. The methylmethacrylate ion-pair will, for instance, not add to styrene monomer, whereas the polystyryl ion-pair adds rapidly to methylmethacrylate [204]. This is a general phenomenon no reaction will occur if the ion-pair is on a monomer unit which has an appreciably higher electron affinity than that of the reacting monomer. The additions are thus extremely selective, more so than in radical copolymerization. There is no evidence that eqn. (26) holds and the approximate agreement with eqn. (25) results from other causes indicated below. 

Yamashita et al. [157] have derived a copolymer composition equation that includes the depropj ation reaction such as might be expected in the cationic copolymerization of BCMO and THF. They consider two models. For the first one it is assumed that monomer M2 adds reversibly to both active chain ends mf and m and that depropagation by detachment of an M1 unit is neglected. The elementary reactions are then... 

Equations 1 to 4 describe the copolymerization composition behaviour based on the following assumptions ... 

In the derivation of copolymer composition equation, Eq. (7.11), we considered only the rates of the four possible propagation steps in a binary system. However, the overall rate of copolymerization depends also on the rates of Initiation and termination. In deriving an expression for the rate of copolymerization in binary systems the following assumptions will be made [25] (a) rate constants for the reaction of a growing chain depend only... 

The copolymerization data for the styrene(Mi)-fumaronitrile(M2) system indicate that there are also effects due to remote monomer units preceding the penultimate unit. The effect of remote units has been treated by further expansion of the copolymer composition equation by the use of greater number of monomer reactivity ratios for each monomer [34]. However, the utility of the resulting expression is limited due to the large number of variables involved. 

For the copolymerization of two monomers by an anionic mechanism, the copolymer composition equation (7.11) or (7.18), derived in Chapter 7 is applicable with the monomer reactivity ratios defined in the same way as the ratios of rate constants r = k /ki2 and T2 = /s22/ 2i> where k and /C22 are the rate constants for the homopropagation reactions ... 

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. ... 

An estimate of ri and r2 can be obtained from the slope of the experimental F versus f plot by comparison with curves based on Eq. (7.17) to choose, by trial and error, the values of ri and r2 for which the theoretical curve best fits the data. A limitation of this method is the relative insensitivity of the curves to small changes in r and r2. Another limitation is the assumption implied in using the differential form of the copolymerization equation [Eq. (7.11) or (7.17)] that the feed composition does not change during the experiment, which is obviously not true. To minimize the error, the polymerization is usually carried out to as low a conversion as possible at which a sufficient amount of the copolymer can still be obtained for direct analysis. The aforesaid limitations can be overcome, however, by the use of an integrated form of the copolymer composition equation, such as Eq. (7.23). In one method, for example, one determines by computational techniques the best values of r and V2 that fit Eq. (7.23) to the experimental curve of /i or /2 versus (1 - N/Nq). 

The existence of an azeotropic composition has some practical significance. By conducting a polymerization with the monomer feed ratio equal to the azeotropic composition, a high conversion batch copolymer can be prepared that has no compositional heterogeneity caused by drift in copolymer composition with conversion. Thus, the complex incremental addition protocols that arc otherwise required to achieve this end, are unnecessary. Composition equations and conditions for azeotropic compositions in ternary and quaternary copolymerizations have also been defined. " ... 

Equation (1.31) gives the molar ratio of A-type to B-type repeat units in the copolymer formed at any instant during the copolymerization when the monomer concentrations are [A] and [B]. Although Eq ion (1.31) is of use, it is more convenient to express conqrositions as mole fiactitxis. The mole fraction /a of monomer A in the comonomer mixture is [A]/([A] + [B]) and that of monomer B is /b = 1 - /a. The mole fracticm Fa of A-lype repeat units in the copolymer formed at a particular instant in time is d[A]/(d(A] + d[B]) and that of B-type repeat units is Fb = 1 — Fa. Additiwi of unity to both sides of Equation (1.31) allows it to be rearranged in terms of Fa (or Fb), /a and /b. The following copolymer composition equations are obtained... 

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