Copolymer composition equations

Combining Eqs. (7.9) and (7.11) yields the important copolymer composition equation ... 

An ideal gas obeys Dalton s law that is, the total pressure is the sum of the partial pressures of the components. An ideal solution obeys Raoult s law that is, the partial pressure of the ith component in a solution is equal to the mole fraction of that component in the solution times the vapor pressure of pure component i. Use these relationships to relate the mole fraction of component 1 in the equilibrium vapor to its mole fraction in a two-component solution and relate the result to the ideal case of the copolymer composition equation. 

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. 

Recognition of these differences in behavior points out an important limitation on the copolymer composition equation. The equation describes the overall composition of the copolymer, but gives no information whatsoever about the distribution of the different kinds of repeat units within the polymer. While the overall composition is an important property of the copolymer, the details of the microstructural arrangement is also a significant feature of the molecule. It is possible that copolymers with the same overall composition have very different properties because of differences in microstructure. Reviewing the three categories presented in Chap. 1, we see the following ... 

Note that pn + pi2 = P22 + P21 = 1- In writing these expressions we make the assumption that only the terminal unit of the radical influences the addition of the next monomer. This same assumption was made in deriving the copolymer composition equation. We shall have more to say below about this so-called terminal assumption. 

Equations (7.40) and (7.41) suggest a second method, in addition to the copolymer composition equation, for the experimental determination of reactivity ratios. If the average sequence length can be determined for a feedstock of known composition, then rj and r2 can be evaluated. We shall return to this possibility in the next section. In anticipation of applying this idea, let us review the assumptions and limitation to which Eqs. (7.40) and (7.41) are subject ... 

Item (2) requires that each event in the addition process be independent of all others. We have consistently assumed this throughout this chapter, beginning with the copolymer composition equation. Until now we have said nothing about testing this assumption. Consideration of copolymer sequence lengths offers this possibility. 

Evaluation of reactivity ratios from the copolymer composition equation requires only composition data—that is, analytical chemistry-and has been the method most widely used to evaluate rj and t2. As noted in the last section, this method assumes terminal control and seeks the best fit of the data to that model. It offers no means for testing the model and, as we shall see, is subject to enough uncertainty to make even self-consistency difficult to achieve. 

The copolymer composition equation relates the r s to either the ratio [Eq. (7.15)] or the mole fraction [Eq. (7.18)] of the monomers in the feedstock and repeat units in the copolymer. To use this equation to evaluate rj and V2, the composition of a copolymer resulting from a feedstock of known composition must be measured. The composition of the feedstock itself must be known also, but we assume this poses no problems. The copolymer specimen must be obtained by proper sampling procedures, and purified of extraneous materials. Remember that monomers, initiators, and possibly solvents are involved in these reactions also, even though we have been focusing attention on the copolymer alone. The proportions of the two kinds of repeat unit in the copolymer is then determined by either chemical or physical methods. Elemental analysis has been the chemical method most widely used, although analysis for functional groups is also employed. 

Still assuming terminal control, evaluate r and T2 from these data. Criticize or defend the following proposition The copolymer composition equation does not provide a very sensitive test for the terminal control mechanism. 

This allows elimination of the radical concentrations from the above equation and the copolymer composition equation (eq. 5),14-16 also known as the Mayo-Lewis equation, can now be derived. 

Other convenient forms of the copolymer composition equation are eq. 8 ... 

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 traditional method for determining reactivity ratios involves determinations of the overall copolymer composition for a range of monomer feeds at zero conversion. Various methods have been applied to analyze this data. The Fineman-Ross equation (eq. 42) is based on a rearrangement of the copolymer composition equation (eq. 9). A plot of the quantity on the left hand side of eq. 9 v.v the coefficient of rAa will yield rAB as the slope and rUA as the intercept. 

The copolymer composition equation only provides the average composition. Not all chains have the same composition. There is a statistical distribution of monomers determined by the reactivity ratios. When chains are short, compositional heterogeneity can mean that not all chains will contain all monomers. 

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. 

Mole fraction of styrene in unreacted monomer Cumulative copolymer composition (Equation 5)... 

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). 

Various methods have been used to obtain monomer reactivity ratios from the copolymer composition data. The most often used method involves a rearrangement of the copolymer composition equation into a form linear in the monomer reactivity ratios. Mayo and Lewis [1944] rearranged Eq. 6-12 to... 

The derivation of the terminal (or hrst-order Markov) copolymer composition equation (Eq. 6-12 or 6-15) rests on two important assumptions—one of a kinetic nature and the other of a thermodynamic nature. The Erst is that the reactivity of the propagating species is independent of the identity of the monomer unit, which precedes the terminal unit. The second is the irreversibility of the various propagation reactions. Deviations from the quantitative behavior predicted by the copolymer composition equation under certain reaction conditions have been ascribed to the failure of one or the other of these two assumptions or the presence of a comonomer complex which undergoes propagation. 

By using the differential form of the copolymer composition equation (26, 28) the products of oxidation of mixtures at low conversions permit comparison of rates of chain propagation in autoxidations of various compounds, essentially free from effects of chain initiation, chain termination, and over-all rates. 

The copolymer composition equation was first applied to co-oxidations in mixtures of aldehydes (25, 39) and later to numerous pairs of hydrocarbons and their derivatives (1, 2, 3, 4, 8, 27, 31, 32, 33). For oxidations of mixtures of A and B, attack by a peroxy radical first gives (by addition or hydrogen abstraction) A and B radicals in the presence of sufficient oxygen all these are then converted to A02 and B02 peroxy radicals. From the relative rates of reaction, A[A]/A[B], of A and B at two or more average feeds [A] / [B], in long kinetic chains, the copolymer composition equation... 

On the basis of these equations and the foregoing assumptions, the classical, copolymer-composition equation (I) is derived that relates the ratio of mers in the copolymer to the monomer concentrations in the feed. 

The simple, copolymer-composition equation should also be applied only to those cases in which the stereoselectivity is very high. If... 

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... 

A number of reactivity ratios have been determined from initial copolymer composition data. These are recorded in Table 2. In view of the difficulties associated with their determination and uncertainty whether the copolymer composition equation is accurate under all conditions, they should be considered as of unknown accuracy. 

Except under high feed ratios of CO, step (3) may be disregarded since homopolymerization of CO occurs at a negligible rate (i.e., k is small). The copolymer composition equation is given below ... 

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 (7.116) is called the copolymer composition equation and can be written in terms of the mole fraction in both the feed and the copolymer ... 

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 ... 

As seen in Equation (7.118), the copolymer and the feed compositions vary with the conversion during the polymerization process. The drift of a feed composition with the conversion favors the less reactive monomer. To obtain F, at different conversions, it is necessary to integrate the copolymer composition equation. Assume that the system contains a total of M moles of monomers and F, > f,. After dM moles of monomers have been polymerized, the copolymer contains F,dM moles of Mj. The feed then contains (M - dM)(fj - dl",) moles of M where the first term is the total monomer concentration and the second term is the mole fraction of M, in the feed. The mass balance of Mj reacted in the system is given by ... 

More recently Durgaryan derived (5) a copolymer composition equation assuming reversibility of all propagation reactions, Equations 1-4. He expressed his solution as a pair of simultaneous equations in which the two unknowns (besides kinetic parameters and monomer feed) were copolymer composition and the ratio [(mi)2 ]/[mi)i S ]. At this writing no experimental tests of Durgaryan s equations have appeared. 

Another reason for errors of the reactivity ratio values are an exactitude in the course of the treatment of the experimental data using the differential or integrated form of the copolymer composition equation. In the first pase, the dependence of X(x°) on the monomer feed composition x° experimentally determined at low conversions is used. In the second case, one should use the data on the dependence of the copolymer composition on conversion p or the current values of x under the measurements of p. 

The copolymer composition equation with reversibility of all propaga-tion steps was derived as a complex function [298,299] ... 

The reactivity ratios should likewise be independent of initiator concentration, reaction rate, and overall extent ol monomer conversion, since no rale constants appear as such in the copolymer composition equation. 

The copolymer composition equation is written in terms of monomer concentrations at the locus of reaction. The same reactivity ratios should apply in principle whether the polymerization is carried out in bulk, solution, suspension, or emulsion systems. In general, the only concentration values available to the experimenter are the overall bulk figures. Deviations of copolymer composition can be expected, therefore, if the concentrations at the polymerization sites differ from these figures. This can occur in emulsion systems, for example, if the monomers differ appreciably in aqueous solubility and diffusion rates. 

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. 

---