Reactivity ratio Copolymer equation

Copolymer Equation tlpA/pB - (tA A/fe + l)/(tB/B//A + 1) fj = mole fraction of monomer A in feed = mole fraction of monomer B in feed Pa = mole fraction of A in copolymer Pb = mole fraction of B in copolymer Ta = reactivity ratio of A Tb = reactivity ratio of B... 

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. 

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 ... 

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. 

Copolymers. Although many copolymers of ethylene can be made, only a few have been commercially produced. These commercially important copolymers are Hsted in Table 4, along with their respective reactivity coefficient (see Co polymers. The basic equation governing the composition of the copolymer is as follows, where and M2 are the monomer feed compositions, and r2 ate the reactivity ratios (6). 

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. 

Wall studied the composition drift and derived what is now called the Wall equation, where n was equal to rx when the reactivity ratio r was equal to the ratio of the propagation rate constants. Thus, r was the slope of the line obtained when the ratio of monomers in the copolymer (M1/M2) was plotted against the ratio of monomers in the feed (mi/ma). The Wall equation is not general ... 

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... 

High-resolution nuclear magnetic resonance spectroscopy, especially 13C NMR, is a powerful tool for analysis of copolymer microstructure [Bailey and Henrichs, 1978 Bovey, 1972 Cheng, 1995, 1997a Randall, 1977, 1989 Randall and Ruff, 1988], The predicted sequence length distributions have been verihed in a number of comonomer systems. Copolymer microstructure also gives an alternate method for evaluation of monomer reactivity ratios [Randall, 1977]. The method follows that described in Sec. 8-16 for stereochemical microstructure. For example, for the terminal model, the mathematical equations from Sec. 8-16a-2 apply except that Pmm, Pmr, Pm and Prr are replaced by p, pi2, p2j, and p22. 

The determination of the reactivity ratios requires a knowledge of the composition of the copolymers made from particular monomer mixtures numerous analytical methods are available (see Sect. 2.3.2). In principle, it is possible to calculate and r2, using Eq. 3.18, from the composition of only two copolymers that have been obtained from two different mixtures of the two monomers M and M2. However, it is more precise to determine the composition of the copolymers from several monomer mixtures and to calculate, for each individual experiment, values of r2 that would correspond to arbitrarily chosen values of r from the rearranged copolymerization equation ... 

Table 1 gives a comparison of Raman and pmr results for a series of copolymers. In the pmr data of Figure the CHg absorption of the polymer backbone at 6O.8 to 3.0 partially overlaps with the CH doublet centered at S2.h and this reduces the accuracy of the integrated intensity of the ester moiety to no better than 25. On the other hand, the accuracy of the Raman data is on the order of 3%, so the two techniques do agree within experimental error. The error associated with the Raman method could be reduced if calibration curves were employed. The weight percent feed and polymer compositions were converted to mole percent and reactivity ratios for MMA and OM were calculated by the Yezrielev, Erokhina and Riskin (YBR) method (9). The following equation, derived from the copolymer... 

If we define the monomer reactivity ratio for monomer 1 and 2, ri and ri, respectively, as the ratio of rate constants for a given radical adding to its own monomer to the rate constant for it adding to the other monomer (ri = fcn/ 12 and ri = 22/ 21 see Table 3.7 for typical values), then we arrive at the following relationship known as the copolymer equation ... 

It was demonstrated that MACROMER will copolymerize with conventional monomers in a predictable manner as determined by the relative reactivity ratios. The copolymer equation ... 

The composition of the copolymer was determined by either NMR analysis at 90 MHz according to the equations derived by Mochel (21) or by infrared. (22) The agreement of these methods was 2% when applied to copolymer taken to 100% conversion. The reactivity ratios were calculated according to the Mayo-Lewis Plot (13,15), the Fineman-Ross Method (14), or by the Kelen-Tudos equation.(16,17,18) The statistical variations recently noted by 0 Driscoll (23), were also considered. 

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. 

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. 

The S-PIB macromonomer was copolymerized by radical copolymerization with MMA and S, and the reactivity ratio of the small comonomer was calculated by a modified copolymer equation [85]. With MMA, rMMA=0.5 was obtained, i.e., close to that reported for conventional S/MMA system. With S however, rs= 2.1 was determined which suggested that the reactivity of S-PIB is lower than that of S, possibly due to steric interference. 

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 ... 

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. 

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. 

So far we have discussed reactivity ratios as if they are known quantities. And many of them are (you can find their values in the Polymer Handbook), thanks to sterling work by many polymer chemists over the years. But what if you re confronted with a situation where you don t have this information - how would you determine the reactivity ratios of a given pair of monomers Essentially, there are two sets of approaches, both of which depend upon using the copolymer equation in one form or another, hence, the assumption that the terminal model applies to the copolymerization we are considering. A form we will use as a starting point was... 

The first method we will consider involves the rearrangement of the copolymer equation (Equation 6-4). The final equation expresses one of the reactivity ratios in terms of the other and the experimental quantities x and y (Equation 6-6). 

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