The Copolymer Equation

To begin to answer the first question, we must estabhsh a suitable kinetic scheme. The following group of homo- and heteropolymerization reactions were proposed by Dostal in 1936 for a radical copolymerization between two monomers M, and Mj and, ultimately, extended and formalized by a number of workers who established a practical equation from the reactions  

Under steady-state conditions, and assuming that the radical reactivity is independent of chain length and depends only on the nature of the terminal unit, the rate of consumption of M, from the initial reaction mixture is then 

The copolymer equation can then be obtained by dividing Equation 5.2 by Equation 5.3 and assuming that 2i[Mi][Mj]= 2[M2][Mi] for steady-state conditions, so that 

The quantities r and r2 are the relative reactivity ratios, defined more generally as the ratio of the reactivity of the propagating species with its own monomer to the reactivity of the propagating species with the other monomer. 

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This is known as the copolymer equation, and is a mathematical expression from which it is possible to determine the ratio of monomer units in the increment of copolymer formed from a given ratio of monomer molecules. 

Identify the different types of copolymers, and use the copolymer equation with appropriate kinetic data to determine which type of copolymer will form. 

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 is important to distinguish between the concentration ratio, /, and the rate of change of concentration, F, since if monomers Mi and M2 are consumed at different rates, then F, fi. Substitution of these ratios Fi, F2, /i, and /2 into the copolymer equation gives... 

This form of the copolymer equation allows us to identify several simplifying cases ... 

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 agreement between heats of fusion of the same polymer is excellent in some cases, but very poor in others. Obviously, in the case of polypropylene more work needs to be done before the heat of fusion of this substance will be known with any certainty. Heats of fusion calculated from the copolymer equation, Eq. (6), are uniformly low, except in the case of Rybnikar s data. As pointed out by Dole and Wunderlich (1957) this is probably due to the failure to measure the maximum melting of carefully annealed samples. Thus, Dole and Wunderlich (1959) found that the calorimetrically estimated melting point in the case of the carefully annealed copolyester, the 80/20 polyethylene terephthalate and sebacate, was 240° C, whereas the value calculated from Eq. (6) using the heat of fusion estimated from the calorimetric data of Smith and Dole (1956) was 245° C. The unannealed sample had a melting point of ca. 210°. 

Where copolymer compositional heterogeneity was a problem, monomer charges and feed ratios were adjusted to produce constant composition homogeneous polymers. While the required composition for the monomer charge and feed ratios may be determined by a random process, it is much more convenient to use an integrated form of the copolymer equation to calculate these quantities (2). 

We should point out that the equation we will derive, the copolymer equation (Figure 6-5), should be applicable to other types of polymerizations, such as those utilizing catalysts. Many commercially used catalysts are heterogeneous, however, meaning that we get polymers with different characteristics (sequence distributions) produced at different sites. The copolymer equation should apply to the polymers produced at each site, but the final product contains all these jumbled up together and there is no way to judge what... 

Obviously, what we would really like to do is not just have a feel for tendencies, useful as this is, but also calculate copolymer composition and sequence distributions, things that can also be measured by spectroscopic methods. We will start by using kinetics to obtain an equation for the instantaneous copolymer composition (it changes as the copolymerization proceeds). Later we will use statistical methods to describe and calculate sequence distributions. In deriving the copolymer equation, we only have to consider the propagation step and apply our old friend, the steady-state assumption, to the radical species present in the polymerization, and... 

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). 

The Fineman-Ross method uses a more conventional plotting procedure, rearranging the copolymer equation into the following form (Equation 6-7),... 

Both the Mayo-Lewis and the Fineman-Ross methods rely on linearizing the copolymer equation. It has been shown that... 

We can now derive the copolymer equation from these simple conditional probabilities. 

Now, if you recall the copolymer equation relating the composition of the copolymer formed at any instant of time (FA, FB), to the monomer feed composition (fA, fB) in a batch copolymerization (Equation 6-5), it should be clear that unless you have rA = rB 1, so that Fa - fA, then one of the monomers is going to be used up faster than the other (unless rA < 1, rB < 1 and you start with a monomer composition corresponding to the azeotrope condition). That means copolymer composition varies with conversion—we say there is compositional drift. 

In other words, you cannot simply use the copolymer equation to calculate copolymer composition and assume [A] and [ ], hence fA orf3 are constant over the entire course of the copolymerization. However, it is reasonable to assume that over some small interval of conversion, say 1%, the monomer concentration in the feed remains essentially constant. Then you can use the simple procedure given in Figure 6-21. Obviously, the smaller you make the conversion interval, the more accurate your calculations will be. Unless you are a screaming masochist, this is not the type of calculation you want to make with your pocket calculator. But it readily lends itself to the construction of a nice little computer program, which can also be used to calculate the other parameters we have mentioned. 

Just as in the derivation of the copolymer equation for the terminal model, we start with a reversibility relationship P3 AAB = P3 BAA. Now we must use second-order Markovian statistics to write this in terms of conditional probabilities (Equation 6-64) ... 

The copolymer equation can be written in a general form (Equations 6-70) ... 

Using the steady-state approximation, derive the copolymer equation for the free radical synthesis of monomer Mt with monomer M,. Express your answer in tenns of the mole fraction of monomer 1 in the copolymer (Fj) and the mole fraction of monomer 1 in the feed (/j). 

As the water solubility of the comonomer decreases, the difference in incorporation of the hydrophobic monomer between the mini- and macroemulsion polymerization becomes more pronounced. This was seen in the copolymerization of VH/MMA. The fraction of the hexanoate in the copolymer formed in the miniemulsion polymerization was substantially higher than that found with the macroemulsion. This incorporation closely follows the copolymer equation. The VEH/MMA miniemulsion copolymerization also followed the copolymer equation. Differences between the mini- and macroemulsion polymerization are not as pronounced in this system. For the VD/MMA and VS/MMA systems there were large differences between the two copolymerizations. In addition, none of the mini- or macroemulsion copolymerizations of vinyl decanoate or vinyl stearate are predicted by the copolymer equation. The miniemulsion copolymerizations fall above the prediction curve (more hydrophobic monomer incorporation than predicted), and the macroemulsions fall below. In these cases, both micellar and droplet nucleation took place in the miniemulsion polymerizations, and the presence of micelles tended to enrich the concentration of the hydrophobic monomer in the droplets, since the micelles would likely be richer in the more water-soluble MMA. 

Samer [104] carried out similar copolymerizations with similar results. An example of his data is given in Fig. 16. Here 2-ethylhexyl acrylate (EHA) was copolymerized with MMA in batch. The miniemulsion polymerizations (two are shown) follow the copolymer equation, while the macroemulsion polymerization gives EHA incorporation that is lower than predicted by the copolymer equation, presumably due to the low concentration of EHA at the locus of polymerization. The dotted hne in Fig. 16 is for a model derived by Samer that accurately predicts the copolymer composition. Samer derived this model by adapting the work of Schuller [149]. Schuller modified the reactivity ratios for the macroemulsion polymerization of water-soluble monomers to take into accoimt that the comonomer concentration at the locus of polymerization is different from the comonomer composition in the reactor due to the water solubilities of the monomers. Samer used the same approach to account for the fact that the comonomer concentration at the locus of polymerization might be different from that of the reactor due to transport limitations of water insoluble comonomers. 

The copolymer equation 10.94 describes the instantaneous copolymer composition at given monomer concentrations. An example is shown in Figure 10.7. 

The reactivity ratios for pairs of given monomers can be very different for the different types of chain-growth copolymerization free-radical, anionic, cationic, and coordination copolymerization. Although the copolymer equation is valid for each of them, the copolymer composition can depend strongly on the mode of initiation (see Figure 10.8). 

While the copolymer equation is universal in that it applies to all kinds of chain-growth copolymerization, an equally universal equation for the polymerization rate cannot be arrived at. For assessing the composition of the copolymer, only the ratio of the monomer consumption rates was needed, and that ratio was found to be a unique function of the monomer concentrations and rate coefficients. In contrast, the polymerization rate is composed of the absolute values of the monomer consumption rates, and these depend also on the concentrations of the propagating centers and thereby indirectly on the mechanism and rate of termination. 

If each radical prefers to add the monomer of tlie opposite type both reactivity ratios will tend to zero, and the copolymer equation becomes... 

The fitting of corresponding feed and copolymer compositions to the copolymer equation to obtain reactivity ratio values is not without pitfalls. Many of the available rj and values in the literature are defective because of unsuspected problems which were involved in estimation procedures, use of inappropriate mathematical models to link polymer and feed compositions, and experimental or analytical difficulties. 

Several procedures for extracting reactivity ratios from difl erential forms of the copolymer equation are mentioned in the following paragraphs. These methods are arithmetically correct, but they do not give reliable results because of the nature of the experimental uncertainties in reactivity ratio measurements. 

Corresponding experimental values of [Mi], [Ma], straight line in the ri a plane and the intersection region of such lines from different feed composition experiments is assumed to give the best values of ri and ri- The same basic technique may be applied to the integrated form of the copolymer equation. The intersection point which corresponds to the best values of f and ri is selected imprecisely and subjectively by this technique. Each experiment yields a straight line, and each such line can intersect one line from every other experiment. Thus n experiments yield (n(n — l)/2 intersections and even one wild experiment produces ( — I) unreliable intersections. Various attempts to eliminate subjectivity and reject dubious data on a rational basis have not been successful. 

Another method [9] involves recasting the copolymer equation in the form... 

The copolymer composition can be estimated usefully in many cases from the composition of unreacted monomers, as measured by gas-liquid chromatography. Analytical errors are reduced if the reaction is carried to as high a conversion as possible, since the content of a given monomer in the copolymer equals the difference between its initial and final measured contents in the feed mixture. The uncertainty in the copolymer analysis is thus a smaller proportion of the estimated quantity, the greater the magnitude of the decrease in the monomer concentration in the feed. It may seem appropriate under these circumstances to estimate reactivity ratios by fitting the data to an integrated form of the copolymer equation. 

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]. 

In order to determine the reactivity of pentachlorophenyl acrylate, 8, in radical initiated copolymerizations, its relative reactivity ratios were obtained with vinyl acetate (M2), ri=1.44 and r2=0.04 using 31 copolymerization experiments, and with ethyl acrylate (M2), ri=0.21 and r2=0.88 using 20 experiments.The composition conversion data was computer-fitted to the integrated form of the copolymer equation using the nonlinear least-squares method of Tidwell and Mortimer,which had been adapted to a computerized format earlier. 

This is the copolymer equation, which may also be derived by statistical means without invoking the stationary-state approximation (Odian, 1991). 

Inspection of the copolymer equation shows the importance of the reactivity ratios in determining the type of copolymerization reaction that will occur. Thus, we consider the following possibilities. 

Table 1.13. Copolymer composition for the ideal copolymerization, for which r- = 1//2 f- and 2 are the molar feed ratios of monomers, Mi and M2, and Fi and F2 are the mole fractions of the repeat units in the copolymer (Equation (1.82))...

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