Copolymers equation

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 

Note that the first and third reactions in the four possible propagation steps shown in 

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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. 

The systematic changes in the glass temperatures illustrated in Table 1 indicate quantitatively the changes in the composition within each phase. The random copolymer equation can be used to estimate the composition within each phase ... 

The most widely used graft copolymer is the styrene-unsaturated polyester copolymer (Equation 7.35). This copolymer, which is usually reinforced by fibrous glass, is prepared by the free radical chain polymerization of a styrene solution of unsaturated polyester. 

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 first approach Is to polymerize small amounts of 4-vlnyl pyridine on to the ends of anionic living polybutadiene, mono- or difunctional, to produce what are essentially AB or ABA block copolymers (equation 5). Materials possessing values of n typically averaging about 3 have been prepared and shown to produce solids when quaternlzed with benzyl bromide. The result of... 

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). 

This reaction can also be applied to achieve step-growth copolymerization of aromatic ketone with a, >-dienes and to synthesize high molecular weight copolymers (equation 135)242,243. 

Ring-opening copolymerization (ROP) of (R)-/3-butyrolactone with (R)-3-methyl-4-oxa-6-hexanolide in the presence of tin(iv) chloride affords a new biodegradable copolymer (Equation 1) <1995MM406>. 

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. 

In random copolymers, there is a high degree of interaction between the different repeat units because they are held together chemically. The glass transition of a random copolymer consequently reflects the composition of the copolymer. Thus, the Tg of random copolymers can usually be represented as a monotonic function of composition by one of the various copolymer equations. Similarly, if blends of polymers are mixed at the segmental level, only one Tg should be evident, and a copolymer equation should be applicable. 

The Gordon-Taylor copolymer equation (Equation 4) has been applied to the Tg data in this study to determine if the blends are single-... 

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... 

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). 

An intersection method is employed, where essentially a first guess or estimate of the value of, say, rx is made and the rearranged copolymer equation used to calculate values of r2 for each value of x and > You end up with a set of straight lines which should intersect at a point defining the actual values of and rv but generally don t. You have to use some method for picking the best point within an area of intersection (Figure 6-11). 

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). 

Both the mini- and macroemulsion copolymerizations of pMS/MMA tend to follow bulk polymerization kinetics, as described by the integrated copolymer equation. MMA is only slightly more soluble in the aqueous phase, and the reactivity ratios would tend to produce an alternating copolymer. The miniemulsion polymerization showed a slight tendency to form copolymer that is richer in the more water-insoluble monomer. The macroemulsion formed a copolymer that is slightly richer in the methyl methacrylate than the co-... 

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