Copolymer equation integrated

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). 

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-... 

Equations (7-13) and (7-14) are alternative versions of the simple copolymer equation. Measurements of corresponding feed and copolymer compositions should yield values of r and r2 which can then be used to predict the relative concentrations of monomer in copolymers formed from any other mixtures of the particular monomers. The equations given are differential expressions and define the composition of the copolymer formed at any instant during the polymerization. They may be integrated, as noted in Section 7.5, to follow reactions in which one monomer is consumed more rapidly than the other. 

Integration of the simple copolymer equation between the limits m , and nil, 2 yields [I] ... 

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. 

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. 

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. 

Comment If one makes the conversion interval smaller and smaller, this corresponds to an integration of the copolymer equation (see below). 

To follow the composition drift of both the comonomer feed and the copolymer formed requires integration of the copolymer equation. This problem is rather complex. The most convenient approach utilizes a numerical or graphical method developed by Skeist [9] for which Eq. (7.18) forms the basis. Consider a system initially containing a total of N moles of the two monomers choose Mi as the monomer in which F) > fi (i.e.. 

To follow the drift in composition of both the feed and the copolymer formed one needs to integrate the copolymer equation. The process being rather complex, the numerical or graphical approach of Skeist (1946) based on Eq. (7.17) provides a simple solution to the problem. Consider a system initially containing a total of N moles of the two monomers and choose Mi as the monomer such that F > f (i.e., the polymer being formed contains more Mi than the feed). Thus at a time when dN moles of the monomer mixture have been converted into polymer, the polymer formed will contain Fi dN moles of M i, while the M i content in feed will be reduced to N - dN)(J df ) moles. Thus a material balance for monomer Ml can be written as (Odian, 1991 Billmeyer, Jr., 1994) ... 

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. ... 

Few studies of reactivity ratios have been undertaken where the copolymerizations have been taken to high conversion. However, Johnson, Karmo, and Smith have demonstrated that the styrene-methyl methacrylate stem shows variations in the copolymer composition from that expected from the integrated copolymer equation. The deviations only occur during gelation or in the presence of precipitants, and the changes in and are reported. A comprehensive review of radical polymerization to high conversion has been produced in the form of a book by Gladysev. Many other comonomer pairs have been studied. ... 

This is shown in Figure 16.1. Consider, for example, an initial monomer composition of /,=/j=0.5. For r =0.1, F, 0.1. Since the polymer is very rich in monomer 2, the monomer composition will move in the direction of higher/,. This will continue until the end of the polymerization, when the polymer will be almost completely made up of monomer 1. Thus, the first chains polymerized will have a very low composition of monomer 1 (/j=0.1) while the last chains polymerized will be homopolymer of monomer 1 (( = 1.0). For living polymerization, the same argument can be made for compositional drift within a single chain. Thus, for free radical polymerization, compositional drift will take the form of a wide distribution of copolymer composition among the chains. The compositional drift can be described by the integrated copolymer equation for batch polymerization [ 12]. 

Only when accurate conversions were reported in the copolymerization data and the method of Tidwell and Mortimer was used for integrating the equations is there any reasonable assurance that the reactivity ratios, if very divergent from unity, do not contain a serious bias because of assumptions made about constancy of feed compositirm. It has bear indicated in Table 10 what method was used to obtain the reactivity ratios in each instance. Abnormalities, such as, for example, an r, product substantially greater than unity, as is seen in some of the data of Brown and Ham (124) in Table 10, can be accounted for on the basis that considerable drift in monomer composition took place during the course of the copolymerization and no correction was made for this by integrating the copolymer equation. 

An exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

By virtue of the conditions xi+X2 = 1>Xi+X2 = 1, only one of two equations (Eq. 98) (e.g. the first one) is independent. Analytical integration of this equation results in explicit expression connecting monomer composition jc with conversion p. This expression in conjunction with formula (Eq. 99) describes the dependence of the instantaneous copolymer composition X on conversion. The analysis of the results achieved revealed [74] that the mode of the drift with conversion of compositions x and X differs from that occurring in the processes of homophase copolymerization. It was found that at any values of parameters p, p2 and initial monomer composition x° both vectors, x and X, will tend with the growth of p to common limit x = X. In traditional copolymerization, systems also exist in which the instantaneous composition of a copolymer coincides with that of the monomer mixture. Such a composition, x =X, is known as the azeotrop . Its values, controlled by parameters of the model, are defined for homophase (a) [1,86] and interphase (b) copolymerization as follows... 

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... 

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... 

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