Copolymer equation determination

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. 

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. 

A mean field approach was applied to determine homopolymer distributions in the lamellar phase of a blend of AB diblock and A homopolymer by Shull and Winey (1992). In the strong segregation limit, complete segregation of the A homopolymer into the A microdomain was predicted. Furthermore, in this limit, the diblocks were treated as brushes , wetted by homopolymer in the A domain. Composition profiles showing the distribution of homopolymer and copolymer were determined by numerical solution of the self-consistent field equations. 

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). 

An aspect of the mechanism of this copolymerization process which, up to this point is unresolved, results from the general observation that the molecular weights of the copolymers, as determined for the purified copolymers, are significantly less than anticipated from a theoretical basis. This discrepancy is generally assumed to be due to a chain-transfer process with trace quantities of water (Equation 8.5). 

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. 

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

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... 

Another important recent contribution is the provision of a good measurement of the precision of estimated reactivity ratios. The calculation of independent standard deviations for each reactivity ratio obtained by linear least squares fitting to linear forms of the differential copolymer equations is invalid, because the two reactivity ratios are not statistically independent. Information about the precision of reactivity ratios that are determined jointly is properly conveyed by specification of joint confidence limits within which the true values can be assumed to coexist. This is represented as a closed curve in a plot of r and r2- Standard statistical techniques for such computations are impossible or too cumbersome for application to binary copolymerization data in the usual absence of estimates of reliability of the values of monomer feed and copolymer composition data. Both the nonlinear least squares and the EVM calculations provide computer-assisted estimates of such joint confidence loops [15]. 

The data analyzed from the UV-LALLS-RI chromatograms by applying equation 7 resulted in a calculated average p for the copolymer of 0.1732 mL/g. This compares well with a theoretical value of 0.1799 mL/g assuming the values of p s and p eo to be 0.185 and 0.050 mL/g, respectively, (5) and a composition of 96.2% PS. The marginal diflFerence could be attributed to error in the value assumed for p eo or error in the copolymer composition determined by NMR. For example, a composition of 91.5% PS would give a theoretical of 0.1735 mL/g. However, in the absence of more experimental results, the original NMR composition has to be accepted. 

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. 

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. 

The copolymer equation enables the mole fractions of the respective monomers that are incorporated into a copolymer to be determined for any feed composition, provided that the reactivity ratios are known. This equation does not indicate the way in which they are incorporated, other than for the cases for which there is a special relationship, such as... 

Several theoretical treatments of cyclocopolymerization have been reported previously (8-11). These relate the compositions of cyclocopolymers to monomer feed concentrations and appropriate rate constant ratios. To our knowledge, procedures for calculating sequence distributions for either cyclocopolymers or for copolymers derived from them have not been developed previously. In this paper we show that procedures for calculating sequence distributions of terpolymers can be used for this purpose. Most previous studies on styrene-methacrylic anhydride copolymerizations (10,12,13) have shown that a high proportion of the methacrylic anhydride units are cyclized in these polymers. Cyclization constants were determined from monomer feed concentrations and the content of uncyclized methacrylic anhydride units in the copolymers. These studies invoked simplifying assumptions that enabled the conventional copolymer equation to be used in determinations of monomer reactivity ratios for this copolymerization system. 

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. ... 

V represents the variance, and R represents the residual of equation 2. The confidence region can be determined by plotting the sum-of-squares contour for several and r values that satisfy the instantaneous copolymer equation for the variables given. 

The monomer ratio of styrene-methacrylate/acrylate copolymers is determined from the intensity ratio of the carbonyl band at 1730 cm for the methacrylate and aromatic ring band at 699 cm for the styrene component. In the measurement, a baseline is drawn between 1990 and 630 cm , and the intensity ratio of the 1730 and 699 cm bands is determined. Styrene content is determined from the empirical equation, percent styrene = 71.4 x ( 599/ 1730) [66,67]. If the methacrylate and acrylate are both present in the same copolymer, the methacrylate content may be determined from the intensity ratio of the ester-ether bands at 1032 cm" (acrylate) and 1021 cm (methacrylate). Figure 38 illustrates the procedure [68] for the measurement of styrene and styrene methacrylate in methyl styrene-acrylate copolymers and methyl methacrylate-modified styrene-acrylate copolymers, respectively. 

Another area in which both proton and carbon-13 nmr have proved very powerful is the determination of the structure of copolymers. This has a long history (ref. 6, Chap. X 15,16)9 beginning with the observations of butadiene-styrene copolymers in 1959, Again, the information content of the spectra has increased remarkably since these early reports. Although compositional sequence lengths and probabilities can be calculated from the copolymer equation using the traditional data of polymer composition vs. monomer feed composition, nmr allows direct measurement of the sequences and gives in addition much structural detail not available from overall composition alone. 

The composition of the surface-attached copolymer is determined by the composition of the monomer feed and the reactivity of the propagating species from the comonomers (homopropagation vs. crosspropagation). The chemical composition represented by the mi/mj molar ratio of monomers 1 and 2 at low degrees of polymerization is described by the well-known copolymerization equation ... 

The analysis of the temperature coefficient data emphasizes the importance of nucleation in the crystallization of copolymers. Equations (10.13) or (10.17) indicate that not all the potentially crystallizable sequences in the untransformed melt can participate in the nucleation act. The thickness, of a critical size nucleus is determined by the copolymer composition and crystalUzation temperature. Only sequences containing or a larger number of units, can be involved in forming a critical size nucleus. A significant number of chain units, therefore, cannot participate. This limitation on the sequences, and thus the crystalUzable units, that can participate in nucleation has important implications for many aspects of the crystallization process. The extent of this limitation is illustrated in Fig. 10.15. Here, as calculated from Eq. (10.17), is plotted against the mol percent of branch points. 

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 ... 

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. 

Changes in heat capacity and measurement of T for blends have been used to determine components of copolymers and blends (126—129), although dynamic mechanical analysis has been found to give better resolution. Equations relating T of miscible blends and ratios of components have been developed from dsc techniques, eg, the Fox equation (eq. 1), where f the blend, or is the weight fraction of component 1 or 2,... 

This equation relates the (instantaneous) copolymer composition with the monomer feed of M and M2. Values for and are usually determined by graphical methods (9,10). Today, with the prevalence of powerful desktop computers, numerical minimisa tion methods are often used (11—14). 

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. 

Several important assumptions are involved in the derivation of the Mayo-Lewis equation and care must be taken when it is applied to ionic copolymerization systems. In ring-opening polymerizations, depolymerization and equilibration of the heterochain copolymers may become important in some cases. In such cases, the copolymer composition is no longer determined by die four propagation reactions. 

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