Copolymerization equation composition distribution

The conditions of the existence of such bimodal distributions are well-known for the binary copolymerization [169-171], since in this case the differential equation proposed by Skeist [12] for the determination of the composition distribution has an explicit solution. Indeed when m = 2, the only one independent equation of the two equations (5.2) has a simple solution [169, 170, 172] ... 

In Refs. [173-176] it was suggested to use the weight composition distributions instead of the molar ones and the results of their numerical calculation for some systems were reported The authors of Ref. [177] carried out a thorough theoretical study of the composition distribution and derived an equation for it without the Skeist formula. They, as the authors of Ref. [178], proposed to use dispersion of the distribution (5.3) as a quantitative measure of the degree of the composition inhomogeneity of the binary copolymers and calculated its value for some systems. Elsewhere [179-185] for this purpose there were used other parameters of the composition distribution. In particular the discussion of the different theoretical aspects of the binary copolymerization is reported in a number of reviews by Soviet authors [186-189], By means of numerical calculations there were analyzed [190-192] the limits of the validity of the traditional assumption which allows to ignore the instantaneous component of composition distribution of the copolymers produced at high conversions. 

The uncertainties and poor j eement in determinations of ethylene/ propene reactivity ratios from monomer and polymer composition [m = (Mj/Mj) monomer p = (Mi/M2) polymer] and the copolymerization equation p = (1 + rjm)/(l + rj/m) give particular interest to approaches based on the analysis of monomer unit distributions in the copolymer. 

Statistical Models. Due to the difficulties involved in calculating the composition distributions by purely deterministic techniques, statistical methods have been developed from which not only the CCD can be obtained but also the sequence length distribution. These methods view the chain growth as an stochastic process having possible states resulting from the kinetic mechanisms. Early work on this approach was reported by Merz, Alfrey and Goldfinger (4) who derived the copolymerization equation and the SLD for the ultimate effect case. Alfrey Bohrer and Mark (19) and Ham (9) formalized this approach. 

Tip 13 (related to Tip 12) Copolymerization, copolymer composition, composition drift, azeotropy, semibatch reactor, and copolymer composition control. Most batch copolymerizations exhibit considerable drift in monomer composition because of different reactivities (reactivity ratios) of the two monomers (same ideas apply to ter-polymerizations and multicomponent cases). This leads to copolymers with broad chemical composition distribution. The magnirnde of the composition drift can be appreciated by the vertical distance between two items on the plot of the instantaneous copolymer composition (ICC) or Mayo-Lewis (model) equation item 1, the ICC curve (ICC or mole fraction of Mj incorporated in the copolymer chains, F, vs mole fraction of unreacted Mi,/j) and item 2, the 45° line in the plot of versus/j. 

Einally, as per Equation 18.27b, if a copolymer has a very narrow composition distribution, will be very close to (M) using a single solvent the same applies to mixed solvents if we follow the procedure mentioned earlier. This can be the case, for instance, of a radical copolymerization in a true azeotrope composition or of a block copolymer synthesized by a controlled anionic or living radical polymerization. 

Not only in the mathematical description of copolymer composition, but also in that of monomer sequence distribution, is it convenient to use so-called conditional probabilities. These conditional probabilities are defined as the chance that a certain event takes place out of aU possibilities at a certain stage. For the purpose of the copolymerization equations, conditional probabilities related to propagation only are considered. In case of the TM, an example of such a conditional probability is the chance that monomer 2 will add to a monomer 1 chain-end radical (P12). In terms of eqns [l]-[4], this probability is the rate of reaction [2] divided by the sum of the rates of reaaions [1] and [2]. The two relevant conditional probabilities are defined as in eqns [6] and [7] ... 

V. Copolvmerization Kinetics. Qassical copolymerization kinetics commonly provides equations for instantaneous property distributions (e.g. sequence length) and sometimes for accumulated instantaneous (i.e. for high conversion samples) as well (e.g. copolymer composition). These can serve as the basis upon whkh to derive nations which would reflect detector response for a GPC separation based upon properties other than molecular weight. The distributions can then serve as c bration standards analagous to the use of molecular weight standards. 

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... 

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... 

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. 

These equations show that the composition of the copolymer formed from a specific comonomer mixture is controlled by the monomer reactivity ratios for the copolymerization. Additionally, they control the sequence distribution of the different repeat units in the copolymer. If ta > 1 then "> A prefers to add monomer A (i.e., it prefers to homopropagate) and extended sequences of A-type repeat units are introduced, whereas if ta < 1 A prefers to add monomer B, i.e., to cross-propagate. In a similar way, ra describes the behaviour of monomer B. The effects of some specific combinations of ta and re values upon copolymer composition and repeat unit sequence distribution are considered in the next section. 

The apparent reactivity ratios that govern the copolymerization in the solvents were determined and are significantly different. Nevertheless, the triad distribution as a funrtion of copolymer composition shows that within experimental error, one set of curves describes all three situations. This again is clear evidence that solvents do not affect the tme monomer reactivity ratios, but only the monomer partitioning. In the derivations by Klumperman and O Driscoll it is clearly shown that these partitioning effects cancel from the sequence distribution versus copolymer composition equations. 

It was found that the composition and the distribution of units in copolymerization is controlled mainly by the propagation process. From this point of view, equations have been formulated (71) concerning how the reactivity ratios depend on the template concentration and individual reactivity rate constants of monomers taking part in the template copolymerization process. However, if long critical length is necessary for the adsorption of the growing macroradical onto the template, any template effect can be destroyed (72). 

For the foUwing estimation of the copolymerization parameters it is useful to discern between the ovmaU or mixed parameters and the true oopolymerization parameters. First we assume that there are only uniform active centres located on the catalyst sur ce, (i.e., one centre model), and use ethene and comonomer peaks in the NMR spectrum of the polymer mixture for the estimation of the oopolymerization parameters according to the Mayo Lewis equqtion This evaluation, via the r versus diagram, leads to the overall or mixed copolymerization parameters. However, for the estimation of the true copolymerization parameters we now use the following considerations. The Mayo-Lewis equation describes the composition of the copolymer as a function of the initial monomers mixture and the oopolymerization parameters. If we know these and the monomers mixture we can calculate not only the copolymer composition but also, by means of statistical considerations, the sequence length distribution of Mj and M2 sequences in the copolymer... 

Taking into account the Gibbs energy of mixing and the entropy of the copolymer units distribution, he obtained the equations that enable us to predict for any initial conditions the composition of the copolymerization system at equilibrium (for the sake of simplicity, he assumed, while deriving the equations, that molar volumes of solvent, comonomers, and copolymer units are equal) ... 

The template effects can be expressed as (1) kinetic effect -usually an enhancement of the reaction rate and change in kinetics equation (2) molecular effect - consisting of an influence of the template on the molecular weight and molecular weight distribution of daughter polymer (3) effect on tactidty -the daughter polymer can have the complementary stmcture to the stmcture of the template used and (4) in the case of template copolymerization, the template effect - deals with the composition and sequence distribution of units. 

As for step copolymerization, differences in monomer reactivity in chain copolymerization affect the sequence distribution of the different repeat units in the copolymer molecules formed. The most reactive monomer again is incorporated preferentially into the copolymer chains but, because of the different nature of chain polymerization, high molar mass copolymer molecules are formed early in the reaction. Thus, at low overall conversions of the comonomers, the high molar mass copolymer molecules formed can have compositions which differ significantly from the composition of the initial comonomer mixture. Also in contrast to step copolymerization, theoretical prediction of the relative rates at which the different monomers add to a growing chain is more firmly established. In the next section a general theoretical treatment of chain copolymerization of two monomers is presented and introduces an approach which can be applied to derive equations for more complex chain copolymerizations involving three or more monomers. 

Monomer reactivity ratios are important quantities since for a given instantaneous comonomer composition, they control the overall composition of the copolymer formed at that instant and also the sequence distribution of the different repeat units in the copolymer. From Equation (2.86), they are the ratios of the homopropagation to the cross-propagation rate constants for the active centres derived from each respective monomer. Thus if a> 1 then prefers to add monomer A (i.e. it prefers to homopolymerize), whereas if rA[Pg.120]

The Mayo-Lewis equation describes the composition of a copolymer as a function of the composition of the monomer mixtime and their respective copolymerization parameters. These parameters not only determine the composition of the resulting copolymer, but also its microstructure, e.g. the distribution of the sequence length. 

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