Copolymer equation instantaneous

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... 

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

The copolymer equations, Eqs. (7.11) and (7.18), give the instantaneous copolymer composition, i.e. the composition of the copolymer formed from a given feed composition at very low degrees of conversion (approximately < 5%) such that the composition of the monomer feed may be considered to be essentially unchanged from its initial value. For all copolymerizations except when the feed composition is an azeotropic mixture or where rj = T2 — the comonomer feed and copolymer product compositions are different. The comonomer feed changes in composition as one of the monomers preferentially enters the copolymer. Thus there is a drift in the comonomer composition, and consequently a drift in the copolymer composition, as the degree of conversion increases. It is important to be able to calculate the course of such changes. 

Equation (7.17), which is also called the copolymer equation, gives the mole fraction of monomer Mi in the copolymer whose feed contained f mole fraction of monomer Ml. It is more convenient to use than its previous form [Eq. (7-H)]. It should be noted, however, that Fi gives the instantaneous copolymer composition and both fi and Fi change as the polymerization proceeds. 

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. ... 

It should be noted that the copolymer equation (7.11) describes the instantaneous copolymer composition on a macroscopic scale, that is, composition in terms of the overall mole ratio or mole fraction of monomer units in the copolymer sample produced, but it does not reveal its microstructure, that is, the manner in which the monomer units are distributed along the copolymer chain. Thus for two monomers Mi and M2, the ratio Fi/(I — F ) gives the overall mole ratio of Mi and M2 units in the copolymer but no information about the average lengths (i.e., number of monomer units) of Mi and M2 sequences, as illustrated (Allcock and Lampe, 1990) for a typical copolymer by... 

The use of a continuous stirred tank reactor permits one to apply the instantaneous copolymer equation for reactivity ratios estimation. 

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. 

Equation 6.7 is known as the copolymerization or the Mayo-Lewis equation. The physical meaning of Equation 6.7 is better appreciated by writing it in terms of mole fractions. If /j is the mole fraction of unreacted monomer i and F is the mole fraction of monomer i in the copolymer formed instantaneously, then... 

The copolymer equation provides a means of calculating the amount of each monomer incorporated in the chain from a givrai reaction mixture or feed when the reactivity ratios are known. It shows that if monomer Mj is more reactive than Mj, then Mi will raiter the copolymer more rapidly consequraitly, the feed becomes progressively poorCT in M, and composition drift occurs. The equation is that an instantaneous expression, which relates only to the feed composition at any given time. 

This equation relates the composition of the copolymer formed to the instantaneous composition of the feedstock and to the parameters rj and r2 which characterize the specific system. Figure 7.1 shows a plot of Fj versus fj-the mole fractions of component 1 in the copolymer and monomer mixture, respec-tively-for several arbitrary values of the parameters rj and r2. Inspection of Fig. 7.1 brings out the following points ... 

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 ratio of these equations provides an expression for the instantaneous copolymer composition (eq. 3). 

The instantaneous copolymer composition is described by the following equation (eq. 12) ... 

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. 

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... 

Even with the Kelen Tudos refinement there are statistical limitations inherent in the linearization method. The independent variable in any form of the linear equation is not really independent, while the dependent variable does not have a constant variance [O Driscoll and Reilly, 1987]. The most statistically sound method of analyzing composition data is the nonlinear method, which involves plotting the instantaneous copolymer composition versus comonomer feed composition for various feeds and then determining which theoretical plot best fits the data by trial-and-error selection of r and values. The pros and cons of the two methods have been discussed in detail, along with approaches for the best choice of feed compositions to maximize the accuracy of the r and r% values [Bataille and Bourassa, 1989 Habibi et al., 2003 Hautus et al., 1984 Kelen and Tudos, 1990 Leicht and Fuhrmann, 1983 Monett et al., 2002 Tudos and Kelen, 1981]. 

An alternative rationale for the unusual RLi (hydrocarbon) copolymerization of butadiene and styrene has been presented by O Driscoll and Kuntz (71). Rather than invoking selective solvation, these workers stated that classical copolymerization kinetics is sufficient to explain this copolymerization. They adapted the copolymer-composition equation, originally derived from steady-state assumptions for free-radical copolymerizations, to the anionic copolymerization of butadiene and styrene. Equation (20) describes the relationship between the instantaneous copolymer composition c/[M,]/rf[M2] with the concentrations of the two monomers in the feed, M, and M2, and the reactivity ratios, rt, r2, of the monomers. The rx and r2 values are measures of the preference of the growing chain ends for like or unlike monomers. 

A number of copolymerizations involving macromonomer(s) have been studied and almost invariably treated according to the terminal model, Mayo-Lewis equation, or its simplified model [39]. The Mayo-Lewis equation relates the instantaneous compositions of the monomer mixture to the copolymer composition ... 

Equation (7.113b) gives the instantaneous copolymer composition in terms of the feed composition and the reactivity ratio. Figure 7.18 shows the copolymer composition for an ideal copolymerization (r,r2 = 1). In this case, the copolymer composition equation becomes ... 

The instantaneous copolymer composition X generally doesn t coincide with the monomer feed composition x from which the copolymer was produced. Such a coincidence X = x can occur only under some special values of monomer feed composition x, called azeotropic . According to definition these values can be calculated in the case of the terminal model (2.8) from a system of non-linear algebraic equations ... 

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. 

When r, r2 values are rather close to unity, one can use for their estimation the so-called approximation method [225, 256-258]. Its idea is based on the fact that if the copolymerization is carried out at low concentrations of one of the monomers, the instantaneous composition of the copolymer depends only on one reactivity ratio. In this case the composition equation in both differential and integrated forms is fairly simple. 

The composition of instantaneously generated chains is not sufficiently described by the copolymerization equation which only informs us about the mean populations of monomers in a statistical copolymer. This problem was studied by Goldfinger and Kane [190] based on the following considerations. 

To solve the kinetics of this four-equation scheme in order to determine the copolymer composition, two assumptions must be made (1) there are only two active sites (Mf and Mf) whose concentrations are at steady state and (2) high polymer is formed which requires that monomer is consumed entirely by propagation. In this case, the instantaneous molar ratio of the two monomer units in the polymer (d[Mi]/d[M2]) is defined by Eq. (23), in which [M ] and [M2] are the concentrations of the two monomers in the polymer feed. 

This equation is often expressed in alternative forms. One such form merely substitutes new variables for the ratio of monomer feed, x - [Mjl/tMj], and the instantaneous copolymer composition, y = d[Mx]/d[M2]. Hence we obtain Equation 6-4. 

It is common for the monomers taking part in copolymerisation reactions to be of different reactivities which leads to a drift in copolymer composition with conversion. The instantaneous copolymer composition can be related to the instantaneous composition of the monomer feed through r5 and r2, the monomer reactivity ratios (10) as shown in Equation (1). 

To overcome the errors brought in by the approximation for instantaneous copolymer composition and the drift in monomer feed ratio, the integration equation for Eq. (5) was used to determine the monomer reactivity ratios, which was given as ° ... 

By simply dividing one of these two equations by the other, it is possible to obtain an expression for the instantaneous composition of the copolymer ... 

Equation (7.18) may be used to calculate the instantaneous composition of copolymer as a function of feed composition for various monomer reactivity ratios. A series of such curves are shown in Fig. 7.1 for ideal copolymerization, i.e., r r2 = 1. The term ideal copolymerization is used to show the analogy between the curves in Fig. 7.1 and Aose for vapor-liquid equilibria in ideal liquid mixtures. The vapor-liquid composition curves of ideal binary mixtures have no inflection points and neither do the polymer-composition curves for random copolymerization in which riV2 = 1. Such monomer systems are therefore called ideal. It does not in any sense imply an ideal type of copolymerization. 

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