Copolymer equation assumptions

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... 

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... 

We have already derived expressions for P and P21 it Eqs. (7-16) and (7-17). These are the respective probabilities that M M] and M2M1 sequences exist in the copolymer. (The assumption implicit here, as in the simple copolymer equations in general, is that the molecular weight of the polymer is fairly large.) The probabilities P22 and P 2 can be derived by the same reasoning, and all four can be expressed in terms of mole fractions /j, in place of the concentrations used to this point ... 

For any specific type of initiation (i.e., radical, cationic, or anionic) the copolymer composition equation is independent of many reaction parameters. Since no rate constants appear as such in the copolymer equation, the copolymer composition is independent of differences in the rates of initiation and termination or of the presence or absence of inhibitors or chain transfer agents. Thus the same copolymer composition is obtained irrespective of whether initiation occurs by the thermal homolysis of initiators (such as AIBN or peroxides), photolysis, radiolysis, or redox systems. Under a wide range of conditions the copolymer composition is also independent of the degree of polymerization. The limitation on the above generalization is that the copolymer be of high molecular weight. It may be recalled that the derivation of Eq. (7.11) involved an assumption that the kinetic chains... 

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. 

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. 

Note that pn + pi2 = P22 + P21 = 1- In writing these expressions we make the assumption that only the terminal unit of the radical influences the addition of the next monomer. This same assumption was made in deriving the copolymer composition equation. We shall have more to say below about this so-called terminal assumption. 

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 ... 

Item (2) requires that each event in the addition process be independent of all others. We have consistently assumed this throughout this chapter, beginning with the copolymer composition equation. Until now we have said nothing about testing this assumption. Consideration of copolymer sequence lengths offers this possibility. 

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. 

The derivation of the terminal (or hrst-order Markov) copolymer composition equation (Eq. 6-12 or 6-15) rests on two important assumptions—one of a kinetic nature and the other of a thermodynamic nature. The Erst is that the reactivity of the propagating species is independent of the identity of the monomer unit, which precedes the terminal unit. The second is the irreversibility of the various propagation reactions. Deviations from the quantitative behavior predicted by the copolymer composition equation under certain reaction conditions have been ascribed to the failure of one or the other of these two assumptions or the presence of a comonomer complex which undergoes propagation. 

On the basis of these equations and the foregoing assumptions, the classical, copolymer-composition equation (I) is derived that relates the ratio of mers in the copolymer to the monomer concentrations in the feed. 

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... 

The problem of predicting copolymer composition and sequence in the case of chain copolymerizations is determined by a set of differential equations that describe the rates at which both monomers, Ma and MB, enter the copolymer chain by attack of the growing active center. This requires a kinetic model of the copolymerization process. The simplest one is based on the assumption that the reactivity of a growing chain depends only on its active terminal unit. Therefore when the two monomers MA and MB are copolymerized, there are four possible propagation reactions (Table 2.17). 

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. 

Again using steady-state assumptions and Equation 10, the copolymer composition is given by (11) ... 

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. 

With the stationary state assumption, they derived and equation for copolymer composition (r, = km/km, r2 = k222Jkm, r = k2n/k2l2, and r2 =... 

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. 

The basic assumption of this approach is that the long and short copolymers segregated at the interface are in equilibrium with the long and short diblocks, respectively, incorporated (at concentration ( >s and ( >L) in the bulk of PS matrix film. This is expressed (see Eq. 64) by equating the chemical potentials of the copolymers at the interface (M-Sbrush an PLbrush) with those in the bulk (psbll k and... 

Assuming that the W/O emulsion behaves as a near hard-sphere dispersion, it is possible to apply the Dougherty-Krieger equation [7, 8] to obtain the effective volume fraction, 4>. The assumption that the W/O emulsion behaves as a near hard sphere dispersion is reasonable as the water droplets are stabihsed with a block copolymer with relatively short PHS chains (of the order of lOnm and less). Any lateral displacement of the polymer will be opposed by the high Gibbs elasticity of the adsorbed polymer layer, and the droplets will maintain their spherical shape up to high volume fractions. 

Problem 7.1 The above derivation of the copolymer composition equation [Eq. (7.11)] involves the steady-state assumption for each type of propagating species. Show that the same equation can also be derived from elementary probability theory without invoking steady-state conditions [6-8]. 

A serious drawback in the use of a differential form of the copolymerization equation [Eq. (7.11) or (7.18)] is the assumption that the feed composition does not change during the experiment, which is obviously not true. One carries out the polymerization to as low a conversion as possible, but there are limitations since one must be able to isolate a suflicient sample of the copolymer for direct analysis, or, if copolymer analysis is done... 

In the derivation of copolymer composition equation, Eq. (7.11), we considered only the rates of the four possible propagation steps in a binary system. However, the overall rate of copolymerization depends also on the rates of Initiation and termination. In deriving an expression for the rate of copolymerization in binary systems the following assumptions will be made [25] (a) rate constants for the reaction of a growing chain depend only... 

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