Penultimate model copolymers

A similar logic can be applied to copolymers. The story is a bit more complicated to tell, so we only outline the method. If penultimate effects operate, then the probabilities Ph, Pi2> and so on, defined by Eqs. (7.32)-(7.35) should be replaced by conditional probabilities. As a matter of fact, the kind of conditional probabilities needed must be based on the two preceding events. Thus reactions (7.E) and (7.F) are two of the appropriate reactions, and the corresponding probabilities are Pj n and V i2 - Rather than work out all of the possibilities in detail, we summarize the penultimate model as follows ... 

Equation (7.32) shows that pjj is constant for a particular copolymer if the terminal model applies therefore the ratio NmjMi/Nmi also equals this constant. Equation (7.49) shows that Pj u is constant for a particular copolymer if the penultimate model applies therefore the ratio NmiMiMi/NmiMi equals this constant, but the ratio NmjMj/Nmj does not have the same value. 

These observations suggest how the terminal mechanism can be proved to apply to a copolymerization reaction if experiments exist which permit the number of sequences of a particular length to be determined. If this is possible, we should count the number of Mi s (this is given by the copolymer composition) and the number of Mi Mi and Mi Mi Mi sequences. Specified sequences, of any definite composition, of two units are called dyads those of three units, triads those of four units, tetrads those of five units, pentads and so on. Next we examine the ratio NmjMi/Nmi nd NmjMiMi/NmiMi If these are the same, then the mechanism is shown to have terminal control if not, it may be penultimate control. To prove the penultimate model it would also be necessary to count the number of Mi tetrads. If the tetrad/triad ratio were the same as the triad/dyad ratio, the penultimate model is proved. 

For many systems, the copolymer composition appears to be adequately described by the terminal model yet the polymerization kinetics demand application of the penultimate model. These systems where rAAB=rliAR and aha bba hut sAfsB are said to show an implicit penultimate effect. The most famous system of this class is MMA-S copolymerization (Section 7.3.1.2.3). 

It has been argued that for a majority of copolymerizations, composition data can be adequately predicted by the terminal model copolymer composition equation (eqs. 5-9). However, in that composition data are not particularly good for model discrimination, any conclusion regarding the widespread applicability of the implicit penultimate model on this basis is premature. 

It is also possible to process copolymer composition data to obtain reactivity ratios for higher order models (e.g. penultimate model or complex participation, etc.). However, composition data have low power in model discrimination (Sections 7.3.1.2 and 7.3.1.3). There has been much published on the subject of the design of experiments for reactivity ratio determination and model discrimination.49 "8 136 137 Attention must be paid to the information that is required the optimal design for obtaining terminal model reactivity ratios may not be ideal for model discrimination.49... 

The implicit penultimate model was proposed for copolymerizations where the terminal model described the copolymer composition and monomer sequence distribution, but not the propagation rate and rate constant. There is no penultimate effect on the monomer reactivity ratios, which corresponds to... 

Figures 6-12 and 6-13 shows plots of copolymer composition and propagation rate constant, respectively, versus comonomer feed composition for styrene-diethyl fumarate copolymerization at 40°C with AIBN [Ma et al., 2001]. The system follows well the implicit penultimate model. The copolymer composition data follow the terminal model within experimental error, which is less than 2% in this system. The propagation rate constant shows a penultimate effect, and the results conform well to the implicit penultimate model with si = 0.055, S2 — 0.32.

Moreover, a whole set of monomers with bulky and polar substitutors is known, the copolymerization of which cannot, be described by the classic scheme (2.1). In this case, in order to calculate the copolymer composition, molecular structure and composition distribution, one should use a penultimate model or the model of complex formation. 

The calculations of the statistical characteristics of such polymers within the framework of the kinetic models different from the terminal one do not present any difficulties at all. So in the case of the penultimate model, Harwood [193-194] worked out a special computer program for calculating the dependencies of the sequences probabilities on conversion. Within the framework of this model, Eq. (5.2) can be integrated in terms of the elementary functions as it was done earlier [177] in order to calculate copolymer composition distribution in the case of the simplified (r 2 = Fj) penultimate model. In the framework of the latter the possibility of the existence of systems with two azeotropes was proved for the first time and the regions of the reactivity ratios of such systems [6] were determined. In a general version of the penultimate model (2.3-24) the azeotropic compositions x = 1/(1 + 0 ) are determined [6] by the positive roots 0 =0 of the following... 

The kinetic copolymerization models, which are more complex than the terminal one, involve as a rule no less than four kinetic parameters. So one has no hope to estimate their values reliably enough from a single experimental plot of the copolymer composition vs monomer feed composition. However, when in certain systems some of the elementary propagation reactions are forbidden due to the specificity of the corresponding monomers and radicals, the less number of the kinetic parameters is required. For example, when the copolymerization of two monomers, one of which cannot homopolymerize, is known to follow the penultimate model, the copolymer composition is found to be dependent only on two such parameters. It was proposed [26, 271] to use this feature to estimate the reactivity ratios in analogous systems by means of the procedures similar to ones outlined in this section. 

When the above-mentioned independence of r, on composition is not the case, it is quite necessary to study different possibilities of the description of the copolymerization in a given system by means of more complicated models. For instance, to establish the applicability of the penultimate model for the copolymers produced at low conversions, one may use the following relations [275] ... 

Table 6.8 Parameters (2.4) of the penultimate model (2.3) describing copolymerization of styrene M, with acrylonitrile M2 in toluene solution at T = 60 °C. The values of reactivity ratios were obtained [283] from the data on copolymer composition (I) and triad distribution (II)...

The simple copolymer model, with two reactivity ratios for a binary comonomer reaction, explains copolymer composition data for many systems. It appears to be inadequate, however, for prediction of copolymerization rates. (The details of various models that have been advanced for this purpose are omitted here, in view of their limited success.) Copolymerization rates have been rationalized as a function of feed composition by invoking more complicated models in which the reactivity of a macroradical is assumed to depend not Just on the terminal monmomer unit but on the two last monomers in the radical-ended chain. This is the penultimate model, which is mentioned in the next Section. 

We have previously reviewed ( 1, 2) the methods used to calculate structural features of copolymers and terpolymers from monomer reactivity ratios, monomer feed compositions and conversions, and have written a program for calculating structural features of copolymers from either terminal model or penultimate model reactivity ratios (3). This program has been distributed widely and is in general use. A listing of an instructive program for calculating structural features of instantaneous terpolymers from monomer feed compositions and terminal model reactivity ratios was appended to one of our earlier reviews (.1). 

Fig. 93 Copolymer composition curve for radical copolymerization of TFMAA with NB (penultimate model) [305]...

Following a procedure similar to that used in deriving Eq. (7.11), the instantaneous copolymer composition equation for the penultimate model is then given (Hamielec et al., 1989) by... 

If the terminal model adequately explains the copolymer composition, as is often the case, the terminal model is usually assumed to apply. Even where statistical tests show that the penultimate model does not provide a significantly better fit to experimental data than the tenninal model, this should not be construed as evidence that penultimate unit effects are unimportant. It is necessary to test for model discrimination, rather than merely for fit to a given model. In this context, it is important to remember that composition data are of very low power when it comes to model discrimination. For MMA-S copolymerization, even though experimental precision is high, the penultimate model confidence intervals are quite large 0.4[Pg.348]

A flrst important question concerns whether the goal is to discriminate between competing models (i.e., terminal vs penultimate model kinetics) or to seek the best parameter estimates. We flrst assume that terminal model kinetics are being considered and later discuss implications regarding the assumption of penultimate model kinetics. As seen in the previous section, for terminal model kinetics, reactivity ratios are typically estimated using the instantaneous copolymer composition equation or the Mayo-Lewis equation, expressed in two common forms. Equations 6.7 and 6.11. 

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