Penultimate model

Since in this model a growing chain is identified by the last two units at the growing chain end, a binary copolymerization system will involve eight propagating reactions which can be represented (Hamielec et al, 1989 Odian, 1991) by 

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 

There are a number of systems where penultimate effects are known to occur (e.g., in the free radical polymerization of methyl 

Vtnylidene Chloride (A) Methyl Acrylate (B) 1.00 ylMyd. 0.50Af 0.5QAf  

Just as in the derivation of the copolymer equation for the terminal model, we start with a reversibility relationship P3 AAB = P3 BAA. Now we must use second-order Markovian statistics to write this in terms of conditional probabilities (Equation 6-64)  

Four independent conditional probabilities can now be written using these four reactivity ratios, in the same way as two conditional probabilities could be written for the terminal model in terms of its two reactivity ratios (Equations 6-62)  

The other four conditional probabilities can simply be determined by subtracting 

The general features of the penultimate model in what have become known as the explicit and implicit fonns are described in Section 7.3.1.2.1. Evidence for remote unit effects coming from small molecule radical chemistry and experiments other than copolymerizalion is discussed in Section 7.3.1.2.2. In Sections 7.3.1.2.3 and 7.3.1,2.4 specific copolymerizations are discussed. Finally, in Section 

we consider the origin of the penultimate unit effects. A general recommendation is that when trying to decide on the mechanism of a copolymerization, first consider the explicit penultimate model.  

From this scheme it can be seen that the copolymer composition is determined by the values of four monomer reactivity ratios. 

Fukuda cf air were the first to recognize that a Further two radical reactivity ratios were required to completely define the polymerization kinetics. 

The rcaclivily ratios taab. bab. bba and taba arc sometimes abbreviated to /-aa, ba, bb ajid Tar or to r., ta, r, fn respectively. The notation used (fAAR, bab, / rba and / aba) is preferred since il allows discussion of situations involving more llian two monomers. 

---

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. 

The early kinetic models for copolymerization, Mayo s terminal mechanism (41) and Alfrey s penultimate model (42), did not adequately predict the behavior of SAN systems. Copolymerizations in DMF and toluene indicated that both penultimate and antepenultimate effects had to be considered (43,44). The resulting reactivity model is somewhat compHcated, since there are eight reactivity ratios to consider. 

The first quantitative model, which appeared in 1971, also accounted for possible charge-transfer complex formation (45). Deviation from the terminal model for bulk polymerization was shown to be due to antepenultimate effects (46). Mote recent work with numerical computation and C-nmr spectroscopy data on SAN sequence distributions indicates that the penultimate model is the most appropriate for bulk SAN copolymerization (47,48). A kinetic model for azeotropic SAN copolymerization in toluene has been developed that successfully predicts conversion, rate, and average molecular weight for conversions up to 50% (49). 

Cases have been reported where the application of the penultimate model provides a significantly better fit to experimental composition or monomer sequence distribution data. In these copolymerizations raab "bab and/or C BA rBBA- These include many copolymerizations of AN, 4 26 B,"7 MAH28" 5 and VC.30 In these cases, there is no doubt that the penultimate model (or some scheme other than the terminal model) is required. These systems arc said to show an explicit penultimate effect. In binary copolynierizations where the explicit penultimate model applies there may be between zero and three azeotropic compositions depending on the values of the reactivity ratios.31... 

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). 

Penpenultiraale and higher order remote unit effect models may also affect the outcome of copolymerizations. However, in most eases, experimental data, that are not sufficiently powerful to test the penultimate model, offer little hope of testing higher order models. The importance of remote unit effects on copolymerization will only be fully resolved when more powerful analytical techniques become available. 

Other experimental data seem to provide support for an implicit penultimate model. Thus, simple (monomeric) model radicals for the propagating radical chain... 

The Chemistry of Radical Polymerization Table 7.5. Implicit Penultimate Model Reactivity Ratios... 

The values of sA and. ru are not well defined by kinetic data.59 61 The wide variation in. vA and for MMA-S copolymerization shown in Table 7.5 reflects the large uncertainties associated with these values, rather than differences in the rate data for the various experiments. Partly in response to this, various simplifications to the implicit penultimate model have been used (e.g. rA3rBA= W- and -Va=- h)- These problems also prevent trends in the values with monomer structure from being established. 

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. 

Mechanisms for copolymerization involving complexes between the monomers were first proposed to explain the high degree of alternation observed in some copolymerizations. They have also been put forward, usually as alternatives to the penultimate model, to explain anomalous (not consistent with the terminal model) composition data in certain copolymerizations.65"74... 

However, these observations are not proof of the role of a donor-acceptor complex in the copolymcrization mechanism. Even with the availability of sequence information it is often not possible to discriminate between the complex model, the penultimate model (Section 7.3.1.2) and other, higher order, models.28 A further problem in analyzing the kinetics of these copolyincrizations is that many donor-acceptor systems also give spontaneous initiation (Section 3.3.6.3). 

With lhe penultimate model, the probability that a chain with a terminal MBA... 

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... 

Similarly, penultimate model reactivity ratios can be estimated from initial monomer feed composition and triad concentrations using eqs. 50-53. 

More recent work has shown that the observed variation in propagation rate constants with composition is not sufficient to define the polymerization rates.5" 161,1152 There remains some dependence of the termination rate constant on the composition of the propagating chain. Thus, the chemical control (Section 7.4.1) and the various diffusion control models (Section 7.4.2) have seen new life and have been adapted by substituting the terminal model propagation rate constants (ApXv) with implicit penultimate model propagation rate constants (kpKY -Section 7.3.1.2.2). 

However, recent work based on the assumption of the implicit penultimate model suggests a value of 0 for S-MMA copolymeri/ation to be in the range 2-3.3"161 T his value is in remarkably good agreement with that suggested by experiments with simple model radicals. These experiments also indicate that cross termination is 2-3 times faster than either homotermination reaction (Section 7.4.3.1). 

More complex models for diffusion-controlled termination in copolymerization have appeared.1 tM7j Russo and Munari171 still assumed a terminal model for propagation but introduced a penultimate model to describe termination. There are ten termination reactions to consider (Scheme 7.1 1). The model was based on the hypothesis that the type of penultimate unit defined the segmental motion of the chain ends and their rate of diffusion. 

Fukuda and coworkers u 2 have recently derived a model equivalent to the Russo-Munari model but where the implicit penultimate model is used to describe the propagation kinetics. 

The important aspect of this problem is that while the penultimate model involves a four dimensional parameter space, the model discrimination problem can be reduced to a two dimensional space by dealing with functions of the original parameters. This approach requires that probabilities for array locations in the four dimensional r, r/, rj, rj ) space be mapped to array locations in the ((ri-r/), (rj-rj )) space. 

Figure 5. Joint 95% posterior probability region for penultimate model. Shimmer bands shown at 95% probability., Hill et al point estimate , point estimate by Monte Carlo.

---