Implicit penultimate model

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. 

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). 

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 original mathematical treatment of the penultimate effect was presented by Merz and coworkers [Barb, 1953 Ham, 1964 Merz et al., 1946], Fukuda and coworkers developed a more extensive treatment, which distinguished between two penultimate models of copolymerization behavior—the explicit penultimate and implicit penultimate models [Coote and Davis, 1999, 2000 Davis, 2001 Fukuda et al., 1985, 1987, 1992, 2002 Ma et al., 2001], The explicit penultimate model for copolymerization involves the use of eight propagation reactions... 

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... 

The reader is cautioned that literature references prior to 1985-1990 did not distinguish between the explicit and implicit penultimate models. The prior penultimate model did not correspond to either the explicit or implicit penultimate models. The pre-1985-1990 penultimate model contained only the four monomer reactivity ratios (Eq. 6-74) with no radical... 

The precision and accuracy of the experimental data must be sufficient to allow one to discriminate between the terminal, explicit penultimate, and implicit penultimate models, [Burke et al., 1994a,b, 1995 Landry et al., 2000]. This has not always been the case, especially in the older literature, and the result has sometimes been contradictory reports. Penultimate effects are most easily detected in experiments carried out by including data at very low or very high f values. 

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.

Fig. 6-13 Plot of propagation rate constant versus f for copolymerization of styrene (M0 and diethyl fumarate (M2). The dotted line represents the terminal model with rj = 0.22, r2 = 0.021 (i.e., Si = S2 = 1). The solid line represents the implicit penultimate model with si = 0.055, s2 = 0-32. After Ma et al. [2001] (by permission of American Chemical Society, Washington, DC) an original plot, from which this figure was drawn, was kindly supplied by Dr. T. Fukuda.

There are two cases to consider when predicting flie effect of solvent polarity on copolymerization propagation kinetics (1) the solvent polarity is dominated by an added solvent and polarity is thus independent of the comonomer feed ratio, or (2) the solvent polarity does depend on the comonomer feed ratio, as it would in a bulk copolymerization. In the first case, the effect on copolymerization kinetics is simple. The monomer reactivity ratios (and additional reactivity ratios, depending on which copolymerization model is appropriate for that system) would vary fi om solvent to solvent, but, for a given copolymerization system they would be constant as a function of the monomer feed ratios. Assuming of course that there were no additional types of solvent effect present, fliese copolymerization systems could be described by their appropriate base model (such as the terminal model or the explicit or implicit penultimate models), depending on the chemical structure of the monomers. 

Examining the composition and kp equations above, it is seen fliat the Bootstrap effect K is always aliased with one of the monomer feed ratios (that is, both equations may be expressed in terms of Kfj and f2). It is also seen that once Kf, is taken as a single variable, the composition equation has the same functional form as the terminal model composition equation, but the kp equation does not. Hence it may seen that, for this version of the Bootstrap effect, the effect is an implicit effect - causing deviation from the terminal model kp equation only. It may also be noted that, if K is allowed to vary as a function of the monomer feed ratios, the composition equation also will deviate from terminal model behavior - and an explicit effect will result. Hence it may be seen that it is possible to formulate an implicit Bootstrap model (that mimics the implicit penultimate model) but in order to do fliis, it must be assumed that the Bootstrap effect K is constant as a function of monomer feed ratios. 

The implicit penultimate model was first suggested by Fukuda and coworkers (7) in 1985, in order to describe their observation that the terminal model could be fitted to the composition data for the copolymerization of styrene with methyl methacrylate, though it could not simultaneously describe the propagation rate coefficients. In this model, the following restriction is placed on the explicit penultimate model. 

However, recent experimental and theoretical work has shown that the assumptions of the implicit penultimate model are unlikely to be applicable to the majority of copol5mierization systems. We have recently published a review of this evidence (37,42), which draws on direct experimental and theoretical measures of reactivity ratios, model testing in a range of copolymerization systems, and other tests of the mechanism of the propagation step via, for example, the examination of solvent effects on reactivity ratios. These studies provide strong evidence for penultimate imit effects but, in all cases where penultimate unit effects have been measured directly, effects on radical selectivity have been shown to be significant. In other words all available evidence contradicts the assumption of the implicit penultimate model that the penultimate unit affects reactivity but not selectivity. 

To summarize, we know firstly from simple model-testing studies spanning the last 20 years that for almost all systems tested, the terminal model can be fitted to (kp) or composition data for a copolymerization system, but not both simultaneously. Secondly, more recent experimental and theoretical studies have demonstrated that the assiunption of the implicit penultimate model— that the penultimate imit affects radical reactivity but not selectivity—cannot be justified. Therefore, on the basis of existing evidence, the explicit penultimate model should replace the terminal model as the basis of free-radical copolymerization propagation kinetics, and hence the failure of the terminal model kp) equation must be taken as a failure of the terminal model and hence of the terminal model composition equation. This means that the terminal model composition equation is not physically valid for the majority of systems to which it has been apphed. 

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