Explicit penultimate model

The general features of the penultimate model in what have become known as the explicit and implicit forms are described in Section 7.3.1.2.1. Evidence for remote unit effects coming from small molecule radical chemistry and experiments other than copolymerization 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 7.3.1.2.5, 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."... 

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... 

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... 

Copolymerization models based upon a Bootstrap effect were first proposed by Harwood and Semchikov" (see references cited therein). Harwood suggested that the terminal model could be extended by the incorporation of an additional equilibrium constant relating the effective and bulk monomer feed ratios. Different versions of this so-called Bootstrap model may be derived depending upon the baseline model assumed (such as the terminal model or the implicit or explicit penultimate models) and the form of equilibrium expression used to represent the Bootstrap effect. In the simplest case, it is assumed that the magni-... 

V ariants of this model may be derived by assuming an alternative basis model (such as the implicit or explicit penultimate models) or by making further assumptions as to nature of the complexation reaction or the behavior of the complexed radical. For instance, in the special case that the complexed radicals do not propagate (that is, Sd = 0 for all i), the reactivity ratios are not affected (that is, r = rj for all i) and the complex formation serves only removal of radicals (and monomer, if monomer is the complexing agent) from the reaction, resulting in a solvent effect that is analogous to a Bootstrap effect (see Section 13.2.3.4). 13.2.3.2.3 Experimentai evidence... 

The Bootstrap model may also be extended by assuming an alternative model (such as the explicit penultimate model) as the baseline model, and also by allowing the Bootstrap effect to vary as a function of monomer feed ratios. Closed expressions for composition and sequence distribution under some of these extended Bootstrap models may be found in papers by Klumperman and co-workers. ... 

In the explicit penultimate model, it is assumed that both the terminal and penultimate units of a polymer radical may affect the rate of the propagation reaction. As in the terminal model, side reactions are considered to be insignificant. The explicit penultimate model was first suggested in 1946 by Merz and co-workers (10), who derived equations for predicting the composition and sequence distribution under this model. A full description of the model—including an expression for ( p)— has since been provided by Fukuda and co-workers (11), and it is their notation that is used below. In addition, penultimate model equations for the case of terpolymerization have been published by Coote and Davis (12). 

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. 

Thus the adjusted monomer reactivity ratios of the penultimate model are replaced simply by their corresponding terminal model values. However, since the penultimate imit effect can remain in the radical reactivity ratios (ie, through values of Si 1), equation 4 does not collapse to its equivalent terminal model form (ie, kii kiii). Since the composition and triad/pentad fraction equations contain only n terms, they collapse to the corresponding terminal model equations. However, since it contains both ft and ku terms, the propagation rate equation, though simpler than that of the explicit penultimate model, continues to differ from that of the terminal model. There is thus an implicit penultimate unit effect— that is, a penultimate unit effect on the propagation rate but not the composition or sequence distribution. 

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. 

When solvent effects on the propagation step occnr in free-radical copolymerization reactions, they result not only in deviations from the expected overall propagation rate, but also in deviations from the ejqiected copolymer composition and microstracture. This may be trae even in bulk copolymerization, if either of the monomers exerts a direct effect or if strong cosolvency behavior causes preferential solvation. A number of models have been proposed to describe the effect of solvents on the composition, microstmcture and propagation rate of copolymerization. In deriving each of these models, an appropriate base model for copolymerization kinetics is selected (such as the terminal model or the implicit or explicit penultimate models), and a mechanism by which the solvent influences the propagation step is assumed. The main mechanisms by which the solvent (which may be one or both of the comonomers) can affect the propagation kinetics of free-radical copolymerization reactions are as follows ... 

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