Bootstrap model

When So was ht (F) to ) , the model that resulted would be noted as M so that when So was applied to the model that resulted was noted as (M = F(So, Dll)). Thus, there were 505 bootstrap models (Mi to M505) that were ht, one for each bootstrap data set. For each of the bootstrap data sets it is necessary to ht the structural model to demonstrate it rehects the underlying structure of the data, which is a basic assumption of the bootstrap. Any bootstrap data set that does not support the structural model should be discarded. 

Figure 7.12 Histogram of bootstrap clearance (top) and between-subject variance estimate for clearance (bottom) for 4-compartment model fit to data in Table 7.4. Bootstrap models fit using first-order approximation. Ninety percent confidence intervals using the percentile method are indicated.

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-... 

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 Bootstrap model may also be extended by assuming an alternative model (such as the explieit penultimate model) as the baseline model, and also by allowing the Bootstrap effeet to vary as a funetion 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 eo - workers. 

Prior to Harwood s work, the existence of a Bootstrap effect in copolymerization was considered but rejected after the failure of efforts to correlate polymer-solvent interaction parameters with observed solvent effects. Kamachi, for instance, estimated the interaction between polymer and solvent by calculating the difference between their solubility parameters. He found that while there was some correlation between polymer-solvent interaction parameters and observed solvent effects for methyl methacrylate, for vinyl acetate there was none. However, it should be noted that evidence for radical-solvent complexes in vinyl acetate systems is fairly strong (see Section 3), so a rejection of a generalized Bootstrap model on the basis of evidence from vinyl acetate polymerization is perhaps unwise. Kratochvil et al." investigated the possible influence of preferential solvation in copolymerizations and concluded that, for systems with weak non-specific interactions, such as STY-MMA, the effect of preferential solvation on kinetics was probably comparable to the experimental error in determining the rate of polymerization ( 5%). Later, Maxwell et al." also concluded that the origin of the Bootstrap effect was not likely to be bulk monomer-polymer thermodynamics since, for a variety of monomers, Flory-Huggins theory predicts that the monomer ratios in the monomer-polymer phase would be equal to that in the bulk phase. 

Based upon the above studies, it may be concluded that there is strong evidence to suggest that Bootstrap effects arising from preferential solvation of the polymer chain operate in many copolymerization systems, although the effect is by no means general and is not likely to be significant in systems such as STY-MMA. However, this does not necessarily discount a Bootstrap effect in such systems. As noted above, a Bootstrap effect may arise from a number of different phenomena, of which preferential solvation is but one example. Other causes of a Bootstrap effect include preferential solvation of die chain end, rather than the entire polymer chain, or the formation of non-reactive radical-solvent or monomer-solvent complexes. In fact, the Bootstrap model has been successfully adopted in systems, such as solution copolymerization of STY-MMA, for which bulk preferential solvation of the polymer chain is unlikely. For instance, both Davis and Klumperman and O Driscoll adopted die terminal Bootstrap model in a reanalysis of die microstructure data of San Roman et al. for the effects of benzene, chlorobenzene and benzonitrile on the copolymerization of MMA-STY. 

Versions of the Bootstrap model have also been fitted to systems in which monomer-monomer complexes are known to be present, demonstrating that die Bootstrap model may provide an alternative to the MCP and MCD models in these systems. For instance, Klumperman and co-workers have successfully fitted versions of the penultimate Bootstrap model to the systems styrene with maleic anhydride in butanone and toluene, and styrene... 

It was reported by Barb in 1953 that solvents can affect the rates of copolymerization and the composition of the copolymer in copolymerizations of styrene with maleic anhydride [145]. Later, Klumperman also observed similar solvent effects [145]. This was reviewed by Coote and coworkers [145]. A number of complexation models were proposed to describe copolymerizations of styrene and maleic anhydride and styrene with acrylonitrile. There were explanations offered for deviation from the terminal model that assumes that radical reactivity only depends on the terminal unit of the growing chain. Thus, Harwood proposed the bootstrap model based upon the study of styrene copolymerized with MAA, acrylic acid, and acrylamide [146]. It was hypothesized that solvent does not modify the inherent reactivity of the growing radical, but affects the monomer partitioning such that the concentrations of the two monomers at the reactive site (and thus their ratio) differ from that in bulk. 

Explanations, other than the inadequacy of the terminal model, have been given to explain potential causes for deviation. Pichot et al. [30] offered several possible explanations for these discrepancies including 1) preferential solvation of one of the monomers in the polymer 2) AN existing as a dimer due to dipole-dipole interactions and 3) terminal radical interaction with the AN nitrile group. Harwood [64] presents evidence indicating that it is the monomer concentrations local to the active radical center that controls the copolymerization and backbone monomer sequence distribution rather than the average monomer concentrations in the reactor. Harwood calls this the bootstrap model because it is the nature of the polymer chain itself that controls the local monomer concentration near its active chain-end. 

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