Miller-Macosko approach

The statistical approach has been applied to systems containing reactants with functional groups of unequal reactivity [Case, 1957 Macosko and Miller, 1976 Miller and Macosko, 1978 Miller et al., 1979]. In this section we will consider some of the results for such systems. Figure 2-15 shows a plot of Mw vs. extent of reaction for the various values of s at r = 1 for the system... 

For example, in recent years Macosko and Miller (MM)37-40 have developed an attractively simple method which at first sight appears to be basically new. However, a closer inspection reveals the MM approach as being a degenerate case of the more general cascade theory. The simplicity is unfortunately gained at the expense of generality, and up-to-date conformation properties are not derivable by the MM-technique. 

Several approaches can be used to obtain statistical parameters of these general systems under ideal conditions (equal reactivities, and absence of substitution effects and intramolecular cycles). In this section we will discuss some of these results. The reader is referred to the papers of Macosko and Miller (1976), Miller and Macosko (1976), and Miller et al. (1979) for the deductions of the equations used in this section. 

Can the system be reacted to complete conversion without gelation If not, what is the extent of conversion of the acid functionality at the gel point calculated from (a) the Carothers equation and from the statistical approaches of (b) Flory-Stockmayer and (c) Macosko-Miller ... 

In addition to being a simpler method for obtaining the average properties such as Mw and M , compared to the Flory and similar approaches (Case, 1958), the recursive approach also more easily allows an evaluation of the effect of unequal reactivity and unequal structural unit molecular weights on the average properties (Macosko and Miller, 1976 Lopez-Serrano et al., 1980 Ziegel et al., 1972). 

UnUke Flory s combinatorial approach, a Markovian analysis such as tot proposed by Macosko and Miller leads to easy derivations of expressions for M and for nonlinear polymers. Before generalizing to a reaction implying multivalent molecules having valence v, the case of a step-growth polymerization involving X4 tetravalent molecules and Ny Y- -Y molecules wiU be considered. 

We now extend this statistical approach to the case of multifunctional monomers in which either a or P2 exceeds 2 so that above a critical gel point conversion, the system forms a cross-linked network that extends throughout space (Figure 7.3). To do so, we use the Macosko-Miller method (Macosko Miller, 1976), in which we compute at a given conversion the average mass IT of a chain that is attached to a randomly-selected monomer unit. The gel point is the conversion at which W diverges to infinity. 

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