Macosko-Miller method

The molecular weight distribution and/or PDI has been described for several cases where the assumption of equal reactivity of functional groups is not valid. Unequal reactivity is easily handled by the Macosko-Miller method. For the A—A + B—B + B B system described in the previous section, we simply redefine the relationship between P and y by... 

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. 

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, (b) the statistical approach of Flory-Stockmayer, and (c) the recursive method of Macosko-Miller ... 

The formation of polymer networks by step-growth polymerization has been modeled using statistical theories, such as the Flory-Stockmayer classical theory [61-64], the Macosko-Miller conditional probability model [65-70], and Gordon s cascade theory [71-74]. However, statistical methods have not been successful for modeling of polymer network formation in chain-growth polymerization systems. 

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. 

Macosko and Miller (1976) and Scranton and Peppas (1990) also developed a recursive statistical theory of network formation whereby polymer structures evolve through the probability of bond formation between monomer units this theory includes substitution effects of adjacent monomer groups. These statistical models have been used successfully in step-growth polymerizations of amine-cured epoxies (Dusek, 1986a) and urethanes (Dusek et al, 1990). This method enables calculation of the molar mass and mechanical properties, but appears to predict heterogeneous and chain-growth polymerization poorly. 

The recursive method of Macosko and Miller [27] has been described earlier for calculating molecular weight averages up to the gel point in nonlinear polymerization. A similar recursive method [34] can also be used beyond the gel point, particularly for calculating weight-fraction solubles (so/) and cross-link density. To illustrate the principles, we consider first the simple homopolymerization, that is, reaction between similar /-functional monomers Ay and then a more common stepwise copolymerization, such as reaction of A/ with B2. 

The physical structure of the polymer is then computed from knowledge of the chemical kinetics, utilizing the statistical methods developed by Macosko and Miller (2A,... 

To derive the molecular weight averages of the polymer, we shall use here the method described by Lopez-Serrano et. al. (1980), which is identical in concept to the recursive method of Macosko and Miller (1976) cited above. Selecting an A group (marked by ) at random, the in direction will be defined as the direction from the chosen A toward the B group of the same mer unit. Out is then the opposite direction from the chosen A, i.e., toward the A side of the mer. The in and out directions associated with B groups will also be defined in the same way. 

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). 

Following the recursive method used above, calculation of P(F ) can now be extended to a general system of Aj s reacting with B/s. For illustration, consider, however, a simple case of A/ reacting with B2, schematically represented (Miller and Macosko, 1976) by... 

As mentioned before, Hory laid out the basic relations for the size distribution of finite macromolecules as a function of the extent of reaction but for cases of practical importance, these distribution functions become quite complex. Macosko and Miller described a simple method for calculating average physical quantities, such as average molar masses, gel point, soluble fraction, and so on. This method is based on an elementary law of conditional probability and on the recursive nature of a step-growth polymerization. 

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