Miller—Macosko theory

Calculating Network Structure Using Miller—Macosko Theory... 

BAUER Calculating Network Structure with the Miller—Macosko Theory 191... 

BAUER Calculating Network Structun with the Miller—Macosko Theory 195... 

BAUER Calculatb Network Structure with the Miller-Macosko Theory 207... 

The theories of Miller and Macosko are used to derive expressions for pre-gel and post-gel properties of a crosslinking mixture when two crosslinking reactions occur. The mixture consists of a polymer and a crosslinker, each with reactive functional groups. Both the polymer and crosslinker can be either collections of oligomeric species or random copolymers with arbitrary ratios of M /Mj. The two independent crosslinking reactions are the condensation of a functional group on the polymer with one on the crosslinker, and the self-condensation of functional groups on the crosslinker. 

Differences in Network Structure. Network formation depends on the kinetics of the various crosslinking reactions and on the number of functional groups on the polymer and crosslinker (32). Polymers and crosslinkers with low functionality are less efficient at building network structure than those with high functionality. Miller and Macosko (32) have derived a network structure theory which has been adapted to calculate "elastically effective" crosslink densities (4-6.8.9). This parameter has been found to correlate well with physical measures of cure < 6.8). There is a range of crosslink densities for which acceptable physical properties are obtained. The range of bake conditions which yield crosslink densities within this range define a cure window (8. 9). 

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. 

More advanced branching models are the so-called recursive theory by Miller and Macosko (Macosko and Miller 1976) and the cascade theory by Gordon (Gordon 1962). Both are able to include nonidealities such as cyclization and long-range substitution effects. All branching theories are mean-field theories and... 

Relations between conversion and molecular parameters In nonlinear radical reactions have been developed by Macosko and Miller (18) using the recursive nature of the branching process and elementary laws of probability. One of the assumptions underlying this theory Is that of no Intramolecular reaction, l.e. no cycllzatlon. As discussed previously, this Is not valid for vlnyl-dlvlnyl co-polymerlzatlon. A revision of this recursive theory to Include the effects of cycllzatlon Is necessary. 

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. 

Macosko, C. W. and Miller, D. R., 1976, Macromolecules, 9,199-206 Naylor, A. W. and Sell, G. R., 1982, Linear Operator Theory In Engineering and Science, second edition. New York Springer-Verlag Nocedal, J. and Wright, S. I, 1999, Numerical Optimization. New York Springer Odian, G., 1991, Principles of Polymerization, third edition. New York Wiley Oran, E. S. and Boris, J. P, 2001, Numerical Simulation of Reactive Flow, second edition. Cambridge Cambridge University Press... 

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