Theories for network formation in multistage processes

Two different theoretical approaches have been developed to describe network formation in a six-component, three-stage process with the stochastic theory of branching processes. Both approaches make use of probability-generating functions and cascade substitutions. In the POLYM approach, the products of the previous stage, namely, complete (branched) molecules, are used as building units in the subsequent stage. The molar masses and unreacted functionalities are identified by means of dummy variables. In the resulting formulas the sequence of the three stages is clearly visible. In the MONOM approach, the system is treated as a quasi-one-stage process; the original monomers are the building units in all three stages, the sequence of which is only apparent from the input parameters. In the MONOM approach the original monomers are used as building units, irrespective of the stage in which they react. A separate dummy variable is chosen for each possible reaction in each stage in the MONOM approach, so that the two approaches give analytically identical results, but along completely different derivations. This avoids correlation errors in the MONOM approach. Both approaches have incorporated kinetic effects by way of substitution effects in two of the monomers. The two approaches are prefaced by the treatment of several paradigms of increasing complexity to show the unity and strength of the theory and to warn against typical pitfalls.