Theory and Mathematical Modeling of Crosslinking

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. 

To address polymer network formation from nonlinear chain-growth polymerization (or copolymerization), kinetic methods are more appropriate [23, 75-83], Some of the most successful kinetic models to address this type of system are based on the method of moments [23, 75-77, 79, 80, 82, 84], Some divergence problems at the vicinity of the gelation point are common with the method of moments, although there are practical ways to avoid this situation [80], A more refined kinetic method to address the issue of modeling the dynamics of gelation in 

It is important to point out that every theory possesses its own advantages and disadvantages the choice of the most suitable method depends on the type of system analyzed, the required information, the level of detail in the reaction scheme, and the computational resources available. Thus, up to now there is not a unified approach that can deal with every crosslinking system, considering all the implementation difficulties and taking into account all nonidealities present in these systems, such as cyclization. A few details about each theory are offered below. 

In general, when a monomer unit is chosen randomly as the root of the tree in an /-functional system, the number of components that can be introduced into generation at conversion a is given by 

The previous observation can also be understood if we look at the expressions for the number and weight-average degrees of polymerization. For this system, these values will be respectively given by Equations 9.14 and 9.15. 

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