Topology evolution in macromolecular networks

Governed by various intermolecular forces, molecular networks tend to evolve from simple to very complex formations that have random structure. This randomness in the connectivity of the basic units can still be captured employing distributional description of the state of the system; the evolution itself by particular stochastic processes, for instance Smoluchowski coagulation. The Smoluchowski coagulation equation can be extended to include collisions of orders distinct from 2, which allows developing a framework that covers various special cases being far beyond the scope of the original application of the coagulation equation. The combination of the population balance equation based on a generalization of the Smoluchowski coagulation and a meshless projection method based on Gaussian basis functions has been found to be a powerful tool allowing to solve many problems of prior unmanageable complexity. For instance, in the case of cross-linking polymerisation in bulk, the population balance model describing a particular set of reaction mechanisms has been present in literature for decades before it has been successfully and to a full extent solved by the numerical method. The other case studies address formation of various molecular networks in a few important cases: polymerisation of AB2 monomers, modification of linear chains into a branched polymer, copolymerisation with a composition drift, and coagulation/ coalescence of colloids of variable fractal dimension.