Random graph modeling of polymer networks

Today’s wide application of polymeric materials became possible through exploiting the fundamental connection between the topology of polymer networks and the physical and mechanical properties of the corresponding materials. Due to limited accessibility of topological properties through an experimental technique, understanding of this causal link has been gained primarily through mathematical models. This thesis provides insight into the effects of several polymerization processes on the evolution of the polymer topology by taking the modern viewpoint of complex networks and random graphs.
On the local level of monomeric units, the network is driven by chemical principles, which defines the statistical number of covalently bonded neighbors of a monomer - its degree. By using the directed random graph with arbitrary degree distribution, we identify step-growth polymerization with the bond percolation process on this graph, whereas the chain-growth polymerization is addressed with the theory of the colored random graphs in combination with a kinetic model of monomer degrees.
Further, we investigate the question of the influence of volume growth on the network topology. Molecular dynamics simulations are used to quantify monomer specific kinetics and the influence of network defects, as well as to critically assess the performance of the random graph model.