Aggregation and Gelation in Random Networks

Book excerpt: In this book the formation, elasticity, and structure of randomly cross-linked networks is investigated analytically and with the help of computer simulations. In the first two chapters, two polymer models are studied by means of equilibrium statistical mechanics and replica theory an isotropic model of randomly cross-linked particles and an anisotropic model of cross-linked directed polymers. For both models, a suitable order parameter describing the gelation transition is developed at first. With it, the elastic properties of the isotropic network is determined for arbitrary cross-link concentration and the extent of the chain fluctuations and the role of the preferred direction are studied for the anisotropic system. As a real life example, the structure of spider silk is analyzed, a material with very high toughness despite its low density. Here nano-crystallites are randomly connected by an amorphous network of chains. A model for the embedding of the crystallites in the amorphous matrix is developed, accounting for the crystallites' structure, arrangement, and their random orientation relative to the fiber axis. With this model, the scattering function S(q) of the crystallites can be calculated and compared to experimental scattering intensities enabling the determination of the geometric and statistical parameters. The rather general model for nano-crystalline materials can also be used to determine the importance of disorder and coherent scattering between different crystallites, which is usually neglected for wide angle X-ray scattering.Random networks of capillary bridges can occur in wet sand. By adding a small amount of a liquid to a granular system, the particles can create liquid capillary bridges between each other. When such a bridge ruptures, a fixed amount of energy is dissipated different from the ordinary granular interaction where a fraction of the kinetic energy is dissipated in a collision. A freely cooling system is shown to undergo a nonequilibrium dynamic phase transition from a state with mainly single particles and fast cooling to a state with growing aggregates, where bridge rupture becomes a rare event and cooling is slow. Initially, the aggregation of particles into clusters is a self-similar growth process, where fractal objects are generated and with a cluster size distribution that can be described by a scaling function. At later times a percolating cluster is found, which gradually absorbs all other particles.