Critical droplets in transient and heterogeneous nucleation

This thesis deals with analytical and computational aspects of the kinetics of phase transitions, the means by which a system initiates a first-order phase transition from a metastable phase. Typically, nucleation is studied in the classical regime, where the driving force to nucleation is small. In this case, thermal fluctuations create a bubble with a stable phase interior (the droplet bulk) and a thin interface. The stable phase interior has a lower free energy than the parent metastable phase, but the interface introduces an additional free energy cost. When the free energy cost of the interface is negligible compared to the free energy saved in the bulk interior, the bubble grows, initiating a decay to the stable phase as its boundary grows to encompass the entire system. We review theoretical aspects of nucleation theory and stochastic processes, with a particular focus on Langer's theory of the critical droplet. We propose and investigate an extension to Langer's theory that extends the definition of the critical droplet to regimes where the system cannot be described by a (metastable) equilibrium theory. We refer to this as the transient regime. We investigate the consequences of our proposal for systems that are described by both non-conserved and conserved order parameters. To better understand nucleation in the laboratory where perfect homogeneity cannot be achieved, we examine critical-droplet profiles and nucleation rates in systems with heterogeneities that both enhance and suppress droplet formation. We also examine aspects of heterogeneities in social systems. Our investigation concludes that in both microscopic nucleation processes and macroscopic social systems, heterogeneities play a dominant role in the observed statistics.