| The phase behavior of systems that are termed thermodynamically small has been the subject of intensive theoretical study over the past two decades. These finite systems consist of order less than 100 particles; as such, they are far removed from the infinite limit of traditional macroscopic thermodynamics. Developing a fundamental understanding of the phase behavior of these systems has direct application for the self- and directed-assembly of structures within materials and devices having wide-ranging technological impact.;In this thesis, we report results of a systematic investigation of phase behavior in two thermodynamically small systems: finite assemblies of colloidal particles interacting via a depletion-attraction potential and the 38-atom Lennard-Jones (LJ38) cluster. In the colloidal system, we have studied the order-to-disorder phase behavior over a range of both system size, expressed by the number of particles N in the assembly, and the inter-particle interaction strength, controlled by the depletant osmotic pressure pi/kT. In the LJ38 cluster, we have focused on the system temperature kT/epsilon--, and studied its effects on polymorphic solid-solid and solid-fluid phase transitions.;To provide a description of the phase behavior of these systems we construct a dynamically relevant coarse grained model. In order to define the coarse grained model we have applied the diffusion mapping approach to both systems of interest. In this coarse variable space we apply kinetic, Smoluchowski equation based, and equilibrium, free-energy landscape based, analyses to describe the phase behavior of these systems. In the colloidal particle assemblies, we find that only a single fluid-like phase is stable for very small clusters, and weak attractive strength. As the cluster size or attractive strength increases, a second ordered phase emerges in coexistence with the fluid-like phase. The onset of stability of this crystalline phase marks the onset of crystallization in colloidal particle assemblies. In the LJ 38 cluster, we find that, at very low temperatures, the system only samples its minimum-energy configuration. As the temperature increases, the system undergoes a polymorphic transition between two different solid phases followed by an order-to-disorder melting-like transition at higher temperatures. We also observe a broken cluster phase which plays an important role in the fluid-solid phase behavior. |
