(214m) Coarse-Grained Modeling of the Phase Behavior of Thermodynamically Small Particle Assemblies

The phase behavior of systems which are termed thermodynamically small has been the subject of intensive theoretical study over the past two decades.  These finite systems consist of a very small number of interacting particles, ~10 to ~100, and, as such, they are far removed from the infinite limit in the sense of traditional macroscopic thermodynamics.  In such small systems, the nature of thermodynamically stable phases and the corresponding phase transitions are of particular interest.  Developing a fundamental understanding of the phase behavior of these systems has direct application for the self- and directed-assembly of colloidal particles into structures within materials and devices as an emerging paradigm with wide-ranging technological impact.  In addition, these finite-size particle clusters may serve as model systems for the types of phase behavior expected and observed in other complex systems, such as proteins, glasses, and gels.

In this poster presentation, we report results of a systematic investigation of phase behavior in two thermodynamically small systems, namely, finite assemblies of colloidal particles interacting via a hard-core and a depletion-attraction potential and the 38-member Lennard-Jones (LJ₃₈) particle cluster.  In the colloidal system, we have studied the order-to-disorder phase behavior over a broad range of parameter space with parameters that include the system size, expressed by the number of particles N in the colloidal assembly, and the inter-particle interaction strength, which is controlled by the depletant osmotic pressure Π/kT.  In the LJ₃₈ cluster, we have focused on a single parameter, the system temperature kT/ε, and studied its effects on polymorphic solid-solid and melting solid-fluid phase transitions.

To provide a description of the phase behavior of these systems, we have constructed free-energy landscapes (FELs) based on Monte Carlo umbrella sampling (MC-US).  A coarse-grained representation of the thermodynamically small system is required for the implementation of MC-US and the construction of meaningful FELs.  This coarse-grained model allows for reducing the description of the system from a 3N-dimensional (3N-D) representation into a much coarser description with a much lower dimensionality.  In order to perform this dimensionality reduction rigorously and systematically, we have applied the diffusion mapping approach to both systems of interest, with data sets provided by Brownian- or molecular-dynamics simulations for the colloidal and LJ₃₈ clusters, respectively.  The effects of varying the system parameters on the FELs have been examined systematically.

In the colloidal particle assemblies, we find that only a single fluid-like phase is stable for very small clusters.  However, as the cluster size increases, a second ordered phase emerges in coexistence with the fluid-like phase; this second stable phase at increased cluster size is a crystalline phase.  The onset of stability of this crystalline phase corresponds to the size of a critical crystalline nucleus and marks the onset of crystallization in colloidal particle assemblies.  In the LJ₃₈ cluster, we find that, at very low temperatures, the system only samples its minimum-energy, octahedral configuration.  However, as the temperature increases, the system undergoes a polymorphic transition between two different solid phases followed by an order-to-disorder, solid-to-fluid melting-like transition at even higher temperatures.