(178m) Toward a Systematic Coarse Grained Model of Phase Behavior in Assemblies of Colloidal Particles

The self- and directed-assembly of finite sets of colloidal (nano- to micro- scale) particles into structures within materials and devices is an emerging paradigm with wide-ranging technological impact. These finite systems consist of a very small number of interacting particles, ~10 to ~100, and, as such, these systems cannot be considered infinite in the sense of traditional macroscopic thermodynamics. Thermodynamically small systems have been the subject of intensive theoretical study over the past two decades. In such small systems, the nature of their phases and the corresponding phase transitions are of particular interest. In this poster presentation, we report results of a systematic investigation of the phase behavior of finite colloidal systems, consisting of particles that interact via a hard-core and a depletion-attraction potential. We have studied the order-to-disorder phase behavior over a broad range of parameter space with parameters that include the system size of the colloidal assembly, expressed by the number of particles N in the assembly, and the inter-particle interaction strength, which is controlled by the depletant osmotic pressure Π/kT. To provide a description of the phase behavior of the system at a single point in parameter space, we have constructed free-energy landscapes (FELs) based on Monte Carlo umbrella sampling (MC-US) according to experimentally derived interparticle interaction potentials.

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 systems of colloidal particle clusters with data sets provided by Brownian-dynamics simulations. We have found that the dynamics of phase behavior of these systems can be described satisfactorily by a reduced dimensionality of two, which enables a 2-D coarse description of the phase behavior of such colloidal clusters. The two coarse variables that we have chosen for our analysis of phase behavior are the radius of gyration of the cluster, R_g^*, and the order parameter <C₆>, which expresses the average number of hexagonally close packed particles around each particle in the cluster; these coarse variables provide good metrics of the density and morphology of the cluster (R_g^*), as well as the cluster order or degree of crystallinity (<C₆>).

We have used this coarse grained model to create FELs over a broad range of parameter space. These computed FELs provide a comprehensive description of the phase behavior of colloidal clusters and allow us to describe the effects of the interparticle interaction strength and the cluster size on the phase behavior as the system approaches the bulk thermodynamic limit. In very small clusters, only a single liquid-like phase is found to be stable. However, as the cluster size increases, a second ordered phase emerges in coexistence with the liquid-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.