(635b) Constructing Free Energy Landscapes in Thermodynamically Small Colloidal Systems Using a Fokker-Planck Formalism

The self- and directed- assembly of finite sets of colloidal particles into crystalline structures within materials and devices is an emerging paradigm with wide-ranging technological impact. However, the ability to create a target structure with an acceptably small level of defects is still lacking; systems too easily become dynamically arrested in undesired disordered, or defect-rich, states. We have been exploring a synergistic combination of recent advances in digital microscopic imaging techniques with free energy calculation methods to create free energy landscapes (FELs) that could be used in monitoring and controlling self-assembly. Following recent computational work by Y.G. Kevrekidis and co-workers, we analyze short-time trajectories of an assembling colloidal system (as might be obtained by a microscopy experiment) and extract the coefficients of a Fokker-Planck equation that describes the underlying probability distribution function and thus the FEL. As a prototype problem, we study a system with a small (n ~ 32) number of Brownian colloidal particles assembling into condensed structures under the influence of a tunable depletion attraction potential. We generate FELs in a small set of order parameters, such as collective radius of gyration and bond order parameter. In addition to serving as maps for design and control of directed assembly processes, the FELs show surprisingly rich features, including signatures of phase transitions characteristic of small systems.