# The Free Energy Landscape of Hard Sphere Clusters

Entry by Leon Furchtgott, APP 225 Fall 2010.

The Free-Energy Landscape of Clusters of Attractive Hard Spheres. Meng, G., Arkus, N., Brenner, M. P., & Manoharan, V. N. (2010). Science, 327, 560-563

## Summary

The paper is interested in the behavior of small (10 or fewer colloidal particles) clusters and their relation to bulk behavior. In particular, the paper discusses the thermodynamics of small clusters: what structures are favored by entropy or by the potential energy, how this competition changes as N grows larger. Through careful experimentation the authors succeed in measuring structures and free energies of small equilibrium clusters. They compare these structures to theoretical predictions and draw conclusions regarding highly favored configurations.

## Experimental Setup

Small numbers of polystyrene (PS) microspheres were placed in cylindrical microwells filled with polyNIPAM nanoparticles (Fig 1A, 1D). The microwells have depth and diameter 30 <math>\mu</math>m, and they are chemically functionalized so that the particles cannot stick to the surfaces. The microspheres have diameter 1.0 <math>\mu</math>m. The nanoparticles have diameter 80 nm and induce a depletion attraction between 2 microspheres (sticky spheres, see Fig 1B). This depletion attraction is very short-ranged (< 1/10 PS sphere diameter) which means that the interactions are pairwise additive (see Fig 1C). Therefore the total potential energy U of a given structure is well approximated by <math>U = CU_m</math>, where <math>C</math> is the number of contacts or depletion bonds and <math>U_m</math> is the depth of the pair potential.

The authors do this for thousands of clusters which they then image using optical microscopy. For each value of N <math>\leq</math> 10 they determine different cluster configurations and their probabilities <math>P_i</math> and thus the free energies <math> F_i = -k_B T ln P_i </math>.

## Results

The cluster classifications can then be compared to experimental predictions, which are found in a previous theoretical PRL paper (Arkus, Manoharan, Brenner. *Phys. Rev. Lett.* **103**, 118303 (2009)).