# The Free-Energy Landscape of Clusters of Attractive Hard Spheres

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Entry: Chia Wei Hsu, AP 225, Fall 2010

Meng, G., Arkus, N., Brenner, M. P., & Manoharan, V. N. (2010). Science, 327, 560-563

## Summary

The study of small clusters can provide a linkage between local geometry and bulk behavior, and also provides insights into nonequilibrium phenomenon such as nucleation and the glass transition. This work examines small clusters (cluster size N ≤ 10) formed by colloidal particles that are essentially "sticky" hard spheres. Experimental measurements of the occurrence probabilities lead to the free energy landscape. For these sticky hard spheres, the lowest free energy states are clusters characterized by lack of symmetry, nonrigid clusters, and clusters with extra bonds.

## Experiment

Using colloidal particles rather than atoms enables direct imaging with optical microscopy. The colloidal particles are polystyrene (PS) spheres with diameter 1 µm, which act at hard spheres. These PS particles are suspended in water and poly(N-isopropylacrylamide (polyNIPAM) nanoparticles. The pressure exerted on the PS particles by the polyNIPAM particles creates a depletion interaction whose range and depth can be tuned with the concentration of PS and polyNIPAM. In this study, the interaction range is 1.05 times the diameter of PS, and energy depth is $U_m=4k_BT$ (fig 1). Because of this very short range of this sticky interaction, the total potential energy U of a given structure is well approximated by $U = CU_m$, where $C$ is the number of contacts or depletion bonds and $U_m$ is the depth of the pair potential.

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These particles are suspended in cylindrical microwells with depth and diameter of 30 µm. The walls of the microwells are functionalized such that particles do not stick to surfaces. This allows clusters to form in the middle of the wells, unaffected by the walls.

After clusters reach equilibrium, optical microscopy is used to scan the wells and identify the geometry of the clusters. This gives the probability $P$ of observing a given cluster, and the free energy $\Delta F=-k_BT\ln P$.

## Results

For N ≤ 5, only one type of cluster is observed (shown in fig 2).

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For N=6, two structures are observed. For N=7, 6 structures. For N=8, 16 structures. The possibilities of observing these structures agree well with previous theoretical study [1], as can be seen in fig 2. These probability distribution is equivalent to the free energy scape (through the relation $\Delta F=-k_BT\ln P$), which completely specifies the equilibrium behavior of these small clusters.

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For N=9 and N=10, there are too many theoretically predicted structures (77 and 393 respectively) than can be categorized experimentally. Instead, the authors

## References

[1] Arkus, N., Manoharan, V. N., & Brenner, M. P. Minimal Energy Clusters of Hard Spheres with Short Range Attractions. Physical Review Letters, 103, 118303-4 (2009).