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


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 and theoretical prediction of the occurrence probabilities yield the free energy landscape for these small clusters. For these sticky hard spheres, the lowest free energy states are clusters characterized by lack of symmetry, nonrigid clusters, and clusters with extra bonds.


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 as 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 (fig 1). In this study, the interaction range is 1.05 times the diameter of PS, and energy depth is <math> U_m=4k_BT</math>. Because of this very short range of this sticky interaction, 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.

Fig. 1. (A) Diagram of experimental system. (B) Illustration of the depletion attraction. (C) Estimated pair potential.

These particles are suspended in cylindrical microwells with depth and diameter of 30 µm. There are 10 PS particles in each well on average. 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. By observing thousands of wells, the authors can calculate the probability <math>P</math> of observing a given cluster, and the free energy <math>\Delta F=-k_BT\ln P</math> associated with that configuration.


The occurrence of clusters is compared to the mechanically stable clusters predicted by previous theoretical study [1]. For N ≤ 5, only one type of cluster is observed, agreeing with theory prediction. These clusters are shown in fig 2.

Fig. 2. Optical micrographs of colloidal clusters in microwells with N = 2, 3, 4, and 5 particles. Scale bar, 1 µm.

For N=6, two structures are observed. The probabilities of observing these two clusters agree well with theory prediction (shown in fig 3). Both structures have 12 contacts, and therefore they have the same potential energy. However, one structure (called polytetrahedron) occurs about 20 times more frequently then the other structure (a octahedron). This difference must come from entropy. The entropy contribution can be separated into the vibrational entropy and the rotational entropy. The vibrational entropy is proportional to the square root of moment of inertia <math>\sqrt{I}</math> and the inverse of the symmetry number <math>\sigma</math>. In this case, the octahedron has a much higher symmetry number (<math>\sigma=24</math>) as compared to the polytetrahedron (<math>\sigma=2</math>). So symmetry accounts for a factor of 12 in <math>P</math>. The remaining factor of 2 comes from moment of inertia and vibrational entropy contribution. This tells us that sticky hard sphere systems favor clusters that are not symmetric.

For N=7, 6 structures are observed (2 of them are chiral). All six structures have 15 contacts, therefore the same potential energy. For N=8, 16 structures are observed; they all have 18 contacts. Again, the possibilities of observing these structures agree well with theoretical predition, as can be seen in fig 3.

Fig. 3. Comparison of experimental and theoretical cluster probabilities <math>P</math> at N = 6, 7, and 8. Structures that are difficult to differentiate experimentally have been binned together at N = 7 and 8 to compare to theory. The calculated probabilities for the individual states are shown as light gray bars, and binned probabilities are dark gray. Orange dots indicate the experimental measurements, with 95% confidence intervals given by the error bars.

For N=9 and N=10, there are too many theoretically predicted stable structures (77 and 393 respectively) than can be categorized experimentally. Instead, the authors examine two types of clusters: (1) nonrigid structures, in which one of the vibrational modes is a large-amplitude, anharmonic shear mode, and (2) structures with more than 3N – 6 bonds. Nonrigid structures arise for N=9 and N=10. They arise when a cluster contains half-octahedra that share at least one vertex, allowing the cluster to twist over a finite distance without breaking or forming another bond. Therefore they have higher vibrational entropy. Indeed, by comparing theory with experiment, the authors estimate non-rigid structures have free energy <math>2k_BT</math> lower than average cluster, for the case N=9. Clusters with more than 3N – 6 bonds occur for N ≥ 10. They have lower potential energy, and thus lower free energy. Many of these clusters with extra bonds are subsets of HCP lattice.

Fig 4 provides the theoretical prediction of free energy for N=6 up to N=10. They completely specifies the equilibrium behavior of these small clusters.

Fig. 4. Calculated minima of the free-energy landscape for 6 ≤ N ≤ 10. The x axis is in units of the rotational partition function. Each black symbol represents the free energy of an individual cluster. The number of spokes in each symbol indicates the symmetry number (dot = 1, line segment = 2, and so on). Orange symbols are nonrigid structures, which first appear at <math>N = 9</math>, and violet symbols have extra bonds, first appearing at <math>N = 10</math>. Vertical gray lines indicate the contribution to the free energy due to rotational and vibrational entropy. The reference states are chosen to be the highest free-energy states at each value of N.


What we've learned is that for these sticky hard sphere, three types of clusters are favored. (1) Rotational entropy favors structures with fewer symmetry elements. (2) Vibrational entropy favors nonrigid clusters, which have half-octahedral substructures sharing at least one vertex. (3) Potential energy favors clusters with both octahedral and tetrahedral substructures, allowing them to have extra bonds.

The situation will be different when interactions are not so short-ranged. For such cases, the potential energy is no longer simply proportional to the number of contacts. Rather, each configuration will have a different energy. This will break the degeneracy we observed in this sticky hard sphere system.


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