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
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.
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>.
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)). Observed structures agree with experimental predictions. For N = 2, 3, 4, 5, one unique structure (dimer, trimer, tetrahedron, triangular dipyramid), as shown in Fig 1E.
The first interesting case is N = 6. Two structures are observed, an octogon and a polytetrahedron. Both have C = 12 contacts and thus have the same potential energy. What explains the 20-fold preference for the polytetrahedron (3 kT difference) is an entropic difference. The entropy can be divided into two factors: a rotational entropy and a vibrational entropy. The rotational entropy makes the largest contribution to the free-energy difference and is proportional to the moment of inertia or equivalently the permutational degeneracy. This contributes a factor of 12, the remaining factor of 2 coming from the vibrational entropy. In the case of N = 6 as for N = 7 and 8, highly symmetric structures are extremely unfavorable among clusters with the same potential energy.
Cases of N = 7 and N = 8 are similar to N = 6, although the number of structures increases to 6 and 16. For all cases, all the structures have the same number of contacts and entropic effects dominate, with highly symmetric structures extremely unfavored. These are all shown in Fig 2.
The landscape undergoes a qualitative change when N reaches 9. The number of structures predicted theoretically reaches 77 for N = 9 and 393 for N = 10, too many to catalog experimentally. The authors concentrate on measuring the probability of having structures of two types:
1. nonrigid structures, in which one of the vibrational modes is a large-amplitude, anharmonic shear mode. These have high vibrational entropy. (Fig 3A, 3B).
2. structures with more than 3N - 6 bonds. These often have high symmetry, but they have extra bonds. The potential-energy gain is therefor large enough to overcome the deficiency in rotational entropy. (Fig 3C).
While the average probabilities at N = 9 and N = 10 should be 1% and 0.25%, the single nonrigid structure at N = 9 has probability 10%, and the 3 extra-bond structures at N = 10 occur 10% of the time. This confirms that nonrigid structures and structures with extra bonds are favored.
The full theoretical free-energy landscape is shown in Fig 4.
The paper shows/confirms that the most stable small clusters of hard spheres with short-ranged attractions can be determined by geometrical rules: (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; and (3) potential energy favors clusters with both octahedral and tetrahedral substructures, allowing them to have extra bonds.
The paper's description of free-energy landscape is still incomplete because of some of the simplifications it makes. In particular, while the interaction energy in the paper is extremely short-ranged, this is not generally true. For longer-range interactions, the effects will no longer be fundamentally entropic, and they will depend on temperature as well.