The Free-Energy Landscape of Clusters of Attractive Hard Spheres

Abstract

One sentence summary: Experiments reveal that small clusters of hard spheres with short-range attractions favor equilibrium structures determined by geometrical rules.The study of clusters has provided the most tangible link between local geometry and bulk condensed matter. But experiments have not yet systematically explored the thermodynamics of even the smallest clusters. Here we present experimental measurements of the structures and free energies of colloidal clusters in which the particles act as hard spheres with short-range attractions. We find that highly symmetric clusters are strongly suppressed by rotational entropy, while the most stable clusters have anharmonic vibrational modes or extra bonds. Many of these are subsets of closepacked lattices. As the number of particles increases from 6 to 10 we observe the emergence of a complex free energy landscape with a small number of ground states and many local minima.An isolated system of 10 interacting atoms or molecules will in general adopt a structure that differs in symmetry and average energy from that of a bulk liquid, solid, or even a system containing 100 particles. Yet the study of such small clusters has shed light on a wide variety of phenomena in condensed matter physics and physical chemistry. Since Frank first predicted (1) that icosahedral short-range order * This is the authors version of the work. It is posted here by permission of the AAAS for personal use, not for redistribution. The definitive version was published

Colloquium : Toward living matter with colloidal particles

“…The results of this calculation for N = 6, 7, 8 are in excellent agreement with the experimentally measured probabilities (Meng et al, 2010), as shown in Figure 2. The main qualitative point from these calculations and experiments is that differences in occupancy between the various clusters are primarily related to differences in symmetry, which comes into the calculation through the rotational partition function (Cates and Manoharan, 2015), see Eqn.…”

“…There is a tremendous opportunity for theory and experiment to work together towards materials with revolutionary properties. The equilibrium probabilities of clusters at N = 6, 7, 8 vary with the structure, even though all structures at a given N have the same total depletion potential (Meng et al, 2010;Perry et al, 2012). For N = 6, the less symmetric cluster, the polytetrahedron, occurs about 24 times as frequently as the more symmetric octahedron, even though both have 3N −6 contacts.…”

“…The attraction is short-ranged, meaning that it extends beyond the particle by only a few percent of the diameter, so that it is effectively pairwise additive. In the experiment, N colloidal particles are placed in each of thousands of microwells, and the strength of the depletion interaction is tuned so that the resulting clusters are in equilibrium (Meng et al, 2010). The clusters that form are then observed directly with an optical microscope as a function of N .…”

Colloids with valence and specific directional bonding

“…Triangle-like AB 3 (Fig. 3c) and tetrahedron-like AB 4 (Fig. 3d) colloidal molecules are similarly obtained by mixing trivalent (3-patch) particles and tetravalent (4-patch) particles, respectively, with monovalent particles (see Supplementary Movie 1 for all colloidal molecules).…”

“…The past decade has seen an explosion in the kinds of colloidal particles that can be synthesized 1,2 , with many new shapes, such as cubes 3 , clusters of spheres [4][5][6] and dimpled particles 7,8 reported. Because the self-assembly of these particles is largely controlled by their geometry, only a few relatively simple crystals have been made: face-centered and body-centered cubic crystals and variants 9 .…”