We describe experimental investigations of the structure of two-dimensional spherical crystals. The crystals, formed by beads self-assembled on water droplets in oil, serve as model systems for exploring very general theories about the minimum energy configurations of particles with arbitrary repulsive interactions on curved surfaces. Above a critical system size we find that crystals develop distinctive high-angle grain boundaries, or scars, not found in planar crystals. The number of excess defects in a scar is shown to grow linearly with the dimensionless system size. The observed slope is expected to be universal, independent of the microscopic potential.Spherical particles on a flat surface pack most efficiently in a simple lattice of triangles, similar to billiard balls at the start of a game. Such six-fold coordinated triangular lattices [1] cannot, however, be wrapped on the curved surface of a sphere; instead, there must be extra defects in coordination number. Soccer balls and C 60 fullerenes [2,3] provide familiar realizations of this fact -they have 12 pentagonal panels and 20 hexagonal panels. The necessary packing defects can be characterized by their topological or disclination charge, q, which is the departure of their coordination number c from the preferred flat space value of 6 (q = 6 − c); a classic theorem of Euler [4,5] shows that the total disclination charge of any triangulation of the sphere must be 12 [6]. A total disclination charge of 12 can be achieved in many ways, however, which makes the determination of the minimum energy configuration of repulsive particles, essential for crystallography on a sphere, an extremely difficult problem. This was recognized nearly 100 years ago by J.J. Thomson [7], who attempted, unsuccessfully, to explain the periodic table in terms of rigid electron shells. Similar problems recur in fields as diverse as multi-electron bubbles in superfluid helium [8], virus morphology [9, 10, 11], protein s-layers [12,13] and coding theory [14,15]. Indeed, both the classic Thomson problem, which deals with particles interacting through the Coulomb potential, and its generalization to other interaction potentials remain largely unsolved after almost 100 years [16,17,18].The spatial curvature encountered in curved geometries adds a fundamentally new ingredient to crystallography, not found in the study of order in spatially flat systems. To date, however, studies of the Thomson and related problems have been limited to theory and computer simulation. As the number of particles on the sphere grows, isolated charge 1 defects are predicted to induce too much strain; this can be relieved by introducing additional dislocations, consisting of pairs of tightly bound 5-7 defects [19] which still satisfy Euler's theorem since their net disclination charge is zero. Dislocations, which are point-like topological defects in two dimensions, disrupt the translational order of the crystalline phase but are less disruptive of orientational order [19]. While they play an essential rol...

We study the assembly of spherical particles with opposite electric charge on both hemispheres, in the case that the particle diameter exceeds the electrostatic screening length. Clusters result, not strings. The cluster shapes are analyzed by combined epifluorescence microscopy and Monte Carlo computer simulations with excellent agreement, indicating that the particles assemble in aqueous suspension to form equilibrated aggregates. The simulations show that charge asymmetry of individual Janus particles is preserved in the clusters.

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Orientation-dependent interactions can drive unusual self-assembly of colloidal particles. This study, based on combined epifluorescence microscopy and Monte Carlo simulations, shows that amphiphilic colloidal spheres, hydrophobic on one hemisphere and charged on the other, assemble in water into extended structures not formed by spheres of uniform surface chemical makeup. Small, compact clusters each comprised of less than 10 of these Janus spheres link up, as increasing salt concentration enhances electrostatic screening, into wormlike strings.

Adding a nonadsorbing polymer to passive colloids induces an attraction between the particles via the "depletion" mechanism. High enough polymer concentrations lead to phase separation. We combine experiments, theory, and simulations to demonstrate that using active colloids (such as motile bacteria) dramatically changes the physics of such mixtures. First, significantly stronger interparticle attraction is needed to cause phase separation. Secondly, the finite size aggregates formed at lower interparticle attraction show unidirectional rotation. These micro-rotors demonstrate the self-assembly of functional structures using active particles. The angular speed of the rotating clusters scales approximately as the inverse of their size, which may be understood theoretically by assuming that the torques exerted by the outermost bacteria in a cluster add up randomly. Our simulations suggest that both the suppression of phase separation and the self-assembly of rotors are generic features of aggregating swimmers and should therefore occur in a variety of biological and synthetic active particle systems. M otile bacteria are simple examples of "living active matter." Passive particles with sizes similar to those of most bacteria, viz., 0.2-2 μm, are colloids. Such particles are in thermal equilibrium with the surrounding solvent and undergo Brownian motion. In contrast, self-propelled bacteria are active colloids. Such particles function far from equilibrium. This renders their physics far richer than that of passive colloids, mainly because they are not subject to thermodynamic constraints such as detailed balance or the fluctuation-dissipation theorem. Thus, bacteria are able to harness their activity to power externally added micro gear wheels (1-3), self-concentrate, and cluster to form a variety of patterns due to geometry, steric effects, or biochemical cues (4-6).Currently, no general statistical mechanical theory relates the microscopic properties of individual active particles to the macroscopic behavior of large collections of such particles. Recent experiments on noninteracting suspensions of synthetic swimmers (7) show that, as in a dilute suspension of passive particles, there is an exponential distribution of particles with height, but with an increased sedimentation length. To date, however, there has been no experiment designed specifically to probe the effect of activity on macroscopic properties that arise from interparticle interaction, such as phase transitions, perhaps the quintessential many-body phenomenon. Here, we report a systematic study of the physics of phase separation and self-assembly in a suspension of interacting active colloids in the form of mutually attracting motile bacteria. Our experimental results, supported by theory and simulations, provide a foundation for general treatments of the statistical mechanics of interacting active particles.Interparticle attraction in passive colloids leads to aggregation and phase separation. Such attraction can be induced by nonadsorbing polymer...