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...

Recent studies aimed at investigating artificial analogues of bacterial colonies have shown that lowdensity suspensions of self-propelled particles confined in two dimensions can assemble into finite aggregates that merge and split, but have a typical size that remains constant (living clusters). In this Letter we address the problem of the formation of living clusters and crystals of active particles in three dimensions. We study two systems: self-propelled particles interacting via a generic attractive potential and colloids that can move towards each other as a result of active agents (e.g. by molecular motors). In both cases fluid-like 'living' clusters form. We explain this general feature in terms of the balance between active forces and regression to thermodynamic equilibrium. This balance can be quantified in terms of a dimensionless number that allows us to collapse the observed clustering behaviour onto a universal curve. We also discuss how active motion affects the kinetics of crystal formation. PACS numbers:Active systems consume energy that keeps them in an out-of-equilibrium state [1,2]. This is typical for many biologically relevant systems which, by exploiting chemical energy, can self-organise into complex structures that lack any equilibrium counterpart. Examples are abundant and exist at different length-scales: from cytoskeleton remodulation during cell mitosis [3] to swarming phenomena in micro-swimmers or flocks of birds [4,5]. The similarity of the patterns displayed by these systems lead many to address the general principles behind their formation using simple models [6]. The reproducibility and robustness of the phenomena under a variety of external conditions motivated a large body of research on selfassembly of active particle as a possible strategy towards the fabrication of new functional nano-and mesoscopic structures [7]. In this respect there have been several efforts to study active self-organisation in a tightly controlled environment; the most studied systems being selfpropelled particles [6,[8][9][10][11] and active gels (e.g. [12]).Recently, two experimental groups have shown how two-dimensional suspensions of self-propelled (SP) colloids (moving through the consumption of an appropriate 'fuel') self-assemble into dynamic clusters that constantly join and split, recombining with each other to reach a steady-state [13,14]. A general understanding of active cluster formation is lacking, but different explanations have been proposed. The authors of Ref.[13] argued that a net attraction between colloids is responsible for clustering. This attraction is due to non-uniformity in the chemical fuel concentration between two colloids. This was confirmed by Ref. [14], that also found a 1/d 2 dependence of the particle-particle attraction ( d being the inter-particle distance). The formation of finite-size clusters has been also observed in low density bacteria/polymer suspensions, at intermediate polymer concentrations just before phase separation [10]. However, in other recent studi...

We used optical tweezers to measure the force-extension curve for the formation of tubes from giant vesicles. We show that a significant force barrier exists for the formation of tubes, which increases linearly with the radius of the area on which the pulling force is exerted. The tubes form through a first-order transition with accompanying hysteresis. We confirm these results with Monte Carlo simulations and theoretical calculations. Whether membrane tubes can be formed in, for example, biological cells, thus depends on the details of how forces are applied.

We compute the free energy of confinement for a flexible self-avoiding polymer inside a spherical cavity. Accurate numerical results allow us to arbitrate between three competing scaling predictions. For moderate confinement, the free energy exhibits a power-law dependence on cavity size that is different from what is observed for planar and cylindrical confinement. At high monomer concentrations, crossover to a different scaling regime is observed, consistent with the screening of the excluded-volume interactions. We demonstrate how our findings lead to a revised prediction for the escape time of a polymer from a spherical confinement.