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...

Engineering synthetic materials that mimic the remarkable complexity of living organisms is a fundamental challenge in science and technology. We study the spatiotemporal patterns that emerge when an active nematic film of microtubules and molecular motors is encapsulated within a shape-changing lipid vesicle. Unlike in equilibrium systems, where defects are largely static structures, in active nematics defects move spontaneously and can be described as self-propelled particles. The combination of activity, topological constraints and vesicle deformability produces a myriad of dynamical states. We highlight two dynamical modes: a tunable periodic state that oscillates between two defect configurations, and shape-changing vesicles with streaming filopodia-like protrusions. These results demonstrate how biomimetic materials can be obtained when topological constraints are used to control the non-equilibrium dynamics of active matter.

No abstract

We propose and analyze an effective free energy describing the physics of disclination defects in particle arrays constrained to move on an arbitrary two-dimensional surface. At finite temperature the physics of interacting disclinations is mapped to a Laplacian SineGordon Hamiltonian suitable for numerical simulations. We then specialize to the case of a spherical crystal at zero temperature. The ground state is analyzed as a function of the ratio of the defect core energy to the Young's modulus. We argue that the core energy contribution becomes less and less important in the limit R ≫ a, where R is the radius of the sphere and a is the particle spacing. For large core energies there are twelve disclinations forming an icosahedron. For intermediate core energies unusual finite-length grain boundaries are preferred. The complicated regime of small core energies, appropriate to the limit R/a → ∞, is also addressed. Finally we discuss the application of our results to the classic Thomson problem of finding the ground state of electrons distributed on a two-sphere.