No abstract

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 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.

The fluctuations of two-dimensional extended objects (membranes) is a rich and exciting field with many solid results and a wide range of open issues. We review the distinct universality classes of membranes, determined by the local order, and the associated phase diagrams. After a discussion of several physical examples of membranes we turn to the physics of crystalline (or polymerized) membranes in which the individual monomers are rigidly bound. We discuss the phase diagram with particular attention to the dependence on the degree of self-avoidance and anisotropy. In each case we review and discuss analytic, numerical and experimental predictions of critical exponents and other key observables. Particular emphasis is given to the results obtained from the renormalization group ε-expansion. The resulting renormalization group flows and fixed points are illustrated graphically. The full technical details necessary to perform actual calculations are presented in the Appendices. We then turn to a discussion of the role of topological defects whose liberation leads to the hexatic and fluid universality classes. We finish with conclusions and a discussion of promising open directions for the future.