2D Crystals on Curved Surfaces

Determining the minimum-energy configuration of repulsive particles on spherical surfaces is a problem whose difficulty was already recognized nearly 100 years ago by J. J. Thomson. Euler's theorem states that twelve pentagons are required to close a hexagonal network. Theory and computer simulation have shown that, with rising system size, the strain induced by single pentagonal disclinations will be lowered by the introduction of additional pairs of bound 5-7 defects. We expect such scars to occur in sufficiently large viral protein capsids, giant fullerenes, bacterial surface layers, and the skeletons of radiolaria. Terminating strings of heptagons and pentagons might serve as sites for chemical reactions or as initiation points for bacterial cell division and will surely influence the mechanical properties of spherical crystalline shells.

An experimental system to test the theoretical predictions has recently been developed. We use surface-modified microspheres self-assembled onto the surface of water droplets in an organic solvent. Crystalline structures are observed as soon as the area density of particles at the interface is high enough. As predicted by theory, we find that these crystals form distinctive high-angle grain boundaries or scars, not found in planar crystals. The number of excess defects in these scars grows linearly with the system size. The observed slope is expected to be universal, independent of the microscopic potential.

Using digital video microscopy, particle tracking algorithms, and triangulation routines, the dynamical behaviour of the defects and the movements of single particles in the spherical lattice are explored. The elastic potential of the lattice, the Young modulus, and the diffusion constant are determined. First results demonstrating typical phase transitions between fluid and crystalline phases are presented. Binary mixtures formed by beads of two different diameters at the interface were investigated. For diameter ratios far from one, disordered states were observed.

Light microscope images of particle-coated droplets. Two droplets (A) and (C) are shown, together with their associated defect structures (B) and (D). (A) An app.13% portion of a small spherical droplet with radius R=12.0 µm and mean particle spacing a = 2.9 µm (R/a = 4.2), along with the associated triangulation (B). Charge +1(-1) disclinations are shown in red and yellow, respectively. Only one +1 disclination is seen. (C) A cap of spherical colloidal crystal on a water droplet of radius R = 43.9 µm with mean particle spacing a = 3.1 µm (R/a = 14.3), along with the associated triangulation (D). In this case the imaged crystal covers app.17% of the surface area of the sphere. Bars [(A) and (C)], 5 µm.
Excess dislocations as a function of system size. The number of excess dislocations per minimal disclination N as a function of system size R/a, with the linear prediction given by theory shown as a solid red line.