Grain Boundary Scars and Spherical Crystallography
by Lidiya Mishchenko
A. R. Bausch, M. J. Bowick, A. Cacciuto, A. D. Dinsmore, M. F. Hsu, D. R. Nelson, M. G. Nikolaides, A. Travesset, D. A. Weitz, Grain Boundary Scars and Spherical Crystallography, Science 299, 1716-1718 (2003).
"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."
Soft Matter Keywords
Colloid, interface, crystal defects, stress/strain on lattice
Soft Matter Example
The repulsive force that shapes two dimensional colloidal crystals on the surface of droplets is still little understood. This paper, however, develops a model based on the formation of defects in these structures that can predict the colloidal surface structure without sensitivity to the exact microscopic potential.
Because a perfect hexagonal array of colloids cannot pack on a spherical surface, point defects are inherent in the system. However, these isolated defects induce a large strain on the crystal, and as the number of particles on the sphere grows (or their interparticle distance shrinks), the excess strain in the crystal is released by the formation of high angle grain boundaries or "scars". Also, because of the spherical geometry, these grain boundaries can actually terminate in the crystal itself (which is energetically unfavorable in planar crystals).
Thermal fluctuations create and destroy these dislocations once every few seconds, thus implying that the observed structure (viewed for minutes at a time), is the equilibrium structure. This equilibrium behavior can be quantified by plotting the number of excess dislocations (per chain in each droplet) that form versus system size. The critical system size that first results in the formation of scars can be predicted with a simple theoretical model. The formation of these defects can be understood as follows: The elastic strain energy associated with a point defect grows with the size of the crystal and can be reduced by the formation of linear dislocation arrays, scaling linearly with system size.
A model can be used to predict the ground state of the crystal structure. The point defects can be seen as the degrees of freedom in the structure. The potential is attractive for oppositely charged defects (i.e. ones that are either 5 (+1 charge) or 7 (-1 charge) coordinated, as opposed to 6 (0 charge)), and repulsive for like-charged defects. The model includes a proportionality constant that actually depends on the microscopic potential, but the model has predictive powers that are insensitive to the microscopic detail.
The only thing missing is that the model does not incorporate thermal fluctuations of the colloids on the surface of the droplet. Otherwise, there is almost an exact match between theory and experiment. This brings us one step closer to understanding the repulsive interactions responsible for the formation of 2D crystals on spherical surfaces. The applications are quite widespread and these findings can be helpful in understanding virus morphology, protein s-layers, etc.