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Negatively-curved interfaces

Pfoam

The densest packing of equal sized disks on an infinite plane is obtained by arranging the disks into a triangular tilling. Similarly, the lowest energy configuration of monodisperse (i.e. equal volume) two dimensional bubbles corresponds to a hexagonal bubble lattice. In both cases the disks and the bubbles occupy the same area on the plane and have the same local environment of six equally-spaced first neighbours which is reproduced all over the plane according to the laws of classical crystallography. 

However, this ideal organization is no longer possible when the disks are assembled onto curved surfaces. Several natural and man-made mechanical systems can be viewed as two-dimensional curved crystals. Examples include architectural structures, viral shells and colloidosomes (crystals formed by beads self-assembled on water droplets in oil). In such systems most of the constituent particles have six nearest neighbours but in addition distinctive high angle grain boundaries are observed. These consist of disclinations (disks or bubbles with five or seven neighbours) and dislocations (a pair of adjacent five-seven disclinations), which together form complex “scar” like arrangements. 

Dislocations have been observed in flat 2D foams but disclinations and scars have not. This project, In collaboration with Gerd Schröder-Turk (Erlangen) and Myfanwy Evans (Erlangen) seeks to elucidate the morphology of quasi 2D foams on familiar curved surfaces (such as the sphere) as well as more exotic surfaces with complex negatively (such as triply-periodic minimal surfaces). We hope to observe disclinations, pleats, scars and other more exotic defect structures….

© Adil Mughal 2012 adil.m.mughal@gmail.com