The best understood crystal ordering transition is that of two-dimensional freezing, which proceeds by the rapid eradication of lattice defects as the temperature is lowered below a critical threshold1,2,3,4. But crystals that assemble on closed surfaces are required by topology to have a minimum number of lattice defects, called disclinations, that act as conserved topological charges—consider the 12 pentagons on a football or the 12 pentamers on a viral capsid5,6. Moreover, crystals assembled on curved surfaces can spontaneously develop additional lattice defects to alleviate the stress imposed by the curvature6,7,8. It is therefore unclear how crystallization can proceed on a sphere, the simplest curved surface on which it is impossible to eliminate such defects. Here we show that freezing on the surface of a sphere proceeds by the formation of a single, encompassing crystalline ‘continent’, which forces defects into 12 isolated ‘seas’ with the same icosahedral symmetry as footballs and viruses. We use this broken symmetry—aligning the vertices of an icosahedron with the defect seas and unfolding the faces onto a plane—to construct a new order parameter that reveals the underlying long-range orientational order of the lattice. The effects of geometry on crystallization could be taken into account in the design of nanometre- and micrometre-scale structures in which mobile defects are sequestered into self-ordered arrays. Our results may also be relevant in understanding the properties and occurrence of natural icosahedral structures such as viruses5,9,10.
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We thank M. J. Bowick and A. Travesset for discussions. This research was primarily supported by the National Science Foundation (NSF) under grant DMR 1105417 and grant GBMF3849 from the Gordon and Betty Moore Foundation. This work was supported partially by the Materials Research Science and Engineering Center (MRSEC) Program of the NSF under award number DMR-1420073 and by the National Aeronautics and Space Administration (NASA) under grant NNX13AR67G. This work has used the NYU IT High Performance Computing resources and services.
Animated Voronoi tessellation of experimentally determined particle positions for Γ=145 sphere, together with renderings of ordinary and cage-relative particle displacements.
Animated Voronoi tessellation of experimentally determined particle positions for Γ=145 sphere described in the text, together with renderings of the ordinary and cage-relative particle displacements.