Freezing on a Sphere

TL;DR

This paper investigates how crystallization occurs on spherical surfaces, revealing a unique defect organization into a single continent and 12 defect seas with icosahedral symmetry, advancing understanding of curved surface crystallization.

Contribution

It introduces a novel order parameter to detect long-range order on a sphere and explains defect organization during freezing on curved surfaces.

Findings

Defects form 12 isolated seas with icosahedral symmetry.

A new order parameter reveals underlying long-range orientational order.

Crystallization on a sphere involves defect sequestration into a single continent.

Abstract

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 threshold. 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 soccer ball or the 12 pentamers in a viral capsid. Moreover, crystals assembled on curved surfaces can spontaneously develop additional defects to alleviate the stress imposed by the curvature. How then can we have crystallization on a sphere, the simplest curved surface where it is impossible to eliminate these defects? Here we show that freezing on a sphere proceeds by the formation of a single, encompassing "continent," which forces defects into 12 isolated "seas" with the same icosahedral symmetry as soccer balls 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. These results further our understanding of the thermodynamic and mechanical properties of naturally occurring structures, such as viral capsids, lipid vesicles, and bacterial s-layers, and show that the spontaneous sequestration and organization of defects can produce mechanical and dynamical inhomogeneities in otherwise homogeneous materials.