Freezing on a Sphere

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.

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• Crystalline Order on a Sphere and the Generalized Thomson Problem
• Nematic textures in spherical shells
• Direct observation of melting in a 2-D superconducting vortex lattice
• Direct Visualization of Dislocation Dynamics in Grain Boundary Scars
• Fractionalization of Interstitials in Curved Colloidal Crystals
• Crystalline Particle Packings on a Sphere with Long Range Power Law Potentials
• Computer simulations of two-dimensional melting with dipole-dipole interactions
• Influence of Dislocations in Thomson's Problem
• Vortices in a Thin Film Superconductor with a Spherical Geometry
• Specific heat in two-dimensional melting
• Charged hydrophobic colloids at an oil/aqueous phase interface
• Phase behavior of charged colloids at a fluid interface
• Orientational Phase Transitions and the Assembly of Viral Capsids

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• Early-Stage Bifurcation of Crystallization in a Sphere
• Interplay of curvature and rigidity in shape-based models of confluent tissue
• Kibble-Zurek mechanism in curved elastic surface crystals
• Glassy dynamics of dense particle assemblies on a spherical substrate
• Spherical structure factor and classification of hyperuniformity on the sphere
• Dislocation screening in crystals with spherical topology
• Bose-Hubbard model on polyhedral graphs
• Conforming nanoparticle sheets to surfaces with Gaussian curvature
• Orientational ordering of lamellar structures on closed surfaces
• Complex Self-Assembly from Simple Interaction Rules in Model Colloidal Mixtures
• Crystal gravity
• Stress-induced ordering of two-dimensional packings of elastic spheres
• Competing Hexagonal and Square Lattices on a Spherical Surface
• Ultracold Bosons on a Regular Spherical Mesh
• Topological defects of dipole patchy particles on a spherical surface
• Locomotion without force, and impulse via dissipation: Robotic swimming in curved space via geometric phase
• Classical and Quantum Gases on a Semiregular Mesh
• Geometric frustration of hard-disk packings on cones
• Hydrodynamic interactions in polar liquid crystals on evolving surfaces
• Global order parameters for particle distributions on the sphere
• Symmetry breaking of dipole orientations on Caspar-Klug lattices
• Scale-Free Crystallization of two-dimensional Complex Plasmas: Domain Analysis using Minkowski Tensors
• Stress driven fractionalization of vacancies in regular packings of elastic particles
• Scanning-probe and information-concealing machine learning intermediate hexatic phase and critical scaling of solid-hexatic phase transition in deformable particles
• Dynamical Spreading and Memory Retention Under Power Law Potential
• Self-assembly of anisotropic particles on curved surfaces
• Properties of surface Landau-de Gennes Q-tensor models
• Data-driven criterion for the solid-liquid transition of two-dimensional self-propelled colloidal particles far from equilibrium