Crystalline Order On Riemannian Manifolds With Variable Gaussian Curvature And Boundary

Abstract

We investigate the zero temperature structure of a crystalline monolayer constrained to lie on a two-dimensional Riemannian manifold with variable Gaussian curvature and boundary. A full analytical treatment is presented for the case of a paraboloid of revolution. Using the geometrical theory of topological defects in a continuum elastic background we find that the presence of a variable Gaussian curvature, combined with the additional constraint of a boundary, gives rise to a rich variety of phenomena beyond that known for spherical crystals. We also provide a numerical analysis of a system of classical particles interacting via a Coulomb potential on the surface of a paraboloid.

arXiv comment12 pages, 8 figures

References in corpus (10)

• Grain Boundary Scars and Spherical Crystallography
• Virus shapes and buckling transitions in spherical shells
• Interacting Topological Defects on Frozen Topographies
• Crystalline Order on a Sphere and the Generalized Thomson Problem
• Crystallography on Curved Surfaces
• Crystalline Particle Packings on a Sphere with Long Range Power Law Potentials
• Grain Boundary Scars on Spherical Crystals
• Ground state of a large number of particles on a frozen topography
• The Geometrical Structure of 2d Bond-Orientational Order
• Solid domains in lipid vesicles and scars

Cited by in corpus (27)

• Two-Dimensional Matter: Order, Curvature and Defects
• Vortices in Superfluid Films on Curved Surfaces
• Theory of defect-mediated morphogenesis
• Extrinsic Curvature, Geometric Optics, and Lamellar Order on Curved Substrates
• Ground States of Crystalline Caps: Generalized Jellium on Curved Space
• Defects in Crystalline Packings of Twisted Filament Bundles: I. Continuum Theory of Disclinations
• Neutral versus charged defect patterns in curved crystals
• Toroidal Crystals
• Bubble Raft Model for a Paraboloidal Crystal
• Curvature-induced frustration in the $XY$ model on hyperbolic surfaces
• Elastic Theory of Defects in Toroidal Crystals
• Topological magnetic solitons on a paraboloidal shell
• Crystalline Particle Packings on Constant Mean Curvature (Delaunay) Surfaces
• Crystalline Order on Catenoidal Capillary Bridges
• Defects in Crystalline Packings of Twisted Filament Bundles: II. Dislocations and Grain Boundaries
• Identifying Phase Space Boundaries with Voronoi Tessellations
• Elasticity in Curved Topographies: Exact Theories and Linear Approximations
• Circle packing in regular polygons
• Geometric Induction in Chiral Superfluids
• Stiffness heterogeneity of small viral capsids
• Dislocation screening in crystals with spherical topology
• Duality and free energy analyticity bounds for few-body Ising models with extensive homology rank
• Exact Solution for Elastic Networks on Curved Surfaces
• A simple, general criterion for onset of disclination disorder on curved surfaces
• Predicting the Characteristics of Defect Transitions on Curved Surfaces
• General Solution for Elastic Networks on Arbitrary Curved Surfaces in the Absence of Rotational Symmetry
• Paraboloidal Crystals