Paraboloidal Crystals

TL;DR

This paper explores how particles arrange themselves on curved paraboloidal surfaces, revealing unique defect structures and configurations influenced by the surface's geometry, advancing understanding of soft matter on curved manifolds.

Contribution

It introduces a detailed analysis of classical charges on paraboloids, illustrating how curvature affects ground state configurations and defect arrangements in 2D crystalline systems.

Findings

Curved surfaces induce topological defects in particle arrangements.

Configurations vary significantly with surface curvature.

Voronoi and Delaunay structures reveal geometric influences on particle order.

Abstract

The interplay between order and geometry in soft condensed matter systems is an active field with many striking results and even more open problems. Ordered structures on curved surfaces appear in multi-electron helium bubbles, viral and bacteriophage protein capsids, colloidal self-assembly at interfaces and in physical membranes. Spatial curvature can lead to novel ground state configurations featuring arrays of topological defects that would be excited states in planar systems. We illustrate this with a sequence of images showing the Voronoi lattice (in gold) and the corresponding Delaunay triangulations (in green) for ten low energy configurations of a system of classical charges constrained to lie on the surface of a paraboloid and interacting with a Coulomb potential. The parabolic geometry is considered as a specific realization of the class of crystalline structures on two-dimensional Riemannian manifolds with variable Gaussian curvature and boundary.