Crystalline Particle Packings on Constant Mean Curvature (Delaunay) Surfaces

TL;DR

This study explores the arrangement and defect structures of crystalline particles on various constant mean curvature surfaces, revealing how surface geometry influences particle configurations and defect evolution during stretching.

Contribution

It provides a numerical analysis of particle packings on Delaunay surfaces, linking surface geometry to defect motifs and validating theoretical predictions about disclination presence.

Findings

Defect motifs evolve from boundary dislocations to interior disclinations.

Ground state configurations depend on surface geometry and stretching process.

Disclinations are present when Gaussian curvature exceeds -pi/3 in a geodesic disc.

Abstract

We investigate the structure of crystalline particle arrays on constant mean curvature (CMC) surfaces of revolution. Such curved crystals have been realized physically by creating charge-stabilized colloidal arrays on liquid capillary bridges. CMC surfaces of revolution, classified by Charles Delaunay in 1841, include the 2-sphere, the cylinder, the vanishing mean curvature catenoid (a minimal surface) and the richer and less investigated unduloid and nodoid. We determine numerically candidate ground state configurations for 1000 point-like particles interacting with a pairwise-repulsive dipole-dipole interaction potential. We mimic stretching of capillary bridges by determining the equilibrium configurations of particles arrayed on a sequence of Delaunay surfaces obtained by increasing or decreasing the height at constant volume starting from a given initial surface, either a fat cylinder or a square cylinder. In this case the stretching process takes one through a complicated sequence of Delaunay surfaces each with different geometrical parameters including the aspect ratio, mean curvature and maximal Gaussian curvature. Unduloids, catenoids and nodoids all appear in this process. Defect motifs in the ground state evolve from dislocations at the boundary to dislocations in the interior to pleats and scars in the interior and then isolated 7-fold disclinations in the interior as the capillary bridge narrows at the waist (equator) and the maximal (negative) Gaussian curvature grows. We also check theoretical predictions that the isolated disclinations are present in the ground state when the surface contains a geodesic disc with integrated Gaussian curvature exceeding -pi/3. Finally we explore minimal energy configurations on sets of slices of a given Delaunay surface and obtain configurations and defect motifs consistent with those seen in stretching.