Geometric frustration of hard-disk packings on cones

TL;DR

This paper investigates how geometric frustration affects the growth of hard-disk packings on conical surfaces, revealing finite-size effects and defect formation that influence crystal growth and structure.

Contribution

It introduces a disk-packing algorithm to analyze crystal growth on cones, highlighting the role of circumference and finite-size effects in defect formation and disorder.

Findings

Defect density increases exponentially with decreasing circumference.

Crystals typically form a seam along the cone's axis, except at special angles.

Finite-size effects cause disorder near the cone tip, independent of seed orientation.

Abstract

Conical surfaces pose an interesting challenge to crystal growth: a crystal growing on a cone can wrap around and meet itself at different radii. We use a disk-packing algorithm to investigate how this closure constraint can geometrically frustrate the growth of single crystals on cones with small opening angles. By varying the crystal seed orientation and cone angle, we find that -- except at special commensurate cone angles -- crystals typically form a seam that runs along the axial direction of the cone, while near the tip, a disordered particle packing forms. We show that the onset of disorder results from a finite-size effect that depends strongly on the circumference and not on the seed orientation or cone angle. This finite-size effect occurs also on cylinders, and we present evidence that on both cylinders and cones, the defect density increases exponentially as circumference decreases. We introduce a simple model for particle attachment at the seam that explains the dependence on the circumference. Our findings suggest that the growth of single crystals can become frustrated even very far from the tip when the cone has a small opening angle. These results may provide insights into the observed geometry of conical crystals in biological and materials applications.