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

Geometric frustration of hard-disk packings on cones