Epitaxial Frustration in Deposited Packings of Rigid Disks and Spheres

TL;DR

This study uses numerical simulations to explore how rigid disks and spheres arrange near incommensurate substrates, revealing density-reducing frustration and potential periodic structures that optimize packing density.

Contribution

It provides new insights into packing frustration and suggests optimal periodic configurations for maximizing density in incommensurate packings.

Findings

Density frustration extends far from substrates.

Periodic configurations may optimize packing density.

Residual disorder persists in sphere packings.

Abstract

We use numerical simulation to investigate and analyze the way that rigid disks and spheres arrange themselves when compressed next to incommensurate substrates. For disks, a movable set is pressed into a jammed state against an ordered fixed line of larger disks, where the diameter ratio of movable to fixed disks is 0.8. The corresponding diameter ratio for the sphere simulations is 0.7, where the fixed substrate has the structure of a (001) plane of a face-centered cubic array. Results obtained for both disks and spheres exhibit various forms of density-reducing packing frustration next to the incommensurate substrate, including some cases displaying disorder that extends far from the substrate. The disk system calculations strongly suggest that the most efficient (highest density) packings involve configurations that are periodic in the lateral direction parallel to the substrate, with substantial geometric disruption only occurring near the substrate. Some evidence has also emerged suggesting that for the sphere systems a corresponding structure doubly periodic in the lateral directions would yield the highest packing density; however all of the sphere simulations completed thus far produced some residual "bulk" disorder not obviously resulting from substrate mismatch. In view of the fact that the cases studied here represent only a small subset of all that eventually deserve attention, we end with discussion of the directions in which first extensions of the present simulations might profitably be pursued.