# Contact-based and spheroidal vibrational modes of a hexagonal monolayer of microspheres on a substrate

## TL;DR

This paper investigates the vibrational modes of a hexagonal monolayer of microspheres on a substrate, revealing contact-based and spheroidal modes, their interactions, and resulting dispersion relations through analytical modeling.

## Contribution

It introduces a comprehensive model accounting for both normal and shear contacts, analyzing vibrational modes and their dispersion in a hexagonal microsphere lattice.

## Key findings

- Identification of three contact-based vibrational modes involving translation and rotation.
- Sphere-substrate contact causes frequency upshift of spheroidal modes.
- Sphere-sphere interactions induce dispersion and propagation of spheroidal modes.

## Abstract

We study acoustic modes of a close-packed hexagonal lattice of spheres adhered to a substrate, propagating along a high-symmetry direction. The model, accounting for both normal and shear coupling between the spheres and between the spheres and the substrate, yields three contact-based vibrational modes involving both translational and rotational motion of the spheres. Furthermore, we study the effect of sphere-substrate and sphere-sphere contacts on spheroidal vibrational modes of the spheres within a perturbative approach. The sphere-substrate interaction results in a frequency upshift for the modes having a non-zero displacement at the contact point with the substrate. Sphere-sphere interactions result in dispersion of spheroidal modes turning them into propagating waves, albeit with a small group velocity. Analytical dispersion relations for both contact-based and spheroidal modes are presented and compared with results obtained for a square lattice.