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

We analytically 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. In addition to contact-based modes, we also study the effect of sphere-substrate and sphere-sphere contacts on spheroidal vibrational modes of the spheres using 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 as well as mode-splitting for some of the degenerate modes of the free sphere. Sphere-sphere interactions result in dispersion of spheroidal modes. Analytical dispersion relations for both contact-based and spheroidal modes are presented and compared with results obtained for a square lattice.