Dynamics of a monolayer of microspheres on an elastic substrate

Abstract

We present a model for wave propagation in a monolayer of spheres on an elastic substrate. The model, which considers sagittally polarized waves, includes: horizontal, vertical, and rotational degrees of freedom; normal and shear coupling between the spheres and substrate, as well as between adjacent spheres; and the effects of wave propagation in the elastic substrate. For a monolayer of interacting spheres, we find three contact resonances, whose frequencies are given by simple closed-form expressions. For a monolayer of isolated spheres, only two resonances are present. The contact resonances couple to surface acoustic waves in the substrate, leading to mode hybridization and “avoided crossing” phenomena. We present dispersion curves for a monolayer of silica microspheres on a silica substrate, assuming adhesive Hertzian interactions, and compare calculations using an effective medium approximation (including elasticity of the substrate) to a discrete model of a monolayer on a rigid substrate. While the effective medium model does not describe discrete lattice effects occurring at short wavelengths, we find that it is well suited for describing the interaction between the monolayer and substrate in the long wavelength limit. We suggest that a complete picture of the dynamics of a monolayer adhered to an elastic substrate can be found by combining the dispersion curves generated with the effective medium model for the elastic substrate and the discrete model for the rigid substrate. This model is potentially scalable for use with nano- to macroscale systems, and offers the prospect of experimentally extracting contact stiffnesses from measurements of acoustic dispersion.