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 closedform 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 with 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.

Vibrational dynamics of a two-dimensional microgranular crystal

“…Figure 2(e) also shows small peaks to either side of the V-mode peak which we assigned to horizontal-rotational modes labeled HR and RH following Ref. [14]. The avoided crossing between the vertical resonance mode and the SAW, studied in previous works [7,9], is just outside the wavevector range of our measurements.…”

“…The interaction between microspheres should result in dispersion, predicted in Ref. [14] but not observed in previous studies due to the lack of long-range order in the samples. As can be seen in Fig.…”

“…2(c). Three different types of acoustic modes can be identified: a mode labeled R with a nearly constant dispersion slope corresponding to the SAW velocity of the substrate, low-frequency modes (HR, V, RH) with weaker frequency dependence, which we identify as contact-based modes [14], and high-frequency, nearly flat branches (S) corresponding to spheroidal vibrational modes of the spheres.…”

“…A model describing vibrations of a monolayer of spheres on a substrate accounting for both sphere-substrate and sphere-sphere contacts was developed for a square lattice [14] and subsequently modified for a hexagonal lattice [22]. The model yields three vibrational modes polarized in the sagittal plane [23]; one of them predominantly involves vertical displacements, while the other two have primarily horizontal-rotational character.…”

“…Spheres also have internal mechanical degrees of freedom; consequently, in addition to contact-based modes, one would expect to see collective modes originating from spheroidal vibrations of the spheres [13]. The presence of the substrate significantly alters the dispersion of contact-based modes [14] and adds Rayleigh SAWs in the substrate, which interact with the vibrational modes of the monolayer [10,14]. In this work, we characterize the dispersion of these three types of modes (contact based, spheroidal, and Rayleigh) and their interaction across the entire Brillouin zone (BZ) for a chosen high-symmetry direction of a 2D microgranular crystal.…”