Quartz crystal microbalance impedance of discrete adsorbates with lubricated contact

We present a thorough theoretical investigation of the shear force exerted on a horizontal plate undergoing fast (MHz range) longitudinal oscillations in presence of a spherical particle suspended in the liquid at close proximity to the plate. This hydrodynamic problem is relevant toward the analysis of frequency response (i.e., complex impedance) of Quartz Crystal Microbalance (QCM) to discrete (particulate) adsorbates. For low areal density of adsorbates, the contributions of individual particles remain additive, rendering the theory applicable towards quantitative interpretation of the experimental measurements. For adsorbates much smaller than the viscous penetration depth, the closed-form analytical expressions for the complex impedance were derived for particles of arbitrary density, assuming a lubricated contact (i.e., non-localized adsorption). The theoretical predictions were shown to be in an excellent agreement with accurate Finite Element numerical computations. The validity of the theoretical approach was confirmed by QCM measurements conducted for polystyrene nanoparticles in a broad range of sizes, from 32 nm to 510 nm, adsorbing onto gold/silica sensor upon varying the oscillation frequency.