Towards vibrational spectroscopy on surface-attached colloids performed with a quartz crystal microbalance

Colloidal spheres attached to a quartz crystal microbalance (QCM) produce the so-called “coupled resonances”. They are resonators of their own, characterized by a particle resonance frequency, a resonance bandwidth, and a modal mass. When the frequency of the main resonator comes close to the frequency of the coupled resonance, the bandwidth goes through a maximum. A coupled resonance can be viewed as an absorption line in acoustic shear-wave spectroscopy. The known concepts from spectroscopy apply. This includes the mode assignment problem, selection rules, and the oscillator strength. In this work, the mode assignment problem was addressed with Finite Element calculations. These reveal that a rigid sphere in contact with a QCM displays two modes of vibration, termed “slipping” and “rocking”. In the slipping mode, the sphere rotates about its center; it exerts a tangential force onto the resonator surface at the point of contact. In the rocking mode, the sphere rotates about the point of contact; it exerts a torque onto the substrate. In liquids, both axes of rotation are slightly displaced from their ideal positions. Characteristic for spectroscopy, the two modes do not couple to the mechanical excitation equally well. The degree of coupling is quantified by an oscillator strength. Because the rocking mode mostly exerts a torque (rather than a tangential force), its coupling to the resonator's tangential motion is weak; the oscillator strength consequently is small. Recent experiments on surface-adsorbed colloidal spheres can be explained by the mode of vibration being of the rocking type.