Finite element simulations of quartz crystal microbalances (QCM): from Sauerbrey to fractional viscoelasticity under water

Abstract 2D finite element simulations are performed on QCM working in the thickness-shear mode and loaded with different homogeneous films. They include a purely elastic film, a viscoelastic Maxwellian liquid, viscoelastic-Voigt solid, and the fractional viscoelastic (power-law) version of each case. Unlike single-relaxation kind models, fractional viscoelasticity considers the relaxation-time spectrum often found in polymeric materials. The films are tested in air or covered with liquids of different viscosities. Two substrate thicknesses are tested: 100 nm and 500 nm, the latter being close to the condition that promotes the resonance of the adsorbed film. In all cases the simulations are compared with small-load approximation theory (SLA). The 100 nm films follow the theory closely, although significant deviations of the SLA are observed as the overtone number n increases, even in purely elastic films. We also show that it is possible to identify the viscoelastic ‘fingerprint’ of the 100 nm films in air using raw data and Sauerbrey’s equivalent thickness obtained with the QCM in the 3 < n < 13 range. These numerical data are validated by experimental measurements of crosslinked polydimethylsiloxane films with thicknesses ∼150 nm. In contrast, the 500 nm films deviate notoriously from the SLA, for all viscoelastic models and overtones, with the largest deviation observed in the elastic film. When a liquid layer covers the QCM without an adsorbed film, the only overtone that numerically reproduces the theoretical value is the fundamental, n = 1. For n > 1, strong coupling between the solid and liquid is detected, and the original vibration modes of the crystal are altered by the presence of the liquid. Finally, the numerical simulations suggest that it is possible to detect whether a viscoelastic film is formed under a liquid layer using only the information from n = 1. In these film/liquid systems we also observe the so-called missing-mass effect, although the theory and simulations exhibit different levels of impact of such effect when the liquid viscosity is high.