Equivalent-circuit model for the thickness-shear mode resonator with a viscoelastic film near film resonance.

We derive a lumped-element, equivalent-circuit model for the thickness-shear mode (TSM) resonator with a viscoelastic film. This modified Butterworth-Van Dyke model includes in the motional branch a series LCR resonator, representing the quartz resonance, and a parallel LCR resonator, representing the film resonance. This model is valid in the vicinity of film resonance, which occurs when the acoustic phase shift across the film is an odd multiple of pi/2 rad. For low-loss films, this model accurately predicts the frequency changes and damping that arise at resonance and is a reasonable approximation away from resonance. Elements of the parallel LCR resonator are explicitly related to film properties and can be interpreted in terms of elastic energy storage and viscous power dissipation. The model leads to a simple graphical interpretation of the coupling between the quartz and film resonances and facilitates understanding of the resulting responses. These responses are compared with predictions from the transmission-line and Sauerbrey models.

Modeling the Responses of Thickness-Shear Mode Resonators under Various Loading Conditions.

We develop a general model that describes the electrical responses of thickness-shear mode resonators subject to a variety of surface conditions. The model incorporates a physically diverse set of single-component loadings, including rigid solids, viscoelastic media, and fluids (Newtonian or Maxwellian). The model allows any number of these components to be combined in any configuration. Such multiple loadings are representative of a variety of physical situations encountered in electrochemical and other liquid-phase applications, as well as gas-phase applications. In the general case, the response of the composite load is not a linear combination of the individual component responses. We discuss application of the model in a qualitative diagnostic fashion to gain insight into the nature of the interfacial structure, and in a quantitative fashion to extract appropriate physical parameters such as liquid viscosity and density and polymer shear moduli.