Physically intuitive continuum mechanics model for quartz crystal microbalance: Viscoelasticity of rubbery polymers at MHz frequencies

Employing a quartz crystal microbalance (QCM) as a MHz-viscoelastic sensor requires extracting information from higher harmonics beyond the Sauerbrey limit, which can be problematic for rubbery polymer films that are highly dissipative because of the onset of anharmonic side bands and film resonance. Data analysis for QCM can frequently obscure the underlying physics or involve approximations that tend to break down at higher harmonics. In this study, modern computational tools are leveraged to solve a continuum physics model for the QCM's acoustic shear wave propagation through a polymer film with zero approximations, retaining the physical intuition of how the experimental signal connects to the shear modulus of the material. The resulting set of three coupled equations are solved numerically to fit experimental data for the resonance frequency Δf_n and dissipation ΔΓ_n shifts as a function of harmonic number n, over an extended harmonic range approaching film resonance. This allows the frequency-dependent modulus of polymer films at MHz frequencies, modeled as linear on a log–log scale, to be determined for rubbery polybutadiene (PB) and polydimethylsiloxane (PDMS) films, showing excellent agreement with time–temperature shifted rheometry data from the literature.