Physically Intuitive Continuum Mechanics Model for QCM Applied to Obtain the Frequency-Dependent Storage and Loss Modulus of PDMS Within the Glass Transition Regime

Quartz crystal microbalances (QCMs) are increasingly applied as MHz-rheometers to measure viscoelastic properties of films beyond the simple Sauerbrey equation relating frequency shifts to mass loading of the crystal. Much QCM modeling in the literature applies mathematical simplifications such as the small load approximation to analytically solve for frequency shifts as a function of harmonic number, which cannot be applied to situations in which the frequency shift is large. In addition, many models utilize equivalent circuit formalisms, losing their physics intuition to the sample’s material properties. We present a continuum mechanics model with no mathematical approximations that numerically solves for the viscoelastic properties of the polymer, while preserving the model’s intuitive physics formulation. By incorporating frequency-dependent moduli, we are able to model polydimethylsiloxane (PDMS) films within the glass transition regime, obtaining good agreement with interpolated literature values of the rubbery and glassy plateau regimes from dynamic mechanical analysis (DMA).