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We have derived the general solution of a wave equation describing the dynamics of two-layere viscoelastic polymer materials of arbitrary thickness deposited on solid (quartz) surfaces in a fluid environment. Within the Voight model of viscoelastic element, we calculate the acoustic response of the system to an applied shear stress, i.e. we find the shift of the quartz generator resonance frequency and of the dissipation factor, and show that it strongly depends on the viscous loading of the adsorbed layers and on the shear storage and loss moduli of the overlayers. These results can readily be applied to quartz crystal acoustical measurements of the viscoelasticity of polymers, which conserve their shape under the shear deformations and do not flow, and layered structures such as protein films adsorbed from solution onto the surface of self-assembled monolayres.
Voinova et al. (Sat,) studied this question.