The problem of uniqueness of fit for viscoelastic films on thickness-shear mode resonator surfaces

We describe a new strategy for interpreting frequency responses of thickness shear mode resonators loaded with spatially uniform viscoelastic films. This procedure leads to unambiguous extraction of the four parameters that characterize such a film: its thickness, density and shear modulus components (storage and loss moduli). The interpretational difficulty is that the experimental frequency response (impedance spectrum) can only provide two parameters; thus, the problem is underdetermined, Previous interpretations employed various approximations and assumptions for two (or more) film parameters to effectively reduce the problem to a two-parameter fit. Such approaches are clearly imperfect. Our new strategy splits the problem into two separate two-parameter subproblems, each of which is solved by the measurement of two different experimental responses. The result is a unique fit to the data without the need to make approximations or assumptions for film parameters. First, in the acoustically thin regime, measured frequency shift and film charge are combined to provide a unique solution for film thickness and density; shear moduli components do not affect the response in this regime. Second, film density is carried forward directly, and the film thickness-charge relationship is extrapolated into the acoustically thick regime. Third, with film density and thickness held fixed, crystal impedance data in the acoustically thick regime provide unambiguous shear modulus components. The method is generalized to any other (nonelectrochemical) probe that provides film thickness data and validated using crystal impedance data for poly(3-methylthiophene) films exposed to propylene carbonate.