The oscillation frequency of a quartz resonator in contact with liquid

A simple relationship is derived which expresses the change in oscillation frequency of a quartz crystal in contact with a fluid in terms of material parameters of the fluid and the quartz. The relationship is Δf = −f320(ηLϱL/πμQϱQ)12, where f0 is the oscillation frequency of the free (dry) crystal, ηL and ϱL are the absolute viscosity and density of the liquid, respectively, and μQ and ϱQ are the elastic modulus and density of the quartz. This relation is obtained from a simple physical model which couples the shear wave in the quartz to a damped shear wave in the fluid. Quantitative comparisons with two test cases, aqueous solutions of glucose and ethanol at various concentrations, demonstrate the accuracy of this model.