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 = ?f^{$\text{3}\text{2}$}₀(η_L?_L/πμ_Q?_Q)^{$\text{1}\text{2}$}, where f₀ is the oscillation frequency of the free (dry) crystal, η_L and ?_L are the absolute viscosity and density of the liquid, respectively, and μ_Q and ?_Q 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.