Kanazawa equation

In passing, we note that this approach, in fact, is the only way to determine the viscoelastic parameters of rubbery polymers with the QCM. hi principle, one might of course also coat the entire crystal with a thick layer of the respective material. If the thickness of the layer is larger than the penetration depth, the sample is acoustically semi-infinite. The Kanazawa equation... 

If the quartz crystal is put in contact with a viscous liquid (Newtonian liquid) at one of its surfaces, then the Kanazawa equation applies ... 

Mason [46] first observed that the viscoelastic properties of a fluid in contact with quartz crystals can affect the resonant properties. However, Mason s work had been forgotten and for a long time there have not been studies of piezoelectric acoustic wave devices in contact with liquids until Nomura and Okuhara [15] found an empirical expression that described the changes in the quartz resonant frequency as a function of the liquid density, its viscosity and the conductivity in which the crystal was immersed. Shortly after the empirical observations of Nomura were described in terms of physical models by Kanazawa [1] and Bruckenstein [2] who derived the equation that describes the changes in resonant frequency of a loss-less quartz crystal in contact with an infinite, non conductive and perfectly Newtonian fluid ... 

Kanazawa et al. [20] and their calculations are valid for rigid films immersed in liquid. However, this work is not valid for non-rigid soft materials. For soft material layers it was even noticed that in applying Sauerbrey s equation the mass of the viscoelastic layer is underestimated and the result is a missing mass which was elucidated in calculations by Voinova et al. [21],... 

The motional inductance La, representing the kinetic energy of die entrained liquid layer (with effective thickness 812), leads to a decrease in the series resonant frequency [14,17] from Equation 3.23 in agreement with the prediction of Kanazawa and Gordon [18] ... 

In addition to deriving the equation that now bears his name, Sauerbrey also developed a method for measuring the characteristic frequency and its changes by using the crystal as the frequency-determining component of an oscillator circuit. It should be mentioned that the Sauerbrey equation was developed for oscillation in air and only applies to rigid masses attached to the crystal. However, Kanazawa and Gordon have shown that quartz crystal microbalance measurements can be performed in liquid, in which case a viscosity-related decrease in the resonant frequency is observed ... 

Perhaps the most general mathematical treatment of the surface mass loading effect on bulk shear wave resonators has been presented by Kanazawa (13). In this work, a wave equation was developed for acoustic wave propagation within the deposited layer, assuming the material had both elastic and viscous properties. Boundary conditions between crystal and deposited mass were established by assuming shear forces and particle displacements were equal for both materials at the interface plane. This approach results in a fairly complex mathematical model, but simplified relationships were derived for purely elastic and purely viscous behaviour. 

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