Quartz Crystal Microbalance - Small-load Approximation

Small-load Approximation

The BvD circuit predicts the resonance parameters. One can show that the following simple relation holds as long as the frequency shift is much smaller than the frequency itself:

ff is the frequency of the fundamental. Zq is the acoustic impedance of material. For AT-cut quartz, its value is Zq = 8.8·106 kg m−2 s−1.

The small-load approximation is central to the interpretation of QCM-data. It holds for arbitrary samples and can be applied in an average sense. Assume that the sample is a complex material, such as a cell culture, a sand pile, a froth, an assembly of spheres or vesicles, or a droplet. If the average stress-to-speed ratio of the sample at the crystal surface (the load impedance, ZL) can be calculated in one way or another, a quantitative analysis of the QCM experiment is in reach. Otherwise, the interpretation will have to remain qualitative.

The limits of the small-load approximation are noticed either when the frequency shift is large or when the overtone-dependence of Δf and Δ(w/2) is analyzed in detail in order to derive the viscoelastic properties of the sample. A more general relation is

This equation is implicit in Δf*, and must be solved numerically. Approximate solutions also exist, which go beyond the small-load approximation. The small-load approximation is the first order solution of a perturbation analysis.

The definition of the load impedance implicitly assumes that stress and speed are proportional and that the ratio therefore is independent of speed. This assumption is justified when the crystal is operated in liquids and in air. The laws of linear acoustics then hold. However, when the crystal is in contact with a rough surface, stress can easily become a nonlinear function of strain (and speed) because the stress is transmitted across a finite number of rather small load-bearing asperities. The stress at the points of contact is high, and phenomena like slip, partial slip, yield, etc. set in. These are part of non-linear acoustics. There is a generalization of the small-load equation dealing with this problem. If the stress, σ(t), is periodic in time and synchronous with the crystal oscillation one has

\frac{\Delta f}{f_f}=\frac 1{\pi Z_q}\,\frac 2{\omega u_0}\left\langle \sigma \left( t\right) \cos \left( \omega t\right) \right\rangle _t

\frac{\Delta (w/2) }{f_f}=\frac 1{\pi Z_q}\,\frac 2{\omega u_0}\left\langle \sigma \left( t\right) \sin \left( \omega t\right) \right\rangle _t

Angular brackets denote a time average and σ(t) is the (small) stress exerted by the external surface. The function σ(t) may or may not be harmonic. One can always test for nonlinear behavior by checking for a dependence of the resonance parameters on the driving voltage. If linear acoustics hold, there is no drive level-dependence. Note, however, that quartz crystals have an intrinsic drive level-dependence, which must not be confused with nonlinear interactions between the crystal and the sample.

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