Chapter

The Quartz Crystal Microbalance in Soft Matter Research

Part of the series Soft and Biological Matter pp 49-123

Date:

Modeling the Resonator as a Parallel Plate

  • Diethelm JohannsmannAffiliated withInstitute of Physical Chemistry, Clausthal University of Technology Email author 

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Abstract

After an introduction to complex resonance frequencies, the chapter provides a thorough discussion of the acoustic impedance, the acoustic wave impedance, and other types of impedances with relevance to either the QCM itself or to related problems. The load impedance (the complex ratio of the amplitudes of periodic stress and periodic velocity, both evaluated at the resonator surface) is, what the QCM measures on a fundamental level. The description continues with three separate but equivalent ways of modeling the viscoelastic plate and its resonances. All three formulations have their benefits and drawbacks. Building on these models, it is proven that the complex frequency shift is proportional to the complex load impedance, which is the essence of the small load approximation. The load impedance can be averaged over area and time. The last section deals with samples, which themselves are small resonators with their own resonance frequency. In the presence of such “coupled resonances”, the frequency shift may be positive or negative, depending on whether the resonance frequency of the coupled resonator is smaller or larger than the resonance frequency of the crystal.