Homogeneous Semi-infinite Samples

  • Diethelm Johannsmann
Part of the Soft and Biological Matter book series (SOBIMA)


The load impedance of a homogeneous, semi-infinite medium in contact with the resonator surface is equal to the material’s shear-wave impedance, which leads to the Gordon-Kanazawa-Mason result. For Newtonian liquids the QCM determines the viscosity-density product. If the density is known independently, one can infer the viscosity. The Gordon-Kanazawa-Mason result can be extended to viscoelastic media, in which case the (complex) viscosity is often converted to the complex shear modulus at MHz frequencies. The formulation can be extended to cover nematic liquid crystals, colloidal dispersions, interfaces with shallow surface roughness, and samples, which touch the resonator in the center, only.


Nematic Liquid Crystal Colloidal Dispersion Effective Medium Theory Newtonian Liquid Complex Shear Modulus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.




Definition (Comments)


Shift factor (Sect.  3.7)


(Effective) area of the resonator plate (Sect.  7.4)


Speed of (shear) sound (c̃ = (G̃/ρ)1/2)


Interparticle distance (Sect.  9.3)


Diffusivity (Sect.  9.3, do not confuse with the dissipation factor (1/Q))


As an index: effective, mostly used in the context of an effective medium




Resonance frequency at the fundamental (f 0 = Z q /(2m q ) = Z q /(2ρ q d q ))


Resonance frequency


An inertial force (Sect.  9.3)

\( \tilde{G} \)

Shear modulus


A sensitivity factor (Sect.  9.4, taking care of an amplitude distribution)


Characteristic vertical scale of roughness (Sect.  9.5)

Wavenumber (k̃ = ω/c̃)


As an index: liquid


Characteristic horizontal scale of roughness (Sect.  9.5)




Overtone order


Pressure (Sect.  9.5)


Wave vector (Sect.  9.5)


As an index: Particle


A position on the resonator surface


Particle radius


As an index: Surface




(Tangential) displacement



Spatial coordinate perpendicular to the surface


Acoustic wave impedance of a liquid ( liq  = (iωρ liq η liq )1/2)


Load impedance


Acoustic wave impedance of AT-cut quartz (Z q  = 8.8 × 106 kg m−2s−1)


Imaginary part of a resonance frequency


As a prefix: a small quantity (Fig. 9.5)


Penetration depth of a shear wave (Newtonian liquids: δ = (2η liq /(ρ liq ω))1/2)


As a prefix: A shift induced by the presence of the sample


Particle volume fraction

\( \tilde{\upeta }_{liq} \,\upeta_{liq} \)





Hydrodynamic time scale (Sect.  9.3)


Momentum relaxation time (Sect.  9.3)


Drag coefficient


Angular frequency


A critical frequency, above which inertial effects are noticeable (Sect.  9.3)


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of Physical ChemistryClausthal University of TechnologyClausthal-ZellerfeldGermany

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