Heterogeneous Samples

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


When the sample is structured in the plane of the resonator with a characteristic scale comparable to the wavelength of sound, analytical predictions of the displacement field and the frequency shift are difficult. Among the samples that are heterogeneous in this sense are nanobubbles, nanodroplets, nanoparticles, vesicles, and biological cells. In analyzing such samples, one can rely on common sense and empirical correlations. If one wants to go beyond those more qualitative pictures, one can calculate the area-averaged periodic stress at the resonator surface numerically. An example of a numerical method is discussed in detail. The finite element method (FEM) is employed to solve the incompressible Stokes problem and to predict the periodic interfacial stress. The frequency shift follows from the area-averaged stress and the SLA.


Frequency Shift Capillary Pressure Capillary Number Slip Length Stoke Flow 
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)


As an index: 3-Phase Line


Area of a droplet


Slip length (see also Sect.  10.7)


Capillary number


Dissipation factor (D = 2Γ/f r )


As an index: Droplet




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


Shear modulus


Height of an adsorbate layer




As an index: liquid


Overtone order


Perimeter of a droplet




Radius of a droplet


Position (a vector)


Radius of a liposome (Fig. 12.12)


As an index: Surface






Melting temperature of a lipid membrane

u, u

Tangential displacement (when bold: a vector)

v, v, \( {\hat{\text{v}}} \), \( {\hat{\mathbf{v}}} \)



As an index: viscous

x, y, z

Spatial coordinates, z: along the surface normal

\( \tilde{Z}_{liq} \)

Shear-wave impedance of a liquid (Z̃ liq  = (iωρ liq η liq )1/2)

\( {\dot{\upgamma }} \)

Shear rate


Surface energy


Imaginary part of a resonance frequency


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


Loss angle


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

\( {\upeta ,\tilde{\upeta }} \)







Tangential stress


Emulsion time (A relaxation time of emulsions and droplets)


Angular frequency


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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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