Stratified Layer Systems

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


Samples, which are homogeneous in the resonator plane, can be modeled as acoustic multilayers. The deformation pattern is a plane wave. Thin films exposed to air behave as predicted by Sauerbrey. For somewhat thicker films, there is a viscoelastic correction scaling as the square of the film’s mass. For films exposed to a liquid, the viscoelastic correction is independent of thickness. If the layer is soft, the correction can be substantial, even for molecularly thin films. Under certain conditions, the film’s elastic compliance, J f ′, can be calculation from the ratio of ΔΓ and (–Δf). Thick films display a film resonance.


Slip Length Simple Liquid Resonator Surface Sauerbrey Equation Polymer Volume Fraction 
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: apparent


Slip length


Acoustic slip length (Eq. 10.7.5)

Speed of (shear) sound (\( \tilde{c} = (\tilde{G}/\uprho )^{1/2} \))


Thickness of a layer


Thickness of the resonator (\( d_{q} = m_{q} /\uprho_{q} = Z_{q} /(2\uprho_{q} f_{0} ) \))




As an index: film


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


As an index: Film Resonance

\( \tilde{G} \)

Shear modulus


Limiting storage modulus at high frequency

\( \tilde{J} \)

Shear compliance (\( \tilde{J} = 1/\tilde{G} \))

\( \tilde{k} \)

Wavenumber (\( \tilde{k} = \upomega /\tilde{c} \))


As an index: liquid


Mass per unit area


Mass per unit area of the resonator (\( m_{q} = \uprho_{q} d_{q} = Z_{q} /(2f_{0} ) \))


Overtone order

\( \tilde{r} \)

Amplitude reflection coefficient (reflectivity, for short)


As an index: reference state of a crystal in the absence of a load or reference frequency for viscoelastic constants (Eq. 10.4.1)


As an index: Surface


As an index: Slipping Layer



\( \hat{u} \)

(Tangential) displacement

\( {\hat{\rm{v}}} \)



Width of a fuzzy interface (Sect. 10.8)


Point of inflection of a segment density profile (Sect. 10.8)

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

Shear-wave impedance of a liquid (\( \tilde{Z}_{liq} = (\text{i}\upomega \uprho_{liq} \upeta_{liq} )^{1/2} \))

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

Load impedance


Limit of integration range (Sect. 10.8)


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

\( \upbeta^{\prime},\upbeta^{\prime\prime} \)

Power law exponents (Eq. 10.4.1)

\( \Gamma \)

Imaginary part of a resonance frequency

\( \updelta \)

Penetration depth of a shear wave (Newtonian liquids: \( \updelta = (2\upeta_{liq} /(\uprho_{liq} \upomega ))^{1/2} \))


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


Polymer volume fraction (Sect. 10.8)

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

Viscosity \( \tilde{\upeta } = \tilde{G}/({\text{i}}\upomega ) \)



\( \hat{\upsigma } \)

(Tangential) stress

\( \hat{\upsigma }_{s} \)

Tangential stress at the surface, also: “traction”


Relaxation time


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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