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Heterogeneous Samples

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The Quartz Crystal Microbalance in Soft Matter Research

Part of the book series: Soft and Biological Matter ((SOBIMA))

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Abstract

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.

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Author information

Authors and Affiliations

  1. Institute of Physical Chemistry, Clausthal University of Technology, Clausthal-Zellerfeld, Germany

    Diethelm Johannsmann

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  1. Diethelm Johannsmann
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Correspondence to Diethelm Johannsmann .

Glossary

Variable

Definition (Comments)

3PL

As an index: 3-Phase Line

A D

Area of a droplet

b S

Slip length (see also Sect. 10.7)

Ca

Capillary number

D

Dissipation factor (D = 2Γ/f r )

D

As an index: Droplet

f

Frequency

f 0

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

G

Shear modulus

h

Height of an adsorbate layer

k

Wavenumber

liq

As an index: liquid

n

Overtone order

L D

Perimeter of a droplet

p

Pressure

r D

Radius of a droplet

r

Position (a vector)

R

Radius of a liposome (Fig. 12.12)

S

As an index: Surface

t

Time

T

Temperature

T m

Melting temperature of a lipid membrane

u, u

Tangential displacement (when bold: a vector)

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

Velocity

vis

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

γ S

Surface energy

Γ

Imaginary part of a resonance frequency

δ

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

δ L

Loss angle

Δ

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

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

Viscosity

θ

Coverage

ρ

Density

σ

Tangential stress

τ r

Emulsion time (A relaxation time of emulsions and droplets)

ω

Angular frequency

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Johannsmann, D. (2015). Heterogeneous Samples. In: The Quartz Crystal Microbalance in Soft Matter Research. Soft and Biological Matter. Springer, Cham. https://doi.org/10.1007/978-3-319-07836-6_12

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