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Homogeneous Semi-infinite 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

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.

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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)

a T

Shift factor (Sect. 3.7)

A

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

c

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

d P

Interparticle distance (Sect. 9.3)

D

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

eff

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

f

Frequency

f 0

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

f r

Resonance frequency

F̂ inertia

An inertial force (Sect. 9.3)

\( \tilde{G} \)

Shear modulus

K A

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

h r

Characteristic vertical scale of roughness (Sect. 9.5)

k̃

Wavenumber (k̃ = ω/c̃)

liq

As an index: liquid

l r

Characteristic horizontal scale of roughness (Sect. 9.5)

M

Mass

n

Overtone order

p

Pressure (Sect. 9.5)

q

Wave vector (Sect. 9.5)

P

As an index: Particle

r S

A position on the resonator surface

R P

Particle radius

S

As an index: Surface

t

Time

û

(Tangential) displacement

v̂

Velocity

z

Spatial coordinate perpendicular to the surface

Z̃ liq

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

Z̃ L

Load impedance

Z q

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

Γ

Imaginary part of a resonance frequency

δ

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

δ

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

Δ

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

φ

Particle volume fraction

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

Viscosity

ρ

Density

τ hyd

Hydrodynamic time scale (Sect. 9.3)

τ MR

Momentum relaxation time (Sect. 9.3)

ξ

Drag coefficient

ω

Angular frequency

ω c

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

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Johannsmann, D. (2015). Homogeneous Semi-infinite 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_9

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