# Homogeneous Semi-infinite Samples

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

## 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.

## Notes

### Glossary

Variable

aT

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)

dP

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

f0

Resonance frequency at the fundamental (f0 = Zq/(2mq) = Zq/(2ρqdq))

fr

Resonance frequency

inertia

An inertial force (Sect. 9.3)

$$\tilde{G}$$

Shear modulus

KA

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

hr

Characteristic vertical scale of roughness (Sect. 9.5)

Wavenumber (k̃ = ω/c̃)

liq

As an index: liquid

lr

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

rS

A position on the resonator surface

RP

S

As an index: Surface

t

Time

û

(Tangential) displacement

Velocity

z

Spatial coordinate perpendicular to the surface

liq

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

L

Zq

Acoustic wave impedance of AT-cut quartz (Zq = 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 = (2ηliq/(ρliqω))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

τhyd

Hydrodynamic time scale (Sect. 9.3)

τMR

Momentum relaxation time (Sect. 9.3)

ξ

Drag coefficient

ω

Angular frequency

ωc

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

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