# Stratified Layer Systems

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

## Abstract

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.

## Keywords

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.

## Glossary

Variable

app

As an index: apparent

bsl

Slip length

bsl,ac

Acoustic slip length (Eq. 10.7.5)

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

d

Thickness of a layer

dq

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

f

Frequency

f

As an index: film

f0

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

FR

As an index: Film Resonance

$$\tilde{G}$$

Shear modulus

G

Limiting storage modulus at high frequency

$$\tilde{J}$$

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

$$\tilde{k}$$

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

liq

As an index: liquid

m

Mass per unit area

mq

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

n

Overtone order

$$\tilde{r}$$

Amplitude reflection coefficient (reflectivity, for short)

ref

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

S

As an index: Surface

SL

As an index: Slipping Layer

t

Time

$$\hat{u}$$

(Tangential) displacement

$${\hat{\rm{v}}}$$

Velocity

w

Width of a fuzzy interface (Sect. 10.8)

zi

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

$$\tilde{Z}_{liq}$$

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

$$\tilde{Z}_{L}$$

zmax

Limit of integration range (Sect. 10.8)

Zq

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: $$\updelta = (2\upeta_{liq} /(\uprho_{liq} \upomega ))^{1/2}$$)

Δ

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

φ

Polymer volume fraction (Sect. 10.8)

$$\tilde{\upeta },\upeta$$

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

ρ

Density

$$\hat{\upsigma }$$

(Tangential) stress

$$\hat{\upsigma }_{s}$$

Tangential stress at the surface, also: “traction”

τ

Relaxation time

ω

Angular frequency

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