Stratified Layer Systems

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.

Glossary

Variable

Definition (Comments)

app

As an index: apparent

b _sl

Slip length

b _sl,ac

Acoustic slip length (Eq. 10.7.5)

c̃

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

d

Thickness of a layer

d _q

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

f

Frequency

f

As an index: film

f ₀

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

m _q

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)

z _i

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

z _max

Limit of integration range (Sect. 10.8)

Z _q

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