The Small Load Approximation Revisited

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

Abstract

The chapter contains two extensions of the SLA. The first (Sect. 6.1) generalizes the SLA to arbitrary resonator shapes and modes of vibration. The load impedance in this formulation is a 3rd-rank tensor. The formalism shows that the statistical weight in area-averaging is the square of the local amplitude. The second extension (Sect. 6.2) is a perturbation analysis, applied to the model of the parallel plate. The perturbation is carried to 3rd order. The 3rd-order result fixes an inconsistency obtained when treating viscoelastic thin films in air with the conventional SLA.

Notes

Glossary

Variable

Definition

\( \left\langle . \right\rangle \)

Average

[.]

As a superscript: perturbation order

A

Effective area of the resonator plate

A

ω2-operator

cijkl

Stiffness tensor

comp

As an index: related to compressional waves

cD

Stiffness tensor at constant electric displacement (Eq. 6.1.33)

cr

As an index: exerted by the crystal

df

Film thickness

diss

As an index: caused by dissipative processes

dq

Thickness of the resonator

\( {\hat{\mathbf{D}}},{\mathbf{D}} \)

Electric displacement

e

As an index: electrode

\( {\hat{\mathbf{E}}},{\mathbf{E}} \)

Electric field

fn

Resonance frequency at overtone order n

fr

Resonance frequency

f0

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

Gq

Shear modulus of AT-cut quartz (Gq ≈ 29 × 109 Pa)

h

One of the tensors quantifying piezoelectric coupling (Eq. 6.1.33)

k

Wavenumber

hq

Half the thickness of the resonator plate

I

Second moment of area

Shear compliance

L

Width of a resonator plate

liq

As an index: liquid

mf

Mass per unit area of a film

mq

Mass per unit area of the resonator (mq = ρqdq = Zq/(2f0))

n

Overtone order

n

ni, Surface normal (a vector)

OC

As an index: resonance condition of the unloaded plate under open-circuit conditions. With no current into the electrodes (more precisely, with vanishing electric displacement everywhere), piezoelectric stiffening is fully accounted for by using the piezoelectrically stiffened shear modulus

q

As an index: quartz resonator

r, ri

Position, a vector and its components

ref

As an index: reference state of a crystal in the absence of a load

rS, rS,i

Position on the resonator surface, a vector and its components

S

As an index: Surface

S

Infinitesimal strain tensor (Eq. 6.1.33)

T

Temperature

T

Stress tensor (Eq. 6.1.33)

û

Displacement (a vector)

Velocity (a vector)

L,ijk

Load impedance in tensor form

mot

Impedance to the left of the transformer in the Mason circuit (Fig. 4.10)

Zq

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

Γ

Imaginary part of a resonance frequency

δαβ

Kronecker δ (δαβ = 1 if α = β, δαβ = 0 otherwise)

δ(.)

Dirac δ-function

εS−1

Inverse dielectric permittivity at constant strain (a tensor) (Eq. 6.1.33)

Δ

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

\( {\hat{\upsigma }}_{S,ij} \)

Stress tensor

ηliq

Viscosity

μ

Non-dimensional mass (Eq. 6.2.9)

ρ

Density

ξliq

Nondimensional shear-wave impedance of the bulk liquid (Eq. 6.2.9)

ω

Angular frequency

ωr

Angular resonance frequency (–ωr2 is the eigenvalue of the ω2-operator)

ζ

Non-dimensional measure of the inverse square shear-wave impedance (Eq. 6.2.9)

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of Physical ChemistryClausthal University of TechnologyClausthal-ZellerfeldGermany

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