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The Quartz Crystal Microbalance in Soft Matter Research pp 313–336Cite as

Nonlinear Interactions

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

Abstract

If the stress at the resonator surface is not proportional to the displacement (if there is nonlinear response), the frequency shift depends on the amplitude of oscillation. Nonlinearities can also be detected in the form of steady forces, steady flows, second-harmonic signals, and third-harmonic signals. The chapter gives particular emphasis to the amplitude dependence of the frequency shift. It is analyzed in the frame of a time-averaged periodic stress inserted into the SLA. An application example is the study of partial slip.

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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)

\( \left\langle \ldots \right\rangle \)

An average

0

As an index: amplitude of a real-valued time-harmonic function

a

Instantaneous amplitude (Eq. 13.3.6)

A

(Effective) area of the resonator plate

f

Frequency

f body

A body force (Eq. 13.2.1)

f d

Damped resonance frequency

f 0

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

F

Tangential force

F +, F –

Force at interface while u S increases (+) and decreases (−)

F N

Normal force

F max

Maximum force in a force-displacement loop

F sam

Force exerted onto a resonator by a sample

F x

Tangential force

m q

Mass per unit area of the plate (m q  = ρ q d q  = Z q /(2f 0))

M R

Mass of a resonator

N P

Number of particles per unit area

R P

Particle radius

p

Pressure

ref

As an index: reference state

S

As an index: Surface

st

As an index: steady

t

Time

T

Period of oscillation

T g

Glass temperature

us

As an index: unsteady

u(t)

Displacement

û S , u S

Tangential displacement

u max

Maximum displacement in a force-displacement loop

u N

Normalized displacement (u N  = u S /u 0)

u 0

Amplitude of oscillation (equal to the absolute value of û S )

v

Velocity

x, y, z

Spatial coordinates

x

Displacement of a simple harmonic resonator

y

Auxiliary variable (Eq. 13.3.3)

Z q

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

α

Ratio of the normal and the tangential component of the plate’s motion (depends on position, x and y; Sect. 13.2)

γ D

Damping factor

Γ

Imaginary part of a resonance frequency

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 p ,φv

Phases (Eq. 13.2.3)

η

Viscosity

κ

Spring constant

κ1, κ2, κ3

Coefficients in a power series of force versus displacement (κ1 is the conventional spring constant)

ϕ

Instantaneous phase (Eq. 13.3.6)

μ

Friction coefficient

μ D

Dynamic friction coefficient

ρ

Density

ω

Angular frequency

ω d

Damped angular resonance frequency

ω0

Undamped angular resonance frequency

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Johannsmann, D. (2015). Nonlinear Interactions. 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_13

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