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The Quartz Crystal Microbalance in Soft Matter Research pp 247–286Cite as

Point Contacts and Contact Stiffness

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

Abstract

If an external object (such as a sphere) touches the resonator surface across a contact with a diameter much below the wavelength of sound and, also, much below the size of the particle, one can infer the stiffness of the contact from the frequency shift. The situation is particularly transparent for particles, which are heavy enough to be clamped in space by inertia, so that they do not follow MHz motion of the resonator. In this case, the frequency shift is positive and proportional to the contact stiffness. Smaller particles give rise to coupled resonances. Coupled resonances can be viewed as absorption lines in shear-wave spectroscopy.

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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)

0

As in index: undamped (Exception: f 0)

a

Contact radius

A

(Effective) area of the resonator plate (See Sect. 7.4)

b̃ sc

Scattering length (Sect. 11.7)

b shear , b bend

Numerical coefficients

(Eq. 11.5.4)

D

Diffusivity (Sect. 11.6)

E

Young’s modulus

E*

Effective Young’s modulus (Eq. 11.2.2)

f

Frequency

f OS

Oscillator strength (a number, not a frequency)

f ZC

Frequency of zero-crossing (Eq. 11.4.7)

f 0

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

F N

Normal force

F x

Tangential force

g

Standard acceleration (g = 9.81 m/s2)

G

Shear modulus

G*

Effective shear modulus (Eq. 11.2.12)

het

As an index: heterogeneously broadened (Sect. 11.4)

intf

As an index: interface (Sect. 11.8)

I

Moment of area (Eq. 11.2.12)

k

Wavenumber

k B T

Thermal energy

liq

As an index: liquid

M

Mass

n

Overtone order

N P

Number of particles per unit area

p

Normal stress

P

As an index: Particle

PD

As an index: Polar Diagram

r

Distance from the center of a contact, distance from a scattering center

rock

As in index: rocking mode

rot

As in index: rotational mode

R P

Particle radius

R PD

Radius of circle in polar diagram (Eq. 11.5.7)

r̃ q,S

Reflectivity evaluated at the resonator surface (Sect. 11.7)

r S

Location at the resonator surface

r S

In-plane distance from the scattering center (Sect. 11.7)

S

As an index: Surface

S̃

Contact stiffness (Sect. 11.8)

s c

As an index: scattered (Sect. 11.7)

û,u

Tangential displacement

v̂

Velocity

Z̃ L

Load impedance

Z q

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

γ P

Damping factor of a coupled resonance (Sect. 11.4)

γ S

Surface energy (Sect. 11.2)

Γ

Imaginary part of a resonance frequency

δ N

Normal compression (Fig. 11.2)

δ scm

Scattering phase (Sect. 11.7)

Δ

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

k

Spring constant

µ

Friction coefficient (µ = F x /F N )

ν

Poisson number

θ

Angle of rotation

ρ

Density

σ

Tangential stress

τ MR

Momentum relaxation time (also: slip time Sect. 11.6)

τ S

A constant tangential stress in the sliding zone of a contact experiencing partial slip (Sect. 11.2)

ξ

Drag coefficient

ω

Angular frequency

ω P

Particle resonance frequency

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Johannsmann, D. (2015). Point Contacts and Contact Stiffness. 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_11

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