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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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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DOI: https://doi.org/10.1007/978-3-319-07836-6_11
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