# Practical Consequences of Piezoelectric Stiffening

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

## Abstract

Piezoelectric stiffening can be exploited to probe the sample’s electrical impedance. Conversely, the electrical impedance of the sample and the circuitry around the crystal can influence on the resonance frequency. Some precautions must be taken to avoid the corresponding artifacts.

## Notes

### Glossary

Variable

A

(Effective) area of the resonator plate

C0

Parallel capacitance

dq

Thickness of the resonator (dq = mqq = Zq/(2ρqf0))

e26

Relevant component of the e-tensor (piezoelectric stress coefficient, e26 = 9.65 × 10−2 C/m2 for AT-cut quartz)

f0

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

Gq

Shear modulus of AT-cut quartz (Gq ≈ 29 × 109 Pa for AT-cut quartz. Gq is the piezoelectrically stiffened modulus)

kt

Piezoelectric coupling coefficient(kt = (e26/(εqε0Gq))1/2, kt2 = 0.8 % for AT-cut quartz)

PE

As an index: PiezoElectric stiffening

q

As an index: quartz resonator

ref

As an index: reference

R

Resistance

Rbulk

Resistance of a bulk liquid

R1, R2, R3

Elements of the pi-network (Fig. 14.2)

$$\tilde{Y}_{ex}$$

External electrical admittance ($$\tilde{Y}_{ex} = 1/\tilde{Z}_{ex}$$)

$$\tilde{Z}_{ex}$$

External electrical impedance

Zq

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

Δ

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

$$\tilde{\upvarepsilon }_{q} ,\,\upvarepsilon_{q}$$

Dielectric constant of AT-cut quartz (εq is the clamped dielectric constant, εq = 4.54 for AT-cut quartz)

ε0

Dielectric permittivity of vacuum (ε0 = 8.854 × 10−12 C/(Vm))

ϕ

Factor converting between mechanical and electric quantities in the Mason circuit (ϕ = Ae26/dq)

ρq

Density of crystalline quartz (ρq = 2.65 g/cm3)

ω

Angular frequency

### References

1. 1.
Shana, Z.A., Zong, H., Josse, F., Jeutter, D.C.: Analysis of electrical equivalent-circuit of quartz-crystal resonator loaded with viscous conductive liquids. J. Electroanal. Chem. 379(1–2), 21–33 (1994)
2. 2.
Shana, Z.A., Josse, F.: Quartz-crystal resonators as sensors in liquids using the acoustoelectric effect. Anal. Chem. 66(13), 1955–1964 (1994)
3. 3.
Zhang, C., Vetelino, J.F.: Chemical sensors based on electrically sensitive quartz resonators. Sens. Actuators B-Chemical 91(1–3), 320–325 (2003)
4. 4.
Gileadi E.: Physical Electrochemistry: Fundamentals, Techniques and Applications: A Textbook for Students of Science and Engineering. Wiley (2011)Google Scholar
5. 5.
Johannsmann, D., Bucking, W., Bode, B., Petri, J.: Simple frequency-based sensing of viscosity and dielectric properties of a liquid using acoustic resonators. IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control 57(3), 677–683Google Scholar