The Sauerbrey equation was developed by the German Günter Sauerbrey in 1959, while working on his doctoral thesis at TTechnische Universität Berlin, Germany. It is a method for correlating changes in the oscillation frequency of a piezoelectric crystal with the mass deposited on it. He simultaneously developed a method for measuring the characteristic frequency and its changes by using the crystal as the frequency determining component of an oscillator circuit. His method continues to be used as the primary tool in quartz crystal microbalance (QCM) experiments for conversion of frequency to mass and is valid in nearly all applications.
The equation is derived by treating the deposited mass as though it were an extension of the thickness of the underlying quartz. Because of this, the mass to frequency correlation (as determined by Sauerbrey’s equation) is largely independent of electrode geometry. This has the benefit of allowing mass determination without calibration, making the set-up desirable from a cost and time investment standpoint.
The Sauerbrey equation is defined as:
\Deltaf=-
| |||||||
A\sqrt{\rhoq\muq |
}\Deltam
where:
f0
\Deltaf
\Deltam
A
\rhoq
\rhoq
\muq
\muq
The normalized frequency
\Deltaf
\Deltaf/f
If the change in frequency is greater than 5%, that is,
\Deltaf/f
\Deltam | |
A |
=
Nq\rhoq | |
\piZfL |
\tan-1\left[Z\tan\left(\pi
fU-fL | |
fU |
\right)\right]
Equation 2 – Z-match method
fL
fU
Nq
\Deltam
A
\rhoq
\rhoq
Z
=\sqrt{\left(
\rhoq\muq | |
\rhof\muf |
\right)}
\rhof
\muq
\muq
\muf
The Sauerbrey equation was developed for oscillation in air and only applies to rigid masses attached to the crystal. It has been shown that quartz crystal microbalance measurements can be performed in liquid, in which case a viscosity related decrease in the resonant frequency will be observed:
\Deltaf={
3/2 | |
- f | |
0 |
(ηl\rhol/\pi\rhoq\muqn)1/2}
where
\rhol
ηl
n