The Sauerbrey equation was developed by the German
Günter Sauerbrey in 1959, while working on his doctoral thesis at the
Technical University of Berlin, Germany. It is a method for correlating changes in the oscillation frequency of a
piezoelectric
Piezoelectricity (, ) is the electric charge that accumulates in certain solid materials—such as crystals, certain ceramics, and biological matter such as bone, DNA, and various proteins—in response to applied mechanical stress. The word '' ...
crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
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:
:
where:
:
–
Resonant frequency of the fundamental mode (Hz)
:
– normalized frequency change (Hz)
:
– Mass change (g)
:
–
Piezoelectrically active crystal area (Area between electrodes, cm
2)
:
–
Density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
of quartz (
= 2.648 g/cm
3)
:
–
Shear modulus
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:
:G \ \stackre ...
of quartz for AT-cut crystal (
= 2.947x10
11 g·cm
−1·s
−2)
The normalized frequency
is the nominal frequency shift of that mode divided by its mode number (most software outputs normalized frequency shift by default). Because the film is treated as an extension of thickness, Sauerbrey’s equation only applies to systems in which the following three conditions are met: the deposited mass must be rigid, the deposited mass must be distributed evenly and the frequency change
< 0.05.
If the change in frequency is greater than 5%, that is,
> 0.05, the Z-match method must be used to determine the change in mass.
The formula for the Z-match method is:
: