Sauerbrey equation
Encyclopedia
The Sauerbrey equation was developed by G. Sauerbrey in 1959 as 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
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:
.
Equation 1 – Sauerbrey’s equation
– Resonant frequency (Hz)
– Frequency change (Hz)
– Mass change (g)
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
< 0.02.
If the change in frequency is greater than 2%, that is,
> 0.02, the Z-match method must be used to determine the change in mass.
The formula for the Z-match method is:
Equation 2 – Z-match method
– Frequency of loaded crystal (Hz)
– Frequency of unloaded crystal, i.e. Resonant frequency (Hz)
– Frequency constant for AT-cut quartz crystal (1.668x1013Hz.Å)
– Mass change (g)
related decrease in the resonant frequency will be observed:
where
is the density of the liquid and 
is the viscosity of the liquid (Kanazawa and Gordon 1985).
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
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
Quartz crystal microbalance
A quartz crystal microbalance measures a mass per unit area by measuring the change in frequency of a quartz crystal resonator. The resonance is disturbed by the addition or removal of a small mass due to oxide growth/decay or film deposition at the surface of the acoustic resonator...
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:
.Equation 1 – Sauerbrey’s equation
– Resonant frequency (Hz) – Frequency change (Hz) – Mass change (g)
- A – Piezoelectrically active crystal area (Area between electrodes, cm2)
– DensityDensityThe mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of quartz (
= 2.648 g/cm3)
– Shear modulus of quartz for AT-cut crystal (
= 2.947x1011 g/cm.s2)
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
< 0.02.If the change in frequency is greater than 2%, that is,
> 0.02, the Z-match method must be used to determine the change in mass.The formula for the Z-match method is:
Equation 2 – Z-match method
– Frequency of loaded crystal (Hz) – Frequency of unloaded crystal, i.e. Resonant frequency (Hz) – Frequency constant for AT-cut quartz crystal (1.668x1013Hz.Å) – Mass change (g)
- A – Piezoelectrically active crystal area (Area between electrodes, cm2)
– Density of quartz (
= 2.648 g/cm3) - Z –
– Shear modulus of quartz (
= 2.947x1011 g/cm.s2)
- Shear modulus of film (Varies: units are g/cm.s2)
Limitations
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 viscosityViscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
related decrease in the resonant frequency will be observed:
where
is the density of the liquid and is the viscosity of the liquid (Kanazawa and Gordon 1985).