Sauerbrey

The Sauerbrey equation is a formula used in quartz crystal microbalance (QCM) measurements to relate the change in resonant frequency of a quartz crystal oscillator to the mass deposited on its surface. It provides a relatively simple and direct method for determining mass changes at the nanogram level.

The equation, named after German physicist Günter Sauerbrey who first derived it in 1959, is expressed as:

Δf = -2f₀² Δm / (A √(ρµ))

Where:

• Δf is the change in resonant frequency (Hz)
• f₀ is the fundamental resonant frequency of the crystal (Hz)
• Δm is the change in mass on the crystal surface (g)
• A is the piezoelectrically active area of the crystal (cm²)
• ρ is the density of quartz (2.648 g/cm³)
• µ is the shear modulus of quartz (2.947 x 10¹¹ g/(cm·s²))

The Sauerbrey equation is valid under certain assumptions:

• The mass change is uniformly distributed on the crystal surface.
• The mass change is rigid and does not significantly alter the mechanical properties of the crystal (i.e., it behaves as a thin, rigid film).
• The frequency change is small compared to the fundamental resonant frequency.
• The deposited material is not viscoelastic (i.e., it doesn't significantly damp the crystal's oscillation).

When these assumptions are not met, the Sauerbrey equation can lead to inaccurate mass determinations. More complex models, such as those accounting for viscoelasticity or non-uniform mass distribution, may be required in such cases. The Sauerbrey equation is widely used in various fields, including surface science, materials science, chemistry, and biology, for monitoring thin film deposition, adsorption processes, and biomolecular interactions.