# QCM: When is the Sauerbrey equation valid?

Latest updated: April 4, 2024

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As previously introduced [1], in 1959, Sauerbrey [2] was the first to establish a relationship between mass change and the resonant frequency change:

$$\Delta f_n=-n\frac{2f_{0,n}^2}{\sqrt{\mu_\mathrm{q} \rho_\mathrm{q}}}\Delta m_\mathrm{a} \tag{1}\label{eq1}$$

With $\Delta f_n$, the change of resonant frequency at the $n^{\mathrm{th}}$ harmonic in $\mathrm{Hz}$, $n$ the harmonic order, $f_{0,n}$ the resonant frequency at the $n^{\mathrm{th}}$ harmonic in $\mathrm{Hz}$, $\mu_\mathrm{q}$ the shear elastic modulus of the quartz in $\mathrm{kg\,m^{-1}\,s^{-1}}$ or $\mathrm{Pa\,s}$, $\rho_\mathrm{q}$ the quartz density in $\mathrm{kg\,m^{-3}}$, and $\Delta m _\mathrm{a}$ the areal mass of the film in $\mathrm{kg\,m^{-2}}$.

At first, used to monitor mass or thickness of films deposited in vacuum, this relationship can also be used when the quartz and the electrodes are exposed to a solution.

Equation $\eqref{eq1}$ is called the Sauerbrey equation is only valid if the film being dissolved or deposited is considered rigid and thin. Such a film is called a Sauerbrey film.

• “Rigid” means that the acoustic wave will propagate elastically in the film, without any energy loss.

• “Thin” means that the film’s acoustic properties (shear wave modulus and density) can be approximated by the quartz crystal properties. Consequently, the wave velocity in the film is the same as in the crystal.
• A “Thick” film means that its properties have to be accounted for and that the velocity of the wave is different in the film compared to that of the crystal.

Please note that the Sauerbrey equation is also valid to study tightly adsorbed nanoparticles. The Sauerbrey equation is valid as long the sample of interest is negligibly deformed. Check out our partners to performed coupled measurements.

How to know if the film is thin or thick ?

If the frequency shift $\Delta f_n$ is over 2% of the initial resonant frequency $f_{0,n}$, the film should be considered thick. The Sauerbrey relationship cannot be used anymore as the film properties need to be accounted for.

In this case a more complicated relationship needs to be used, that involves a different wave velocity in the quartz and in the film [3]:

$$\Delta m_\mathrm{a}=\frac{\rho_\mathrm{film} v_\mathrm{film}}{2\pi f}\text{arctan}\left(\frac{\rho_\mathrm{q} v_\mathrm{q}}{\rho_\mathrm{film} v_\mathrm{film}} \text{tan}\left(\pi \frac {f_0-f}{f_0}\right) \right) \tag{2}\label{eq2}$$

With $\Delta m_\mathrm{a}$ the areal mass of the film in $\mathrm{kg\,m^{-2}}$, $\rho_\mathrm{film}$ and $\rho_\mathrm{q}$ the density of the film and the quartz, respectively in $\mathrm{kg\,m^{-3}}$, $v_\mathrm{film}$ and $v_\mathrm{q}$ the wave propagation velocity in the film and the quartz, respectively in $\mathrm{m\,s^{-1}}$, $f_0$ and $f$ the resonant frequency of the quartz and the quartz+film composite resonator in $\mathrm{Hz}$.

#### How to know if the film is rigid or not?

Dissipation measurement

To evaluate the rigidity or elasticity of the film, one should look at the change of half bandwidth shift $\Delta \Gamma$ in $\mathrm{Hz}$ between a clean and a coated sensor as shown in Figure 2 of the topic Quartz crystal Microbalance: Measurement principles [4]. A bandwidth shift is considered small when it is smaller than the resonant frequency shift $\Delta f$ as it is the case in Figure 2 of the topic Quartz Crystal Microbalance: Measurement principles [4].

Instead of the half bandwidth change, the dissipation factor change $\Delta D$, expressed as a ratio and not a frequency, is measured using:

$$\Delta D=\frac{2\Delta \Gamma}{f_{01}}\tag{3}\label{eq3}$$

With $f_{01}$ the initial fundamental frequency in $\mathrm{Hz}$ as shown in Figure 1 and Figure 2 in the topic Quartz crystal Microbalance: Measurement principles [4].

In the case of dissipation measurement the criterion for rigidity is:

$$\frac{\Delta D}{\Delta f}≪\frac{1}{f_{01}}\tag{4}\label{eq4}$$

#### Measurements at harmonics/overtones

It is also possible to measure the resonant frequencies at higher harmonics rather than the fundamental one. In the field of acoustic waves, only odd harmonics are measured.

Measuring at harmonics give another way of ensuring that the film coating the bare electrode is rigid. If the value $\Delta f_n/n$ is constant for each harmonic, the film can be considered rigid. More information on overtones measurements and their use and interest are given in the topic Quartz Crystal Microbalance: Why measuring at overtones? [5].

[2] G. Sauerbrey Z. Phys. 155 (1959) 206

[3] T. Pauporté, D. Lincot, in : Microbalance à cristal de quartz, Techniques de l’Ingénieur, (2006) P 2 220.

Quartz Crystal Microbalance Sauerbrey equation gravimetry dissipation overtones

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