# Sauerbrey Equation

## Introduction

The Sauerbrey equation is a linear relation between the changes in the resonant frequency of a quartz crystal and the mass of a thin rigid film added to its surface [1]. It is widely used in the quartz crystal microbalance applications because the frequency to mass conversion depends only on the geometrical and physical characteristics of the quartz crystal.

The Sauerbrey equation is based on the assumption that the added mass causes a thickness increase of the quartz crystal itself, in other words it treats the added mass and the quartz crystal as a whole element. The linearity of the Sauerbrey equation is valid if:

• the added mass and its thickness are much smaller than those of the quartz crystal;
• the added mass is uniform, rigid and integral to the quartz crystal;
• the quartz crystal vibration takes place in vacuum or in air.

Increasing the quartz crystal thickness causes the increase in the stationary wavelength travelling through the quartz crystal, resulting in the decrease of the resonant frequency.

The Sauerbrey equation relates the decrease in resonant frequency $\Delta f$ to the added mass $\Delta m$ as follows:

$\Delta f = -\frac{2f_0^2}{\sqrt{ \rho_Q \mu_Q } }\cdot \frac{\Delta m}{A} = -\frac{2f_0^2}{\rho_Q v_Q}\cdot \frac{\Delta m}{A} \qquad (1)$

where $f_0$ is the resonant frequency, $A$ is the active electrode area, $\rho_Q$ is the density, $\mu_Q$ is the shear modulus and $v_Q$ is the shear wave velocity of the quartz crystal. In particular for an AT-cut quartz crystal we have:

Density $\rho_Q = 2.643 \ g \ cm^{- 3}$
Shear modulus $\mu_Q = 2.947 \times 10^{11} \ g \ cm^{- 1} \ s^{-2}$
Wave velocity $v_Q = \sqrt{\frac{\mu_Q}{\rho_Q}}=3.340 \times 10^5 \ cm \ s^{-1}$

The proportionality factor in Sauerbrey equation depends only on density and shear modulus of the quartz crystal. The Sauerbrey equation is the main tool in quartz crystal microbalance because it has the benefit of weighing mass without calibration with an easy experimental setup. But the frequency to mass linear relationship as some limitations. The Sauerbrey equation does not hold for viscous medium deposited on the quartz crystal. In this case the frequency variation is mainly due to the energy losses caused by the viscous friction.

## Simple Derivation of Sauerbrey Equation

The resonance phenomena in quartz crystal correspond to a stationary shear wave propagating along the quartz thickness, the resonance occurs when the quartz thickness is equal to the half length of the stationary wave:

$t_Q = \frac{\lambda_0}{2}$

The shear wave velocity in quartz crystal is given by:

$v_Q = \sqrt{\frac{\mu_Q}{\rho_Q}}$

where $\mu_Q$ is the shear modulus and $\rho_Q$ is the density of the quartz crystal. The resonance frequency is given by:

$f_0 = \frac{v_Q}{\lambda_0} = \frac{1}{2t_Q}\sqrt{\frac{\mu_Q}{\rho_Q}}$

A thin rigid layer added on the surface increases the total thickness of the quartz crystal causing an increase of the stationary wavelength in order to propagate inside the rigid layer. As final result, the deposition of a thin rigid mass on the surface of the quartz crystal causes the decrease of the resonant frequency. A simple calculation can lead to a linear relationship between the added mass and the change in the resonant frequency. By differentiating the last equation respect to $\Delta t$ one has:

$\Delta f = - \frac{1}{2t_Q^2}\sqrt{ \frac{\mu_Q}{\rho_Q} }\Delta t = - \frac{\Delta t}{t_Q}f_0$

By supposing valid the approximation that the thickness relative variation is equal to that of the quartz crystal mass, by defining $A$ the surface and $\rho_Q$ the density of the quartz crystal one can write:

$\Delta f = - \frac{\Delta m}{A \rho_Q t_Q}f_0 = - \frac{\Delta m}{A \rho_Q t_Q} \cdot \frac{ f_0^2 }{ \frac{1}{2t_Q}\sqrt{ \frac{\mu_Q}{\rho_Q} } } \quad \Rightarrow$

$\Delta f = -\frac{2f_0^2}{\sqrt{ \rho_Q \mu_Q } }\cdot \frac{\Delta m}{A}$

the last relation is known as Sauerbrey equation, it directly relates the change in resonance frequency to the added mass.

## Mass Sensitivity

Sensitivity of quartz crystal microbalance is measured using laser ablation along the quartz diameter. a) images of 12 laser ablation b) the corresponding frequency variations can be fitted by Gaussian function (Costas B. Grigoropoulos, Berkeley 1997, Applied Physics A, DOI: 10.1007/s003390050514)

In quartz crystal microbalance applications it is common use to define the frequency to mass conversion factor $S_Q$ as Mass Sensitivity:

$S_Q = \frac{2f_0^2}{\sqrt{ \rho_Q \mu_Q } } = \frac{2f_0^2}{\rho_Q v_Q}\qquad (2)$

which is measured in CGS unit $[S_Q] = cm^2 \ g^{-1} \ s^{-1}$. By inserting eq. (2) in (1), the Sauerbrey equation can be written as:

$\Delta f = -S_Q \cdot \frac{\Delta m}{A}\qquad (3)$

By means of the eq. (3) a quartz crystal microbalance can directly relate the frequency shift to the mass per unit area $\frac{\Delta m}{A}$. For example, an At-cut quartz crystal vibrating at $f_0 = 10 MHz$ has a mass sensitivity equal to $S_Q = 2.26 \times 10^8cm^2 \ g^{-1} \ s^{-1}$. A frequency variation $\Delta f = 1 Hz$ corresponds to an added mass per unit area equal to $\frac{\Delta m}{A} = 4.42 \ ng \ cm^{-2}$

The mass sensitivity is not uniform over the entire quartz surface, it has a maximum at the centre and decreases near the the edges of the electrodes. The sensitivity of quartz crystal microbalance can be investigated by laser ablation along the diameter of the quartz [2].

Moreover the spatial distribution of the local mass sensitivity can be measured using calibration beads or by sensing the quartz surface with an AFM - Atomic Force Microscope. The experimental results show that the mass sensitivity has the same spatial distribution of the quartz vibration amplitude and that they are well fitted by Gaussian functions.

In particular, the mass sensitivity $S_Q$ is proportional to the root of the radial amplitude $u(r)$:

$S_Q(r) \propto \left\vert u(r) \right\vert^2$

The radial amplitude is well fitted by a Gaussian function:

$u(r) = u_{max} \exp \left ( -a \frac{r^2}{R^2} \right )$

$R$ is the electrode radius, $r$ is the radial coordinate, $u_{max}$ is the maximum amplitude at the centre of the crystal $r = 0$ and $a$ is the coefficient measuring the distribution width. [3]

## References

1. Sauerbrey G., "Verwendung von Schwingquarzen zur Wägung dünner Schichten und zur Mikrowägung", Zeitschrift Für Physik, vol. 155, no. 2, pp. 206-222, 1959. DOI 110.1007/BF01337937
2. Zhang X. et al. "Excimer laser ablation of thin gold films on a quartz crystal microbalance at various argon background pressures", Applied Physics A, vol. 64, pp. 545-552, 1997. DOI 10.1007/s003390050514
3. Mecea, V. M. “From Quartz Crystal Microbalance to Fundamental Principles of Mass Measurements” Analytical Letters, vol. 38, no. 5, pp. 753-767, 2005. DOI 10.1081/AL-200056171