3.4 Stability measurements 61
stress in a crystal generates an electric potential due to a net change in dipole
moment. The opposite eect is the generation of atomic displacement due to
an applied electric eld. This means that an oscillating potential can induce
acoustic resonances with a specic frequency. This frequency can be determined
with high precision. The relationship between the frequency change, f, and
mass, m, is described in the Sauerbrey equation 202:
f = ��
2f2
0
A
p
qq
m (3.8)
where f0 is the nominal frequency of the crystal, A the piezo-active area, q
the density of quartz and q the shear modulus of an AT-cut quartz crystal. In
practice a simplied version of the equation can be used:
f = ��Cf m (3.9)
where Cf is a constant that can be calibrated. The crystals used for this thesis
had a nominal Cf of 56.6 Hz/gcm2. Calibration of Cf was carried out using
silver electrodeposition 150, 203. A clean EQCM substrate was immersed in
a 0.5 M HNO3 solution with 50 mM AgNO3 and -50 A were passed until a
frequency change of 60 Hz was reached. For silver deposition it can be assumed
that one electron is passed per Ag atom deposited. Therefore, the frequency
change can be correlated to the mass of Ag deposited. The silver deposition
was repeated 6 times and the Cf value obtained was 57 1 Hz/gcm2, which
compared very well to the expected value.
The method is well suited for investigating mass changes for electrochemical
electrodes, however, it is important to emphasize that the frequency change is
measured, not the mass. Besides change of mass, there are other factors that
can lead to a change in frequency. Among the most important are 200:
Viscosity. The crystal oscillations are inuenced by the media in which
the crystal is immersed. For liquid media the interaction with the crystal
can cause a dampening, which can be expressed as the following equation
204:
3
2
0
f = ��f
r
ll
qq
(3.10)
where is the viscosity of the liquid and the other symbols represent the
same as in equation 3.8. It is clear that a higher viscosity results in a
dampening in the frequency. For water there is furthermore a temperature
dependence of the viscosity. As an example a change from 20 to 30
oC results in a frequency change of 85 Hz. It is therefore important to
maintain a stable temperature while using an EQCM setup. In our laboratory
the temperature could be monitored and in short term testing it
was never observed to have an impact.