W. Richard Bowen and Nidal Hilal 4

256 9. APPLICATION OF ATOMIC FORCE MICROSCOPy
where k is the cantilever force constant, � the ratio of cantilever to substrate
modulation amplitude and φ the phase angle between substrate
and cantilever response. It is important to note that the equivalence of
the phase terms φ and � is not always physically representative, and a
quantitative description of the rheological properties of a homogeneous
fluid sample may be verified under some conditions; however, when the
sample fluid is non-uniform, e.g. an adsorbed polymer layer is present,
only qualitative information can be obtained.
9.4 DETERMInATIon oF RhEoLogICAL PRoPERTIES
FRoM RESonAnCE SPECTRA
The resonance characteristics of microcantilevers immersed in fluids
have long been recognised as a potential basis for rheometrical microsensor
technology as the resonance spectra of a vibrating cantilever are
influenced by the viscosity and density of the surrounding medium.
Compared to vibration in air, the fundamental resonant frequency of a
rectangular cantilever is reduced when operated in liquid; immersion
in increasingly more viscous liquids reduces the resonant frequency
further – this is accompanied by a broadening of the resonant peak and a
decrease in peak amplitude (Figure 9.5).
The shift in the mechanical response is attributed to the viscous
drag and inertial effects caused by the fluid adjacent to the cantilever. The
resonance characteristics are often derived from the effect of environmental
stimuli – most commonly thermally induced vibrations. The thermal
noise spectrum is derived from the Fourier transform of the displacement
of the cantilever. Fast Fourier transforms (FFTs) are routinely used
in sound and vibration analysis to convert a signal from a time domain to
A
High viscosity
ω
Low viscosity
FIguRE 9.5 Resonance frequency shift due to immersion in viscous fluids.