W. Richard Bowen and Nidal Hilal 4

254 9. APPLICATION OF ATOMIC FORCE MICROSCOPy
The application of hydrodynamic lubrication approximations (equation
(9.4)) is illustrated by the viscous damping coefficient described in
the work of Friedenberg and Mate (1996) [63] who studied the amplitude
and phase response of a colloid probe in contact with a thin film of lowmolecular-weight
poly(dimethylsiloxane) (PDMS). The PDMS film was
constrained between a sphere attached to a tungsten cantilever and a flat
substrate – the substrate being subjected to oscillatory motion. Both the
separation distance between the sphere and the surface h and the vibration
frequency � were varied (Figure 9.4).
A simple viscous model incorporating the contribution of meniscus
forces is considered, which relates both amplitude and phase to a single,
viscous damping coefficient b p. Equations (9.5) and (9.6) show the simplest
form of the derived model when capillary forces are neglected.
� φ
� � bp
k
sin
tanφ
b
p
�
�
k
bp 2
6π�R
�
h
(9.5)
(9.6)
(9.7)
where � is the ratio of cantilever and drive amplitudes, k and φ the spring
constant and the phase difference between the substrate and cantilever
motion and R the radius of the sphere.
PDMS fluid
Silicon
substrate
Tungsten
cantilever tip
FIguRE 9.4 Schematic diagram of the probe–fluid geometry (Friedenberg and Mate,
1996, the illustrated geometry has been rotated 90°). As the probe is not fully immersed,
capillary forces affect the contact area.
h
dz