W. Richard Bowen and Nidal Hilal 4

258 9. APPLICATION OF ATOMIC FORCE MICROSCOPy
where A 0 is the zero-frequency amplitude response, � and � R are the
radial and radial resonant frequencies, respectively [70],
and
�
R
�
�
⎡
⎢
π�x
⎢
1 �
⎣ 4�
vacuum
2
⎤
� r( �R)
⎥
⎦
4� � �r(
�
2 R)
π�x
Q �
� ( � )
i R
1 2
(9.16)
(9.17)
where � (�� cxt) is the mass per unit length of the cantilever of density � c,
width x and thickness t. � r and � i are the real and imaginary parts of the
aforementioned hydrodynamic function �(�).
For an unknown fluid, the fundamental resonance profile is used to
determine � R and Q from equation (9.15). The method is not restricted to
the use of the fundamental resonance mode, and higher resonance peaks
may be used although the accuracy of the result is reduced. Provided
� vacuum is known, the experimentally determined values of the resonance
frequency and the quality factor may then be used to determine the viscosity
and density by numerically solving equations (9.16) and (9.17).
The SHO model is useful for the determination of both viscosity and
density provided Q � 1. If the cantilever response is heavily damped,
Q � 1, then the SHO model is not valid. However, this may in some
instances be circumvented by the selection of a cantilever with different
mechanical properties.
Maali et al. (2005) [71] have evaluated a simplified approximation for
the hydrodynamic function �(�) and propose that
and
�r( �)
� a � a
1 2
�
�
� ⎛�
⎞
�i( �)
� a3 � a ⎜ 4
� ⎝⎜
�⎠⎟
2
(9.18)
(9.19)
where a 1, a 2, a 3 and a 4 are constants and � is the characteristic thickness of
fluid surrounding the cantilever equivalent to an exponential damping
length �, where
� �
2�
� �
fluid