W. Richard Bowen and Nidal Hilal 4

The value of the viscous damping coefficient b p is then determined by
fitting the measured values of both amplitude and phase as a function of
separation distance h to equations (9.5) and (9.6). The apparent viscosity
of the fluid � may then be calculated from the damping coefficient and
equation (9.7). Using this method reasonable agreement between the calculated
viscosity and measured ‘bulk’ viscosity was found; however, the
results also indicated that the surface roughness of the sphere may cause
deviations from the expected zero-intercept relationship inferred from
equation (9.7) (i.e. 1/b p vs. h). Choi and Kato (2003) [64] have extended
this approach to investigate the shear properties of perfluoropolyether
liquid bridges formed between two glass spheres by inducing lateral oscillations
(as opposed to the normal perturbations depicted in Figure 9.4).
Suraya et al. (2008) [62] have reported the use of dynamic AFM to
investigate the viscoelastic response of adsorbed hydroxypropyl guar
(HPG) layers. The results identified dominant viscous properties at large
surface separations and viscoelastic behaviour at closer separations
where the polymer layers interact. The adsorbed polymer also exhibited
viscoelastic frequency dependence and ‘shear thinning’ characteristics.
Similar studies were performed by Braithwaite and Luckham (1999)
[65] who studied the viscoelastic response of adsorbed layers of gelatine
under compression using dynamic modulation. The system consisted
of a colloid probe interacting with a hard substrate where both surfaces
were coated with a thin film of adsorbed gelatine. The model employed
considers the characteristics of the viscoelastic materials as discussed by
Radmacher et al. (1993) [66].
In such studies, as the actual values of stress and strain in the sample
cannot be determined directly, they are instead speculatively related to
the force F and displacement z of the cantilever through the use of an
‘apparatus coefficient’ b such that for a given frequency:
�
ε
� � G
1 F
*
b z
(9.8)
From which an alternative definition of the storage and loss moduli may
be described [65, 66] wherein the equations are of the form:
and
9.3 dyNAMIC MOdULATION STUdIES ON CONFINEd FLUIdS 255
�k (cos φ � �)
G′
( �)
�
2 b � � 2 � cosφ
� 1
�k sin φ
G′′
( �)
�
2 b � � 2 � cosφ
� 1
(9.9)
(9.10)