W. Richard Bowen and Nidal Hilal 4

where � is the shear stress, ε the shear strain and � the frequency. The
complex modulus may be described as the summation of an in-phase
elastic component and an out-of-phase viscous component, i.e.
G*( �) � G′ ( �) � i G′′
( �)
(9.2)
and
9.3 dyNAMIC MOdULATION STUdIES ON CONFINEd FLUIdS 253
G′
( �)
� tan �
G′′
( �)
(9.3)
where G� is the storage (elastic) modulus, G�″ the loss (viscous) modulus
and � the phase angle between stress and strain.
The analogy between bulk oscillatory and dynamic scanning probe
microscope techniques is discussed by Suraya et al. (2008) [62], where
for the latter, the motion of one surface imparts oscillatory motion to
the fluid due to hydrodynamic forces, which in turn invokes oscillation of
the ‘receiver’ surface, i.e. the colloid probe. However, although the receiver
surface will oscillate at the same frequency, viscous effects are such that the
oscillation amplitude is attenuated and the phase is shifted (relative to the
driven surface). For an inelastic liquid, the detected signal will be 90° outof-phase,
whereas an elastic response is expected to be in-phase with the
driven surface. When oscillating surfaces bearing adsorbed polymer layers
are immersed in dilute polymer solutions, the vibrating motion produces
a viscous response at large surface separations. As the separation distance
is reduced, the polymer layers interact and deform, and the viscoelastic
properties of the material become apparent as the receiver amplitude
increases and the phase shift reduces.
Suraya et al. (2008) note that the measurement of the effect of the hydrodynamic
force upon the phase and amplitude is insufficient to directly
measure the stress and strain in the fluid layer. To translate the measured
parameters into terms which are physically representative of the stress
and strain, a mechanical model is required. A common approach employs
hydrodynamic lubrication equations, which conveniently describe the
hydrodynamic force that arises from the flow of a purely viscous liquid
between a flat surface and a colloid sphere, the force F is given by
F U R
� 6 π η , ( h