W. Richard Bowen and Nidal Hilal 4

66 2. MEASUREMENT OF PARTICLE ANd SURFACE INTERACTIONS
2.3.7 Effect of Hydrodynamic drag on AFM Force
Measurements
When performing measurements with particles in liquid, the effects of
hydrodynamic drag on both the particle and also the cantilever may be significant,
depending on the velocities at which measurements are taken. In
particular, the effects of confinement of the liquid between two particles or
between two surfaces are important. As the distance between the two surfaces
decreases, the finite drainage time for the confined liquid produces a
force dependent on the separation distance, the velocity of the approach as
well as the viscosity and density of the fluid medium. Ignoring any contribution
of drag on the cantilever to this effect, the hydrodynamic force, F H,
for a sphere approaching a plane surface is given by a modified version of
Stoke’s law for the drag force on a sphere [148, 149]:
F � 6πν�r �
H s c
(2.58)
where υ is the velocity of the probe, � is the dynamic viscosity of the
surrounding fluid, r s is the radius of the spherical particle and � c is a correction
applied to Stoke’s law to account for the presence of the confining
wall in close proximity.
where
� c
4
� � sinh�
3
∞
∑
c c
n�1
n( n � 1)
( 2n � 1)( 2n � 3)
⎡ 2sinh( 2n � 1) �c
+ ( 2n + 1)
sinh2�
⎤
⎢
c
− 1⎥
⎢ 2 2
⎣⎢
4sinh ( n � 0. 5) �c � ( 2n � 1) sinh2�
⎥
c ⎦⎥
�1
D � rs
D r
� cosh �ln
rs
rs
⎛
⎧
⎞
⎪
⎪⎛
⎞
⎜ ⎨
⎪ ⎜
⎝⎜
⎠⎟
⎪
⎪⎝⎜
⎠⎟
⎩⎪
� s
�
⎡
⎢⎛D
� r ⎞
⎜ s
⎢⎜
⎢⎜⎝
rs
⎠⎟
⎣
2
⎤⎫⎪
⎥
⎪
� 1⎥
⎬
⎪
⎥⎪
⎦⎪
⎭⎪
(2.59)
(2.60)
where D is the distance between the plane surface and closest point of the
sphere. For measurements where r � �h, then this can be simplified to:
F
H
rs
�
D
6πν�
2
(2.61)
In most cases, contributions due to interactions between the surface
and the cantilever itself can be neglected. However, depending on the
size of the colloid probe and the speed at which the experiment is carried
out, this may not always be the case. Vinogradova et al. [150] studied the