W. Richard Bowen and Nidal Hilal 4

42 2. MEASUREMENT OF PARTICLE ANd SURFACE INTERACTIONS
Equation (2.12) is the expression for the non-retarded Hamaker constant.
When comparing Hamaker constant data with literature values,
this equation should be used, as the values given in the literature are for
non-retarded Hamaker constants. However, in calculations where particles
in an electrolyte solution are considered, and for some cases at large
interparticle separations, the effects of retardation [18] and screening [19]
on the Hamaker constant calculation need to be taken into account. This
can be done by modifying equation (2.12) to:
where
A 131 �A ( 1�2κD) e +
A F( H)
ξ
o
�2κD
F H
H
⎡
( ) � ⎢ ⎛ π ⎞
1�⎜
⎢ ⎝⎜
4 2 ⎠⎟
⎣⎢
3/ 2
�2/
3
D
H �no ( n o �n
3 1 o )
3 c
/ w w
⎤
⎥
⎦⎥
2 2 1 2 1 3
(2.19)
(2.20)
(2.21)
2.3.1.4 Calculation of van der Waals Forces from
Force Distance Curves
Measurement of the van der Waals interactions between different
materials in different media is possible using the AFM. By changing
the sharp probe with a colloidal particle, a wide range of different
interactions can be measured. However, much care must be taken with
the design of experiments. In many situations, forces other than van
der Waals interactions may be present, such as double layer interactions
and hydrodynamic effects, which may make it difficult or impossible
to extract accurate estimations of van der Waals forces or Hamaker
constants. In addition, any calculations made must take into account the
probe sample interaction geometry. In the case of a spherical particle versus
a plane surface or other spherical particles, this is relatively straightforward.
However, for an irregularly shaped particle or one exhibiting
significant surface roughness, then this is not a straightforward matter.
This subject is dealt with in more detail in Section 2.5 of this chapter.
There are several ways in which Hamaker constants may be estimated
from AFM force measurements. First, one of the power laws, appropriate
for the interaction geometry used in the experiment, from Table 2.1 may
be fitted to the part of the force distance measurement obtained when
the probe is approaching the surface just prior to any jump-to-contact.
During the jump-to-contact event, the cantilever system is out of equilibrium
and consequently, the power laws may not be fitted to this part
ξ
�1