W. Richard Bowen and Nidal Hilal 4

of the force curve. Researchers have overcome this problem by the use
of magnetically actuated cantilevers operating under a feedback mechanism
to maintain the stability of the system [20].
Butt et al., in a comprehensive review of force measurement techniques
[21], describe a method for calculating the Hamaker constant from both the
jump-in distance and the deflection of the cantilever due to the jump-in. This
requires knowledge of the radius of curvature of the probe tip (or spherical
particle replacing the probe) along with the effective stiffness of both the
cantilever and total system. For a sphere approaching a plane surface,
A
H
2.3 INTERACTION FORCES 43
3
jtc
3
kc
⎟ keff
2
3k D ⎛
eff jtc 2k ⎞
⎜ 3k x
C eff 24
� � ⎜ �
Rt
⎝⎜
keff
⎠ Rt
Rt
(2.22)
where D jtc is the jump-in distance, x jtc is the cantilever deflection due to
the jump-in, R t is the radius of curvature of the probe tip, k c is the spring
constant of the cantilever and k eff is the effective spring constant of the
cantilever-sample system. The effective stiffness can be calculated from
considering the contribution to the stiffness of the system of the cantilever
and sample surface interacting in series:
1 1 1
� �
keff kc ks
(2.23)
where k s is the stiffness of the sample, assuming negligible deformation
of the probe. For hard samples or soft cantilevers, k eff � k c. Das and colleagues
[22] used a similar approach to measure the Hamaker constant
between silicon nitride AFM probes and a number of surfaces from the
jump-in distance using the following relationship:
A
H
k D
�
R
24⎛
⎜
27⎝⎜
c jtc
t
⎞
⎠⎟
(2.24)
Measurements were carried out on a SiO 2 surface, freshly cleaved
mica and on a silver metal film in air under ambient conditions. Values
obtained were of the same magnitude, but not identical, to literature values
obtained from Lifshitz theory and measurements with the surface
forces apparatus (SFA). One possible reason for this divergence may be
the presence of a contaminating water layer present on most surfaces
under ambient conditions.
The Hamaker constant for an interaction may also be calculated from the
work of adhesion, W a, between the two bodies. The work of adhesion may
be obtained from the adhesion as measured during the retract cycle of a force
distance measurement and then applying the appropriate contact mechanics