W. Richard Bowen and Nidal Hilal 4
W. Richard Bowen and Nidal Hilal 4
W. Richard Bowen and Nidal Hilal 4
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of the force curve. Researchers have overcome this problem by the use<br />
of magnetically actuated cantilevers operating under a feedback mechanism<br />
to maintain the stability of the system [20].<br />
Butt et al., in a comprehensive review of force measurement techniques<br />
[21], describe a method for calculating the Hamaker constant from both the<br />
jump-in distance <strong>and</strong> the deflection of the cantilever due to the jump-in. This<br />
requires knowledge of the radius of curvature of the probe tip (or spherical<br />
particle replacing the probe) along with the effective stiffness of both the<br />
cantilever <strong>and</strong> total system. For a sphere approaching a plane surface,<br />
A<br />
H<br />
2.3 INTERACTION FORCES 43<br />
3<br />
jtc<br />
3<br />
kc<br />
⎟ keff<br />
2<br />
3k D ⎛<br />
eff jtc 2k ⎞<br />
⎜ 3k x<br />
C eff 24<br />
� � ⎜ �<br />
Rt<br />
⎝⎜<br />
keff<br />
⎠ Rt<br />
Rt<br />
(2.22)<br />
where D jtc is the jump-in distance, x jtc is the cantilever deflection due to<br />
the jump-in, R t is the radius of curvature of the probe tip, k c is the spring<br />
constant of the cantilever <strong>and</strong> k eff is the effective spring constant of the<br />
cantilever-sample system. The effective stiffness can be calculated from<br />
considering the contribution to the stiffness of the system of the cantilever<br />
<strong>and</strong> sample surface interacting in series:<br />
1 1 1<br />
� �<br />
keff kc ks<br />
(2.23)<br />
where k s is the stiffness of the sample, assuming negligible deformation<br />
of the probe. For hard samples or soft cantilevers, k eff � k c. Das <strong>and</strong> colleagues<br />
[22] used a similar approach to measure the Hamaker constant<br />
between silicon nitride AFM probes <strong>and</strong> a number of surfaces from the<br />
jump-in distance using the following relationship:<br />
A<br />
H<br />
k D<br />
�<br />
R<br />
24⎛<br />
⎜<br />
27⎝⎜<br />
c jtc<br />
t<br />
⎞<br />
⎠⎟<br />
(2.24)<br />
Measurements were carried out on a SiO 2 surface, freshly cleaved<br />
mica <strong>and</strong> on a silver metal film in air under ambient conditions. Values<br />
obtained were of the same magnitude, but not identical, to literature values<br />
obtained from Lifshitz theory <strong>and</strong> measurements with the surface<br />
forces apparatus (SFA). One possible reason for this divergence may be<br />
the presence of a contaminating water layer present on most surfaces<br />
under ambient conditions.<br />
The Hamaker constant for an interaction may also be calculated from the<br />
work of adhesion, W a, between the two bodies. The work of adhesion may<br />
be obtained from the adhesion as measured during the retract cycle of a force<br />
distance measurement <strong>and</strong> then applying the appropriate contact mechanics