W. Richard Bowen and Nidal Hilal 4

22 1. BAsIC PRINCIPLEs OF ATOMIC FORCE MICROsCOPy
and for torsional forces experienced by a V-shaped cantilever:
k
φ �
3
2
Et w w ⎛2w
l ⎞ 3⎛
b⎞
1
1 � 2log⎜
� 2 � ⎜
3( 1 � v) l b ⎝⎜
b ∆l⎠⎟
8 ⎝⎜
l ⎠⎟
1 �� b
⎧⎪
⎡
2
⎛ ⎞ ⎤⎫
⎪
⎪
⎪ ⎢
⎜
⎥⎪
⎪ ⎢
⎜
⎥
⎪
⎨
⎪ ⎢
⎜
⎢
⎜
⎥
⎜ 2
⎢
⎜
⎥
⎬
⎪
⎪
⎪
⎪
2
⎢
⎝⎜
4l
⎠⎟
⎥⎪
⎪
⎥
⎪
⎩⎪
⎣
⎦⎭⎪
−1
(1.13)
where �l is the distance of the base of the probe tip from the apex of the
lever.
For relevant dimensions of the levers see Figures 1.5 and 1.6. In terms
of measured interactions, k lat can be defined in similar terms to Hooke’s
law for the normal spring constant of the lever [73]:
Flat � klat∆ x
(1.14)
where F lat is the force experienced by the tip and �x the lateral movement
of the tip along the x-direction (i.e. at right angles to the major axis
of the cantilever).
.7 ColloId probes
By attaching a microsphere to the end of a tipless cantilever, the geometry
of interactions between the probe and the surface can be greatly simplified,
allowing the AFM to be used to probe surface forces, much akin to the
surface force apparatus (SFA). Microparticles can also be used as probes,
which will allow particle to particle adhesion forces to be measured.
An example of a colloid probe is illustrated in Figure 1.8. Here a scanning
fIgure .8 SEM image
of a colloid probe created
by the attachment of a silicon
dioxide sphere to the
apex of an AFM microcantilever
using an epoxy
resin. The scale bar shown
is 1 �m long, with the particle
approximately 5 �m in
diameter.