W. Richard Bowen and Nidal Hilal 4

2.2 COLLOId PROBES 33
FIGuRE 2.1 SEM image of a colloid probe created by the attachment of a glass sphere
to the apex of an AFM microcantilever using an epoxy resin. The scale bar shown is 5 �m
long, with the particle approximately 7.5 �m in diameter.
where R 1 and R 2 are the radii of the two spheres and W(D) is the interaction
energy per unit area as a function of distance [1, 2]. This relationship
is referred to as the Derjaguin approximation. In the case of a sphere
approaching a flat surface (or where one sphere is much greater in size than
the other), this is further simplified to create:
F( D) � 2πR1W( D)
(2.2)
Note that the forces of interaction are proportional to the radius or radii
of the interacting spheres. For this reason, it is typical in colloidal probe
measurements to normalise forces by dividing by the radius of curvature
of the probe. This generally leads to forces being measured in units of the
form of mN m �1 . It should be noted that one of the assumptions on which
the Derjaguin approximation is based is that the range of the interaction
forces is much less than the radii of the interacting spheres. For small
spheres with radii in the nanometre range, this approximation is likely
to be invalid, a situation that has been recently experimentally verified
by carrying out measurements using AFM tips terminated with particles
with radii of 10 and 20 nm [3]. Dividing the forces by the particle radii was
insufficient to superimpose the force traces obtained with these particles.
Colloidal probes are most commonly prepared by attaching the particle
to the apex of the AFM cantilever with an appropriate glue using a