W. Richard Bowen and Nidal Hilal 4

2.4 AdHESION FORCES MEASUREd By AFM 67
effect of sphere size and probe speed on the contribution of drag on the
AFM cantilever to measured hydrodynamic forces. They found that for
the cantilever used that at speeds of 7.5 �m s �1 and for spheres of radii
less than 3 �m, the cantilever did measurably contribute to the observed
effects. Obviously, for faster speeds, the minimum sphere size to prevent
cantilever contributions confounding sphere–plane measurements will
increase. For most operating conditions, the authors recommended using
colloid probes of no less than 5 �m in radius.
2.4 AdHESIon FoRCES MEASuREd by AFM
Adhesive forces measured as the pull-off force in AFM measurements
represent the sum of all the interaction forces occurring in the contact
regime. This includes long-range forces such as van der Waals and electrostatic
forces; capillary forces (if measurements are undertaken in air);
solvation forces; hydrophobic interactions and steric interactions as well
as any chemical bonding between groups present on the surfaces.
2.4.1 Contact Mechanics and Adhesion
The magnitude of the adhesion between two surfaces is dependent
upon the contact area at the junction between the surfaces as well as
the interaction forces themselves. The contact area will depend upon
the mechanical deformation of the materials due to the applied force
and material properties. The understanding of the relationship between
adhesive forces and contact mechanics as generally applied in the field of
force microscopy is dependent upon the works of Johnson, Kendal and
Roberts (JKR theory) [151] and Derjaguin, Muller and Toporov (DMT
theory) [152] some decades ago. According to the JKR model of contact
mechanics, for a sphere–plane system, the adhesion (pull-off) force can
be related to the work of adhesion, contact area and mechanical compliance
of the interacting surfaces by:
JKR 3
ac
2
FAd � πRpWa
�
2
R 3
2
p
6πW
E*
a
(2.62)
where R p is the radius of the probe sphere, W a is the work of adhesion per
unit area and a c is the contact radius. The parameter a 2 c/R is the sample
deformation and �. E* is the reduced Young’s modulus for the system:
2
1 3 1 1
s
E* 4 Es E f
�
⎛
⎜ � ν � ν
⎜ �
⎜
⎝⎜
2
f
⎞
⎠⎟
(2.63)