Biomedical Engineering Reference
In-Depth Information
electrostatic, magnetic, or capillarity forces determine the interaction between the
tip and the sample.
The, generally attractive, Van der Waals force is strong enough to deflect the
AFM cantilever at tip-sample distances of few nanometers. The van der Waals
potential of two interacting molecules separated by a distance
z
has the following
form:
U
vdW
C
1
=
z
6
;
(3.1)
where C
1
is the London coefficient. For shorter tip-sample distances, the wave
function of electrons situated at the cantilever tip overlaps the wave function of
electrons on the sample surface, generating a significant repulsive Born force, with
a potential
U
Born
D
C
2
=
z
12
;
(3.2)
where C
2
is a constant. The total intermolecular potential, called also Lennard-Jones
potential, is then
U
D
C
2
=
z
12
C
1
=
z
6
:
(3.3)
The AFM detects the sum of all repulsive and attractive forces acting between the
tip and the sample.
At the macroscopic scale, the attractive van der Waals potential between two
spheres with radius R and densities
1
and
2
is given by
U
vdW
D
.
AR
=6
z
/;
(3.4)
and the corresponding van der Waals force is then
F
vdW
D
dU
vdW
=d
z
D
AR=6
z
2
;
(3.5)
where A
/
2
C
1
1
2
is the so-called Hamaker constant.
The repulsive force between macroscopic objects, called Hertz force or indenta-
tion force, can be approximated as the force on a sphere of radius R that is pushed
against a flat surface and has the expression
F
Hertz
D
.K
2
Rh
ind
/
1=2
;
(3.6)
where h
ind
is the indentation depth.
The forces exerted between the tip and the sample has a key role in determining
the various types of AFM operations. The three basic AFM operation modes,
dynamical, static, and tapping modes, are displayed in Fig.
3.1
.
In the
dynamical mode
, termed also as noncontact imaging mode or attractive
force imaging, the tip of the cantilever makes no contact with the tested surface but
is situated above it at all times during the scanning of the surface. The cantilever
is driven by a piezoelectric actuator situated over it and vibrates mechanically at
a frequency f . The vibrations of the AFM cantilever can be subjected either to
frequency modulation (FM) or amplitude modulation (AM).
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