Biomedical Engineering Reference
In-Depth Information
However, in many practical cases, it is sufficient to assume that the geometri-
cal shape of tip and sample does not change until contact has been established at
z
z
0
and that afterwards, the tip-sample forces are given by the
DMT-M theory
,
denoting Maugis' approximation to the earlier Derjaguin-Muller-Toporov model. In
this approach, an offset
F
vdW
=
is added to the well-known Hertz model, which
accounts for the adhesion force between tip and sample surface. Therefore, the DMT-
M model is often also referred to as
Hertz-plus-offset model
(Schwarz, 2003). The
resulting overall force law is given by
(
z
0
)
−
A
H
R
6
z
2
for
z
≥
z
0
3
E
∗
√
R
F
DMT
(
z
)=
(2.9)
−
M
A
H
R
6
z
0
4
3
/
2
(
z
0
−
z
)
−
for
z
<
z
0
Figure 2.3b displays the resulting tip-sample forces curves for the DMT-M model
for a hard and soft sample, respectively. The parameters are given in the figure cap-
tion. They represent typical values for AFM measurements under ambient condi-
tions.
Frictional forces.
During the scanning of the tip on the sample surface, there are
counteracting frictional forces. These forces dissipate the kinetic energy of the mov-
ing tip-sample contact into the surface or tip material. This can be due to permanent
changes in the surface itself, by scratching or indenting, or also by the excitation of
lattice vibration (i.e., phonons) in the material.
Chemical binding forces.
Due to the overlap of molecular orbitals there might arise
specific bonding states between the tip and the surface molecules. These forces are
extremely short-ranged and can be exploited to achieve atomic resolution imaging
of surfaces (mostly in vacuum). Since these forces are also specific to the chemical
identity of the molecules, it is conceivable to identify the chemical character of the
surface atoms with AFM scans (Sugimoto et al., 2007).
Magnetic and electrostatic forces.
Long-range magnetic or electrostatic force might
be attractive or repulsive. They are usually measured when the tip is not in contact
with the surface (i.e., “noncontact” mode). For magnetic forces, magnetic materials
have to be used for tip or tip coating. Well-defined electrical potentials between tip
and sample are necessary for the measurement of electrostatic forces.
More detailed information on intermolecular and surface forces relevant for AFM
measurements can be found in the monographs of Israelachvili (1992), Sarid (1994),
and Butt & Kappl (2010). Figure 2.4 nicely summarizes the most important ones. In
principle, every type of force can be measured with an AFM.
2.2 CONTACT MODE
An AFM can be driven in different modes of operation. First, we introduce the
contact mode, which is the historically the oldest one. To distinguish it from the
later introduced dynamic modes, the contact mode is also sometimes referred to as
static mode. However, due to its straightforwardness it can be used to easily obtain
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