Biomedical Engineering Reference
In-Depth Information
Pauli or ionic repulsion.
Repulsive forces are the most important forces in conven-
tional contact mode AFM. The Pauli exclusion principle forbids that the charge
clouds of two electrons showing the same quantum numbers can have some sig-
nificant overlap; first, the energy of one of the electrons has to be increased. This
yields a repulsive force. In addition, overlap of the charge clouds of electrons can
cause an insufficient screening of the nuclear charge, leading to ionic repulsion of
Coulombic nature. The Pauli and the ionic repulsion are nearly hard wall poten-
tials. Thus, for tip and sample in intimate contact, most of the (repulsive) interaction
is carried by the atoms directly at the interface. The Pauli repulsion is of purely
quantum mechanical origin and semiempirical potentials are mostly used to allow an
easy and fast calculation. A well-known model is the
Lennard-Jones
potential, which
combines short-range repulsive interactions with long-range attractive van der Waals
interactions:
E
0
r
0
z
6
12
2
r
0
z
V
LJ
(
)=
−
z
(2.6)
where
E
0
is the bonding energy and
r
0
is the equilibrium distance. In this case, the
repulsion is described by an inverse power law with
n
=
12. The term with
n
=
6
describes the attractive van der Waals potential between two atoms/molecules.
Elastic forces.
If the tip is in contact with the sample, elastic deformations can occur.
Since this deformation affects the effective contact area the knowledge about the
elastic forces and the corresponding deformation mechanics of the contact is an
important issue in AFM. The repulsive forces occurring during elastic indentation
of a sphere into a flat surface were already analyzed in 1881 by H. Hertz (Johnson,
1985; Landau & Lifschitz, 1991)
3
E
∗
√
R
4
3
/
2
F
Hertz
(
)=
(
−
)
≤
z
z
0
z
for
z
z
0
(2.7)
where the effective elastic modulus
E
∗
μ
t
μ
s
E
∗
=
(
1
1
−
)
+
(
1
−
)
(2.8)
E
t
E
s
depends on the Young's moduli
E
t
s
of tip and surface,
respectively.
R
is the tip radius and
z
0
is the point of contact. Figure 2.3a shows two
curves following this force law for a soft and hard sample, respectively.
However, this model does not include adhesion forces, which have to be con-
sidered at the nanometer scale. Two extreme cases were analyzed by Johnson et al.
(1971) and Derjaguin et al. (1975). The model of Johnson, Kendall, and Roberts
(JKR model) considers only the adhesion forces
inside
the contact area, whereas
the model of Derjaguin, Muller, and Toporov (DMT model) includes only the adhe-
sion
outside
the contact area. Various models analyzing the contact mechanics in the
intermediate regime were suggested by other authors (see, e.g., Schwarz, 2003 for
an overview).
s
and the Poisson ratios μ
t
,
,
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