234 6.5 Scanning Probe Microscopy and Force Spectroscopy
where Ebond is the bonding energy, with a decay length κ. Quantum mechanics can pre
dict an exact potential energy curve for the H2
+ diatomic molecule system, which the
Morse potential fits very well, whose form exhibits the qualitative features at least of
the real AFM chemical potential energy function. However, for more complex chem
ical interactions of higher atomic number atoms involving anisotropic molecular orbital
effects, as occurred in practice with the tip–sample system for AFM, empirical models are
used to approximate the chemical potential energy, including the Stillinger–Weber poten
tial and Tersoff potential, with the Stillinger–Weber potential showing most promise from
ab initio calculations for interactions involving silicon-based materials (as is the case
for AFM tips). The functional form of this potential energy involves contributions from
nearest-neighbor and next nearest-neighbor atomic interactions. The nearest-neighbor
component UNN is given by
(6.23)
U
E
A B
z
z
z
a
NN
bond
p
q
=
−
]
[
−
′
′
′
−
−
σ
σ
σ
exp
/
1
where A, B, a, p, and q are all constants to be optimized in a heuristic fit. The next nearest-
neighbor component, UNNN, is a more complex formulation that embodies angular depend
ence of the atomic orbitals:
(6.24)
U
E
h x z
h x
z
h x
z
NN
bond
ij
ik
jik
ji
jk
ijk
ki
kj
ijk
=
(
)+ (
)+ (
)
,
,
,
,
,
,
θ
θ
θ
such that
(6.25)
h x z
z
a
z
a
ij
ik
jik
ij
ik
ijk
,
,
/
θ
λ
σ
σ
θ
(
) =
−
(
)
+
−
(
)
′
′
exp
/
cos
1
1
+
1
3
2
where
λ is a constant
i, j, and k are indices for three interacting atoms
If an AFM tip is functionalized to include electrostatic components, these can interact
with electrostatic components on the biological sample surface also. The functional
form can be approximated as U
R V
z
ES
rip
≈πε
2
2
2
/
where ε is electrical permittivity of
the aqueous solvent surrounding the tip and sample, Rtip is again the AFM tip radius of
curvature and V is the electrical potential voltage across a vertical distance z between tip
and sample.
The most significant of the interaction forces for AFM are the vdW forces, modeled as the
Lennard–Jones potential (also known as the 6–12 potential, introduced in Chapter 2):
(6.26)
U
E
z
z
LJ
bond
= −
−
2
6
6
12
12
σ
σ
The vdW forces arise from a combination of the fluctuations in the electric dipole moment,
and the coupling between these fluctuating dipole moments, and the exclusion effect between
paired electrons. The longer-range ~z6 dependence is the attractive component, while the
shorter-range ~z12 component results from the Pauli exclusion principle between paired
electrons that prohibits electrons with the same spin and energy state from occupying the same
position in space and thus results in a repulsive force at very short tip–sample separations.