Biomedical Engineering Reference
In-Depth Information
“
+
”and“
−
” for easy distinction. The integral
I
is a weighted average of the tip-
+
sample forces (
F
is directly connected
to Δ
E
,whichreflects the energy dissipated during an individual oscillation cycle.
Consequently, this integral vanishes for purely conservative tip-sample forces, where
F
↓
+
F
). On the other hand, the integral
I
↑
−
are identical. A detailed discussion of these integrals was already given by
Durig, 2000b and Sader et al., 2005.
By now combining Equations 2.15b and 2.22b, we get a direct correlation between
the phase and the energy dissipation.
*
and
F
↓
↑
A
A
0
f
d
f
0
+
Q
0
Δ
E
π
k
cant
A
0
A
sinφ
=
−
(2.23)
This relationship can also be obtained from the conservation of energy principle
(Cleveland et al., 1998; Tamayo & Garcıa, 1998; Garcia et al., 2006) and demon-
strates that the phase signal in tapping mode is directly related to the energy dissipa-
tion caused by the tip-sample interaction.
Equation 2.21 can be used to calculate the resonance curves of a dynamic force
microscope including tip-sample forces. The results are
a
d
1
A
=
(2.24a)
Q
0
2
2
f
d
2
f
0
2
f
d
−
−
I
+
(
d
,
A
)
+
f
0
+
I
−
(
d
,
A
)
f
d
1
Q
0
f
0
+
I
−
(
d
,
A
)
tanφ
=
(2.24b)
f
d
2
f
0
2
1
−
−
I
+
(
d
,
A
)
Equation 2.24a describes the shape of the resonance curve, but it is an implicit func-
tion of the oscillation amplitude
A
and cannot be plotted directly.
Figure 2.11a contrasts the solution of this equation (solid lines) with numerical
solution (symbols). As pointed out by various authors (Gleyzes et al., 1991; Kuhle
et al., 1998; Wang, 1998; Aime et al., 1999; Sasaki & Tsukada, 1999; Nony et al.,
2001; Lee et al., 2002; San Paulo & Garcıa, 2002), the amplitude versus frequency
curves are multivalued within certain parameter ranges. Moreover, as the gradient
of the analytical curve increases to infinity at specific positions, some branches are
unstable. The resulting instabilities are reflected by the “jumps” in the simulated
curves (marked by arrows in Figure 2.11), where only stable oscillation states are
obtained. Obviously, they are different for increasing and decreasing driving fre-
quency. This is a well-known effect frequently observed in nonlinear oscillators that
leads also to a hysteresis in amplitude versus distance curves (Figure 2.11b). As
a rule of thumb, it can be said that the tip-sample forces in AM-mode are in the
attractive range before the jump and repulsive after the jump (Holscher et al., 2006).
Therefore, it is highly advantageous to scan delicate surfaces with a high amplitude
set-point before the jump. The increase in resolution caused by the reduction of the
tip-sample forces by this procedure has been nicely demonstrated by San Paulo and
Garcıa (2000).
*
The “-” sign on the right side of this equation is due to our definition of the phase φin Equation 2.14.
Search WWH ::
Custom Search