Biomedical Engineering Reference
In-Depth Information
Figure 2.12 displays the resulting amplitude and phase versus distance curves
during approach, respectively. The assumed parameters and the conservative force
are the same as in Figure 2.13b while the following parameters have been used for
the dissipative force:
F
0
10
−
6
Ns/m and ζ
0
5 nm. Again the amplitude curve
shows the previously discussed discontinuity caused by an instability.
The subsequent reconstruction of
F
ts
and Δ
E
based on the data provided by the
amplitude and phase versus distance curves is presented in Figures 2.12c and 2.12d.
The assumed tip-sample force and energy dissipation are plotted by solid lines, while
the reconstructed data is indicated by symbols; the excellent agreement demonstrates
the reliability of the method. Nonetheless, it is important to recognize that the often
observed instability in amplitude and phase versus distance curves affects the recon-
struction of the tip-sample force. If such an instability occurs, experimentally acces-
sible κ
=
=
0
.
values will feature a “gap” at a specific range of tip-sample distances
D
. This issue is illustrated in Figure 2.12e, where the gap is indicated by an arrow.
As a consequence, one might be tempted to interpolate the missing κ-values in the
gap. This is a workable solution if, as in our example, the accessible κ-values appear
smooth and, in particular, the lower turning point of the κ
(
D
)
(
D
)
values is clearly visi-
ble. In most realistic cases, however, the κ
values will not look so smooth as in
our simulation and/or the lower turning point might not be reached, and uttermost
caution is advised when applying any inter- or extrapolation.
Finally, let us note two more issues: (a) The reconstruction of the energy dissi-
pation does not require the continuous knowledge of κ-values. Thus, it is not influ-
enced by the instability and gives reliable values also after the jump (Figure 2.12d).
(b) The “large amplitude approximation” is not a prerequisite for the inversion of
the tip-sample forces from the amplitude and phase data. The application of other
numerical methods (Lee & Jhe, 2006; Hu & Raman, 2008; Holscher, 2008) where
the amplitude is not restricted to large values is also possible.
(
D
)
2.4.5 T
HEORY OF
FM-M
ODE
After the analysis of the AM-mode, we now give the solution of the equation of
motion Equation 2.12 for the FM-mode. As before we assume that the tip-sample
force is so small and the
Q
-factor so high that, as a consequence, higher harmonics
can be neglected. Inserting again the ansatz Equation 2.17 leads now to a set of two
coupled trigonometric equations
f
2
f
0
a
exc
A
−
cos
(
2π
ft
0
)=
−
I
+
(
d
,
A
)
(2.31a)
f
0
a
exc
A
1
Q
f
f
0
−
(
sin
2π
ft
0
)=
I
−
(
d
,
A
)
(2.31b)
where we again defined the two integrals
I
Equation 2.22b.
Both equations can be simplified for the conditions typically found in DFM
experiments, where the FM-mode is applied. First, we assume that the frequency
Equation 2.22a and
I
+
−
Search WWH ::
Custom Search