Biomedical Engineering Reference
In-Depth Information
=
(
Here,
z
)
are the spring constant, the effective mass, and the eigenfrequency of the cantilever,
respectively. Somewhat simplifying, it is assumed that the quality factor
Q
0
unites
the intrinsic damping of the cantilever and all influences from surrounding media
such as air or liquid, if present, in a single overall value. The equilibrium position of
the tip is denoted as
d
. The (nonlinear) tip-sample interaction force
F
ts
is introduced
by the first term on the right side of the equation.
The two driving mechanisms are considered by the distinction on the right side
of the equation. The external driving force of the cantilever is used for the AM-
mode. Here, the driving signal is modulated with the CE amplitude
a
exc
at a fixed
frequency
f
d
. The self-excitation of the cantilever used in the FM-mode is described
by the retarded amplification of the displacement signal, that is, the tip position
z
is
measured at the retarded time
t
(
t
)
is the position of the tip at the time
t
;
k
cant
,
m
,and
f
0
k
cant
/
m
)
/
(
2π
t
0
. Nonetheless, a consideration of the time shift
by a phase differenceθ
0
is also possible, giving equivalent results. Therefore, we use
“time shift” and “phase shift” as synonyms throughout this review and notice that
both parameters are scaled by θ
0
−
2π
f
d
t
0
.
Before finishing this section, we would like to add some words of caution regard-
ing the validity of the equation of motion Equation 2.12, as it disregards two effects
that might become of importance under specific circumstances. First, we describe the
cantilever by a spring-mass-model and neglect in this way higher modes of the can-
tilever. This is justified in most cases as the first eigenfrequency is by far dominant
in typical AM-AFM experiments (see, e.g., Fig. 1 in Cleveland et al., 1998). Thus, a
mathematical treatment that ignores higher modes is still able to describe and explain
all major general features experimentally observed in standard DFM imaging, which
is the limited goal of this review. Comparison with studies that include higher har-
monics by numerical means (Stark & Heckl, 2000; Rodrıguez & Garcıa, 2002; Lee
et al., 2002; Stark et al., 2004) confirms this statement. It might, however, not apply
if advanced signal analysis in certain DFM spectroscopy modes is intended.
Second, we assume in our model equation of motion that the dither piezo applies
a sinusoidal force to the spring, but do not consider that the movement of the dither
piezo simultaneously also changes the effective position of the tip at the cantilever
end by the current value of the excitation
a
exc
cos
=
(Lee et al., 2002; Legleiter
& Kowalewski, 2005; Legleiter et al., 2006). This effect becomes important when
a
exc
is in the range of the cantilever oscillation amplitude. Fortunately, for conditions
characterized by sufficiently high-quality factors, this effect can be neglected. This
is usually safely the case for measurements in air, where oscillation amplitudes typ-
ically exceed excitation amplitudes by several hundred times. During operation in
liquids, however, the
Q
factor is low, and the oscillation amplitudes might be com-
parable with
a
exc
(Legleiter et al., 2006).
Finally, to avoid confusion with other literature, we would like to mention some
words regarding the terminology used throughout this review. Due to the frequently
occurring intermitted contact between tip and sample at the lowest point of the oscil-
lation, the AM mode introduced above has often been denoted as
tapping mode
(Zhong et al., 1993). Over the years, use of the term “tapping mode” has then evolved
(
2π
f
d
t
)
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