Biomedical Engineering Reference
In-Depth Information
f
i
K
rec
= 0.1906
r
b
2
LQ
r
w
G (
w
f
)
2
(2.11)
f
d
-
3
-
1
D
0
d
2L
3
(2.12)
K
tri
=
1
+4
b
3
where:
f
G(
w
f
) = imaginary part of
hydrodynamic function
k
rec
= rectangular cantilever force constant
k
tri
= trianguar cantilever force constant
r
4L
3
=
D
k
= density of the medium
0
rec
b
Q
r
= Quality factor at resonance
d
=
d
b
2
w
f
= Resonant frequency
1
+
4L
2
Fig. 2.33. Equations for calculating force constants of rectangular and triangular cantilevers with the
Sader method [66].
cantilevers (see, for example references [60-65]). The most commonly used method used
for to calculate the normal force constant is the known as the Sader method [66-68]. In this
method, the physical geometry of the cantilever is measured, typically with an optical
microscope and the quality factor,
Q
, is measured. The quality factor is of the cantilever is
essentially a measure of the bandwidth of the oscillation compared to the exciting
frequency), and is simple to measure with the AFM. With the physical measurements,
and the
Q
of the cantilever, Equations 2.11 and 2.12 can be used to calculate the spring
constant, see Figure 2.33. This method has been later extended to the calculation of
k
tor
,
the torsional constant [61], as well as to cantilevers of arbitrary geometry [69].
Several other methods for cantilever calibration are also widely used. The added mass
method, first described by Cleveland [70], takes advantage of the change in frequency of a
lever when a mass is added to the end. Essentially, the method consists of physically
adding a small well characterized particle to a probe tip, for example a tungsten or polymer
microsphere. This method is at least as accurate as the Sader method [63, 71], and does not
require measurement of cantilever dimensions, but although described as non-destructive,
