Supplementary Information
This supplement provides additional information related to the manuscript
“Compositional contrast of biological materials using the momentary excitation of
higher eigenmodes in dynamic Atomic Force Microscopy”. In what follows, we
compare theory and experimental data from three different soft cantilevers to
demonstrate the broad applicability of the momentary excitation phenomenon for local
compositional contrast. We also provide a simple explanation for the observed
inversion of contrast observed between the second harmonic and the ME harmonic
images.
Additional experimental results demonstrating the wide applicability of the results
To further confirm the validity of our experiments, measurements of the amplitude
of the higher harmonics for the Mica/PM (Purple Membrane) system were performed
using three different soft microcantilevers. The properties of these cantilevers are
listed in Table I. The specific conditions for each experiment are given in Table II.
Numerical simulations of the amplitude contrast were performed for each of the
three cantilevers using Eq. (1) in the main text with the properties given in Table I. The
conditions of each experiment, as listed in Table II, were used to numerically simulate
the waveform of the cantilever oscillation as the tip interacts with the sample. When
the probe tip interacts with the hard mica surface (Young’s modulus 60 GPa,
Poisson’s ratio 0.3), we model the tip-sample interaction force by the Hertz contact
1
model [2]: Fts ( z)  43 E* Rz 3/ 2 for z  0 and otherwise zero, where R is the tip radius,
z is the instantaneous tip-sample separation, E * is the effective elastic modulus of
the tip/sample system. For the soft purple membrane (Young’s modulus 100 MPa),
the tip-sample interaction force is modeled by Chadwick’s theory for thin membranes
on rigid surfaces [3]: Fts ( z)   2 ER / 3h  z 2 for z  0 and otherwise zero, where h
is the membrane thickness.
The calculated amplitude contrast as a function of higher harmonic for each
cantilever is compared to the experimental values as plotted in Fig. S1.
2
Table I
Rectangular cantilever properties (in liquid), the mode dependent
parameters are acquired by measuring the thermal spectrum of the cantilevers when
they are within imaging distance of the sample while the stiffness I calibrated using
Sader’s method from the thermal spectrum in air [1]
Cantilever A
Cantilever B
Cantilever C
350×35×1.5
300×35×1.5
130×35×1
Stiffness (N/m)
0.11
0.22
0.82
Tip radius (nm)
50
50
50
1st Mode Resonance Frequency (kHz)
3.93
8.99
22.54
1st Mode Q-factor
1.85
2.0
2.49
2nd Mode Resonance Frequency (kHz)
30.00
66.10
187.20
2nd Mode Q-factor
4.29
2.1
2.65
Young’s Modulus (GPa)
169
169
169
L(μm)×b(μm)×h(μm)
Table II
Experiments
List of experiments
(a)
(b)
(c)
Cantilever A
Cantilever B
Cantilever C
Operating frequency (kHz)
3.29
7.98
20.06
Initial amplitude (nm)
11.4
5.6
5.8
Amplitude setpoint
90%
83%
89%
12
4
4
Cantilever
Repeat times
3
Fig. S1 Comparison of the experimental results with numerical simulation – contrast
of higher harmonics. Three different cantilevers have been investigated here
(parameters and operating conditions are listed in Table I and II). The inserts are the
corresponding 2nd harmonic and ME harmonic images of PM samples on mica using
each specific cantilever. Both experiments and simulations show that the ME
harmonics are most sensitive to the local material elasticity, nearly an order of
magnitude more than the second harmonics. The experimental error bars are based
on many repeats of the experiments as described in Table II.
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Explanation of inversion of second harmonic and ME harmonic image contrasts
As seen from Fig. S1 and form the results in the main text, we find that under
gentle imaging conditions (setpoint amplitudes>85-90%) softer materials appear
darker in the ME harmonic images while they appear brighter in the second harmonic
images. This can be easily understood in the context of the approximate theory
developed in the main text in Eqs. (2)- (4).
Second harmonic contrast derives from the sensitivity to local elasticity of
higher harmonics in the motion of fundamental eigenmode [4]. At high setpoint
amplitudes (gentle imaging conditions), the second harmonic amplitude is inversely
proportional to local elastic modulus: i.e. greater in magnitude for softer materials
compared to stiffer materials [4]. On the other hand, the ME harmonic contrast derives
from the sensitivity of the momentary excitation of the second eigenmode.
As we
have seen in the main text, for small contact times (gentle imaging conditions), the
magnitude of momentary excitation scales inversely with contact time. The contact
time in turn is inversely proportional to local elastic modulus, so that the ME
harmonics are proportional to local elasticity, i.e. greater in magnitude on harder
samples than on softer ones. This explains the contrast reversal observed between
the second and ME harmonics images.
Reference:
[1]
Sader, J.E., J.W.M. Chon, and P. Mulvaney, Review of Scientific Instruments, 70:
3967-3969, 1999.
[2]
H. Hertz, Journal für die reine und angewandte Mathematik, 92: 156-171, 1882.
[3]
R.S.
Chadwick, SIAM Journal on Applied Mathematics, 62(5): 1520-1530,
2002.
[4]
J. Preiner, J.L. Tang, V. Pastushenko, and P. Hinterdorfer, Physical Review
Letters, 99(4): 046102, 2007.
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