Supplementary material
Energy dissipation in a dynamic nanoscale contact
Sergio Santos1,2, Neil H Thomson1,3
1
School of Physics and Astronomy, University of Leeds, LS2 9JT, UK. 2Faculty of
Biological Sciences, Structural Molecular Biology, University of Leeds, 9JT, UK.
3
Department of Oral Biology, Leeds Dental Institute, University of Leeds, LS2 9JT,
UK.
A. Model
We have used a point mass model as the equation of motion Eqn. (S1) and
incorporated the tip-surface forces Fts Eqns. (S2-S3).
m
d 2 z m0 dz

 kz  Fts  F0 cos t
Q dt
dt 2
(S1)
Fts (d )  
HR
6d 2
d  a0
(S2)
Fts (d )  
HR 4 *
 E R(a0  d )3
6a02 3
d  a0
(S3)
1
Here a0 is an intermolecular distance with a typical value of 0.165nm1 and E* is the
effective elastic modulus typically used in contact mechanics models2, 3.
The tip-
surface forces can be modelled with the long range van der Waals4 force for a sphere
of radius R and an infinite surface Eqn. (3) and the Derjaguin-Muller-Toporov
(DMT)5 model of contact mechanics Eqn. (4) can be used for the short range
interactions. These two forces are prevalent in any AFM experiment and have been
shown to be powerful for predicting the dynamics of the system6. No dissipation is
added in the model for simplicity and for another two reasons. First, the main aim is
to find the effective area of interaction per cycle <Area> in the steady state for a
stable system. This area can be shown to be similar (data not shown) when dissipation
is accounted for by following the method presented here, Eqns. (2-7) and
incorporating several relevant dissipation forces that have been shown to be relevant
in AM AFM7. Second; dissipative forces are difficult to model and are a challenge in
their own right8. Thus, since these would simply provide a slightly more accurate vale
of <Area>, the benefits of incorporating these forces do not justify the difficulty. The
equation of motion Eqn. (S1) has been solved by implementing it Matlab 9 and using
an standard fourth order Runge Kutta algorithm.
We have used the DMT theory of contact mechanics2 to calculate the contact area aDMT Eqn.
(S4) as a function of indentation δ and tip radius R. This is an equivalent expression to Eqn.
(3) in the main article.
a 2 ( DMT )  R
(S4)
It is also useful2 to introduce the concept of mean normal pressure p m Eqn. (S5) acting
between the tip and the sample. This is the term we have used in this article throughout. Note
that the peak pressure refers to the largest value of Pm during one cycle whereas the average
2
pressure is the time average of Pm per cycle. The maximum normal stress, that is, the stress at
the centre of contact (r=0), can be expressed in terms of p m as shown in Eqn. (S6). The
negative sign accounts for compression. It is observed that the maximum value of stress is
only 1.5 times the mean value.
pm 
4 E*
3 
 N ( MAX )  

(S5)
R
3
pm
2
r=0
(S5)
Finally, as stated, the modelling the effective area of interaction <Area> in the
dynamic mode can prove challenging but we use a method that produces reasonable
results10. First, the effective area for the vdW and the DMT forces are given by Eqn.
(2)10 and Eqn. (3)2,
5
respectively. These areas are valid for the static case and, in
particular, we have taken the effective area of interaction for the vdW accounting for
95% of the energy of interaction (note the index 0.95). The equation is derived by
assuming that the interaction is taking place between an infinitely thick cylinder of
radius r (surface) and a sphere of radius R (tip). The reference energy is that for an
infinite surface and it is used to normalise the energy of interaction. Then to account
for 95% of the vdW energy of interaction r is chosen so that the energy of interaction
between the cylinder and the tip is 95% that of the infinite surface and the tip. We use
Eqns. (4-6) to finally obtain <Area> in the dynamic mode (7). The index i stands for
the iteration parameter or instantaneous value. The final value <Area> Eqn. (7) is the
effective area of interaction per cycle for a given equilibrium cantilever-surface
separation zc. DNA on mica has been used a model system to investigate the
3
outcomes experimentally. Details on sample preparation can be found in the
literature11.
B. Full description of Figures 1 and 2.
In Figure S1 we show a nc mode image of the dsDNA molecules (m3-m7) prior to the
sequence described in the main article. The average height of the molecules in the
image is approximately 1.1nm and the molecules show a characteristic semi-flexible
polymer shape. The complete sequence of the scans described in the main article and
corresponding to the description of Figs. 2 and 3 is shown in Fig. S2. In Fig. S3 the
scans for low (A0=4.5nm), high (A0=95nm) and very high values of A0 (A0=165nm)
are shown for comparison. The result of the sequence can be observed in a nc mode
image obtained immediately after the above sequence and the molecular damage is
shown in zoomed views.
FIG. S1. Topography of the molecules submitted to the sequence described in the
main article and obtained in the nc mode with A0=3nm and Asp/A0=0.90 at resonance.
4
FIG. S2. Full topographic sequence of the dsDNA molecules on a mica surface
described in Fig. 2. The values of free amplitude used in the sequence are shown in
the bottom left corner. The size of the scan is 1μm2 and the scan rate 2Hz.
5
FIG. S3. Three of the scans in Fig. S2 are reproduced here to allow easy comparison
for the smallest (a) A0=4.5nm to (b) high values A0=95nm and (c) the highest value
A0=165nm used in the sequence. Here (a) looks like a typical nc mode image and (b)
like a typical tapping mode image obtained in the repulsive regime (Ref. 12). In (c)
the molecules are completely damaged. (d) A larger scan of the area (zoom out) taken
in the nc mode (A0=4.5nm and Asp/A0=0.9) shows the results. Only the molecules
submitted to the sequence are labelled; m3 to m7 (region inside the dashed rectangle).
Here m3 to m5 have been submitted to the higher free amplitudes (A0≤165nm)
whereas m6 and the left part of m7 to A0≤138nm only (compare with Figure S2). (e)
Zoomed view of m1 in the nc mode before the sequence and (f) after the sequence.
The molecular damage is clearly observed. (g) Zoom of molecule m 6 where
substantial molecular damage is also observed, even though it has been submitted to a
maximum of 138nm. As in Figure 3, this implies that molecular damage was
occurring already for A0≤138nm.
6
C. The relevance of peak pressure in the contact and its relationship to imaging
soft matter in the repulsive regime
Similar experiments with IgG antibodies can be conducted. In particular, we have
previously shown13, 14 that it is possible to image antibodies in the repulsive regime
and relatively large values of A0 with considerably high resolution without causing
molecular damage, for the full range of Asp/A0. As we discuss in the main article, the
fact that sometimes it is possible for a range of operational parameters to image soft
matter without inducing sample deformation whereas other times it is not is a
consequence of variations in the pressure in the tip-sample interaction due to the
dynamic character of R and the induced variations in <Area>. That is, the concepts of
<Area>, <Ets>/nm2 and pressure are much more fundamental than the more intuitive
concept of force, Fts. For example, in Fig. S4, we show simulated predictions of the
average (black continuous lines) and peak (dashed blue lines) tip-sample force Fts (the
top row in Figure S4) and the average (black dashed-continuous lines) and peak
(dashed blue lines) pressure (Pm) (bottom row in Figure S4) for R=6, 8, 10 and
R=20nm, for a relatively compliant sample (E=1.2GPa). Briefly, the average force is
almost independent of R, and the peak force (repuslive) only slightly dependent on R.
The highest dependence on R is found in Pm and it dramatically falls with increasing
R. This figure exemplifies that in terms of tip and/or sample damage the relevance lies
in the dependency of the pressure on R rather than of Fts on R.
7
FIG. S4. Relationships between peak forces and average forces per cycle (top), and peak and
average pressure per cycle as function of normalised tip-sample separations zc/A0 for a range
of A0 and R. In the top row, average and peak forces are shown for A0 = (a) 10nm, (b) 20nm
and (c) 30nm. Both the maximum (repulsive) and minimum (attractive) peak forces are
shown. The average force values are also shown represented by continuous lines for all R for
simplicity since these almost completely overlap. The respective average and peak pressure
per cycle is shown in the bottom row in (d), (e) and (f). In the case of A 0=10nm, there is
switch from the attractive to the repulsive regime for R=6nm while the attractive prevails for
the rest. For higher values of A0, all the interactions for Asp/A0≤0.8 take place in the repulsive
regime. Recall that Asp/A0=0.8 has been used in the experimental sequence in Figs. 2 and S3.
Average forces range from approximately 0.2nN in (a) to 1.5nN in (c). Nevertheless, it is the
behaviour of the force and pressure in the figure more than the actual values that are relevant
in this work. The parameters are: f=f0=300kHz, k=40N/m, Q=500, γ=40mJ, E=1.2GPa,
Et=120GPa and the rest as detailed in the figures.
8
We further show (Fig. S5) a sequence of topographic images of IgG antibodies on a
mica surface imaged with relatively high values of A0 where the antibodies can be
imaged in the repulsive regime. From Fig. S4, the highest peak pressure is reached for
0.6<Asp/A0<0.8. In Fig. S5 the IgG antibodies have been imaged at resonance and the
set-point has been gradually reduced. While it is clear from Fig. S4 that average and
peak forces can increase with reducing set-point, provided R is sufficiently large, the
pressure in the contact region and the energy dissipated per atom (Fig. 3(c)) remains
small enough to not cause molecular damage. In terms of resolution and comparing
attractive with repulsive imaging, the increase in <Area> with these larger values of R
can be compensated by the fact that <Area> significantly decreases in the repulsive
regime relative to the attractive (data not shown). These agrees with our previously
published results10, 13, 14.
FIG. S5. Sequence of topographic images of IgG antibodies on a mica surface in the
(a) and (b) attractive and (c) to (h) repulsive regime taken by gradually decreasing
Asp/A0 as shown. The bottom images are zoomed views. No apparent severe
molecular damage is observed by reducing the set-point and reaching the repulsive
9
regime in the hard tapping region (e.g. 0.8<Asp/A0, see Fig. S4). The parameters are:
f=f0=305kHz, k=40N/m and Q=550 and A0=27nm.
Imaging small scan areas with relatively sharp tips in the repulsive regime can lead to
severe molecular damage before R has had time to broaden and reach mechanical
stability. For example, in Fig. S6, a dsDNA molecule has first been imaged in the nc
mode (Figure S6(a)). Then a second scan has been performed where the Asp has been
periodically decreased and increased along the length of the molecule (slow scan
direction is upwards in Figure S6(b)) to reach the attractive and the repulsive regimes
respectively. This molecular scission scan is not shown.
Figure S6(b) was
subsequently obtained in the nc mode to observe and verify where damage occurred.
The regions where the attractive and the repulsive regime were reached can be clearly
distinguished by the intermittent cuts in the length of the molecule. However, for
clarity the regions where the repulsive regime was reached are marked by horizontal
arrows; the repulsive regime was reached three times and the three regions are clearly
distinguished. A second experiment with a nearby molecule with this tip and using the
same operational parameters did not induce observable damage on the molecule (data
not shown). In this case, A0 had to be increased to over 80nm to produce similar
effects. Then the outcomes were reproducible. This type of experiment is highly
reproducible and can be interpreted from Figs. 3 and S4 and understanding that the
broadening of the tip is ongoing in the repulsive regime until a given value of A0L is
reached. The meaning of A0L is discussed in the main article. These results have direct
consequences for nanomanipulation.
10
FIG. S6. Topographic images of dsDNA molecules on a mica surface obtained in the
nc mode (a) before and (b) after selectively imaging regions of the molecule in the
attractive and repulsive regimes (white horizontal arrows) by decreasing Asp/A0 from
0.95 to 0.4 and back respectively. The long vertical arrow shows the scan direction in
the slow axis. The parameters are: f=f0=312kHz, k=40N/m, Q=600 and A0=25nm.
Finally, for very high values of A0 (e.g. A0>100-150nm) holes can be produced on the
harder background mica surface and/or the tip might fracture mechanically even when
the radius is relatively large already (e.g. R>20-30nm). In Figure S7(a), a topographic
image of dsDNA molecules on mica is shown where A0 is already very large relative
to the standard values used in AM AFM. Furthermore the set-point corresponds to
hard tapping regions (Asp/A0=0.40, compare with Fig. S5) and the energy dissipated
per cycle, from the experimental value of the phase angle and using (1) is of the order
of keV. Interestingly, under these conditions the average height of the molecules is
still approximately 0.4nm, which is a typical value in AM AFM for dsDNA12. It is
remarkable that it is possible to image soft matter with these large values of A0
without causing severe molecular plastic deformation provided R is large enough.
Overall, this also argues in favour of a reduction in pressure and the spreading out of
the energy dissipated per atom with increasing R. Further increasing A0 to 200nm
(Figure S7b) while keeping the same normalised set-point produces a hole in the mica
11
surface. The hole can be seen in lower magnification topography (Figure S7(c)) and
phase (Figure S7(d)) images obtained immediately after the scan in Figure S7(b).
From bottom to top, first the tip is in the repulsive regime, then, in the middle of the
scan and where the hole is, there is much noise and the tip switches to the attractive
regime at the end. Even though there are background features and contamination on
the surface after the hole has been made, the molecules can still be resolved. A 3D
image of the hole is shown in Figure S7(e) and a cross-section in Figure S7(f). The
depth is approximately 40nm and material has been removed from the bottom layers
of the mica sheet and deposited on the top layer (approximately 80nm of height). This
is an example of the non-linear relationship between increasing R with A0, plastic
deformation and spreading out of energy in <Area>. But as stated in the main article,
this behaviour occurs either when suddenly increasing A0 from small to large values
(i.e. A0>50-100nm) when the tip is sharp (i.e. R<10-20nm) and/or when increasing A0
to very large values (i.e. A0>150-200nm) when the tip is relatively large already (i.e.
R>20-30nm). Recall that these numbers have physical significance in terms of plastic
deformation and nanofracture.
12
FIGURE S7. (a) Hard tapping topographic image of dsDNA molecules on a mica
surface. (b) Further increasing A0 produces a hole on the mica surface. The hole can
be observed in a subsequent scan in (c) topography, (d) phase and (e) 3D. (f) A crosssection of the region where the hole is (dashed line in Figure (e)) shows the depth of
the hole. The parameters are: f=f0=307kHz, k=40N/m, Q=450 and the rest as shown
in the other figures.
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