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EPAPS Supplementary Material
An Energy-Based Model to Predict Wear in Nanocrystalline Diamond AFM Tips
R. Agrawal, N. Moldovan, H. D. Espinosa*
Department of Mechanical Engineering, Northwestern University,
2145 Sheridan Road, Evanston, IL 60208-3111
*corresponding author
 Currently
at Advanced Diamond Technologies, Inc., Romeoville, IL 60466
1. Mathematical method (MM) to estimate tip radius based on surface roughness
measurements
A mathematical algorithm developed by Villarrubia [1] was used for de-convolution of
tip radii. A 2.5 X 2.5 m2 area on the UNCD surface was characterized in terms of
roughness and a calibration curve was obtained. This calibration curve relates the
roughness of the “characterized surface” as measured from an AFM image, to the radius
of the tip used to scan the surface. The details of the method follow.
To accurately determine the surface morphology of the substrate, a sharp Silicon Nitride
(Si3N4) probe (tip radius < 10 nm, as specified by the vendor (Budget sensor Inc.)) was
first characterized. The tip was examined under SEM to obtain an estimate of its shape
and radius. The sequence of images shown in Figure S1, with increasing magnification,
reveal that imaging at very high magnification results in blurriness due to thermal
vibration and carbon (or other contaminants) deposition during scanning with the electron
beam. In this particular case, the color contrast between Si3N4 and contamination
confirms that the tip radius is certainly below 10 nm. To obtain a better estimate of the tip
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radius, a 1.0 X 1.0 m2 area was scanned on a UNCD surface. The AFM image obtained
with this tip (Figure S2a) was later used to predict the tip shape, using the “blind tip
estimation” subroutine [1]. A tip radius of 3.6  0.2 nm was calculated, consistent with
the SEM metrology (Figure S1) and a tip shape (T) as shown in Figure S2b was
predicted.
1 m
50 nm
100 nm
Figure S1 SEM images of a commercial Si3N4 probe (from Budgetsensors, Inc.)
(a)
(b)
R = 3.5 nm
Figure S2 (a) A 1x1 um2 AFM scan of UNCD substrate obtained with the probe shown in Figure S1;
(b) Characterized Tip shape (T) estimated from (a) using the Blind Tip Estimation algorithm [1].
Note that each curve corresponds to a tip cross-section in the xz plane.
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The tip shape (T) was also used to characterize a larger area (A), 2.5 X 2.5 m2, on the
substrate. This characterized area (A) was subsequently scanned before and after wear
with other probes to measure its roughness. The estimated tip shape (T) was used to erode
the AFM image of area A, to remove the artifact of tip shape (T) from the obtained image.
This step yielded the actual surface morphology: Aactual  A  T , where the ‘-‘ symbol
here denotes the erosion operation.
Since any arbitrary tip produces images different than Aactual, due to its particular shape,
we mathematically perform a dilation operation for parabolic tips of different radius (Tr)
to obtain the corresponding surfaces (Ar). In languages of sets, a dilation operation is
defined as: Ar  Aactual  Tr , where the  symbol denotes the dilation operation, i.e.,
adding the artifact of tip Tr to the actual area Aactual
The RMS roughness of these mathematically generated surfaces was then calculated and
plotted as a function of tip radius. This resulted in a calibration curve (shown in Figure
S3) for roughness of the surface measured as a function of the tip radius used to scan the
surface.
From this calibration curve, the radius of the tip can be estimated based on the roughness
of the area (A) measured by the tip. It is important to note that as the tip radius becomes
larger, the measured roughness decays abruptly reducing the resolution of the calibration
curve. The autocorrelation length () of the surface was measured to be ~180  20nm.
From the RMS roughness-tip radius plot, it is apparent that the obtained curve flattens as
the tip radius reaches half of the correlation length. Since the larger tip radii can anyways
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be measured with reasonable accuracy through SEM imaging, the calibration curve
reported here is used only when the tip radius was measured to be less than ~90 nm
(</2).
Figure S3 Calibration curve obtained after dilation of actual UNCD surface to obtain roughness of
the surface as a function of the tip radius used to scan the surface. The red line denotes a tip radius
equal to half of the surface autocorrelation length.
2. Calculation of Wear Volume
Table S1 summarizes the tip radii as measured via SEM imaging and as predicted by the
deconvolution method described above. It is noteworthy that the “before wear”
measurements of tip radius from both methods are in agreement within ~10% for most of
the cases. After wear, the MM remains useful only when the wear is such that the tip
maintains its parabolic shape. For example, we found that for an applied contact load of
70 nN, doped and undoped UNCD probes undergo changes too small to be captured by
SEM imaging; however the MM showed a change of ~ 2-3 nm. A limitation of the MM is
related to the formation of large and sharp asperities during wear as these asperities can
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produce a very fine AFM image, which might not represent the actual worn tip radius.
Likewise, in high wear cases, where the tip shape deviates significantly from being
parabolic or the tip diameter becomes comparable to the autocorrelation length (~180 
20nm) of the surface (see supporting information), SEM images were used to calculate
wear volume.
A schematic of the change in profile of an AFM probe before and after wear is shown in
Figure S4. Knowing the initial tip radius (Ri), the final tip radius (Rf) after wear, and the
tip shape, the volume removed during wear can be calculated using the following
formula:
V 

3
( R f  Ri ) ,
3
3
(S1)
where  is a geometrical factor derived for the conical tip shape and is defined as:
  cot  cos 3   (1  sin  ) 2 (2  sin  ) , where ( ) is the tip’s half-angle. At high applied
contact forces of 200 and 280 nN, the Si3N4 probes were found to be flat after wear and
no final radius could be associated with the tips (as shown schematically in Figure S4b).
In these cases, the volume lost due to wear was calculated based on a truncated pyramidal
geometry with the initial tip radius set to Ri. Note that the geometrical factor  and prefactor (/3) are based on the tip shape and instead of being constant they can be a
function of tip height; for example, in case of oxide sharpened AFM tips.
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Table S1 Tip Radius measured before and after wear using two different methodologies. Tip radius
using de-convolution method is not reported (represented by a ‘-‘ in the table) wherever tip shape
deviated from being parabolic. For doped UNCD, three probes were tested which are reported here.
Applied Before
Contact or After
Force
Wear
(nN)
70
Before
After
100
Before
After
130
Before
After
175
Before
After
200
Before
After
280
Before
After
Silicon Nitride
SEM
MM
(nm)
(nm)
Doped UNCD
SEM
MM
(nm)
(nm)
Undoped UNCD
SEM
MM
(nm)
(nm)
75
125
42
154
83
~170
47
~375
45
Flat
40
Flat
64
65
x
x
x
x
65
204
x
x
107
375
32
32
32
88
62
125
38
148
88
175
225
280
68
38
74
42
42
-
61
63
x
x
x
x
63
x
x
-
24
27
27
46
50
17
80
-
Rf
Ri
Ri
(a)
2
(b)
2
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Figure S4 (a) Schematic of an AFM probe showing the initial and final tip radius and the
corresponding volume removed during wear (dark gray). (b) Schematic of an AFM probe with
higher contact force such that the tip apex flattens after wear. Dark grey corresponds to
removed material.
3. Wear of the substrate
The wear of the substrate was found to be negligible, by subsequent measurements with
an unworn tip. For example, AFM images in Figure S5 are taken on an area on which
wear test was performed with an undoped UNCD probe at 200 nN load. These images
were acquired before and after wear experiment, using an unworn tip. Regions 1 and 2 (in
dashed lines) show that the topographic features of the substrate are not substantially
altered, which confirms that there was no wear of the substrate.
Figure S5 AFM images taken with unworn sharp AFM tip before (a) and after (b) wear test. The test
in this case was performed with an undoped UNCD probe at applied force of 200 nN. Two regions 1
and 2 are highlighted to show that the substrate wear is negligible as all the features are intact. The
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offset in the two images is due to the difficulty in finding the exact same area on the substrate after
changing AFM tips.
4. Separating the contribution of Fmen and Fadh from pull-off forces
Pull-off forces were measured before each wear test and the two contributions were
separated. The meniscus force was calculated based on initial tip radius, contact angles
and surface tension of water. The difference between pull-off force and meniscus force
was attributed to the adhesion force, which was then used to calculate the “work of
adhesion”. These results for undoped UNCD probes are given in Table 2
Table 2 Pull-off force measurements before the wear test and the calculation of work of adhesion
after subtracting the meniscus forces.
Tip
Material
Case
No.
1
2
3
Undoped
4
UNCD
5
6
Initial
Tip
Radius,
Ri (nm)
24
27
50
38
87
225
Pull-off
Force,
Fpull-off
(nN)
18.3
20.9
40.7
32.0
68.0
167.5
Calculated Estimated
Meniscus adhesion force,
force, Fmen Fadh = Fpull-off (nN)
Fmen (nN)
8.9
9.39
10.0
10.81
18.6
22.15
14.1
17.91
32.4
35.67
83.7
83.76
Average value of w
Work of
adhesion
(mJ/m2),
w = (Fadh/2Ri)
62.3
63.8
70.5
75.0
65.3
59.3
66.0 ± 5.8
5. Dependence of frictional force on contact area
In our study, we verified that the frictional force varies linearly with the contact area.
Figure S6 shows the graph of frictional force as a function of contact area for Silicon
Nitride and undoped UNCD probes.
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Friction Force (nN)
2.0
y = 0.3436x + 0.0659
R2 = 0.9386
y = 0.1988x - 0.0335
R2 = 0.9638
1.5
1.0
Silicon Nitride
undoped UNCD
Linear (Silicon Nitride)
Linear (undoped UNCD)
0.5
0.0
3
4
5
6
Contact area (nm2)
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Figure S6 Plot of frictional force as a function of contact area
Reference
Villarrubia, J.S., Algorithms for Scanned Probe Microscope Image Simulation, Surface
Reconstruction, and Tip Estimation. Journal of Research of the National Institute of
Standards and Technology, 1997. 102(4): p. 425.
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