Supplementary materials
Morphology dependence of radial elasticity in multiwalled Boron Nitride
nanotubes
H.-C. Chiu1, S. Kim1, C. Klinke2 and E. Riedo1a)
1
School of Physics, Georgia Institute of Technology, Atlanta, GA, 30329, USA
2
Institute of Physical Chemistry, University of Hamburg, 20146 Hamburg, Germany
1. Characterization of the Boron Nitride nanotubes (BN-NT) and sample preparation:
(1) Transmission Electron Microscope (TEM):
Transmission electron microscopy imaging was performed with a Philips CM-300 UT
microscope at 200 kV and a JEM-Jeol-1011 microscope at 100 kV with a CCD camera
(Gatan694).
Figure S1. TEM images of the BN-NT and Carbon Nanotube (C-NT) used in this work.
(2) Preparation of BN-NT samples:
The CVD grown BN-NTs are purchased from NanoTechLabs, Inc. (Yadkinville, NC, USA). The
BN-NTs were first suspended in IPA (Isopropyl alcohol) and then agitated in an ultrasonic bath
for about an hour. Next, a droplet of this solution was drop-casted on an untreated, clean silicon
wafer (~1 cm2). After 30 seconds, the remaining solution was blown away with a nitrogen pistol.
To achieve higher nanotube density on the surface, this procedure can be repeated several times.
The BN-NTs will be adsorbed on the silicon substrate via van der Waals force and are ready for
the Modulated Indentation (MoNI) experiments with an Atomic Force Microscopy (AFM).
1
(3) The statistics of Rext/Rint as obtained by TEM from 33 MW BN-NTs. The center from the
Gaussian fit is 2.0±0.48, where the error here is the Full Width Half Maximum (FWHM) of the
Gaussian distribution. The insect shows a typical AFM image of BN-NTs deposited on a silicon
substrate.
Count
15
10
5
0
1
2
3
4
5
6
Rext/Rint
Figure S2. The statistics of Rext/Rint as obtained by TEM from 33 MW BN-NTs. The insect shows
a typical AFM image of BN-NTs deposited on a silicon substrate.
2. Force-distance curve on a BN-NT with RNT ≈ 25 nm:
(b)
15
10
5
Deflection (nm)
Deflection (nm)
(a)
BN-NT: Loading
BN-NT: Unloading
Substrate: loading
0
-5
60
70
80
2
0
-2
-4
-6
78
90
zpiezo (nm)
BN-NT: Loading
BN-NT: Unloading
Substrate: loading
80
82
84
86
88
zpiezo- Cantilever bending (nm)
Figure S3. (a) Raw force-distance curve obtained from a MW BN-NT with a cantilever of spring
constant klev ≈ 0.2 N/m ;(b) Zoom-in and calibrated force distance curve in (a). The spring
constant of the cantilever used in these measurements is about 0.2 N/m. The steep contact line
shows that this nanotube has similar stiffness to the silicon substrate.
2
3. Modulated Indentation (MoNI) with AFM:
When performing MoNI, the force between the AFM tip and the sample is maintained constant
by the feedback loop while a small oscillation is applied to the piezo plate attached to the tip for
small indentation. In this configuration, the feedback gain parameters have to be large enough to
maintain the setpoint but also need to be small enough “not to see“ the small oscillation we
applied externally. Thus, for a given oscillation frequency, we change the feedback settings until
a correct Young’s modulus can be obtained from the reference substrates such as fused silica or
ZnO used in this work. The calculation of the contact area and contact stiffness between the AFM
tip and the sample will be discussed in the next section. A typical experimental data of kcont vs. F
is shown in Fig. S4.
kCont (N/m)
80
60
40
20
 1
F
1 
 ktot ( F )  


z piezo
k
k
cont 
 lev
1
0
-20
-10
0
10
20
F (nN)
Figure S4. A typical measured contact stiffness kcont v.s. normal force F obtained from a BN-NT
of Rext =13.7±0.4 nm. The average measured ERadial of this BN-NT is 32.0±3.8 GPa.
4. Calculation of the contact area and contact stiffness between a spherical tip and a flat
surface or a cylinder:
To perform MoNI, we need to know the contact area between the AFM tip and the sample. This
can be calculated with the Hertz model presented below1:
(1) For flat sample substrate:
The contact area between a spherical AFM tip and a flat sample surface can be determined with
the Hertz theory modified with the contribution of the adhesion force FAdh:
3 R

Area    r    Tip* ( FN  FAdh )
4 E

2
2/3
2
2
1   Tip

1   sample

, where E  

Esample 
 ETip
1
*
is the effective modulus. E and  are the Young’s modulus and Poisson’s ratio of the respective
materials.
3
The contact stiffness in this case will be:
kcont  2rE *  2rE * r 2
1
r
2

2

 Area  E *
(2) For a cylindrical object such as a BN-NT:
The contact area between a spherical AFM tip and a cylindrical BN-NT is expected to be an
ellipse with semi-major axis a, and semi minor axis b:
Area    a  b
, where b is determined by
b3
, where 1 = 1 + 1
R
Rtip
3k  ( 1  k 2 )
R  ( FN  FAdh )  ka
2E *
and k is the solution to the following equation
2RNT
1
RTip
RNT
1
 ( 1  k 2 )   ( 1  k 2 )
2
k
 ( 1  k 2 )  ( 1  k 2 )
;  (k ' ) and  (k ' ) are the complete elliptic integrals of the second kind:
( k ' )  
 /2
0
 (k ' )  
 /2
0
1/ 2
(1  k '2 sin 2  ) d
(1  k '2 sin 2  )
1 / 2
d
From equation (2) in the manuscript, the contact stiffness in this case is
4 
kcont     E * 
3 
2/3
R  F  FAdh 1/ 3
, where     
 2k 
2/3
 (1  k '2 )1/ 3 
  (1  k '2 ) 


We can further rewrite kcont as
2/3
1
4 
1/ 3
kcont     E *   R  F  FAdh  ab
   Area  E * , where  is a function of Rtip and
3
ab


RNT, and can be calculated with the equations provided above.
4
5. Error in the radial elasticity resulting from the substrate deformation:
For the smallest and stiffest BN-NT we investigated, the obtained radial elasticity is 192.2±46.8
GPa. In the following we estimate the error in the radial elasticity resulting from the substrate
deformation, which is not accounted for in the equation (1) in the manuscript.
Using the Hertz theory, and a first order approximation (dF/dz = F/z), the deformation at the NTsubstrate contact can be described as2

F
z1 

k NT  substrate  E1 2 RNT
F




2/3
1
2
 3 1  vsubstrate

 . Here we assume the NT as
, where E1  
 4 Esubstrate 
a rigid rod to avoid double accounting of the deformation of nanotube 2, which will be accounted
for in z2, deformation at the tip-NT contact in the following.
At the tip-NT contact, the deformation can be estimated to be z2 = F/kcont , where kcont is defined
as equation (2) of the manuscript. Thus,
z2 
F
kcont
1  3F 
  *
  4 E R 
2/3
, where E* and R are defined in the manuscript.
Therefore by comparing z1 and z2, we will have k Nt  substrate 
z2
kcont and equation (1) will
z1
1
 1  z2  1 
F
 with

 ktot ( F )  
 1 
now become
z piezo
z1  kcont 
 klev 
z2 1  3 
  
z1   4 
2/3
 E1 
 *
E 
2/3
1/ 3
 2 Rext 
 * 
 R  .
We use this equation to repeat fitting to the data acquired from the BN-NT with RNT=3.7 nm. The
radial elasticity now becomes 233.5±54.2 GPa, which has a 21.5% increase than the value with
the consideration of substrate deformation. Such error is smaller than the error bar of 24%
reported in Fig. 3 of the manuscript.
5
6. A simplified model to demonstrate how the in-plane extension in layered material
influences the out of plane extension when a force is applied normally to the layers.
(a)
(b)
Consider two systems shown in figure (a) and (b) consisted of different number of atoms with
springs connecting one with the other. These springs have the same initial length l0 when no force
is applied. The in-plane spring constant k// is much larger than the out of plane spring constant k.
From figure (a), the simplest configuration, and (b), a system containing n atoms and springs, we
2
2
2
will have the following two equations L1  z  l0 and (
n
L )
i 1
i
2
 z 2  (nl0 ) 2 .
A straightforward calculation will show that the in-plane extension for the configuration (a) is
z 2
L1  l0 
; and for configuration (b), the in-plane extension will be
L1  l0
n
 L  nl
i 1
i
0

z
n
 L  nl
i 1
i
n
n
i 1
i 1
. Since  Li  nl0  L1  l0 , we will have L1  l0   Li  nl0 .
0
This shows that with a increasing number of springs in the plane, the total extension along the
plane becomes smaller and has less influence on the out of plane extension. Because of this effect
and k// >> k, it is expected that the radial modulus of the nanotubes (when ovalization is absent)
is larger for smaller nanotubes (less springs can be stretched on the parallel plane) and it
converges to the E33 modulus of the 2D film for large and thick nanotubes.
Reference
1. K.L. Johnson, Contact Mechanics, Cambridge University Press (1987)
2. Y. H. Yang and W. Z. Li, Appl. Phys. Lett. 98, 041901 (2011)
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