Sub-surface AFM imaging using tip generated stress and electric fields
Supplementary Material
S1. Finite element analysis (FEA)
We used the capabilities of FEA built into COMSOL Multiphysics 4.4.
a. Electrostatic force calculation
The Electrostatics interface, available in the AC/DC module, is used to solve the
electrostatics problem with boundary conditions specified by i) a constant electric potential
(   V 0 ) applied to the probe, ii) a constant electric potential (   0 , ground) at the bottom
surface of the sample (polymer matrix with dielectric constant  PI ), iii) a floating potential on the
cylinder surface (CNT bundle), and iv) zero charge ( n  D  0 ) in the air region surrounding the
probe and the sample. Charge conservation was enforced everywhere. We use the floating
potential condition assuming the CNT bundle is conductive or has a conductivity many orders of
magnitude larger than the surrounding medium and is not connected directly to ground. The tip
of the probe is modeled as a hemisphere (apex) with radius Rtip attached to a cone with a half
angle  and length Ltip . The cantilever is modeled as a disk of radius R cant and thickness Tcant
approximately equal to the given specifications of the FORTA probe (AppNano). The
surrounding air is defined as an infinite element domain. The values of relevant parameters are
given in Table I.
To compute the electrostatic force on the probe an integration of the Maxwell’s stress tensor
is performed on the external surface
F      Τ dV 
V
S Τ  dS ,
(S1)
where the stress tensor T is given by
T
0
E  E   0EET .
2
(S2)
Numerous cross-checks were made on the code to insure that calculated forces and
capacitances for well-defined geometries matched known analytical solutions. The convergence
criteria of the employed stationary solver is defined by a relative tolerance equal to 0.001.
b. Contact stiffness calculation
A similar procedure is followed to simulate the stress field generated when the tip exerts a
force on the polymer composite. The contact stiffness that results when an AFM tip scans over
a nanocomposite is analyzed using the Solid Mechanics module. The tip is modeled as a silicon
hemisphere. The top of the sample and the bottom of the tip are defined as a contact pair while
the CNT-polymer interface is treated as perfectly bounded. Automatic global fine meshing is
applied on all FEA elements, with further adaptive refinement around the tip-sample contact area
and the CNT-polymer interface.
At each point of contact, the FEA calculates the sample deformation dFn for an applied
normal load Fn. The contact stiffness is then calculated by,
1 ( F  F)  Fn F(n  Fn ) F 
k (* F)n   n

.
2  d Fn F  d Fn
d Fn  d Fn F 
(S3)
Here, F is an incremental force step (5 nN in our calculations). Before computations, we
validated the 3D FEA model by comparing with the Hertz theory in the case of an AFM tip
contacting an elastic half-space. The relative deviations of the contact stiffness were found to be
less than 5% for a load up to 100 nN. The relevant parameters used in these simulations are
listed in Table I.
S2. Experimental AFM details
A block diagram illustrating the set-up for both the 2nd-harmonic KPFM and CR-AFM
techniques is provided in Fig. S1.
FIG. S1. Schematic of the experimental setup for sub-surface imaging of polymer-CNT
composites. In (a) the diagram of the basic modules used in 2nd harmonic KPFM. The A 2
e
map is used for mapping sub-surface CNTs. In (b), the diagram of the setup for CR-AFM.
The CR-Freq map is monitored to map sub-surface CNTs.
a. 2nd-harmonic KPFM (see Fig. S1.a)
The rationale for using 2nd-harmonic KPFM technique to probe subsurface features has been
established in previous works1,2. Considering the cantilever-sample system as a small vibrating
capacitor, the electrostatic force between the tip and sample with the tip positioned a few
nanometers above the sample is given by
Fe 
where
C z
is
V Vdc Vcp V ac e
the
i e t
1 C
 V
2 z
change
in

2
,
(S4)
capacitance
with
separation
distance
z
and
. V dc is a dc voltage applied to cancel Vcp , the contact potential
difference between the tip and the sample and V ac is an ac voltage applied to the tip at a
frequency e . Using V in equation S4 and expanding, three separate contributions to the
electrostatic force are found 3
Fe  Fdc  Fe  F2e .
(S5)
In this work, we focus on the last term in equation S5, the component of the electrostatic
force at the 2nd harmonic of e which is given by
F2e 
1 C
4 z
i 2 
Vac2 e  e e  .
(S6)
2e
The electrostatic force F 2e is mapped by the output of Lock-in 3 which measures the amplitude
( A2e ) and phase ( 2e ) of the detected force on the cantilever at the frequency 2e . In essence,
A2e tracks the spatial dependence of C z across the sample. To convert the measured A2e
to C z , we use
C
z

2e
4 k A2
e
Vac2 | H |
,
(S7a)
with H given by
H 
1
1 r    r Q 
2
2
,
(S7b)
where r  2e 0 , k and Q are the spring constant and quality factor of the cantilever,
respectively. Equation S7b is the magnitude of the cantilever transfer function at 2e assuming
a point mass model as described elsewhere4.
To implement this scheme in double-pass KPFM mode, a conductive cantilever is
mechanically excited at a frequency m approximately equal to the first eigenmode (mechanical
resonance) frequency of the cantilever ( 0 ). The signal detected by Lock-in 1 serves to map out
the surface topography of the sample using a conventional non-contact mode which keeps the
distance between the tip and the sample constant. To avoid inadvertent jump-to-contact while
imaging topography, we used the second eigenmode frequency of the cantilever. In the double
pass mode, the tip is withdrawn by an amount
z lift , and an electrical excitation
(V V ac sin e t Vdc ) is applied to the tip with e close to 0 2 . The tip is scanned across the
sample surface taking into account the geometry of the sample that was just previously
measured. Lock-in 2 detects the ωe signal which maps the surface potential across the sample.
Lock-in 3 detects the cantilever response at 2e which is proportional to the 2nd harmonic of the
electrostatic force (equation S6) using resonance-enhanced mode ( 2e  0 ). Some of the
parameters used in this study are: V ac  1.7 V, z lift 10 nm, and 2e  53.4 kHz . The typical
image size was 10 μm x 10 μm and contained 512 x 512 points. The scan rate was 1.0 Hz and
it required ~ 8.5 minutes to acquire a single image.
The minimum resolvable force is indicated by the horizontal threshold line marked in Fig. 2a.
The force sensitivity is defined as the minimum detectable force and is mainly limited by the
noise level which is dominated by i) the thermal noise of the cantilever ( th ) due to Brownian
motion and ii) the detector noise with the optical beam deflection sensor ( obd ) as the major
contributor5. In Cypher AFM obd  25 fm/Hz1/2 .
The thermal noise is given by
th 
2k BT
 f 0kQ
H ,
(S8)
where k B is the Boltzmann constant and f 0 , k , Q and H are the resonance frequency, spring
constant, quality factor and transfer function of the cantilever, respectively. Within the frequency
bandwidth range of the lock-in amplifier ( B ), the minimum detectable amplitude due to thermal
noise is then given by Ath  th 2B , which depends on the cantilever geometry and the
environment.
With the parameters from the experiment, f 0  53475 Hz , k 1.23 N/m ,
Q  99.05 , B  1000 Hz , at an excitation frequency of 55000 Hz (assumed for A2e detection),
Ath is equal to 15.14 pm. Then the minimum detectable force due to thermal noise ( Fth ) is 1.09
pN. On the other hand, the minimum detectable amplitude and force due to obd is 1.12 pm and
2
0.081pN, respectively. Therefore, the total minimum force is Fmin  Fth2  Fobd
 1.09 pN .
b. CR-AFM (see Fig. S1.b)
After mapping the KPFM signal, the same region of the sample is studied using CR-AFM
using the same probe. Ultrasonic excitation is generated by a piezoceramic transducer (Steminc,
Miami, FL, USA), placed in close contact to the bottom of the sample. The transducer has a
resonance frequency of 4.25 MHz and a nominal piezoelectric constant of 450 pm/V. The
excitation waveform is generated by the internal direct digital synthesizer of the AFM controller
and the cantilever oscillation signal is analyzed using the ARC2 controller of the AFM. Subsurface imaging is realized by either driving at a single frequency or employing the dual AC
resonance tracking (DART) mode6.
The cantilever is first brought into contact with the sample at a preset normal force. By using
a frequency sweep, the contact resonance frequencies are determined. Then, dual excitations
are applied to modulate the tip-sample contact at two frequencies ( 1 and 2 ). These two
frequencies are centered near the contact resonance with one below resonance and another
above. The corresponding cantilever amplitudes ( A1 and A2 ) are analyzed by lock-ins 1 and 2
respectively. During scanning, the resonance frequencies are tracked by changing the excitation
frequencies via a feedback loop to maintain the amplitude difference constant 6. In our
experiments, the applied normal force was approximately 65.0 nN. The 3 rd eigenmode with
resonance frequency of ~1.035 MHz was used because it demonstrated the best frequency
sensitivity to the contact stiffness variations among the first three eigenmodes. The typical image
size was 10 μm x 10 μm and contained 512 x 512 points. The scan rate was 1.0 Hz and it
required ~8.5 minutes to acquire a single image.
The acquired resonance frequency as a function of
lateral position is needed to infer the contact stiffness
k * , using a model that includes the lateral force, the tip
position offset and cantilever tilt7. The cantilever is
approximated as a rectangular beam with uniform cross
section. As shown in Fig. S2, the tip with a length h is
located at position L1 from the cantilever base. The total
length of the cantilever is L and its tilt angle is  . The
tip-sample contact is represented by a normal contact
stiffness k * and a lateral stiffness k L* . Owing to the high
quality factor of the contact resonance, the influence of
damping on the frequency shift is neglected.
FIG. S2. Schematic of the
analytical model of a cantilever in
contact with the surface of the
sample, used to find k * 7.
The equation for flexural vibration of the cantilever is then,
EI
 4 y( ,x )t
( 2, y) x t


A
 0,
x 4
t 2
(S9)
where E is the elastic modulus, I is the area moment of inertia,  is the mass density and A
is the cross-sectional area. The cantilever deflection is y( x) , y x is the slope, EI  2 y / x 2
is the torsional moment and EI  3 y / x 3 is the shear force. At the clamped position, the
deflection and the slope have to be zero. At the free end, no moment and shear force can be
present. A general solution of the above differential equation can be expressed as,
x
y( ,x )t ()y x( e i t  Ae
 A2e x  A3e i x )A4e i x e i t .
1
(S10)
Substituting the general solution into equation (S9), we have the dispersion relation with the
wave number  ,
EI  4   A 2  0 .
(S11)
The above dispersion equation is used to calculate the resonance frequencies. The resonance
frequency of the nth mode normalized by the free resonance frequency is,
2
n  n L 

 .
n0  n0L 
(S12)
Here, superscript 0 denotes the free resonance and the subscript n denotes the eigenmode
number.
To relate resonance frequencies to contact stiffness, we need to build the characteristic
function, which is obtained from the general solution by taking the boundary conditions into
account7. For convenience, we can define the mode shapes y1 ( x1 ) and y2 ( x2 ) for the two
cantilever sections of length L1 and L2  L  L1 , y1 ( x1 ) is the same as y ( x) . Function y2 ( x2 ) begins
at the free end of the cantilever and terminates at the tip position. At the tip position, we have
the corresponding boundary conditions,
y  y2,
(S13)
y  x 
y
 2 2 ,
x
x 2
(S13’)
  2 y  2 y 22 
EI  2 
 Fx h ,
2 


x

x

2 
(S13’’)
 3 y 3 y 2 
EI  3 
  Fy ,
x 23 
 x
(S13’’’)
where
Fx  h
y *
( ksin
x
Fy  h
y
sin cos ( k L*  k)* ( ysink *
x
2
*
  kcos
L
2
) sin
y cos
 (  k L* ) k * ,
2
 cos
 k L* ) 2  .
(S14)
(S14’)
The boundary conditions are then used to determine the unknown constants in the mode
shape function. After a rather tedious process, we finally obtain 8,
C kc
k*

B

3
A
 1,
3 k*
kc
where
(S15)
2
h 
A     1  cos n L1 cosh n L1 1  cos n L2 cosh n L 2  ,
 L1 
2
 
k L*
,
k*
(S16)
(S16’)
B  B1  B 2  B 3 ,
(S16’’)
C  2  n L1  1  cos n L cosh n L  ,
(S16’’’)
4
and
2
h 
3
B1     n L1  sin2 a   cos2 a
 L1 
 1  cos n L2 cosh n L2  sin n L1 cosh n L1  cos n L1 sinh n L1  ,


(S17)
 1  cos n L1 cosh n L1  sin n L2 cosh n L2  cos n L2 sinh n L2  
h 
2
B 2  2    n L1     1 cos a sin a
 L1 
,
 1  cos n L2 cosh n L2  sin n L1 sinh n L1
(S17’)
 1  cos n L1 cosh n L2  sin n L2 sinh n L2 

B 3  n L1 cos 2 a   sin2 a

 cos n Lcosh
 (1
n L)s2 in
2
cosh
L
n 1
n L1
 cos n L1 sinh n L1   1  cos n L1 cosh n L1 
.
(S17’’)
  sin n L2 cosh n L2  cos n L2 sinh n L2  
Combining equations S12 and S15, we can thus relate the measured resonance frequency
to contact stiffness. The main parameters used in recovering the stiffness are listed as follows:
cantilever spring constant of 1.23 N/m, 1st mode free resonance frequency of 53.48 kHz,
cantilever length of 225 μm, length ratio L1 L of 0.96, tip height of 15 μm, cantilever tilt angle of
11, lateral to normal stiffness ratio of 0.85. These parameters are chosen in accordance with
experimental measurements.
Contact resonance sensitivity is further analyzed from the frequency-stiffness relation9. The
sensitivity can be defined as the frequency change with respect to the contact stiffness change,
that is,
Sn  ( n/ /)10( / k *) kc . The calculated sensitivity ratio defined in this way was
compared with the measured frequency shift ratio among the first three eigenmodes. Within our
experimental settings, the 3rd eigenmode resonance demonstrated the best sensitivity to contact
stiffness. The experimental frequency resolution is roughly estimated as 1 kHz. Using tip radius
of 12 nm, normal force of 65 nN, cantilever spring constant of 1.23 N/m and 1 st free resonance
frequency of 53.48 kHz, the corresponding stiffness resolution is determined to be approximately
0.84 nN/nm.
S3. Depth estimation: comparison between CR-AFM and KPFM
As a further validation of depth estimation, an additional region shown in figures S3(a) for CRAFM and (b) for KPFM, was analyzed following the same procedure described in the manuscript
(figure 4). It is noted that the results in figure S3(c) follows the same trend of over- and underprediction of CNT depth using CR-AFM and KPFM, respectively.
Figure S3: Comparison of depth estimation of CNT bundle using CR-AFM and KPFM. In (a) and
(b) are the sub-surface maps from the 3rd eigenmode CR-Freq (CR-AFM) and the A2ωe map
(KPFM), respectively. In (c), the estimation of CNT depth at positions numbered from 1 to 7 in
the two maps, using the results from the FEA in Fig. 2(a) of the manuscript.
S4. Polyimide-SWCNTs Sample Preparation
SWCNTs with diameters of approximately 1.5 nm (Carbon Nanotechnologies Inc.) are bath
sonicated in dymethylacetamide (DMAc, Acros organics) for 2 hours before adding 44'Oxydianiline (Chriskrev), 3-3', 4-4'benzophenone tetracarboxylic dianhydride (Acros Organics)
and sodiumdodecylbenzene sulfonate surfactant (Aldrich, 5 weight percent (%wt) with respect
to polymer). After 3 hours of polymerization under sonication, films are cast on glass plates,
DMAc is removed at 80oC under vacuum and the polymer is thermally imidized by a series of
isothermal steps from 150 to 350oC, thereby obtaining PI-SWCNT films between 25 and 35 μm
thick. The flexible polymer films produced in this way appear yellow or opaque depending on the
SWNT concentrations. From parallel SEM studies, there is good evidence that the individual
CNTs form twisted ropes of approximate diameter of ~20 nm with a typical length of a few 100
nm10. These twisted ropes of SWCNTs are referred to as CNT bundles and are modeled by
cylinders using FEA.
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