Supporting information part:
Detecting single molecules inside a carbon nanotube to control molecular
sequences using inertia trapping phenomenon
ZL Hu1,2, Gustaf Mårtensson1, Murali Murugesan1, Yifeng Fu1, Xingming Guo3, and Johan
Liu1,3*
1. Chalmers University of Technology, Address: Kemivägen 9, Se 412 96 Göteborg, Sweden
2. Shanghai Institute of Applied Mathematics and Mechanics No 149, Yan Chang Road,
Shanghai 200072, China
3. Key Laboratory of New Displays and System Integration, SMIT Center and School of
Mechatronics and Mechanical Engineering, Shanghai University No 149, Yan Chang Road,
Shanghai 200072, China
*E-mail: jliu@chalmers.se
A. Numerical simulations:
A molecular dynamics (MD) simulation is performed to verify if the molecule can remain
in the carbon nanotube (CNT) and influence the vibrational amplitude. Consistent with a
previous study1, a (5,5) CNT and Kr based atoms will be utilized in the simulations.
To model the interaction of the carbon-carbon atoms in CNT, the adaptive
intermolecular reactive empirical bond order (AIREBO) potential2 was used. AIREBO is a
generalization of the reactive empirical bond order (REBO) potential, also called Brenner
potential3. The AIREBO-potential describes both the short-range (covalent sp2) interactions
(EREBO), as well as the long-range Van der Waals (vdW)- (Lennard-Jones (ELJ)) interaction
between the carbon atoms. For interactions between the CNT and the trapped particles, as
well as between trapped particles, only the Lennard-Jones 6, 12 potential was applied
1
which has parameters depth of the potential well  and the finite distance at which the interparticle potential is zero . And the cut-off distance is rc=10.0 Å. For the carbon atoms, the
values 1/kB=28 K and 1=3.4 Å4,5, where kB is the Boltzmann constant were used. For the
krypton atoms, on the other hand, the values 2/kB=171 K and 2=3.6 Å5,6 were
implemented. The interaction between carbon and krypton atoms can to a good
approximation be modeled by taking  to be the arithmetic average of 1 and 2, while  is
set as the geometric average of 1 and 27. This leads to a value of  = 3.5 Å, however 
will be taken as 3.4 Å, to fit the small diameter of a (5,5) CNT. In this paper ‘model’
particles with masses larger than the Kr atom will also be chosen while vdW coefficients
were kept the same.
(a)
(b)
Vb
I
Fix
Fix
+
+
+
+
+
Vb
-
e-
VRF
I
VRF
Figure S1. Schematics of the actuation/detection setup of (a) “” encoders (see also Figure 2a of reference
8); (b) “” encoders (see also Figure 1 of reference 15). Only the beam part of the CNT is shown.
Generally a voltage Vb is applied to the device. Then physically one would drive a CNT electrostatically to
oscillate by coupling to the CNT with radio-frequency (RF) signal VRF with a driving frequency f. The
amplitude of the vibration can be detected by measuring the current I (see references 9,10,15) since there
is a coupling between the current and the amplitude. Notice that for (b) this I is field emission current.
Since the largest amplitude of the vibration happens only with the resonance frequency, by finding the
peak of the function of I(f) it is possible to get the resonance frequency (see references 8,15).
2
During the simulations, the CNT were put in a Berendsen thermostat to keep the system
at a stable temperature of T0=30 K mainly or 77 K for encoder +, in which the amplitude
is too big to let the system to be at 30 K. Note that trapped particle(s) were excluded from
the thermostat since coupling with the thermostat would alter the motion of the inner
particles. Throughout the simulations, the instantaneous total energy was monitored
continuously. The time step was taken to be 0.5 fs to consist with1.
Encoder + was studied first. Figure 1 shows the initial state of the molecular encoder
+. The corresponding experimental schematic of the CNT resonator is shown in Figure
S1a. In the simulations, a 65 ring (5,5) CNT was used, see Figure 2a. The length of 65 rings
was chosen because if the CNT is too long it may be too expensive to take the simulation
and if it is shorter the CNT may not be well treated as a beam. The coordinate system of the
model, with the z-direction parallel to the main axis of the CNT, is shown in Figure 2a and
the origin was set to be the left end of the CNT axis. The left 6 rings and the 55th-60th rings
of CNT were clamped. The right 5 rings were under a radial pressure of 21 GPa, which is
twice the critical radial pressure for radial buckling11. Due to the radial pressure, the right
tip was flat and the passage for particle was closed. This radial buckling will be taken to
mimic the switch in Figure 1. The overall structure of the CNT is a double-clamped beam
with a prolongation to the right boundary. To begin with the eigenfrequencies for this
system were calculated using MD method. To find the resonance frequencies in the case
that a krypton atom is trapped at an antinode, the procedure was under the assumption that
the particle was initially pinned to the location of the specific antinode. The pinning of the
particle implies that the particle’s z-coordinate was fixed, while its x and y coordinates were
left free. Thereafter, the trajectories (y-coordinates) of atoms at the center of the beam were
recorded for 5 ns at a sampling rate of 2 THz and the spectrum was obtained by using the
3
fast Fourier transform (FFT). Following this procedure, the resonant frequency of the first
mode with zero amplitude of the vibration were turned out to be f = 139.5 GHz, which has
one antinode at z=70 Å. This frequency, as well as several close values, would be chosen as
driving frequencies in order to excite the eigenmode of the beam. Vibrations were excited
by putting F=F0 sin(2ft) with F0=5 meV/Å on each carbon atom of the beam on ydirection. This excitation is different from that shown in Figure S1, where the force
concentrates somehow near the CNT center. Luckily, since excited modes are determined
only by the driving frequency, the conclusions will not be affected. During the simulations,
the y-coordinate at the center of the beam was recorded with a sampling rate of 2 THz
during the elapsed time from 2.5 ns to 5 ns. The variance VyT of this data was subsequently
calculated. Hereafter, VyT would be used to quantitatively describe the amplitude of the
vibration of the beam. With the premise that the apex has a sinusoidal trajectory, the
amplitude of the vibration is A  2V yT . After the series of driving frequencies near the
predicted driving frequency were tested with a precision of 0.01 ps for their periods, the
frequency with the largest VyT was chosen for next simulations. The driving frequency was
finally 157.2 GHz with
2VyT =8.99 Å to excite the lowest mode, although the inertia
trapping can happen with another mode. Thereafter a Kr-atom was initially be place at z = –
10 Å, and shot into the CNT with an initial speed of 1 m/s after 0.5 ns of excitation on the
beam. This initial speed was increased to ~750 m/s once the particle entered the CNT.
Figure 2b shows trapping of a Kr-atom into the CNT, during which that >700 m/s speed
gradually decreased and the particle gradually approached the center of the beam.
According to inertia trapping, the particle will finally stay near the antinode of the CNT
resonator. Notice that the amplitude of the vibrational amplitude of 8.99 Å is exaggerated.
If this were not so, the process of trapping will take too long and would not be able to be
shown here. In conjunction with the first bouncing of the atom, its speed dropped from 753
4
m/s to 718 m/s at z = 10 Å, in which the loss of kinetic energy is equivalent to the kinetic
energy of the particle with a speed of 227 m/s. This suggests that even if the initial speed of
the particle is much larger than 1 m/s, the particle will not be shot back to z <–10 Å. To
assess the amplitude of a hollow CNT, an extra simulation was perfromed with a frequency
f = 157.2 GHz, where the amplitude was
2VyT = 8.80 Å. On the other hand, the amplitude
of a CNT with a particle inside was not calculated from the same simulation for exhibiting
the trapping process which has not reach the equilibrium. Instead, it was a separate
calculation where the particle initially stays at the antinode, which reaches equilibrium
faster. From this calculation, a driving frequency of 157.2 GHz was obtained. Notice that in
what follows, the amplitudes of vibration would always be evaluated from an separate
simulation, since it would otherwise be far from a state of equilibrium. Eventually, it may
be concluded that the amplitude increases from 8.80 Å to 8.99 Å as shown in table 1 due to
the insertion of the particle, which verifies that the particle has indeed entered the CNT. A
further simulation was taken in which F0 will be replaced with -F0/10 after 0.5 ns excitation
with the Kr-atom initially at z=70 Å, showing that if the excitation stops, the particle will
still stay inside the CNT, although the inertia trapping will deteriorate. After this, the switch
(red part in Figure 2a) will be open and methods such as temperature gradients12,13 can be
used to transport the particle through the CNT to the storage area to the right of the switch.
Hereafter, encoder - was studied in an analogue fashion. In the simulations, the
molecular model was the same as encoder +. Using MD simulations and FFT analysis,
resonant frequency for hollow CNT was extracted and found to be f = 146.4 GHz for the
primary mode. Vibrations were excited by applying a force F=F0 sin(2ft) with F0 = 1
meV/Å on each carbon atom of the beam in the y-direction. The excitation is weaker than
the encoder +, otherwise the particle may be shot out by the fierce vibration. During the
5
simulation, VyT was calculated as described above, with
2V yT = 5.03 Å after exciting the
lowest mode. A Kr-atom was initially placed at z = –10 Å, and subsequently be shot into
the CNT with an initial speed of 1 m/s after 1 ns of excitation on the beam. The excitation
was within 1 ns instead of 0.5 ns because in the first attempt of simulation where 0.5 ns is
taken the particle was ejected back out of the CNT. This relatively low initial speed was
increased to ~750 m/s once the particle entered the CNT as above. In contrast, the speed of
the atom did not decrease substantially, but with a final speed of ca 550 m/s at t = 5 ns, the
system is far from equilibrium. This amplitude of the CNT with the Kr-atom inside was
found via an extra simulation, where the particle cannot be trapped, but is in a ballistic
motion, 1.65 Å, which once again verifies that the particle is in the CNT. In addition, this
change in the amplitude is much larger than that in encoder +, because of the nonlinearity
of Duffing oscillator14 in encoder + caused by excessively large amplitude, which is
unlikely to happen with real resonators.
For encoder +, a 25 ring (5,5) CNT was used. The corresponding experimental
schematic of the CNT resonator is shown in Figure S1b. The reason for chosen of the CNT
length is the same as that stated in the section of encoder . The right tip is capped. The
coordinate system of the model, with the z-direction parallel to the main axis of the CNT, is
shown in Figure 2c. The origin of the system was set as the left end of the CNT axis. The
left 6 rings of CNT were clamped. The left part is a cantilever with a length of 51 Å.
Vibrations in the CNT were then excited by setting the y-coordinate of each atom of the 6th
ring of the CNT in each time step to be y = y0 + A - Acos(2ft), where y0 is the initial value
of y, f is the driving frequency, t is elapsed time and A = 0.02 Å is the amplitude of the
wiggle. The driving frequency f was initially 103.6 GHz which is the eigenfrequency for
the CNT in its fundamental vibrational mode with one Kr-atom pinning at the tip. The
eigenfrequencies were obtained using a similar strategy as for the encoder +. A Kr-atom
6
was initially be place at z = -10 Å, and subsequently shot into the CNT with an initial speed
of 1 m/s after 0.5 ns of excitation on the beam. This initial speed of the atom increased
to >700 m/s once the particle enters the CNT, as mentioned above. After the Kr-atom
entered the CNT, it would be trapped near the CNT tip in the same way as for the +
encoder. The process is shown in Figure 2d. As a result, the amplitude increased from 1.36
Å to 6.95 Å, which was used for detection purposes. The driving frequency is then 101.2
GHz, which is the eigenfrequency for the CNT in its fundamental cantilever vibrational
mode with a Kr-atom near a model particle 50 percent heavier than the Kr-atom at the right
tip of the CNT. Here the model particle is used to distinguish it from the first particle.
Using a similar process as outlined above, the model particle was inserted into the CNT
which has a previously trapped Kr-atom. The amplitude increases from 1.66 Å to 7.35 Å
due to the insertion and a consequent trapping. Hence, a molecule chain of two molecules
was encoded.
Finally, encoder - was studied in the same way. The molecular model for encoder - is
the same as for encoder +. The driving frequency f is initially the eigenfrequency for the
CNT in its fundamental vibrational mode 106.5 GHz. A Kr-atom was initially be place at z
= -10 Å, and will subsequently be shot into the CNT with an initial speed of 1 m/s after 0.5
ns of excitation on the beam. As a result, the vibrational amplitude decreased from 6.29 Å
to 1.98 Å, which was used as a detection signal. The driving frequency is 103.6 GHz with
physical meaning mentioned above. Using the same process as above, a model particle that
is 50 percent heavier than the Kr-atom used earlier was inserted into the CNT at t=0.5 ns
after a Kr-atom was already trapped at the tip. The vibrational amplitude then decreased
from 6.95 Å to 2.35 Å. Hence again a molecule chain of two molecules was encoded.
B. Feasibility
7
The scheme proposed above is not limited to an interesting theoretical exercise, but
should be studied from a practical viewpoint. First, it should be ensured that the molecule
can be shot into the CNT one at a time. In order to do this, an estimation of the interval
between the insertions of a molecule is needed. Neglecting the interaction between the
CNT and the particles, the interval can be calculated from the equation that governs the
balance of pressure,
APt  2 N  vm 
(S1)
where A is the cross-section of the CNT, P is the gas pressure, t is the time duration, N is the
number of molecule shot into the inlet, v is speed of molecule in z-direction, and m is the mass
of molecule. The number of molecules shot into the CNT per second will be
N /t 
AP
2
with  v 
2vm

2 RTg
(S2)
M
where R=8.31 J/(mol k) is the gas constant and M is the molar mass of the substance which
is 83.9 g/mol for krypton gas. With P=1 Pa, <v>=87 m/s for krypton gas at Tg=30 K, and
assuming A =r2, where r = 3.38 Å is the radius of the CNT, the number of molecules per
time, N/t, is found to be 15000. The interval between the insertion of molecule is t/N = 0.07
ms. This should be quite enough for step 1-3 shown in Figure 1, let alone that in the lab the
P can be several orders lower than 1 Pa15, resulting in a much longer time interval. Also,
the simulations reveal that the time for either inertia trapping of particles or the equilibrium
of vibration is within several ns, hence the time for detection can be much shorter than the
above-mentioned interval. Therefore the possibility that more than one molecule get inside
the CNT can be omitted by a fast shuttering of the slot.
For "+" encoders, the process of inertia trapping should be verified. To do that, a brief
deduction is presented here based on reference 1, 17 with cases where particles are near an
antinode, which is not an apex of the beam. The first goal is to estimate V, the variance of z
8
coordinates of the particle. The Fokker-Planck equation16 can be used to evaluate
probability density function of position of a moving particle in a given potential well. The
specific form of the equation confined in one dimension with strong friction limit (i.e.
neglecting time variations in the velocity) can be found in reference 16 as,
Π d 2 P
dP Π g  2 Π
 2 Π


dz z 2γ z 2
dz
where Π  Π(z,τ) is the probability density of the particle,=t/ is the rescaled time,  is the
friction constant, t is the time, P=P(z) is the potential well, z is the position of the particle, T
is the temperature of the system and g=2kBT is the noise strength. By taking    , this
equation will be
d 2P
dP Π g  2 Π
Π


dz z 2γ z 2
dz 2
which governs the probability density of the particle’s position after equilibrium. This
equation has an analytical solution of
 P( z )
Π ( z )  C exp(
)
(S3)
k BT
where C is constant for normalization.
0
It can be seen from the simulations in reference 1 that the particle(s) is rotating in the x-y
plane. Therefore the potential well will be twice that of reference 17, where the motion of
nanobeam is supposed to be in a plane. As a result, P( z )  (1 / 2) 2 A 2 ( z ) 2 m  C1 , where
=2f is the angular frequency, A is the vibrational amplitude,  (z ) is the spatial profile of
the mode, m is the mass of the particle and C1 is some constant. To get an analytical form of V,
 (z ) needs to be simplified. This can be done if the particle moves close to the antinode. If the
antinode is in a half wave between [-z0+z*, z0+z*], the potential well can be simplified to be
harmonic, such that the potential
P(z) = 0.5k(z-z*)2
(S4)
2 2
with P(-z0+z*) = P(z0+z*) =  and P(z*) =0, where  0  (1 / 2) rp m  (
9
2
rp ) 2 m / 2 is the
p
depth of potential well caused by the inertia force near the antinode, rp is the amplitude of this
circular motion which can be estimated as
2V y .The constant k is evaluated as k 
8 2 rp2 m
2 z 02 p 2
.
By substituting equation (S4) into equation (S3), the probability density of the particle  can
be solved as
k Tz 2
 P( Z  z*)
1
Z2
)  exp(
) with B  k BT / k  B2 2 0
k BT
B
2B
 rp m
2V
where Z = z - z*. The V of the particle is

k BTz 02
2
V   Π ( Z ) Z dZ  B  2 2
(S5)
 rp m

Π (Z ) 
1
exp(
This expression can be applied to cases when particles are near the apex of a cantilever
by mirroring it to the other side of the apex. Presuming a sinusoidal trajectory, the
amplitude of the particle in axial direction can then be estimated as 2V . Now the ratio
between the amplitude of the particle and half length of the half wave is
c  2V / z 0 
2k B T
 2 rp2 m
(S6)
Therefore if c<1, the particle is likely to move between its two adjacent nodes and the
trapping happens. This c is a tool for assessing the feasibility of the trapping. It can be seen
from (S6) that the amplitude of particle’s circular motion rp, the mass of the particle m and
the angular frequency  are critical to the trapping. It is noteworthy that the z0, which
reflects directly the CNT length, is canceled in (S6). Therefore although lengths of the
simulated CNTs (e.g. about 15 nm for “” encoders) are much shorter than those in recent
papers (e.g. about 110 nm in reference 13), the assessment of feasibility of the trapping can
be without caring about the CNT length.
Now what remains to do is to take experimental data into (S6). Here takes reference 18
as an example and considers a Kr particle, where CNT resonators can reach frequencies
10
f=4.2 GHz to the fundamental mode and f = 11 GHz for the 2nd or 3rd mode18 at T = 4 K.
Then with a rp = 10.6 Å, c is 0.15 for the higher mode (i.e. f = 11 GHz). This rp =10.6 Å is
close to experimental values e.g. in reference 19. Therefore for the higher mode the inertia
trapping is likely to happen. It is interesting to assess fundamental mode which more
controllable in experiment. Then with f=4.2 GHz, c is 1, Considering that T can be as low
as 100 mK20 in lab noticing that T is not Tg, c can be quite small hence also a trapping.
Therefore for encoder + the trapping is potentially difficult, but still realizable.
The assessment of feasibility of inertia trapping is problematic since the literatures for
cantilever CNT resonator are relatively rare. However this trapping should still be feasible
even with a smaller frequency f since amplitudes of vibration which reflects rp can be one
order larger than those in encoder +22,23, which can effectively decease the value of c.
Another issue that is of interest is naturally the sensitivity of the CNT resonator for the
mass of the particle, and this should naturally be verified. The sensitivity can be expressed
by the equivalent mass of the particle17
mr  m
 p( z)
2
(S7)
( z )dz
beam
where p(z) is the probability that the particle stays at z, (z) is the profile of displacement of
the axis. mr≈m if c<<1. With respect to the issue of sensitivity, it has been reported that the
mass of 2/5 Au atom (roughly a Kr-atom), or the mass of a proton can be detected by a
cantilever resonator15, or a double clamped resonator21, respectively. It is therefore feasible
that single atom molecule e.g. Kr or organic molecule like nucleic acid should be detectable
for two "+" encoders if good inertia trapping, i.e. c<<1, can be achieved.
For the "-" encoders, the particle is not likely stay in a stable state in the vicinity of the
antinode because it will not be driven with its eigenfrequencies1. This as well as cases when
"+" encoders can not achieve a good trapping will result in a smaller mr. And in the worst
case, when p(z) = const, the mr can be estimated to be 0.4m for “” encoders and 0.25m for
11
“” encoders, respectively, if particles are restricted only inside the beam. Considering that
in reality the particles can move in extended parts of the beam (blue parts in Figure 2a &
Figure 2c), mr will be prominently smaller than m.
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