Mathematical Model of a Proposed Carbon Nanotube Sensor for Ultra Sensitive Acetone Sensing
Ming Xia, Xingguo Xiong
Department of Electrical and Computer Engineering,
University of Bridgeport, Bridgeport, CT 06604
Abstract: In recent years, MEMS/NEMS (Micro-/Nano Electro Mechanical Systems) sensors have attracted tremendous interest among
researchers due to their low cost, quick response time, as well as high sensitivity and selectivity. In this paper, an ultra sensitive acetone sensor
based on carbon nanotube (CNT) structure is proposed. In this device, a carbon nanotube is anchored to a substrate in one end, and the other end
is free-standing. Piezoelectric activation is used to activate the vibration of such a carbon nanotube cantilever structure. If an acetone molecule is
attached to the carbon nanotube, the resonant frequency of the cantilever will be changed for a certain amount. By sensing this certain resonant
frequency change, the existence of a single acetone molecule can be detected. A theoretical model is developed to describe the vibration of the
carbon nanotube cantilever structure. The resonant frequency change of the cantilever due to attached mass is analyzed. The proposed acetone
sensor can achieve extremely high sensitivity in molecular level. It can be potentially used for sensing the trace acetone concentration in human
breath, which leads to a quick, convenient, accurate and painless breath diagnosis of diabetics. Such breath diagnosis can greatly reduce the risk
of blood-transmitted diseases in the traditional blood testing of diabetic’s diagnosis.
I. Introduction
Since their discovery in 1991, carbon nanotubes (CNTs) have been recognized as one of the most promising nanomaterials and attracted
tremendous interest among researchers around the world. Carbon nanotubes have many unique mechanical, electrical, thermal and chemical
properties [1]-[2]. For example, as allotropes of carbon, CNTs have high aspect ratio, large surface-to-volume ration and high elastic module.
CNTs are the strongest and stiffest material on the earth, with mechanical strength much stronger than steel. On the other hand, CNTs are also
highly resilient and can be easily bent or twisted. Depending on the chirality, CNTs can be either metallic or semiconductor. They have excellent
chemical and thermal stability, but also can be modified by chemical reactions [2]. As a result, CNTs have found broad applications [2], such as
scanning probe tips, hydrogen storage based fuel cells, water/air filters, catalyst support, NEMS sensors, field emission devics, nanoelectronics,
etc. Furthermore, there has been a growing interest in using CNTs for biomedical applications. Due to their extremely tiny size, CNTs are suitable
for developing ultra-sensitive nano-biosensors, electro-analytical devices, as well as electromechanical actuators for artificial muscle applications.
For example, CNTs have been reported to be successfully used for expanding a blood vessle in vascular stent procdure, or for high precision, low
dose nano drug delivery system.
Carbon nanotubes have also been used for ultra-sensitive mass sensor application [3][4]. For this purpose, a carbon nanotube is anchored to
substrate in one end or both ends to form a cantilever or double-clamped beam structure. It can be activated to vibrate and the resonant frequency
of the nanotube structure can be measured. The resonant frequency of the nanotube structure depends on the spring constant of the nanotube and
the mass distribution along the nanotube. If another atom or molecule is attached to the carbon nanotube, the mass distribution along the beam
will be changed, and hence the resonant frequency of the nanotube will be shifted. By measuring the resonant frequency shift of the carbon
nanotube resonator, the mass of the attached atom/molecule can be precisely measured. Since such mass sensor is based on an individual carbon
nanotube, extremely high resolution can be achieved. It can be used to measure the mass of an individual atom or molecule. Various work about
using carbon nanotube resonator as a mass sensor and its theoretical model analysis have been reported [5]-[13]. In this paper, an ultra sensitive
acetone sensor based on carbon nanotube (CNT) structure is proposed. The carbon nanotube based mass sensor is designed specifically to target
the sensing of acetone molecules, which exist in the breath sample of diabetic patients and can be used for diabetes diagnosis. A theoretical model
is developed to describe the vibration of the carbon nanotube cantilever structure. The relationship between the resonant frequency of the
cantilever and the attached mass is also analyzed. ANSYS FEM simulation is also used to verify the results. The proposed acetone sensor can
achieve extremely high sensitivity in molecular level. It can be potentially used for sensing the trace acetone concentration in human breath,
which leads to a quick, convenient, accurate and painless breath diagnosis of diabetics. Such a breath diagnosis can also greatly reduce the risk of
blood-transmitted diseases in the traditional blood testing of diabetic’s diagnosis.
II. Device Design and Analysis
The proposed carbon nanotube based acetone sensor is shown in Figure 1. In this device, a carbon nanotube is anchored to a substrate in one end,
and the other end is free-standing. Piezoelectric activation is used to activate the vibration of such a carbon nanotube cantilever structure. The
resonant frequency of the carbon nanotube cantilever is measured through a resonance measurement system. There are many identical carbon
nanotube cantilevers forming a sensing array. The whole array is sealed inside a filter membrane. When the breath sample of a patient is
introduced, the filter membrane repels all other molecules and only allows acetone molecules to pass through. Some special radicals are
pre-attached to the end of the carbon nanotubes, and they are designed to attract the acetone molecules. If an acetone molecule is attached to the
carbon nanotube, the mass distribution of the carbon nanotube structure is changed, hence the resonant frequency of the cantilever will be shifted
for a certain amount. By sensing this certain resonant frequency change, the existence of a single acetone molecule can be detected. Due to the
carbon nanotube structure, the mass sensor can achieve extremely high sensitivity so that it can detect the existence of an individual acetone
molecule. It is known that the acetone level in the breath of diabetic patients is much higher than that from a healthy person due to the
glucose-related metabolism disorder [14]. This can be used as an important biomarker for diabetes diagnosis. However, traditional sensors lack
the enough sensitivity to detect the trace amount of acetone level in breath. The proposed nanotube based acetone sensor has extremely high
sensitivity and can detect the existence of even individual acetone molecules. This can be used for quick acetone sensing in breath sample for
diabetes diagnosis. Compared to the traditional blood glucose test, such breath test is non-invasive, painless, low cost and convenient.
Furthermore, it is safe, and greatly reduces the risk of blood-transmitted diseases (such as HIV, hepatitis, etc.) in blood test.
Figure 1. Carbon nanotube based acetone sensor
As shown in Figure 2, one end of a carbon nanotube is anchored to the substrate, and the other end is free-standing. The carbon nanotube
cantilever is activated to vibrate along Y direction freely. Assume small-angle deflection, under the given Cartesian coordinate system, the
vibration of the nanotube can be described with following equation [4]:
(1)
where E is Young’s modulus of the carbon nanotube, I is the inertial moment of the nanotube, ρ is the density of the carbon nanotube, A is the
corss-section area of the carbon nanotube at location x, y is the deflection of nanotube at location x. The carbon nanotube cantilever can be
treated as a simplified spring-mass model. Hence the resonant frequency of the cantilever can be calculated as
(2)
where k is the effective spring constant of the cantilever, and m eq is
the equivalent vibration mass of the carbon naotube with the attached mass
at its end tip. From this equation, we can see that the resonant frequency of the carbon nanotube resonator is closed related to the spring constant
of the nanotube and the mass distribution along the nanotube. A certain mass attached to the end tip of the carbon nanotube will introduce a
certain resonant frequency shift to the nanotube resonator structure.
Figure 2. Cantilevered nanotube resonator with an attached mass at the tip
2.1
Spring Constant of the Carbon Nanotube Resonator
In order to calculate the resonant frequency of the nanotube cantilever, the equivalent spring constant of the cantilever need to be derived. As
shown in Figure 3, a carbon nanotube cantilever with length L bends under the localized force F at its end tip.
Figure 3. Deformation of a cantilever under load at its end tip
Define w(x) as the deflection of the cantilever along Z direction, according to Figure 3, we have [4]
We try to set w(x) as 3rd order polynomial:
, where A, B, C and D are coefficients to be decided. Putting w(x)
into above equations, we have
Thus the bending shape of the nanotube cantilever can be described as:
The maximum bending deflection occurs at the end of the cantilever, which can be calculated as
As a result, the effective spring constant of the carbon nanotube cantilever is found to be
(3)
2.2
The Equivalent Mass of Carbon Nanotube Resonator
Figure 4. Bending shape of carbon nanotube resonator due to attached mass at the end tip
In simplified spring-mass model, the mass of the spring is omitted. However, the above nanotube resonator, the mass of the carbon nanotube is
even larger than the attached acetone mass and should not be omitted. As shown in Figure 4, the carbon nanotube cantilever bends under the load
of attached acetone mass at its end tip. Assume the cantilever-mass system is activated to vibrate, the total kinetic energy of the system is the sum
of the kinetic energy of the carbon nanotube (T SWCNT ) and that of the attached mass (T mass ) [4]
Among it, the kinetic energy of spring is:
. Since
, we have
(4)
The mass of the carbon nanotube can be calculated as:
,
Where ρ is the density of carbon nanotube, A is the cross-section area of the nanotube, and L is the length of the nanotube. Assume the acentone
molecule with mass m is attached at the end tip of the carbon nanotube, the total kinetic energy of the system can be rewritten as
(5)
Substitute w(x) and w(L) from previous results into above equation, we have [4]
Compare the above results with
1
T = ω 2meq w2 ( L)
2
We can see that the above carbon nanotube resonator can be treated as a spring connected with an equivalent mass m eq at its end tip:
(6)
2.3 Frequency Shift Due to Attached Mass
Using simplified spring-mass model, the resonant frequency of the carbon nanotube resonator with acetone mass attached at its end tip is:
(7)
When no mass is attached, the original resonant frequency of the carbon nanotube resonator is:
(8)
The resonant frequency of carbon nanotube resonator with attached acetone mass can be re-written as
(9)
Thus the resonant frequency shift due to attached acetone mass is
The percentage of resonant frequency shift compared to original resonant frequency is:
(10)
Thus, by measuring percentage resonant frequency shift (Δf/f 0 ), the attached mass can be calculated as:
(11)
That is, by measuring the relative resonant frequency shift (Δf/f 0 ), we know the value of the attached acetone mass. This is the working principle
of using carbon nanotube resonator as a mass sensor.
The mass of one acetone (CH 3 COCH 3 ) molecule is 58.08×1.66×10-27kg=9.64×10-14ng. In the above carbon nanotube resonator, if only one
acetone molecule is attached at the end tip of the carbon nanotube, the resulted percentage frequency shift is
That is, the resonant frequency of the carbon nanotube resonator will shift for 1.87% due to the attachment of one single acetone molecule with
mass of 9.64×10-14ng. However, in the real device operation, there may be more than one acetone molecules attached to the end of the carbon
nanotube. As a result, the resonant frequency shift will be even larger. The relationship between the attached acetone mass and the percentage
1.0
X: 1.869
Y: 9.646e-014
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0
0.2
0.4
0.6
0.8
1
1.2
Frequency Shift Percentage: Δf/f (%)
1.4
1.6
1.8
2
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Frequency Shift percentage: ∆f/fo (%)
(a)
2.5
-8
x 10
2.5
2
2
Attached mass of Acetone: m (ng)
-13
x 10
1.2
Attached mass of acetone: m (×10-8ng)
1.2
Attached mass of Acetone: m (ng)
Attached mass of acetone: m (×10-13ng)
resonant frequency change is plotted with Matlab, as shown in Figure 5.
1.5
1.5
1
1
0.5
0.5
0
X: 89.91
Y: 2.439e-010
X: 1.889
Y: 9.77e-014
0
0
0
10
20
10 20
30
40
50
60
Frequency Shift Percentage: Δf/f (%)
30 40
50
70
80
90
100
60 70 80 90 100
Frequency Shift percentage: ∆f/fo (%)
(b)
Figure 5. Relationship between percentage resonant frequency change and the attached mass.
(a) Frequency shift from 0~2%, cursor at (1.869%, 9.646×10-14ng);
(b) Frequency shift from 0~100%, cursors at (1.889%, 9.77×10-14ng) and (89.91%, 2.439×10-10ng)
From Figure 5, we can see that if the attached mass is more than 2.439×10-10ng, which corresponds to about 2530 acetone molecules, the
percentage resonant frequency shift is around 90%. However, this will not happen in the real device operation, because the sensitive head group
in carbon nanotube mass sensor may not be able to attract so many acetone molecules. In the real device operation, we only need to consider the
case when only a few acetone molecules are attached. For this range, the relationship between percentage resonant frequency change and the
number of attached acetone molecules is approximately linear, as shown in Figure 5(a).
III. ANSYS Simulation of Carbon Nanotube Resonator
In order to verify the effectiveness of the above model, ANSYS modal analysis has been used to simulate the resonant frequencies of the carbon
nanotube resonator, and the simulation results are compared to the values predicted by the theoretical model. The details are discussed as below.
3.1
Case #1: CNT Resonator without Attached Mass
ANSYS simulation has been used to simulate the resonant frequency of a SWCNT cantilever without attached mass. The material properties of
SWCNT used in simulation are shown in Table 1.
Table 1. Material properties of SWCNT used in ANSYS simulation [4]
Material Properties
Values
15
Young's Modulus E
1.0×10 ng/nm·s
Density ρ
1.4×10-12ng/nm3
Radius r
0.55nm
Length L
8nm
Gravity Acceleration g
9.8×109nm/s2
Poisson’s ratio
0.2
2
Based on theoretical model, the resonant frequency of SWCNT cantilever resonator can be calculated as
,
The SWCNT has a circular cross section. As a result, its moment I can be calculated as [11]
Thus its resonant frequency is
(12)
In ANSYS simulation, we use rectangle 2D model instead of the columniform 3D model to simplify the question. We use a square cross section
with equivalent thickness of t to represent the circular cross section of the SWCNT. The equivalent thickness t can be calculated as
ANSYS simulation result gives the resonant frequency of the SWCNT resonator as 6.52×1010 Hz, as shown in Figure 6. This is exactly the same
as the theoretical value given by Equation (12).
(a)
(b)
Figure 6. ANSYS modeling and simulation with no mass attached. (a) ANSYS model (thickness 0.976nm, length 8nm); (b) ANSYS simulation
(resonant frequency 6.52×1010 Hz)
3.2
Case #2: CNT Resonator with Attached Mass
For verifying the cases of CNT resonator with attached mass, we randomly select 4 points in Fig. 6 to compare them with ANSYS simulation
results, as shown in Figure 7.
-13
7
x 10
6
6
5
5
Attached mass of Acetone: m (ng)
Attached mass of acetone: m (×10-13ng)
7
4
X: 6.784
Y: 3.785e-013
4
3
X: 5.036
Y: 2.732e-013
3
2
X: 3.396
Y: 1.795e-013
2
X: 1.874
Y: 9.674e-014
1
1
01
0
1
2
3
2 1 3
4
4
5
6
Frequency Shift Percentage: Δf/f (%)
5
6
7
7
8
8
Frequency Shift percentage: ∆f/fo (%)
9
9
10
10
Figure 7. Frequency shift (1~10%) of CNT resonator due to attached mass. Cursors are set at four data points: (1.874%, 9.674×10-14ng); (3.396%,
1.795×10-13ng); (5.036%, 2.732×10-13ng); (6.784%, 3.785×10-13ng)
Take the first data point of (1.874%, 9.674×10-14ng) as an example, the attached mass of 9.674×10-14ng is simulated as a rectangle attached to the
end of CNT cantilever with length and thickness as 0.2nm and 0.3455nm separately. The ANSYS simulation results are shown in Figure 8.
ANSYS simulation results show that the resonant frequency of CNT with attached mass of 9.674×10-14ng is 6.4087×10-10Hz, which gives the
percentage frequency shift Δf/f 0 =1.71% due to the attached mass. Since a simplified model is used in theoretical analysis, ANSYS simulation
results are more accurate. Compared to the ANSYS simulation result, the theoretical result of Δf/f 0 gives a relative error of
|(1.874%-1.71%)/1.71%|×100%=9.59%, which verifies the effectiveness of the theoretical model. Similarly, the ANSYS simulation results and
theoretical prediction of other 3 cases can be verified as well, and the results are listed in Table 2. From Table 2, we can see that for all the 4
cases, the theoretical prediction of percentage frequency shift (Δf/f 0 ) gives a small error of ~10%. This verifies the effectiveness of above
theoretical model. Furthermore, the theoretical model tends to underestimate the percentage frequency shift (Δf/f 0 ) due to attached mass.
(a) ANSYS model
(b) ANSYS modal simulation result
Figure 8. ANSYS model and simulation result of SWCNT with attached mass of 9.674×10-14ng. (a) ANSYS model (Attached mass thickness
0.3455nm, length 0.2nm); (b) ANSYS simulation (resonant frequency is 6.41×1010Hz)
Table 2. ANSYS simulation results v.s. theoretical predictions of frequency shift ∆f/f o
Case
1
2
3
4
Attached mass m
9.674×10-14 ng
frequency shift ∆f/f o
frequency shift ∆f/f o
(Theoretical)
(ANSYS simulation)
1.874%
1.71%
9.59%
Relative Error
-13
ng
3.396%
3.1%
9.55%
-14
ng
5.036%
4.63%
8.77%
-14
ng
6.784%
6.213%
9.19%
1.795×10
2.732×10
3.785×10
Based on above results, we can see that the predicted percentage frequency shift due to attached mass of the SWCNT is very close to the ANSYS
results, with a small relative error of ~10%. This proves the effectiveness of the theoretical model. The small relative error is due to the simplified
spring-mass model used for the resonant frequency calculation.
IV. Conclusions and Future Work
In this paper, the mathematical model of an ultra sensitive acetone sensor based on carbon nanotube (CNT) cantilever structure is proposed. The
carbon nanotube cantilever is activated by piezoelectric actuation. If any acetone molecule is attached to the end of the CNT, it changes the mass
distribution of the SWCNT resonator, hence changing the resonant frequency of the cantilever. By measuring the percentage frequency shift of
the SWCNT cantilever, the attached mass of the acetone molecules can be derived. The numerical results indicate that the mass sensitivity of
carbon nanotube based mass sensor can achieve resolution up to10-26 kg. A theoretical model is developed to describe the vibration of the carbon
nanotube cantilever structure. The resonant frequency change of the cantilever due to attached mass is analyzed. The effectiveness of the model is
further verified with ANSYS simulation results. The ANSYS simulation results show that the percentage frequency shift ∆f/f
o
based on
theoretical model leads to a small percentage error of ~10% compared to ANSYS results. This is reasonable considering the simplified spring
mass model is used in theoretical analysis. The proposed acetone sensor can achieve extremely high sensitivity in molecular level. It can be
potentially used for sensing the trace acetone concentration in human breath, which leads to a quick, convenient, accurate and painless breath
diagnosis of diabetics. Breath diagnosis can also reduce the risk of blood-transmitted diseases in the traditional blood testing of diabetic’s
diagnosis. In the future, we will further look into the influence of chiralty/diameter of SWCNT on the frequency shift due to the same attached
mass. We will also analyze the cases when the acetone mass is attached to location other than the end point of the SWCNT cantilever structure.
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