Modeling and validation of the electromechanical behavior of
highly stretchable cnt/polymer composites through network-based
percolation model
Seong-Won Jin1)**, Myeong-Seok Go**, Jae Hyuk Lim1)*, and Youngu Lee2)
1)
Department of Mechanical Engineering, Jeonbuk National University, 567 Baekje-daero,
Deokjin-gu, Jeonju-si, Jeollabuk-do 54896, Republic of Korea
2)
Department of Energy Science & Engineering, Daegu Gyeongbuk Institute of Science and
Technology (DGIST), Daegu 42988, Republic of Korea
* Author to whom any correspondence should be addressed.
**1st author equally contributed
Abstract
In this study, a program was developed to predict the electromechanical behavior of highly
stretchable CNT-polymer composites in which the electrical properties change in response to
uniaxial deformation. RVE was generated by randomly distributing CNTs in a polymer matrix
using the Monte-Carlo simulation algorithm. In the case of tensile strain, the CNT and RVE
configurations and network models were updated considering the tension and shrinkage by
Poisson's ratio, and the electrical conductivity was calculated. Trade-off studies were
conducted on RVE size, geometric periodicity of CNT fibers, and tensile strain to examine the
appropriateness of the proposed methodology. In addition, the electromechanical behavior of
CNT mixtures with different aspect ratios was confirmed and compared with the experimental
data. When the size of RVE was more than three times larger than the length of CNT and CNT
1
with different aspect ratios were mixed, it was similar to the experimental data. This confirms
that the proposed model adequately predicts the electrical conductivity of the uniformly
dispersed CNT-polymer composite.
Keywords: Carbon nanotubes, Electrical conductivity, Percolation threshold, Modified nodal
analysis, Monte-Carlo simulations, Periodic conditions
* Corresponding author: jaehyuklim@jbnu.ac.kr
2
1. Introduction
Demand for flexible and stretchable strain sensors is increasing in various fields, such as
structural health monitoring and soft robotics. One of the materials for highly stretchable strain
sensors is carbon nanotube (CNT) based polymer composites [1]. Polymer materials are
lightweight and have excellent mechanical properties but poor conductivity. However, the
conductivity of polymer composites is greatly improved when carbon nanofillers such as
carbon black, CNT, and graphite nanoplatelets with high conductivity are added [2-4]. The
enhancement of electrical conductivity can explain by direct contact and the formation of a
percolation network based on tunneling adjacent conductive fillers. Here, the tunneling effect
is a quantum phenomenon that allows electrical conduction across isolating barriers between
two highly conductive fillers [5].
When filler with a sufficient amount is evenly dispersed in the polymer, the overall
conductive pathway begins to form. The amount of filler required to generate such a pathway
is called the percolation threshold. The percolation threshold is affected by aspect ratio,
tunneling mechanism, and physical contact. At the percolation threshold, the electrical
conductivity of the nanocomposite rises rapidly, so research is being conducted to form it at a
low fiber volume ratio [6].
In particular, low aspect ratio fillers such as carbon black cause percolation at a fiber volume
fraction of 10 to 16% [7]. On the other hand, nanowires and CNTs have a high aspect ratio of
20 to 10,000 [8] and a one-dimensional conductive structure, which efficiently induces
percolation even at a low fiber volume fraction of 0.2 to 1.5% [9-11]. In addition, it is possible
to secure high electrical conductivity by maintaining percolation for a long time, even when a
high strain rate is applied [5,6].
For developing these highly conductive materials, many studies have been conducted on
analytical, numerical, and experimental approaches to calculate the equivalent electrical
3
conductivity of carbon nano-filler composites. [5, 6, 9, 12-16]. Since the experimental process
requires a high cost and a lot of labor, many analytical/numerical methods have been developed
and used as preliminary studies for material development. For example, Kim et al. [16]
generated a finite element (FE) model of the carbon black composite, added an interface to
consider the tunneling effect and electrical network, and calculated the equivalent electrical
conductivity. Alian et al. [10] generated a FE model of CNT composites and applied an axial
load. Then, the electrical conductivity of the nanocomposite was calculated by updating the
equivalent electrical model with the displacement of the CNT in the modified FE model. This
method can accurately predict the equivalent electrical conductivity for mechanical strain and
residual stress. However, when the shape of the filler is complex or the number is large, it isn't
easy to create an FE model, and a lot of calculation time is required. Accordingly, it is
impossible to process using multiple samples due to the predictive characteristics of electrical
conductivity in which several samples are analyzed, and the average is processed.
Jung et al. [3] developed a network model for a curved CNT-polymer highly stretchable
composite and studied its electrical properties under large deformation. Using Poisson's ratio
definition of the polymer material, the geometry of CNT and representative volume element
(RVE) for the strain was updated, and the electrical conductivity was calculated. Haghgoo et
al. [15] studied the effect of temperature and thermal expansion on the resistance of CNTpolymer composites. The numerical analysis method using such an electrical network has the
effect of reducing the calculation time because it does not require the creation of an FE model.
In this study, a program for predicting the electromechanical behavior of highly stretchable
CNT-polymer composites in response to uniaxial strain is developed and verified. Trade-off
studies on RVE size, geometric periodicity of CNT fibers, and tensile strain were conducted to
review the appropriateness of the methodology. In addition, verification of the
electromechanical behavior of CNT mixed states with different aspect ratios is also conducted.
4
To this end, Section 2 describes generating RVEs by randomly distributing CNTs using the
RSA (Random Sequential Adsorption) algorithm. And considering the contact and tunneling
of the filler, the CNT network formation and the calculation process of updating the position
of the CNT before and after the deformation of the RVE according to the tensile load are
described. In addition, the method of calculating the CNT network's electrical conductivity by
calculating the CNT's specific and tunneling resistance is explained. Section 3 compares the
electrical conductivity calculation results according to the RVE size, the geometric periodicity
of the CNT fiber, and the tensile strain. The program is finally verified by comparing them
with the experimental results. Finally, Section 4 concludes the thesis with a conclusion.
5
2. Simulation procedure
In this study, the model generation process and electrical conductivity calculation are shown
in Figure 1. First, input parameter values such as CNT diameter and CNT length are determined.
Next, CNTs are randomly distributed using the RSA algorithm, and the CNTs are updated
according to whether periodicity is applied. Next, a network model is calculated by applying
contact and tunneling resistance between neighboring CNTs considering the distance between
CNTs. Finally, when there is a tensile strain, the CNT and RVE configurations and network
models are updated considering the tension and shrinkage due to Poisson's ratio, and their
electrical conductivity is calculated.
START
Before applying uniaxial strain
Generate random RVE
Generate network model considering
tunneling effect
Calculate electrical conductivity
After applying uniaxial strain
Applying strain in uniaxial direction
Update CNT and RVE
configurations and network models
Calculate electrical conductivity
END
Figure 1. Flow chart of calculate electrical conductivity
6
2.1 RVE modeling
The RSA algorithm was used as a method for generating CNTs. The RSA algorithm
randomly distributes fibers through Monte Carlo simulation [17, 18]. The shape of the CNT is
a cylinder with a long aspect ratio, as shown in Figure 2. First, the coordinate ( x0 , y0 , z0 ) of
the starting point of the CNT is randomly determined inside the cube of size Lx  Ly  Lz .
Determine the coordinate ( x1 , y1 , z1 ) of the endpoint of the CNT as shown in Eq. (1) below.
 x1   x0  l cos  cos  
  

 y1    y0  l cos  sin  
 z   z  l sin  
0
 1 

(1)
Here, l is the length of the CNT,  is the elevation angle,  is the azimuth angle, and 
and  are expressed as Eq. (2, 3).



(1  2  rand ),    
2
2
2
  2  rand , 0    2

(2)
(3)
Where, rand represents an arbitrary number in the interval [0, 1].
Figure 2. Configuration of the CNT
The above process is repeated as often as the number of CNTs with a target fiber volume
ratio ( V f ) to produce CNTs. When generating CNTs, overlapping of CNTs was allowed [14].
7
In addition, in this study, two cases of RVE were classified to compare the electrical
conductivity according to the application of the periodic boundary condition. In the nonperiodic case, which is the first case, as shown in Figure 3, when CNTs are placed across the
boundary of the window during the RVE creation process, the part that crosses the edge is cut
off, and only the region within the window is finally placed. On the other hand, in the case of
periodic, the second case, geometric periodicity is satisfied by arranging the part that crosses
the window boundary of the CNT on the opposite side of the window, as shown in Figure 4.
Next, regions outside the edges of the RVE were deleted. However, periodic boundary
conditions were given, so the flow of electricity continued. It is known that using this method,
the equivalent electrical conductivity can be effectively and accurately predicted even if the
RVE size is small and the number of CNTs is small compared to a model that does not
implement periodic fiber arrangement. This will be further discussed in the numerical analysis
results in Section 3.Ошибка! Источник ссылки не найден.
(a)
(b)
Figure 3. Configuration of the RVE with the geometric non-periodicity
(a) ISO view, (b) Front view
8
(a)
(b)
Figure 4. Configuration of the RVE with the geometric periodicity
(a) ISO view, (b) Front view
For the RVE generated through the RSA method to have randomness, it is necessary to
ensure that the distribution of the azimuth and elevation angles of the cylindrical filler is
uniform. This is known to be related to the number of fillers and the relative length ratio of
fillers to RVE size [19-21]. We review this in this section.
To verify the randomness of the generated RVE, the azimuth and elevation angles of the
cylindrical fillers were measured, and their distribution was checked and compared with the
utterly random case. If the cylindrical filler has a directionality, the distribution of angles will
appear biased to one side.
The directional distribution integral of the filler in 3D space is calculated as in Eq. (4) [21].

2

  0  
2
  ( p)dp 
s


 ( , ) cos  d d
(4)
Here,  ( p ) is the directional distribution function of the filler, and when the cylindrical filler
is evenly spread and randomly distributed, the same value as Eq. (5) can be obtained.
 ( p) 
1
2
(5)
Therefore,  ( ) is derived as in Eq. (6).
1
cos( )
cos( )d 
  0 2
2
 ( )  

(6)
9
Where the elevation angle distribution f ( ) of the filler when the section of [0,  ] is divided
into 18 is as shown in Eq. (7).
f ( ) 
cos( )   cos( )
 
2
18
36
(7)
By repeating the above procedure, the azimuth distribution f ( ) of the filler can be obtained
as in Eq. (8), and the result of the angle distribution of the pillar is shown in Figure 5.

f ( )   2
 
1
 1  1
cos( )d    
18  18 18
2 2

(8)
In this study, CNTs with a length of 5μm were generated for RVE size L . Currently, CNTs
are approximately 380, 3,000, and 10,300, respectively. Currently, the number of samples of
each RVE used is 300. At this time, the error of the CNT shape regarding the ideal azimuth
and elevation angles were defined as Eq. (9) and (10) and shown in Figure 5.
R =
1 18
1
 ( f (i )  18 )2 , i  [1 ,2 ,
18 i 1
R =
 cos(i ) 2
1 18
( f (i ) 
) , i  [1 , 2 ,

18 i 1
36
,18 ]  [5,15,
,175]
, 18 ]  [85, 75,
(9)
,85] (10)
The error of the sample derived through this process is ( R , R ) = (8.04%, 10.75%) in the
case of L / l  1, and it was confirmed that the error decreases as L / l increases. Therefore,
in this study, analysis was performed in consideration of L / l  3 .
10
(a)
(b)
(c)
Figure 5. Azimuth and elevation distribution of cylindrical fillers with V f  3% :
(a) L / l  1 (b) L / l  2 (c) L / l  3 for l  5
2.2 Percolation modeling
11
The total electrical resistance of the CNT-polymer composite can be expressed as the sum
of the CNT resistivity and the tunneling resistance. The specific resistance RCNT in the
longitudinal direction of the CNT can be calculated as in Eq. (11).
RCNT 
4l
 CNT D 2
(11)
Here, l is the length of the CNT, D is the diameter of the CNT, and  CNT is the intrinsic
electrical conductivity of the CNT.
The interconnect features between CNTs are not in direct contact and are several nanometers
apart due to a thin layer of insulating polymer preventing direct contact between CNTs [14].
Tunneling resistance considers the resistance between CNTs that are several nanometers apart.
Tunneling resistance Rtunnel between two CNTs is as Eq. (12) [9].
Rtunnel 
h 1
2e 2 M 
(12)
Where h is Planck’s constant, e is electron charge,
h
2 e2
 12.9504 kΩ, M is the number
of conduction channels, and T is the transmission probability of electron charges between
contacting surfaces [13]. T can be estimated by solving the Schrödinger equation for a
rectangular potential barrier, as shown in Eq. (13) [9, 14, 15].

 d vdW 
0  d  D  d vdW
 exp  

d

 tunnel 
(13)
T 


d

D
exp 

 D  d vdW  d  D  d cutoff

 dtunnel 

Here, d vdW is the Van der Waals distance, d is the minimum distance between two CNT,
d cutoff is the cutoff distance, and dtunnel represents the characteristic tunneling length as shown
in Eq. (14).
12
dtunnel 
Where
(14)
8me ΔE
is the Dirac constant, me is the electron mass, and ΔE is the height of the barrier.
D
RCNT
l
d
Tunneling
Rtunnel
Figure 6. Example of CNT resistance and tunneling
The effective resistance can be obtained as Eq. (15) using the modified nodal analysis (MNA)
method [10].
 I external   K 1 Vnodal 
V
   1 0   I

  source 
source

 
(15)
Here, I external is the external current; the value is 0, Vsource is the voltage between the two
opposite sides of RVE, Vnodal is the node voltage, I source is the current passing through the
circuit due to Vsource , and K is the global coefficient matrix that can be obtained as Eq. (16)
m
K   [ Kije ]
(16)
e 1
Where m is the total number of resistors in the percolating network, i and j are the end nodes
of the resistor, and K ije is the element conductance. Electrical conductivity can be calculated
by obtaining the equivalent resistance Reffective as shown in Eq. (17) using Ohm’s law.
Reffective 
Vsource
I source
(17)
2.3 Realization of the electromechanical behaviour
13
The electrical resistance changes according to the strain rate when using nanocomposite as
a strain sensor. To this end, when a significant strain is applied in the uniaxial direction, the
shape and position of the CNTs are updated, and the network connection shape is also changed.
To simplify the calculation process, uniaxial strain is applied in the z-axis direction, and at this
time, the x, y, and z-axis strains are expressed as Eq. (18) below.
d x 
dx
dy
dz
, d y  , d z 
x
y
z
(18)
Here,  x ,  y , and  z are strains in the x, y, and z axes. Integrating Eq. (18) and applying
Poisson's ratio definition of the matrix as in Eq. (19) below is expressed in Eq. (20) below. This
assumption is valid for CNT composites with V f within 5%.
 
 
d
d  trans
d
 x  y
d  axial
d z
d z
L  L
L
L   L dx
L   L dy
dz


L
L
z
x
y
(19)
(20)
Where  is Poisson’s ratio, L is the RVE size, L is the length of the RVE increses. By
solving this and exponentiating it, the relationship between the strains in the x, y, and z axes
can be defined as in Eq. (21) below [3].
1   z 

 1  x  1  y
(21)
In conclusion, the lengths Lx , L y , and Lz of RVE for deformation in the z-axis direction
are changed as shown in Eq. (22-24).
Lx  (1   x )  Lx  (1   z )   Lx
(22)
Ly  (1   y )  Ly  (1   z )  Ly
(23)
Lz  (1   z )  Lz
(24)
14
In addition, the elevation angle, azimuth angle, and center point ( X c , Yc , Z c ) of CNT are
expressed as in Eq. (25-27).
  
(25)
   tan 1[(1   z )1 tan  ]
(26)
( X c , Yc, Z c )  [(1   z )  X c , (1   z )  Yc , (1   z ) Z c ] (27)
The shapes of CNT and RVE updated by Z-axis deformation are shown in Figure 7 and
Figure 8.
Figure 7. Configuration of CNTs under tensile strain
Tensile strain
Figure 8. Configuration of RVE under tensile strain
15
3. Results and discussions
In this study, the filler was assumed to be multi-walled CNTs (MWCNTs), and the RVE size,
geometric periodicity of CNT fibers, electrical conductivity concerning tensile strain, and
resistance to deformation were predicted. Table 1 shows the geometrical and physical
properties of CNTs and polymers.
Table 1 Instrinsic and phsycal properties of CNT and polymer [9]
Property
Value
5, 7, 8, 9
CNT length l
CNT diameter D
50, 230
1.4
Cutoff distance d cutoff
Unit
μm
nm
nm
Conduction channels M
Intrinsic conductivity of CNT  CNT
460
10,000
S/m
Polymer Poisson’s ratio 
Energy barrier height E
0.33
1
eV
16
3.1 Periodicity
Figure 9 shows the predicted electrical conductivity and the experimental results depending
on whether the periodic boundary condition was applied. The length of CNT is 5μm, the
diameter of CNT is 50nm, the size of RVE is 15 15 15 m3 , and the number of CNT is about
3,438 per V f  1% . Each point on the graph represents the average value of 300 samples. As
a result of the prediction, the percolation threshold value was around 0.625% [refer]. The
predicted electrical conductivity values in the vicinity of the percolation threshold differ
somewhat depending on whether the CNT arrangement is periodic. Still, after that, it was
confirmed that the electrical conductivity monotonically increased to a similar value regardless
of whether or not the periodic boundary condition was applied.
(a)
(b)
Figure 9. Prediction results according to (a) periodic boundary conditions, (b) RVE size
3.2 RVE size effect
As mentioned earlier in subsection 2.3, the size of RVE compared to the length of CNTs is
crucial to realize the ideal random CNT distribution, which also affects the effective electrical
conductivity. To check such effect, a set of evaluations on the effective electrical conductivity
was conducted by increasing the RVE size from 5 μm to 15 μm by 2.5 μm increments with the
length of CNTs fixed at 5 μm. The number of CNTs for each RVE size is 127, 430, 1019, 1989,
17
and 3438, respectively. As shown in Fig. 10, the prediction results show that the slope of the
curve became steeper as the size of the RVE increased around the percolation threshold.
Figure 10. Prediction results according to RVE size
3.3 Tensile strain effect
In addition, the variation in the electrical resistance was investigated according to the tensile
strain. For simulation the test results and data set for the simulation are given in Table 2.
Reportedly, the length of the CNTs was 5~9 μm in the test samples; for simulation, the average
length of 7 μm of the CNTs was chosen. The diameter of CNT is 230nm, the size of RVE is
20  20  20 m3 , the V f of Test Sample#1 is 2.6%, and the V f of Test Sample#2 is 4.2%.
When the RVE size was 20  20  20 m3 , the resistance curve was bent as the tensile strain
increased, which was determined to be due to the insufficient number of CNTs. Therefore, the
number of CNTs was increased by increasing the size of the RVE to 30  30  30 m3 , and as
a result, the resistance curve did not bend. Compared to the test results, it was confirmed that
Test Samples #1 and #2 had similar resistance curves at 2.6% and 5% fiber volume ratios,
respectively.
Table 2 Test results and data set for the simulation
Case
CNT length l RVE size L
A-3
A-4
5~9 μm
5~9 μm
7 μm
7 μm
7 μm
7 μm
7 μm
A-5
A-6
7 μm
7 μm
A-7
7 μm
Test Sample #1 [3]
Test Sample #2 [3]
A-1
A-2
20 μm
20 μm
20 μm
20 μm
20 μm
30 μm
30 μm
35 μm
30 μm
30 μm
35 μm
30 μm
18
Vf
Number of CNT
2.6%
4.2%
2.5%
3%
5%
2.5%
2.6%
2.6%
2.7%
3%
3%
4%
688
825
1,375
2,321
2,414
3,833
2,507
2,785
4,423
3,713
7 μm
A-8
30 μm
35 μm
5%
5%
4,642
7,371
(a)
(b)
Figure 11. The relative resistance change curve (a) , (b)
3.4 Mixed CNT effect
As shown in Figure 12, the change in resistance according to the tensile strain was confirmed
by mixing the CNT length. Table 3 shows the network model for resistance prediction and the
information on Test Sample [3]Table 2. RVE was set to the same size as 30  30  30 m3 for
all of them. At the same fiber volume ratio, it was confirmed that the increase in resistance
decreased as the length of the CNT compared to the size of the RVE increased. In addition, by
mixing the same number of CNTs with 5, 7, and 9 μm lengths, the test sample could be
accurately simulated. It was confirmed that Test Samples #1 and #2 had similar resistance
curves to the network model at 2.5% and 5% fiber volume fractions, respectively.
Table 3 Test and network model data
Case
CNT length l
Test Sample #1 [3]
Test Sample #2 [3]
B-1
B-2
B-3
B-4
B-5
5~9 μm
5~9 μm
7 μm
8 μm
9 μm
5,7,9 μm
5,7,9 μm
RVE size L
Vf
Number of CNT
20 μm
20 μm
30 μm
30 μm
30 μm
30 μm
30 μm
2.6%
4.2%
2.5%
2.5%
2.5%
2.5%
3%
2,321
2,031
1,805
2,321
2,785
19
B-6
B-7
B-8
B-9
5,7,9 μm
5,7,9 μm
5,7,9 μm
5,7,9 μm
30 μm
30 μm
30 μm
30 μm
3.5%
4%
4.5%
5%
3,249
3,714
4,178
4,642
(a)
(b)
Figure 12. The relative resistance change curve
최종 정리 Table
TEST Sample 1
(Reference)
Tensiel Strain
7μm, 30μm, 2.5%
w/ CNT Mixed effect
5,7,9μm, 30μm, 2.5%
TEST Sample 2
(Reference)
Tensiel Strain
7μm, 30μm, 5%
w/ CNT Mixed effect
5,7,9μm, 30μm, 5%
10%
0. 6084
20%
2.7376
30%
6.5779
40%
12.6236
50%
24.7909
60%
-
0.9626
(58.23%)
0.8346
(37.18%)
3.1318
(14.40%)
2.5101
(-8.31%)
10.0629
(52.98%)
6.3194
(-3.93%)
26.8728
(112.9%)
12.2384
(-3.05%)
69.9114
(182.0%)
29.5224
(19.09%)
-
10%
0.0760
20%
0.2281
30%
0.6084
40%
1.0266
50%
1.6730
60%
2.6996
0.3351
(340.7%)
0.3380
(344.5%)
0.7322
(221.0%)
0.7231
(216.9%)
1.2130
(99.39%)
1.1847
(94.73%)
1.8047
(75.79%)
1.7281
(68.33%)
2.4775
(48.08%)
2.3722
(41.79%)
3.2645
(20.93%)
3.1290
(15.90%)
-
4. Conclusions
In this study, a program was developed and verified to predict the electromechanical
behavior of highly stretchable CNT-polymer composites in which the electrical properties
20
change in response to uniaxial deformation. Monte-Carlo simulation method was used to
generate RVEs with randomly distributed CNTs. Monte-Carlo simulation method was used to
generate RVEs with randomly distributed CNTs. In the case of tensile strain, the CNT and
RVE shapes and network models were updated by considering the shrinkage by Poisson's ratio.
To examine the appropriateness of the proposed methodology, trade-off studies were
conducted on RVE size, geometric periodicity of CNT fibers, and tensile strain. When the RVE
size was more than three times the aspect ratio of CNT, the CNT showed a uniform distribution
and accurately expressed the increase in resistance according to tensile strain. Next, the
electromechanical behavior of CNT mixtures with different aspect ratios was confirmed and
compared with the experimental data. When the aspect ratio was mixed rather than one, the
analysis result accurately expressed the experimental data. In conclusion, the proposed model
accurately predicts the percolation of CNT-polymer composites and helps predict electrical
properties under extensive strain conditions. Also, from the expected results, it was confirmed
that the RVE size, the geometric periodicity of the CNT fiber, and the degree of CNT dispersion
affect the resistance. As a future study, we plan to conduct a study to predict the electrical
conductivity of composite materials containing complex-shaped fillers.
Acknowledgements
This study was conducted with the support of the National Research Foundation of Korea,
funded by the Ministry of Science and ICT. (NRF-2020R1F1A1075588).
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