12. - 14. 10. 2010, Olomouc, Czech Republic, EU
EFFECT OF UNIT CELL LENGTH ON YOUNG’S MODULUS OF ZIGZAG SINGLE WALLED
CARBON NANOTUBES
Mehrdad ARJMAND, Ali SHOKUHFAR, Mohammad AMINI SARABI
Department of Mechanical engineering, K.N.TOOSI University of Technology, Tehran, Iran,
mehrdadarjmand@sina.kntu.ac.ir
Abstract
A new theoretical method has been implemented to study the effect of length on Young’s modulus of zigzag
carbon nanotubes (CNTs). In this model, a combination of continuum mechanics and molecular mechanics
methods has been used to find the Young’s modului of intact and defective CNTs. The Young’s modulus of
an intact CNT unit cell, length = 6.39 angstrom, was calculated about 940 GPa that shows a good agreement
with experimental and theoretical results. Increasing the length of unit cell to 7.81nm causes a reduction in
the modulus and it becomes 887 GPa, while a smaller unit cell with length of 4.97 angstrom has the modulus
about 1040 GPa. By increasing the diameter, the huge difference between the young’s moduli decreases
and for long nanotubes, the moduli of all unit cells with different lengths approaches to 940 GPa.
Keywords: SWCNT, Young's modulus, length, vacancy
1. INTRODUCTION
During recent years, mechanical characterization of carbon nanotubes (CNTs) has been one of the top
issues in mechanical and materials engineering. In order to predict the mechanical behavior of CNTs,
various methods have been developed. Theoretical studies [1-4] such as continuum mechanics (CM) [5-7],
molecular dynamics, molecular mechanics [8] and quantum mechanics [9] have been improved in parallel
with numerical computational simulations [10,11], semi-empirical and the experimental methods [12-15].
Scientists have always had an eye on theoretical approaches in comparison with numerical computational
simulations and empirical methods, because theoretical methods are easy to use due to their parametric
structure in spite of high computational costs of numerical simulation methods and difficult procedures of
experimental methods for nanomaterials. Therefore, new improvements in this field are always welcomed.
The outstanding mechanical characteristics of CNTs hold for nearly perfect CNTs. However, if CNTs have
defects in their atomic network, one can expect that due to their quasi-one dimensional atomic structure even
a small number of defects will result in some degradation [16] in their characteristics. Therefore, to fully
understand the behavior of CNTs with defects, it seems essential to know the effect of defects on mechanical
properties of CNTs, especially the Young’s modulus. Until now, numerous researches have been done to
predict the elastic modulus of intact CNTs. But no major results have been reported yet indicating
development of CM theory for CNTs containing vacancies.
Vacancies are the most common defects in CNTs and therefore have attracted the attention of many
scientists [17-19]. Lu and Pan [17] performed a calculation on single vacancies in SWCNTs and obtained a
relation between the formation energy of defective CNT and radii of vacancy. Sammalkorpi et al. [20] derived
an analytical expression, with coefficients parameterized from atomistic computer simulations, which relates
the Young’s modulus and defect density in CNTs.
In this article, we are going to study the effects of length on the elastic modui of CNTs. We assume that a
CNT is consisted of several unit cells. Studying a unit cell in order to estimate the modulus of a long CNT
requires knowing the effect of length on modulus of each section. Xiao et al. [21] have already studied the
case and have shown that the length of nanotube affects the modulus.
12. - 14. 10. 2010, Olomouc, Czech Republic, EU
2. MODELING
Some theoretical calculations have previously been conducted to predict the moduli of perfect CNTs
[1,22,23]. Calculating the modulus for a section instead of whole CNT seems to be a wise solution which has
been proposed by Sammalkorpi et al. [20]. But the effect of cell’s length on the Young’s modulus has never
been studied. Are we allowed to select any part of a CNT as a unit cell to find its modulus? Does the
variation in length have any effect on Young’s modulus of a CNT? Xiao et al. [21] have shown that there is a
relation between the modulus and length for small lengths.
(1)
It is assumed that a CNT has vertical bonds and diagonal bonds as shown in Fig. 1. It is obvious that
in pristine CNTs, all the bonds have the same length b. shows the total length of each CNT while equals
to . Fig. 1 shows the unrolled sections with different lengths:
(2)
(3)
(4)
a
b
c
Fig. 1 Structure of unrolled sections (a) basic model T (b) reduced model Max (c) extended model Min
12. - 14. 10. 2010, Olomouc, Czech Republic, EU
All the three possible configurations of CNTs have been shown in Fig. 1. First picture explains a case in
which there are equal numbers of vertical and diagonal bonds and we call it the basic model T. In second
picture we have one more diagonal bond than vertical bonds that represents model Max, while in third
picture there is one more vertical bond named model Min. The relations between total displacement and
displacement of bonds are indicated below.
(5)
(6)
Where shows total displacement of the tube and is the elongation of each bond. Here, total
displacements of three models are shown below.
(7)
(8)
(9)
By substituting the Eqs. (7-9) in Eqs. (2-4) and then in Eqs. (8-10), we have:
(10)
(11)
(12)
Where . represents the chirality, shows the thickness, is
the radius and Is the modulus of the CNT. It is obvious in these formulations that for short CNT sections,
even one carbon bond is important.
Results and discussion
Already mentioned in the modeling section, there were no suggestions available for the length of CNT
sections. Based on our research, it is not always correct to divide a CNT with arbitrary length to N sections.
What we suggest is that a CNT should be built by putting together specific unite cells called sections.
Knowing that each section only consists of a few carbon atoms, we can almost build any desired CNT. The
main model T in Fig. 1 has already been studied. By adding and removing n atoms at the end of the section
(n is the chirality), we have two new structures with different lengths shown in Fig. 1. Remarkable difference
between their moduli as shown in Fig. 2 approves the assumption that the length of each cell plays an
12. - 14. 10. 2010, Olomouc, Czech Republic, EU
important role. The black lines in Fig. 2 show the moduli of intact CNT sections. The greatest modulus, 1040
GPa, belongs to the section with smallest length (4.97 angstrom) shown in Fig. 1b, while the longest CNT
section (7.81 angstrom) has the least modulus about 887 GPa demonstrated in Fig. 1c. The moduli of
defective sections are also shown in Fig. 2. As expected, the moduli of defective sections tend to the moduli
of intact sections by increasing the diameter. Fig. 2 also shows that effect of defects are more remarkable for
shorter cells.
Fig. 2 moduli of different CNT models containing vacancies varying with diameter
Table 1. has classified some details about different vacancy types and also shows the differences between
sections. The Young’s moduli for (4,0) and (20,0) CNTs clearly explain the fact that defects have stronger
effect on CNTs with smaller diameters. In additions, it is shown that the difference between elastic moduli of
intact and defective CNTs for Max model is more than the difference for T model while the difference is the
least for the Min model.
Model Length of the No. of transverse Modulus (GPa) Modulus (GPa)
section vacancies/section (4,0) (20,0)
(Angstrom)
T-0 6.39 0 940.9 940.9
T-1 6.39 1 901.1 932.9
T-2 6.39 2 861.3 925
T-3 6.39 3 NA 917
Max-0 4.97 0 1040 1040
Max-1 4.97 1 981.6 1028.3
Max-2 4.97 2 923.2 1016.6
12. - 14. 10. 2010, Olomouc, Czech Republic, EU
Min-0 7.81 0 887.1 887.1
Min-1 7.81 1 867 883.1
Min-2 7.81 2 846.8 879.1
Table 1. Comparison between (4,0) and (20,0) nanotubes using three models
After studying three different models with different lengths and finding out the variations in modulus caused
by length, a question may come to the minds that which one is correct and how this difference could be
explained. To answer this question, a study has been done on three intact CNTs based on the three
proposed models. We have chosen three intact CNTs with different lengths as shown in Fig. 1. It is shown in
Fig. 3 how the moduli of all models approach to that of the main model, T, with increase in length. It also
shows that the effect of added or reduced carbon bonds smoothly decreases for longer CNTs , for lengths
more than 50 angstrom, while the effect is remarkable for smaller ones. It can be concluded that the effect of
some added or removed carbon bonds is negligible for long CNTs which is really in good agreement with our
common sense.
Fig. 13 moduli of intact CNTs with different models varying with length
Conclusions
A theoretical formulation has been derived to predict the Young’s modulus of intact and defective CNTs.
Combining CM with atomistic potential functions has made this research unique. Proposing 941 GPa
obtained for the Young’s modulus of an intact CNT is in good agreement with most recent research and is an
indication of the value of the theory. This model predicts that the elastic modulus of a CNT decreases with
vacancies, which has also been predicted in the literature. To define a unit cell, it is important to know effect
of length on modulus of CNTs. The elastic modulus of a defective CNT greatly changes by adding or
removing one carbon atom to the boundary atoms, especially for small CNTs. Min model predicts the
modulus of a (5,0) CNT with a T-2 vacancy about 855 GPa. Meanwhile, Max model calculates 947 GPa as
the modulus and T model approximates the modulus about 877 GPa. But for long CNTs, Max and Min
models approach to the main model T. All the defective CNTs have the moduli less than an intact one, but by
increasing the length, effect of defects reduces and modulus of a defective CNT approaches that of an intact
one.
12. - 14. 10. 2010, Olomouc, Czech Republic, EU
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