Computation simulation is a powerful tool for predictiong the mechanics models of elastic properties of armchair and zigzag single-walled nanotubes.  The aim of this work is investigation and comparison of Young’s modulus, shear modulus and Poisson’s 
ratio variations of armchair and zigzag tubes as functions of diameter. We obtained a set of concise, closed form expressions for the size-dependent elastic modulus, shear modulus and Poisson’s ratio of armchair (n, n) and zigzag (n, 0) nanotubes, which are basic for constructing mathematical models of elastic properties of SWNTs. We investigated armchair nanotubes with chirality (3, 3)–(40, 40) and zigzag (3, 0)–(40, 0) with diameters 4,2–54,2 Å and 2,4–31,3 Å respectively. We calculated Young’s modulus to be 0,26–2,95 TPa for armchair and 0,5–3,7 TPa for zigzag nanotubes. The shear modulus calculated for armchair nanotube appeared to be in the range of 0,2–2,0 TPa and for 
zigzag one in the range of 0,2–2,7 TPa. Specifically, it was inverse dependences of Young’s modulus and shear modulus on diameter. The Poisson’s ratio was in range from 0,28 to 0,42 and from 0,27 to 0,39, respectively. Results of this research can be 
used for design, analysis and evaluating of nanotubes unctioning and creating new materials based on CNTs.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/2063

Protsenko, Olena Borysivna

Yemelianenko, Viktoriia Volodymyrivna

Емельяненко, Виктория Владимировна

Проценко, Елена Борисовна

Проценко, Олена Борисівна

Ємельяненко, Вікторія Володимирівна

Electronic Sumy State University Institutional Repository

130                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I             THE ANALYSIS OF THE ELASTIC PROPERTIES OF ARMCHAIR AND ZIGZAG SINGLE-WALLED CARBON NANOTUBES Victoriya V. Yemelyanenko*, Olena B. Protsenko  Sumy State University, R-Korsakov 2, 40007, Sumy, Ukraine  ABSTRACT Computational simulation is a powerful tool for predicting the mechanical proper-ties of carbon nanotubes. In this paper we present analytical molecular mechanics mod-els of elastic properties of armchair and zigzag single-walled nanotubes. The aim of this work is investigation and comparison of Young’s modulus, shear modulus and Poisson’s ratio variations of armchair and zigzag tubes as functions of diameter. We obtained a set of concise, closed-form expressions for the size-dependent elastic modulus, shear modu-lus and Poisson’s ratio of armchair (n, n)  and zigzag (n, 0) nanotubes, which are basic for constructing mathematical models of elastic properties of SWNTs. We investigated armchair nanotubes with chirality (3, 3)–(40, 40) and zigzag (3, 0)–(40, 0) with diame-ters 4,2–54,2 Å and 2,4–31,3 Å respectively. We calculated Young’s modulus to be 0,26–2,95 TPa for armchair and 0,5–3,7 TPa for zigzag nanotubes. The shear modulus calculated for armchair nanotube appeared to be in the range of 0,2–2,0 TPa and for zigzag one in the range of 0,2–2,7 TPa. Specifically, it was inverse dependences of Young’s modulus and shear modulus on diameter. The Poisson’s ratio was in range from 0,28 to 0,42 and from 0,27 to 0,39, respectively. Results of this research can be used for design, analysis and evaluating of nanotubes functioning and creating new materials based on CNTs.  Key words: carbon nanotubes, Young’s modulus, shear modulus, Poisson’s ratio  INTRODUCTION The advancement of science and technology has evolved into the era of nanotechnology. Carbon nanotubes (CNTs) have stimulated great interest and extensive research with regard to the measurement of their exact mechanical properties [1, 2] and the search for potential structural applications [1] ever since their discovery by Iijima [3]. Specific characteristics, such as the exceptionally high stiffness and strength, which are in the range of TPa, the extreme resilience, the ability to sustain large elastic strain as well as the high aspect ratio and low density [1, 4], make CNTs the ideal reinforcing material for a new class of superstrong nano-composites. That is why the accurate assessment of the mechanical properties                                                * e-mail: v.v.yemelyanenko@id.sumdu.edu.ua, tel: (+38)0503078188 Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   131  of the nanotubes is an important first step towards the potential development of the structural composites. Investigation of the properties of those structures has a lot of experimental difficulties arising from their nanoscale, necessity for complicated and expensive equipment and apparatus with large resolution. Computational simulation is a powerful tool relative to the experimental diffi-culties. In this paper we present two mathematical models, based on classical mechanics and molecular mechanics approach, for investigating the difference between properties of armchair and zigzag single-walled CNTs (SWNTs). METHODS OF INVESTIGATION The methods of molecular dynamics (MD) and molecular mechanics (MM) based on the molecular nature of nanotube structure are often used for simulation [4-6] of their mechanical properties. In this work we used MM ap-proach based on modern continuum mechanics and elasticity theory. It allows to calculate the geometry of the frame with sufficient accuracy and to model some physical processes in nanotubes under influence of external factors: the deforming forces, external electromagnetic fields, etc.  We used some basics of molecular mechanics [4, 5] which is based on the concept of molecular force field for this investigation. This approach was based on a link between molecular and solid mechanics. Using the harmonic energy functions the nanotube was modeled as a frame structure and a closed-form elastic solution was obtained. Let a1 and a2 be unit vectors of the two-dimensional graphene sheet, (n, m)  be  a  pair  of  integers  that  indexes  the  atomic  structure  of  CNT  uniquely determining the size of the SWNT. Then a pair (n, m) corresponds to a lattice vector Ch = na1 + ma2 on the graphite plane. If m = 0, then such a tube is called zigzag, and when n = m it is an armchair tube. Based on rolling graphene sheer model the diameter D of nanotube can be determined as follows:     2 23S  bD n m mn ,  (1)  where b is the carbon-to-carbon bond length. This parameter was taken as 0,66 Å [4]. For armchair and zigzag tubes diameter is equal to D1 and D2 respectively:  213 , bnDʌ23 bnDʌ. (2)  Given the effective wall thickness t of SWNTs which corresponds to the thickness of the graphene layer, the effective diameter Deff is equal to Deff = Di + t, where the parameter Di (i = 1, 2) is taken by D1 for an armchair tube, and D2 for a zigzag tube. Effective thickness t was taken as 0,66 Å [4]. 132                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I             We have analyzed the forces and geometrical relations for an armchair tube. Also we have used the assumption that all atomic interactions in a molec-ular structure of nanotubes satisfy the potential laws, so they can be described by using molecular mechanics. Regarding aforementioned we have build the following relations which comprise our model.  Young’s modulus of armchair and zigzag nanotubes is expressed as fol-lows:  2 216 33 9§ · ¨ ¸¨ ¸ © ¹ieffș ȡi iȡ șȜ K K DEb K ȜK D, (3)  where parameters Ȝi,  Di, Deff i (i   1, 2) are taken by Ȝ1,  D1, Deff 1 for an armchair tube, and Ȝ2,  D2, Deff 2 for a zigzag tube; Ȝi is the elongation, equaled to the ratio of total nanotubes elongation įl to its length l0 before de-formation:  21 216 2cos4 cosO  ȖȖ, 22 218 2cos4 3cos ȖȜȖ, (4)  where Kȡ, Kș are force constants, which depend on the force field. Kȡ = 97800 kcal/mole/nm2, Kș = 126  kcal/mole/rad2, Ȗ = ʌ/2n is the angle, related to the curvature effect. Shear modulus is given by  444 2162 33UU§ · ¨ ¸ § ·© ¹ ¨ ¸© ¹effieff șiișeffiDD tȜ K KGtD b K ȜK. (5)  Poisson’s ratio is defined as:   22 3U TU TOQ O iib K Kb K K. (6)  Mathematical models of elastic properties of armchair and zigzag tubes have been obtained in the form of equations (2), (3), (5) and (6) with taken i = 1 for an armchair tube, and i = 2 for a zigzag tube. RESULTS AND DISCUSSION The elastic properties of two main types of CNT — armchair and zigzag – were investigated and discussed in this paper. We can say that calculated values of elastic properties largely depend on the assumption of the wall thickness t of SWNT. There are other information about different wall thickness such as 6,9 Å, 5,7 and 3,4 Å [4-6]. In this investigation t was taken as 0,66 Å [4]. Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   133  It was build application for investigation elastic properties using software Delphi 7 and mathematical models of SWNTs. It makes possible to estimate Young’s modulus E, shear modulus G and Poisson’s ratio Ȟ of  armchair  (n, n) or zigzag (n, 0) tubes with any specified diameter D.  For example, we investigated armchair tubes with chirality from (3, 3) to (40, 40) and zigzag tubes from (3, 0) to (40, 0). Those armchair tubes had di-ameter from 4,2 to 54,2 Å and zigzag tubes – from 2,4 to 31,3 Å. The values of Young’s modulus ranged from 0,26 to 2,95 TPa for armchair and from 0,5 to 3,7 for zigzag nanotubes. We calculated that shear modulus of armchair nano-tubes was in the range of 0,2–2,0 TPa and of zigzag was in the range of 0,2–2,7 TPa. Young’s and shear moduli have shown inverse chirality and diameter dependences. The Poisson’s ratio was ranging from 0,28 to 0,42 and from 0,27 to 0,39 for an armchair and zigzag tubes, respectively. Those results agree well with reported results in literature [3, 4, 6].  CONCLUSIONS In this paper we present the approach for investigation the elastic proper-ties of SWNTs, based on molecular mechanics. The mathematical models for simulation Young’s modulus, shear modulus and Poisson’s ratio of armchair and zigzag tubes were built. Those parameters were investigated as functions of the nanotube size and structure. According to the presented models we calcu-lated values of Young’s modulus, shear modulus and Poisson’s ratio for arm-chair and zigzag CNTs. It can be seen that predicted values of Young’s and shear modulus for zigzag tube are larger than for armchair tube, especially for smaller tubes, but they both decrease rapidly while the diameter increases. With diameter increasing Young and shear modulus of the both type of tubes begin to have the same value. As for Poisson’s ratio, we can say that zigzag tube is more sensitive to the diameter than that of the armchair tube. REFERENCES [1] V.N. Popov. Mater. Sci. Eng., 2004, R43, P. 61-102. [2] Y. Jin, F.G. Yuan. Compos. Sci. Technol., 2003, No. 63, P. 1507-1515. [3] S. Iijima. Nature, 1999, Vol. 354, No. 6348, P. 56–58. [4] T. Natsuki, K Tantrakarn, M. Endo. Appl. Phys., 2004, No. 79, P. 117-124. [5] T. Chang, H. Gao. J. Mech. Phys. Solid., 2003, No. 51, P. 1059-1074. [6] M. Meo, M. Rossi. Compos. Sci. Technol., 2006, No. 66, P. 1597-1605.    

The analysis of the elastic properties of armchair and zigzag single-walled carbon nanotubes

SocioEconomic Challenges Journal (Sumy State University)

130                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I            
 
THE ANALYSIS OF THE ELASTIC PROPERTIES OF 
ARMCHAIR AND ZIGZAG SINGLE-WALLED CARBON 
NANOTUBES 
Victoriya V. Yemelyanenko*, Olena B. Protsenko 
 
Sumy State University, R-Korsakov 2, 40007, Sumy, Ukraine 
 
ABSTRACT 
Computational simulation is a powerful tool for predicting the mechanical proper-
ties of carbon nanotubes. In this paper we present analytical molecular mechanics mod-
els of elastic properties of armchair and zigzag single-walled nanotubes. The aim of this 
work is investigation and comparison of Young’s modulus, shear modulus and Poisson’s 
ratio variations of armchair and zigzag tubes as functions of diameter. We obtained a set 
of concise, closed-form expressions for the size-dependent elastic modulus, shear modu-
lus and Poisson’s ratio of armchair (n, n)  and zigzag (n, 0) nanotubes, which are basic 
for constructing mathematical models of elastic properties of SWNTs. We investigated 
armchair nanotubes with chirality (3, 3)–(40, 40) and zigzag (3, 0)–(40, 0) with diame-
ters 4,2–54,2 Å and 2,4–31,3 Å respectively. We calculated Young’s modulus to be 
0,26–2,95 TPa for armchair and 0,5–3,7 TPa for zigzag nanotubes. The shear modulus 
calculated for armchair nanotube appeared to be in the range of 0,2–2,0 TPa and for 
zigzag one in the range of 0,2–2,7 TPa. Specifically, it was inverse dependences of 
Young’s modulus and shear modulus on diameter. The Poisson’s ratio was in range 
from 0,28 to 0,42 and from 0,27 to 0,39, respectively. Results of this research can be 
used for design, analysis and evaluating of nanotubes functioning and creating new 
materials based on CNTs. 
 
Key words: carbon nanotubes, Young’s modulus, shear modulus, Poisson’s ratio 
 
INTRODUCTION 
The advancement of science and technology has evolved into the era of 
nanotechnology. Carbon nanotubes (CNTs) have stimulated great interest and 
extensive research with regard to the measurement of their exact mechanical 
properties [1, 2] and the search for potential structural applications [1] ever 
since their discovery by Iijima [3]. 
Specific characteristics, such as the exceptionally high stiffness and 
strength, which are in the range of TPa, the extreme resilience, the ability to 
sustain large elastic strain as well as the high aspect ratio and low density [1, 4], 
make CNTs the ideal reinforcing material for a new class of superstrong nano-
composites. That is why the accurate assessment of the mechanical properties 
                                               
* e-mail: v.v.yemelyanenko@id.sumdu.edu.ua, tel: (+38)0503078188 
Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   131 
 
of the nanotubes is an important first step towards the potential development of 
the structural composites. Investigation of the properties of those structures has 
a lot of experimental difficulties arising from their nanoscale, necessity for 
complicated and expensive equipment and apparatus with large resolution. 
Computational simulation is a powerful tool relative to the experimental diffi-
culties. In this paper we present two mathematical models, based on classical 
mechanics and molecular mechanics approach, for investigating the difference 
between properties of armchair and zigzag single-walled CNTs (SWNTs). 
METHODS OF INVESTIGATION 
The methods of molecular dynamics (MD) and molecular mechanics 
(MM) based on the molecular nature of nanotube structure are often used for 
simulation [4-6] of their mechanical properties. In this work we used MM ap-
proach based on modern continuum mechanics and elasticity theory. It allows 
to calculate the geometry of the frame with sufficient accuracy and to model 
some physical processes in nanotubes under influence of external factors: the 
deforming forces, external electromagnetic fields, etc.  
We used some basics of molecular mechanics [4, 5] which is based on the 
concept of molecular force field for this investigation. This approach was based 
on a link between molecular and solid mechanics. Using the harmonic energy 
functions the nanotube was modeled as a frame structure and a closed-form 
elastic solution was obtained. 
Let a1 and a2 be unit vectors of the two-dimensional graphene 
sheet, (n, m)  be  a  pair  of  integers  that  indexes  the  atomic  structure  of  CNT  
uniquely determining the size of the SWNT. Then a pair (n, m) corresponds to a 
lattice vector Ch = na1 + ma2 on the graphite plane. If m = 0, then such a tube is 
called zigzag, and when n = m it is an armchair tube. Based on rolling graphene 
sheer model the diameter D of nanotube can be determined as follows: 
   ? ?2 23?? ? ?
bD n m mn ,  (1) 
 
where b is the carbon-to-carbon bond length. This parameter was taken as 
0,66 Å [4]. 
For armchair and zigzag tubes diameter is equal to D1 and D2 respectively: 
 
2
1
3 ,? bnD
?
2
3? bnD
?
. (2) 
 
Given the effective wall thickness t of SWNTs which corresponds to the 
thickness of the graphene layer, the effective diameter Deff is equal to 
Deff = Di + t, where the parameter Di (i = 1, 2) is taken by D1 for an armchair 
tube, and D2 for a zigzag tube. Effective thickness t was taken as 0,66 Å [4]. 
132                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I            
 
We have analyzed the forces and geometrical relations for an armchair 
tube. Also we have used the assumption that all atomic interactions in a molec-
ular structure of nanotubes satisfy the potential laws, so they can be described 
by using molecular mechanics. Regarding aforementioned we have build the 
following relations which comprise our model.  
Young’s modulus of armchair and zigzag nanotubes is expressed as fol-
lows: 
 
2 2
16 3
3 9
? ?
? ? ?? ?? ? ?ieff
? ?
i i
? ?
? K K DE
b K ?K D
, (3) 
 
where parameters ?i,  Di, Deff i (i ? 1, 2) are taken by ?1,  D1, Deff 1 for 
an armchair tube, and ?2,  D2, Deff 2 for a zigzag tube;? ?i is the elongation, 
equaled to the ratio of total nanotubes elongation ?l to its length l0 before de-
formation: 
 
2
1 2
16 2cos
4 cos
? ??
?
?
?
, 
2
2 2
18 2cos
4 3cos
??
?
?
?
?
, (4) 
 
where K?, K? are force constants, which depend on the force field. 
K? = 97800 kcal/mole/nm2, K? = 126  kcal/mole/rad2, ? = ?/2n is the angle, 
related to the curvature effect. 
Shear modulus is given by 
 
4
4
4 2
16
2 3
3
?
?
? ?? ?? ? ? ?? ?? ? ??? ?
effi
eff ?i
i
?
effi
D
D t
? K KG
tD b K ?K
. (5) 
 
Poisson’s ratio is defined as:  
 
2
2 3
? ?
? ?
?? ?
??
?
i
i
b K K
b K K
. (6) 
 
Mathematical models of elastic properties of armchair and zigzag tubes 
have been obtained in the form of equations (2), (3), (5) and (6) with taken i = 1 
for an armchair tube, and i = 2 for a zigzag tube. 
RESULTS AND DISCUSSION 
The elastic properties of two main types of CNT — armchair and zigzag – 
were investigated and discussed in this paper. We can say that calculated values 
of elastic properties largely depend on the assumption of the wall thickness t of 
SWNT. There are other information about different wall thickness such as 
6,9 Å, 5,7 and 3,4 Å [4-6]. In this investigation t was taken as 0,66 Å [4]. 
Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   133 
 
It was build application for investigation elastic properties using software 
Delphi 7 and mathematical models of SWNTs. It makes possible to estimate 
Young’s modulus E, shear modulus G and Poisson’s ratio ? of  armchair  (n, n) 
or zigzag (n, 0) tubes with any specified diameter D.  
For example, we investigated armchair tubes with chirality from (3, 3) to 
(40, 40) and zigzag tubes from (3, 0) to (40, 0). Those armchair tubes had di-
ameter from 4,2 to 54,2 Å and zigzag tubes – from 2,4 to 31,3 Å. The values of 
Young’s modulus ranged from 0,26 to 2,95 TPa for armchair and from 0,5 to 
3,7 for zigzag nanotubes. We calculated that shear modulus of armchair nano-
tubes was in the range of 0,2–2,0 TPa and of zigzag was in the range of 0,2–
2,7 TPa. Young’s and shear moduli have shown inverse chirality and diameter 
dependences. The Poisson’s ratio was ranging from 0,28 to 0,42 and from 0,27 
to 0,39 for an armchair and zigzag tubes, respectively. Those results agree well 
with reported results in literature [3, 4, 6].  
CONCLUSIONS 
In this paper we present the approach for investigation the elastic proper-
ties of SWNTs, based on molecular mechanics. The mathematical models for 
simulation Young’s modulus, shear modulus and Poisson’s ratio of armchair 
and zigzag tubes were built. Those parameters were investigated as functions of 
the nanotube size and structure. According to the presented models we calcu-
lated values of Young’s modulus, shear modulus and Poisson’s ratio for arm-
chair and zigzag CNTs. It can be seen that predicted values of Young’s and 
shear modulus for zigzag tube are larger than for armchair tube, especially for 
smaller tubes, but they both decrease rapidly while the diameter increases. With 
diameter increasing Young and shear modulus of the both type of tubes begin 
to have the same value. As for Poisson’s ratio, we can say that zigzag tube is 
more sensitive to the diameter than that of the armchair tube. 
REFERENCES 
[1] V.N. Popov. Mater. Sci. Eng., 2004, R43, P. 61-102. 
[2] Y. Jin, F.G. Yuan. Compos. Sci. Technol., 2003, No. 63, P. 1507-1515. 
[3] S. Iijima. Nature, 1999, Vol. 354, No. 6348, P. 56–58. 
[4] T. Natsuki, K Tantrakarn, M. Endo. Appl. Phys., 2004, No. 79, P. 117-124. 
[5] T. Chang, H. Gao. J. Mech. Phys. Solid., 2003, No. 51, P. 1059-1074. 
[6] M. Meo, M. Rossi. Compos. Sci. Technol., 2006, No. 66, P. 1597-1605. 
 
  


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The analysis of the elastic properties of armchair and zigzag single-walled carbon nanotubes

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