A cellular automata simulation for the tension behavior of carbon nanotubes

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World Journal Of Engineering


453

A cellular automata simulation for the
tension

behavior of carbon
nanotubes

X. H. Huang and
X.

Q. He

Department of Building and Construction, City University of Hong Kong, Kowloon,
Hong Kong

Email:
bcxqhe@cityu.edu.hk


Introduction


Carbon nanotubes (CNTs
) are currently studied
by two types of widely used theoretical methods,
i.e. atomistic
-
based methods [1,

2] and
continuum mechanics
[3]
.
The atomistic based
methods are currently limited to very small size
and time
scale
because the computational time
exp
onentially explodes with increase in the
number of atoms. The simulation of larger
systems or longer period is currently achieved
with continuum methods. However, the existing
continuum models are inapplicable to CNTs
where the dimension is in nanometer sc
ale and
the continuum assumption breaks down
.
In order
to overcome these shortcomings,

t
he
cellular
automata
algorithm is

appli
ed to
investigate

the
tension

characteristic of
CNT
s

[4
]

in this study
.
Tersoff
-
Brenner’s potential is adopted to
describe the in
teraction between atoms.
The
stress
-
strain response from elastic to fracture is
simulated to obtain the tension behavior of
CNT
s.
Our studies show that the
atomistic
-
based

cellular automata
algorithm is an efficient
method for
investigating
the
tension
cha
racteristic

of CNTs.


Cellular automata approach



Consider an initial equilibrium configuration
of CNT which consists of
a system of
N

atoms
,
as shown in Fig. 1
.








The cellular automaton approach is adop
ted to
describe the equilibrium configuration of CNTs.
The atomic
-
based cellular automaton is a
discrete

system which consists of a hexagonal grid of
cells,
and eac
h atom is taken as a cell.
The
Tersoff
-
Brenner’s many
-
body potential

U

stored in the atomic bonds is used to obtain
the equilibrium equation:



Ku=P

(1)


where
u

the displacement vector of a
ll atoms,
K

is the stiffness matrix which is expressed as




2
2
( )
U



x
K
x


(5)

and
P

is a non
-
equilibrium force vector





( )
U

 

x
P F
x



(6)

Simulations




To characterize the tensile behavior of CNTs,
we consider (10, 10) single
-
walled CNTs with
the length L=
12.3668

nm. Figure 2

shows
the
stress
-
strain relationships

of (10, 10) CNTs
from elastic deformation to rapture
.
It is shown
from these figures that w
hen
0 0.04

 
, the
stress
-
strain relationship follows Hooke’s law,
i.e.


E

.


















With a further increase in strain, the stress
-
strain curve becomes nonlinear and has
Fig.
1.

Cellular space of a carbon nanotube.


Fig. 2 Stress
-
strain curve of a SWCNT
(10, 10) under axial tension

World Journal Of Engineering


454

smaller and smaller slope until the curve

becomes
horizontal
when
0.2 0.27

 
.

Beyond the
point

0.2


, considerable plastic deformation occurs
without noticeable increase in the stress, i.e. in
the strain scope of
0.2 0.27

 
, the stress is
approximately
expressed as
5
1.4 10

 

MPa
.
This ductile deformation

displays the re
-
arrangement of configuration of
CNT
s for the
strain release via the formation of the Stone
-
Wales defects.
Further increase of the strain

after
0.2


, the p
henomenon of the break and the
reform of bonds occurs at the strain

0.22



with the formation of the 5
-
7
-
7
-
5 defects
, as

shown in Fig. 3(a)
. With the strain increase after
0.22


, more
(5
-
7
-
7
-
5)
vacancie
s are gene
rated
and formed the two helical strips
,

as shown in
Fig. 3(b)
. The vacancy turns to larger and larger
and the CNT ruptures very quickly when
reaching the maximum strain
,

as shown in Fig.
3(c)
.












Fig. 3. The brittle fracture
process of (10,10) CNTs.
(a) Formation of initial
5
-
7
-
7
-
5 defect
; (b)
Formation of

the two helical strips
; (c)
Breaking
completely.




Conclusion


In this paper,
an atomic
-
based cellular
automata
algorithm is presented for atomic
-
scale simulation of CNTs.
The ACAA employs
the
inter
-
atomic p
otential to account for multi
-
body interactions and is as accurate as the MM
method, but much faster.

Our simulations show
that the Stone
-
Wales defect leads to the plastic
deformation and the brittle fracture
. The
numeric
al results show that the ACAA is a
effective method for the simulation of large
system of CNTs.


Acknowledgement

The work described in this paper
was

fully
support
ed by a research grant from the
Research Grants Council of the Hong
Kong Special Administrat
ive Region,
China

[Project No.
CityU 114010
].


References

[1]

B.I. Yakobson, C.J. Brabec, J. Bernholc,
Nanomechanics of carbon tubes:
instabilities beyond linear response
,

Physical Review Letters
76

(1996) 2511
-
2514
.


[2]

K. M. Liew, X.Q. He, and C. H. Wong,

On
the study of elastic and plastic properties of
multi
-
walled carbon nanotubes under axial
tension using molecular dynamics
simulation
,
Acta Materialia

52

(2004)

2521
-
2527
.

[3] S. Govindjee, and J. L. Sackman,
On the
use of continuum mechanics to estimate th
e
properties of nanotubes
, Solid State
Communications
110

(1999) 227
-
230
.

[4]

X.Q. He,

X.H. Huang
,
On the use of
cellular automata
algorithm

for the atomic
-
based simulation of carbon nanotubes
,
PROCEEDINGS OF THE ROYAL
SOCIETY A
-
MATHEMATICAL PHYSICAL
AND E
NGINEERING SCIENCES

465

(
2009
)

193
-
206.



(a) Strain = 0.223



(b) Strain = 0.271




(c) Strain = 0
.275