mechanics of defects in carbon nanotubes s namilae, c shet and n chandra
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Mechanics of defects in Carbon nanotubes
S Namilae, C Shet and N Chandra
Defects in carbon nanotubes (CNT)
Point defects such as vacancies
Topological defects caused by forming pentagons and heptagons e.g. 5-7-7-5 defect
Hybridization defects caused due to fictionalization
Sp3 Hybridization here
Role of defects
Mechanical properties Changes in stiffness observed.
Stiffness decrease with topological defects and increase with functionalization
Defect generation and growth observed during plastic deformation and fracture of nanotubes
Composite properties improved with chemical bonding between matrix and nanotube
Electrical properties Topological defects required to join
metallic and semi-conducting CNTs Formation of Y-junctions End caps
Other applications Hydrogen storage, sensors etc
1Ref: D Srivastava et. al. (2001)
1
Mechanics at atomic scale
Physical Problem
Molecular Dynamics-Fundamental quantities (F,u,v)
Born Oppenheimer
Approximation
Compute Continuum quantities-Kinetics (,P,P’ )-Kinematics (,F)-EnergeticsUse Continuum Knowledge- Failure criterion, damage etc
Stress at atomic scale
Definition of stress at a point in continuum mechanics assumes that homogeneous state of stress exists in infinitesimal volume surrounding the point
In atomic simulation we need to identify a volume inside which all atoms have same stress
In this context different stresses- e.g. virial stress, atomic stress, Lutsko stress,Yip stress
Virial Stress
1 1 1
2 2
N Ni j
ij i j
r r Vm v v
r r
Stress defined for whole system
For Brenner potential:
1 1 1
2 2
N N
ij i j i jm v v f r
Total Volume
if Includes bonded and non-bonded interactions
(foces due to stretching,bond angle, torsion effects)
BDT (Atomic) Stresses
Based on the assumption that the definition of bulk stress would be valid for a small volume around atom
1 1 1
2 2
N
ij i j j imv v r f
Atomic Volume
- Used for inhomogeneous systems
Lutsko Stress
1 1
1 1 1
2 2
N Nlutskoij i j j iLutsko
mv v r f
r
- fraction of the length of - bond lying inside the averaging volume
Averaging Volume
-Based on concept of local stress in statistical mechanics-used for inhomogeneous systems-Linear momentum conserved
l
Averaging volume for nanotubes
No restriction on shape of averaging volume (typically spherical for bulk materials)
Size should be more than two cutoff radii
Averaging volume taken as shown
Averaging Volumefor Lutsko stress
ZY
X
Strain calculation in nanotubes
Defect free nanotube mesh of hexagons
Each of these hexagons can be treated as containing four triangles
Strain calculated using displacements and derivatives shape functions in a local coordinate system formed by tangential (X) and radial (y) direction of centroid and tube axis
Area weighted averages of surrounding hexagons considered for strain at each atom
Similar procedure for pentagons and heptagons
G
Y’X’
Z’ Z
X
Y
i
j
l
Updated Lagrangian scheme is used in MD simulations
Conjugate stress and strain measures
Stresses described earlier Cauchy stress Strain measure enables calculation of and F, hence
finite deformation conjugate measures for stress and strain can be evaluated
Stress Cauchy
stress 1st P-K stress
2nd P-K stress
Strain Almansi strain Deformation
gradient Green-
Lagrange strain
Elastic modulus of defect free CNT
Strain
Stress
(GPa)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
10
20
30
40
50
60
Bulk Stress (E=1.002 TPa)
Lutsko Stress (E= 0.997 TPa)
BDT Stress (E= 1.002 TPa)
-All stress and strain measures yield a Young’s modulus value of 1.002TPa
-Defect free (9,0) nanotube with periodic boundary conditions
-Strains applied using conjugate gradients energy minimization
-Values in literature range from 0.5 to 5.5 Tpa. Mostly around 1Tpa
Strain in triangular facets
strain values in the triangles are not necessarily equal to applied strain values.
The magnitude of strain in adjacent triangles is different, but the weighted average of strain in any hexagon is equal to applied strain.
Every atom experiences same state of strain.
The variation of strain state within the hexagon (in different triangular facets) is a consequence of different orientations of interatomic bonds with respect to applied load axis.
e q=0 .0 6353 3 = 0 .077 71 3 = 0 .011 =-0 .0 0427
e q = 0 .0 6 0 9
3 3 = 0 .0 6 8 3
1 3 = -0 .0 2 11 1 = -0 .0 0 4 2 7
e q = 0 .0 6 2 4
3 3 = 0 .0 6 8 3
1 3 = - 0 .0 2 4 1
1 1 = -0 .0 0 4 2 73
1
e q = 0 .0 7 9 0 8
3 3 = 0 .0 9 6 7
1 3 = 0 .0
1 1 = -0 .0 0 4 2 7
7.5 % Applied Strain
CNT with 5-7-7-5 defect
Lutsko stress profile for (9,0) tube with type I defect shown below
Stress amplification observed in the defected region
This effect reduces with increasing applied strains
In (n,n) type of tubes there is a decrease in stress at the defect region
z - position
Str
ess
(Gp
a)
-20 -10 0 10 2010
20
30
40
50
60
3 % Applied Strain
0 % Applied Strain
1 % Applied Strain
5 % Applied Strain
7 % Applied Strain
8 % Applied Strain
Strain profile
Longitudinal Strain increase also observed at defected region
Shear strain is zero in CNT without defect but a small value observed in defected regions
Angular distortion due to formation of pentagons & heptagons causes this
z positions
Str
ain
-30 -20 -10 0 10 20 300
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09 8% strain
7% Strain
5% strain
3% strain
1% strain
Local elastic moduli of CNT with defects
Strain
Stress
(GPa)
0 0.025 0.05 0.075 0.10
10
20
30
40
50
60
(9,0) CNT no defect
Type I defect
Type II defect
(a)
(b)
(c)
-Reduction in stiffness in the presence of defect from 1 Tpa-Initial residual stress indicates additional forces at zero strain-Analogous to formation energy
-Type I defect E= 0.62 TPa
-Type II defect E=0.63 Tpa
Evolution of stress and strain
Strain and stress evolution at 1,3,5 and 7 % applied strainsStress based on BDT stress
Bond angle variation
Applied strain
Bondangle
0 2 4 6 8112
113
114
115
116
117
118
119
120
121
122
123
QPY, PQR, SRQ, QVW
UPQ, TSR, PQV
U
P
QR
Strains are accommodated by both bond stretching and bond angle change
Bond angles of the type PQR increase by an order of 2% for an applied strain of 8%
Bond angles of the type UPQ decrease by an order of 4% for an applied strain of 8%
Applied strain
Bond
angle
0 2 4 6 885
90
95
100
105
110
115
120
125
130
135
140
145
ABC
BCD
GABCDE BAJ
BHI ABH
Bond angle variation contdA
BC
D
E F
G
HI
J
For CNT with defect considerable bond angle change are observed
Some of the initial bond angles deviate considerably from perfect tube
Bond angles of the type BAJ and ABH increase by an order of 11% for an applied strain of 8%
Increased bond angle change induces higher longitudinal strains and significant lateral and shear strains.
Bond angle and bond length effects
Pentagons experiences maximum bond angle change inducing considerable longitudinal strains in facets ABH and AJI
Though considerable shear strains are observed in facets ABC and ABH, this is not reflected when strains are averaged for each of hexagons
Effect of Diameter
Strains
Str
ess
(GP
a)
0 0.01 0.02 0.03 0.04 0.05 0.060
10
20
30
40
50
(9,0) at defect
(10,0) at defect
(11,0) at defect
(13,0) at defect
(15,0) at defect
(9,0) no defect
(10,0) no defect
(11,0) no defect
(13,0) no defect
(15,0) no defect
stress strain curves for different (n,0)tubes with varying diameters.
stiffness values of defects for various tubes with different diameters do not change significantly
Stiffness in the range of 0.61TPa to 0.63TPa for different (n,0) tubes
Mechanical properties of defect not significantly affected by the curvature of nanotube
Effect of Chirality
Strain
Str
ess
(GP
a)
0 0.01 0.02 0.03 0.04 0.05
0
5
10
15
20
25
30
35
40
45
(5,5) no defect
(5,5) at defect
(6,4) no defect
(6,4) at defect
(7,3) no defect
(7,3) at defect
(9,0) no defect
(9,0) at defect
Chirality shows a pronounced effect
Functionalized nanotubes
Change in hybridization (SP2 to SP3) Nanotube composite interfaces may consist of bonding with
matrix (10,10) nanotube functionalized with 20 Vinyl and Butyl groups
at the center and subject to external displacement (T=77K)
Strain
Stress
(Gpa)
0 0.01 0.02 0.03 0.04
5
10
15
20
25
30
35
(10,10) CNT 0.84 T Pa
(10,10) CNT with vinyl 0.92 T Pa
(10,10) CNT with butyl 1.03 T Pa
Functionalized nanotubes contd
Increase in stiffness observed by functionalizing Stiffness increase more with butyl group than
vinyl group
Summary
Local kinetic and kinematic measures are evaluated for nanotubes at atomic scale
This enables examining mechanical behavior at defects such as 5-7-7-5 defect
There is a considerable decrease in stiffness at 5-7-7-5 defect location in different nanotubes
Changes in diameter does not affect the decrease in stiffness significantly
CNTs with different chirality have different effect on stiffness
Functionalization of nanotubes results in increase in stiffness
Volume considerations
Virial stress Total volume
BDT stress Atomic volume
Lutsko Stress Averaging
volume
Bond angle and bond length effects
Bond angle variation contd
Applied strain
Bondangle
0 2 4 6 8112
113
114
115
116
117
118
119
120
121
122
123
QPY, PQR, SRQ, QVW
UPQ, TSR, PQV
Applied strain
Bond
angle
0 2 4 6 885
90
95
100
105
110
115
120
125
130
135
140
145
ABC
BCD
GABCDE BAJ
BHI ABH
Some issues in elastic moduli computation
Energy based approach Assumes existence of W Validity of W based on potentials
questionable under conditions such as temperature, pressure
Value of E depends on selection of strain
Stress –strain approach Circumvents above problems Evaluation of local modulus for defect
regions possible
2
2
WE