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