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

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

1

1Ref D Srivastava et. al. (2001)

Mechanics at atomic scale

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

Stress defined for whole system

For Brenner potential

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 ?

- Used for inhomogeneous systems

Lutsko Stress

- fraction of the length of ?-? bond lying inside

the averaging volume

- Based on concept of local stress in
- statistical mechanics
- used for inhomogeneous systems
- Linear momentum conserved

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

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

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

-Defect free (9,0) nanotube with periodic

boundary conditions

-Strains applied using conjugate gradients

energy minimization

- All stress and strain
- measures yield a Youngs
- modulus value of 1.002TPa

- 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.

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

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

Local elastic moduli of CNT with defects

- Type I defect ? E 0.62 TPa

- Type II defect ? E0.63 Tpa

- Reduction in stiffness in the presence of defect

from 1 Tpa - -Initial residual stress indicates additional

forces at zero strain - -Analogous to formation energy

Evolution of stress and strain

Strain and stress evolution at 1,3,5 and 7

applied strains Stress based on BDT stress

Bond angle variation

- 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

Bond angle variation contd

- 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

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

stress strain curves for different (n,0) tubes

with varying diameters.

Effect of Chirality

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 (T77K)

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

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