Mechanics of defects in Carbon nanotubes

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Mechanics of defects in Carbon nanotubes

• S Namilae, C Shet and N Chandra

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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

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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

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1Ref D Srivastava et. al. (2001)
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Mechanics at atomic scale
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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

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Virial Stress
Stress defined for whole system
For Brenner potential
Includes bonded and non-bonded interactions
(foces due to stretching,bond angle, torsion
effects)
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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
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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

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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

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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
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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

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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

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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.

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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

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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

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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

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Evolution of stress and strain
Strain and stress evolution at 1,3,5 and 7
applied strains Stress based on BDT stress
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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

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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.

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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

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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.
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Effect of Chirality
Chirality shows a pronounced effect
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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)

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Functionalized nanotubes contd

• Increase in stiffness observed by functionalizing
• Stiffness increase more with butyl group than
  vinyl group

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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

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Volume considerations

• Virial stress
• Total volume
• BDT stress
• Atomic volume
• Lutsko Stress
• Averaging volume

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Bond angle and bond length effects
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Bond angle variation contd
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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