Atomic Simulation of deformation of Carbon nanotubes with defects

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Atomic Simulation of deformation of Carbon
nanotubes with defects

• S Namilae, C Shet and N Chandra

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Carbon Nanotubes

• Carbon Nanotubes ? Graphite sheet rolled into a
  tube
• Length 100 nm to few ?m Diameter 1 nm
• Chirality based on rolling direction
• Found as single wall and Multi wall tubes

• E 1 TPa Strength 150 GPa
• Conductivity depends on chirality

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Outline

• Deformation of carbon nanotubes (CNT) using
  molecular dynamics/statics
• Stress and strain measures
• Local elastic moduli of CNT containing
  topological defects
• Interaction between topological defects
• Functionalized nanotubes
• Inelastic deformation
• MD-FEM study of Interfaces in nanotube composites

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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
Stress-strain required for local behavior
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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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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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Stiffness reduction

• Decrease in elastic properties applies to other
  topological defects

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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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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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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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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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Residual stress at zero strain

• Stress is present at zero strain values.
• This corresponds to stress due to curvature
• It is found to decrease with increasing diameter
• Basis for stress calculation ? graphene sheet
• Brenner et. al.1 observed similar variation in
  energy at zero strain

Robertson DH, Brenner DW and Mintmire 1992
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Effect of Chirality
Chirality shows a pronounced effect
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Interaction of Topological Defects
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Interaction between defects

• Stiffness reduction in defects placed at
  different distances along length
• Defects placed close to each other experience
  more loss of stiffness
• If distance between the defects is greater than
  42 A i.e. 30l , it does not effect stiffness

Variation of stiffness loss with spacing of two
interacting defects, placed along the length of
the tube.
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Stress and strain profiles

• Defects placed at distance show two independent
  peaks
• These peaks coalesce as distance between them
  decreases

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Non-interacting Defects

• Reduction in stiffness increases linearly with
  number of defects when distance between them gt
  30l
• Linear damage model predicts this

Variation in stiffness loss for with the number
of non-interacting defects placed along the
length of the carbon nanotube.
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Interacting defects

• When distance between defects lt 30 l
• Reduction in stiffness linear non linear
• Change from non linear to linear occurs if of
  defects such that distance between 1st and nth gt
  30 l

Variation in stiffness loss for with the number
of interacting defects placed along the length of
the carbon nanotube.
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Prediction of Reduction in stiffness
A simple damage like model predicts reduction in
stiffness when number of defects are present
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Functionalized Nanotubes and Nanotube composite
Interfaces
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Role of Interfaces in Composites

• Critical issues in nanotube composites
• Alignment
• Dispersion
• Load Transfer
• Load transfer effected by interface
  characteristics
• Interfaces further effect strength, fracture and
  fatigue properties
• Interface ? Bounding surface with physical /
  chemical / mechanical discontinuity
• Strengthening of interfaces in conventional
  composites
• Chemical reaction e.g SiC in Ti
• Surface asperities on fiber
• In CNT composites ? chemical attachments

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Chemical bonding at CNT- matrix interfaces

• Wagner et.al. (1998) indirectly measured
  interface strength as high as 500MPa, suggest
  chemical bonding at interfaces
• Gong and coworkers (2000) improved composite
  properties by coating nanotubes with surfactants
• Qian et.al. report significant load transfer
  across interface in CNT-polystyrene composites
• Eitan et. al (2003) report carboxylization of CNT
  and predict covalent bonding with epoxy matrix
• Frankland et. al.(2002) based on MD simulation
  suggest increased interface strength with
  chemical bonding

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Functionalized nanotubes

• Objective
• Study Deformation of functionalized nanotubes
• Study deformation on nanotubes in composites

• Change in hybridization (SP2 to SP3)
• Experimental reports of different chemical
  attachments
• Application in composites, medicine, sensors

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Simulation Parameters

• Various nanotubes with hydrocarbon attachments,
  vinyl and butyl in central portion
• Temperature 77K and 3000K
• Applied displacement at both ends at 0.05A/1000
  steps
• Lutsko stress computed on volume shown

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Functionalized nanotubes contd

• Increase in stiffness observed by functionalizing
• 20 attachments show about 10 increase in local
  elastic modulus

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Local Stiffness of functionalized CNTs
 

• Stiffness increase is more for higher number of
  chemical attachments
• Stiffness increase higher for longer chemical
  attachments

 
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Contour plots
Stress contours with one chemical attachment.
Stress fluctuations are present
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Radius variation

• Increased radius of curvature at the attachment
  because of change in hybridization
• Radius of curvature lowered in adjoining area

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Defect evolution and onset of plastic deformation

• Yakobson et al show that onset of yielding is
  accompanied by formation of topological defects

• Defect evolution starts at about 7 strain
• (10,10) CNT T3000K

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Evolution of defects in functionalized CNT

• Defects Evolve at much lower strain of 4.5 in
  CNT with chemical attachments
• Onset of plastic deformation at lower strain.
  Reduced fracture strain

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Nanotube Composite Interfaces
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Multi scale model (I Molecular dynamics)
(10,10) CNT with varying no of hydrocarbon
attachments Applied displacements at 300K
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Interfacial shear
Interfacial shear measured as reaction force of
fixed atoms
Max load
Typical interface shear force pattern. Note zero
force after Failure (separation of chemical
attachment)
After Failure
250,000 steps
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Debonding and Rebonding of Interfaces
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Variation in interface behavior
Homogenization ?
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Force distribution along the interface
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Multi scale approach (II FEM/cohesive zone
model)

• Assumptions
• Nanotubes deform in linear elastic manner
• Interface character completely determined by
  traction-displacement plot

CZM enables modeling of surfaces before and after
fracture
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Preliminary result Numerical pull-out test

• Pull out of nanotube from polymer matrix modeled
  with CZM/ atomically informed interface
  characteristics

Scaled up version of the previous test nanotube
length 2 microns
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Applications of the multiscale model
Composite effective properties
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Summary

• Local kinetic and kinematic measures are
  evaluated for nanotubes at atomic scale
• There is a considerable decrease in stiffness at
  topological defect location in different
  nanotubes
• Reduction in stiffness when more defects are
  present can be predicted by simple models
• Functionalization of nanotubes results in
  increase in stiffness
• Onset of inelastic deformation characterized by
  evolution of topological defects occurs at lower
  strains in functionalized tubes
• Interface constitutive behavior has been modeled
  using MD
• The chemical attachments exhibit bonding
  rebonding
• Interface behavior and elastic moduli computed
  using MD are passed to continuum model using CZM.
  This can be used to solve larger problems

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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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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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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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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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Defects placed along diameter

• No loss in stiffness when 2 defects placed at
  different distances along diameter
• Orientation of loading and defects is an
  important criterion

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