1. Mechanics of defects in Carbon nanotubes S Namilae, C Shet and N Chandra

2. 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

3. 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 Ref: D Srivastava et. al. (2001) 1

4. Mechanics at atomic scale Molecular Dynamics -Fundamental quantities (F,u,v) Compute Continuum quantities -Kinetics ( ,P,P ’ ) -Kinematics ( ,F) -Energetics Use Continuum Knowledge - Failure criterion, damage etc

5. 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

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

7. 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

8. 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

9. 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

10. 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

11. 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 1 st P-K stress 2 nd P-K stress Strain Almansi strain Deformation gradient Green-Lagrange strain

12. Elastic modulus of defect free CNT -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

13. 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.

14. 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

15. 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

16. Local elastic moduli of CNT with defects -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

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

18. 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%

19. 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.

20. 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

21. Effect of Diameter 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

22. Effect of Chirality Chirality shows a pronounced effect

23. 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)

24. Functionalized nanotubes contd Increase in stiffness observed by functionalizing Stiffness increase more with butyl group than vinyl group

25. 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

26. Volume considerations Virial stress Total volume BDT stress Atomic volume Lutsko Stress Averaging volume

27. Bond angle and bond length effects

28. Bond angle variation contd

29. 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