Nano Mechanics and Materials: Theory, Multiscale Methods and Applications

1. Nano Mechanics and Materials:Theory, Multiscale Methods and Applications by Wing Kam Liu, Eduard G. Karpov, Harold S. Park

2. 3. Lattice Mechanics

3. (n,m) 3.1 Elements of Lattice Symmetries The term regular lattice structure refers to any translation symmetric polymer or crystalline lattice 1D lattices (one or several degrees of freedom per lattice site): 2D lattices: …n-2 n-1 n n+1 n+2 … …n-2 n-1 n n+1 n+2 … n-2 n-1 nn+1 n+2 …

4. Regular Lattice Structures The 14 Bravais lattices: 3D lattices (Bravais crystal lattices) Bravais lattices represent the existing basic symmetries for one repetitive cell in regular crystalline structures. The lattice symmetry implies existence of resonant lattice vibration modes. These vibrations transport energy and are important in the thermal conductivity of non-metals, and in the heat capacity of solids.

5. 3.2 Equation of Motion of a Regular Lattice …n-2 n-1 n n+1 n+2 … Equation of motion is identical for all repetitive cells n Introduce the stiffness operator K

6. Periodic Lattice Structure: Equation of Motion …n-2 n-1 n n+1 n+2 … Equation of motion is identical for all repetitive cells n Introduce the stiffness operator K

7. y=f(x) y x X Y F=A{f} f F Xf XF 3.3 Transforms Recall first: A functionf assigns to every element x (a number or a vector) from set X a unique elementy from set Y. Function f establishes a rule to map set X to Y Examples: y=xn y=sin x y=Bx A functional operator A assigns to every function f from domain Xf a unique functionF from domain XF . Operator A establishes a transform between domains Xf and XF Examples:

8. Functional Operators (Transforms) Inverse operatorA-1 maps the transform domain XF back to the original domain Xf f=A-1{F} f F Xf XF Linear operators are of particular importance: Examples:

9. Integral Transforms Laplace transform (real t, complex s) Fourier transform(real x and p) Linear convolution with a kernel function K(x): Important properties

10. Laplace Transform: Illustration Laplace transform gives a powerful tool for solving ODE Example: Solution: Apply Laplace transform to both sides of this equation, accounting for linearity of LT and using the property y(t) t

11. Discrete Fourier Transform (DFT) Motivation: discrete Fourier transform (DFT) reduce solution of a large repetitive structure to the analysis of one representative cell only. Discrete functional sequences DFT of infinite sequences p–wavenumber, a real value between –p and p DFT of periodic sequences Here, p– integer value between –N/2 and N/2 Discrete convolution

12. DFT: Illustration Transform p-sequence Original n-sequence

13. 3.4 Standing Waves in Lattices

14. continuum Wave Number Space and Dispersion Law λ = 10d, p = π/5 Wave numberp is defined through the inverse wave length λ (d – interatomic distance): The waves are physical only in the Brillouin zone (range), The dispersion law shows dependence of frequency on the wave number: λ = 4d, p = π/2 λ = 2d, p = π λ = 10/11d, p = 11π/5 (NOT PHYSICAL)

15. continuum continuum Phase Velocity of Waves The phase velocity, with which the waves propagate, is given by Dependence on the wave number: Value v0 is the phase velocity of the longest waves (at p 0).

16. 3.5 Green’s Function Methods

17. Periodic Structure: Response (Green’s) Function …n-2 n-1 n n+1 n+2 … Dynamic response functionGn(t) is a basic structural characteristic. G describes lattice motion due to an external, unit momentum, pulse:

18. Lattice Dynamics Green’s Function: Example Assume first neighbor interaction only: …n-2 n-1 n n+1 n+2 … Displacements Velocities Illustration (transfer of a unit pulse due to collision):

19. Time History Kernel (THK) The time history kernel shows the dependence of dynamics in two distinct cells. Any time history kernel is related to the response function. f(t) …-2 -1 01 2 …

20. Elimination of Degrees of Freedom Domain of interest …-2 -1 01 2 … Equations for atoms nr1 are no longer required

21. 3.6 Quasistatic Approximation • Miultiscale boundary conditions • Applications • Conclusions

22. Quasistatic MSBC All excitations propagate with “infinite” velocities in the quasistatic case. Provided that effect of peripheral boundary conditions, ua, is taken into account by lattice methods, the continuum model can be omitted Standard hybrid method Multiscale boundary conditions The MSBC involve no handshake domain with “ghost” atoms. Positions of the interface atoms are computed based on the boundary condition operators Θ and Ξ. The issue of double counting of the potential energy within the handshake domain does not arise.

23. Multiscale BC f f … … a 0 1 2 a–1 0 1 … MD domain Coarse scale domain 1D Illustration 1D Periodic lattice: Solution for atom 0 can be found without solving the entire domain, by using the dependence This the 1D multiscale boundary condition

24. Application: Nanoindentation: Problem description: R C - Au L-J Potential Diamond Tip Au FCC Au - Au Morse Potential:

25. x,n y,m z,l (2,1,1) (1,0,1) (1,2,1) (0,1,1) (2,0,0) (2,2,0) (1,1,0) (0,0,0) (0,2,0) Face centered cubic crystal Bravais lattice Numbering of equilibrium atomic positions (n,m,l) in two adjacent planes with l=0 and l=1. (Interplanar distance is exaggerated).

26. x,n y,m z,l Atomic Potential and FCC Kernel Matrices Morse potential K-matrices Fourier transform in space Inverse Fourier transform for r(evaluated numerically for all p,q and l):

27. Atomic Potential and FCC Kernel Matrices Boundary condition operator in the transform domain is assembled from the parametric matrices G (a – coarse scale parameter): Inverse Fourier transform for p and q Final form of the boundary conditions Qn,m , element (1,1) redundant block, if This sum can be truncated, because Θ decays quickly with the growth of n and m (see the plot).

28. FCC gold Method Validation a 1/4 Karpov, Yu, et al., 2005.

29. Fixed faces Multiscale BC at five faces Compound Interfaces Problem description Edge assumption

30. MSBC: Twisting of Carbon Nanotubes The study of twisting performance of carbon nanotubes is important for nanodevices. The MSBC treatment predicts u1 well at moderate deformation range. Efforts on computation for all DOFs in the range between l = 0 and a are saved. Load a = 20 (13,0) zigzag Fixed edge Large deformation MSBC l = a l = 0 Qian, Karpov, et al., 2005

31. MSBC: Bending of Carbon Nanotubes The study of bending performance of carbon nanotubes is important for nanodevices. The MSBC treatment predicts u1 well at moderate deformation range. Efforts on computation for all DOFs in the range between l = 0 and a are saved. Computational scheme l = a Qian, Karpov, et al., 2005 l = 0

32. MSBC: Deformation of Graphene Monolayers The MSBC perform well for the reduced domain MD simulations of graphene monolayers Problem description: red – fine grain, blue – coarse grain. Coarse grain DoF are eliminated by applying the MSBC along the hexagonal interface Indenting load Tersoff-Brenner potential Medyanik, Karpov, et al., 2005

33. MSBC: Deformation of Graphene Nanomembranes Shown is the reduced domain simulations with MSBC parameter a=10; the true aspect ration image (non-exaggerated). Error is still less than 3%. Deformation Comparison (red – MSBC, blue – benchmark) Shown: vertical displacements of the atoms

34. Conclusions on the MSBC • We have discussed: • MSBC – a simple alternative to hybrid methods for quasistatic problems • Applications to nanoindentation, CNTs, and graphene monolayers • Attractive features of the MSBC: • – SIMPLICITY • – no handshake issues (strain energy, interfacial mesh) • – in many applications, continuum model is not required • – performance does not depend on the size of coarse scale domain • – implementation for an available MD code is easy • Future directions: • Dynamic extension • Passage of dislocations through the interface • Finite temperatures