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

3. Lattice Mechanics

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 … 3.1 Elements of Lattice Symmetries

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. The 14 Bravais lattices: Regular Lattice Structures

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

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

3.3 Transforms Recall first: A function f assigns to every element x (a number or a vector) from set X a unique element y from set Y. Function f establishes a rule to map set X to Y A functional operator A assigns to every function f from domain X f a unique function F from domain X F. Operator A establishes a transform between domains X f and X F XY x y y=f(x) Examples: y=x n y=sin x y=B x XfXf XFXF f F F=A{f} Examples:

Linear operators are of particular importance: Examples : Inverse operator A -1 maps the transform domain X F back to the original domain X f XfXf XFXF f F f=A -1 {F} Functional Operators (Transforms)

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

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 t y(t)y(t) Laplace Transform: Illustration

Discrete convolution 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 Motivation: discrete Fourier transform (DFT) reduce solution of a large repetitive structure to the analysis of one representative cell only. Discrete Fourier Transform (DFT)

Original n-sequence Transform p-sequence DFT: Illustration

3.4 Standing Waves in Lattices

Wave Number Space and Dispersion Law Wave number p 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: continuum λ = 10d, p =  π/5 λ = 4d, p =  π/2 λ = 2d, p =  π λ = 10/11d, p =  11π/5 (NOT PHYSICAL)

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

3.5 Green’s Function Methods

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

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

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) Time History Kernel (THK)

Equations for atoms nr1 are no longer required … … Domain of interest Elimination of Degrees of Freedom

3.6 Quasistatic Approximation Miultiscale boundary conditions Applications Conclusions

All excitations propagate with “infinite” velocities in the quasistatic case. Provided that effect of peripheral boundary conditions, u a, is taken into account by lattice methods, the continuum model can be omitted Quasistatic MSBC 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. Standard hybrid method

1D Illustration f a–1 a … 210 MD domain Coarse scale domain … … f 10 Multiscale BC 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

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

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

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): z,l y,m x,n

Atomic Potential and FCC Kernel Matrices Q n,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). 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

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

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

Performance of Multiscale Boundary Conditions 1/18 of the original volume Back half-domains are shown Radius of Diamond Tip: 1 nm Full MD domain size: atoms Reduced domain size: 3600 atoms For 1/18 of the original volume: Computation time has been reduced from 73 hours to 1 hour Lattice deformation pattern is similar for the benchmark and the multiscale simulations:

MSBC: Twisting of Carbon Nanotubes The study of twisting performance of carbon nanotubes is important for nanodevices. The MSBC treatment predicts u 1 well at moderate deformation range. Efforts on computation for all DOFs in the range between l = 0 and a are saved. Fixed edge Load (13,0) zigzag a = 20 l = 0 l = a Large deformation MSBC Qian, Karpov, et al., 2005

MSBC: Bending of Carbon Nanotubes The study of bending performance of carbon nanotubes is important for nanodevices. The MSBC treatment predicts u 1 well at moderate deformation range. Efforts on computation for all DOFs in the range between l = 0 and a are saved. l = 0 l = a Qian, Karpov, et al., 2005 Computational scheme

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 Tersoff-Brenner potential Indenting load Medyanik, Karpov, et al., 2005

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

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