Nano Mechanics and Materials:Theory, Multiscale Methods and Applications

byWing 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 …

n-2 n-1 n n+1 n+2 …

(n,m)

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

int

int1 1

2 2

1 1 2 2

( ) ( ) ( )

( )

...

2 2 ...

n n n

n n n n n

n n n n

n n n n n n

Mu t f t f t

f t k u u k u u

u u u u

k u u u u u u

int' ' 0 1 2

'

( ) ( ) ( ), 2( ), , , ...

n

n n n n nn n

f t K u t K u t K k K k K

( ) ( ) ( )n n nMu t K u t f t

… n-2 n-1 n n+1 n+2 …

Equation of motion is identical for all repetitive cells n

Introduce the stiffness operator K

int

int1 1

2 2

1 1 2 2

( ) ( ) ( )

( )

...

2 2 ...

n n n

n n n n n

n n n n

n n n n n n

Mu t f t f t

f t k u u k u u

u u u u

k u u u u u u

int' ' 0 1 2

'

( ) ( ) ( ), 2( ), , , ...

n

n n n n nn n

f t K u t K u t K k K k K

( ) ( ) ( )n n nMu t K u t f t

… n-2 n-1 n n+1 n+2 …

Periodic Lattice Structure: Equation of Motion

3.3 TransformsRecall 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 fromdomain Xf a unique function F from domain XF .Operator A establishes a transform between domains Xf and XF

X Yx y

y=f(x) Examples:y=xn

y=sin xy=B x

Xf XF

f FF=A{f}

Examples:

0

( ) ( ( ))

( ) ( )t

F x f f x

F t f d

Linear operators are of particular importance:

{ } { }{ } { } { }

A C f C A fA f g A f A g

Examples:

2 2

2ˆ ˆ, ( )

2

f ga f bg a bx x x

H E H V xm x

Inverse operator A-1 maps the transform domain XF back to the original domain Xf

Xf XF

f Ff=A-

1{F} 1{ { }}A A f f

Functional Operators (Transforms)

Linear convolution with a kernel function K(x):

( ) ( ') ( ') 'K f x K x x f x dx

Important properties

2

0

( ) ( ) ( )'( ) ( ) (0) ( )( ) ˆˆ''( ) ( ) (0) '(0) ( ) ( ) ( )

t K f t K s F sf t s F s f F sf df t s F s s f f s K f x K p f p

LL LL F

Laplace transform (real t, complex s)

1

0

1: ( ) ( ) : ( ) lim ( )2

a ibst st

ba ib

F s f t e dt f t F s e dsi

L L

Fourier transform (real x and p)

1 1ˆ ˆ: ( ) ( ) : ( ) ( )2

ixp ixpf p f x e dx f x f p e dp

F F

Integral Transforms

Laplace transform gives a powerful tool for solving ODE

Example: 2( ) ( ) 0(0) 1(0) 0 ( ) ?

y t y tyy y t

Solution: Apply Laplace transform to both sides of this equation, accounting for linearity of LT and using the property

2( ) ( ) (0) (0) :y t s Y s s y y L

2 2 2

2 2

1 12 2

( ) ( ) 0 ( ) ( ) 0

( )

( ) ( )

y t y t s Y s s Y ssY s

ssy t Y s

s

L

L L( ) cosy t t 0 2.5 5 7.5 10 12.5 15

-1

-0.5

0

0.5

1

t

y(t)

Laplace Transform: Illustration

Discrete convolution ' ''

ˆ ˆ( ) ( )n n n n nn

K u K u K u K p u p F

Discrete functional sequencesInfinite: ( / ), 0, 1, 2, ...Periodic: , is integer, 0, 1, 2, ...

n

n kN n

u f nx a nu u N k

1ˆ ˆ( ) ( )2

ipn ipnn n

n

u p u e u u p e dp

DFT of infinite sequences

p – wavenumber, a real value between –p and p

2 2/ 2 1 / 2 1

/ 2 / 2

1ˆ ˆ( ) ( )i pn i pnN N

N Nn n

n N p N

u p u e u u p eN

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

-20 -10 0 10 20

-1

-0.5

0

0.5

1

-20 -10 0 10 20

-1

-0.5

0

0.5

1

-20 -10 0 10 20

-1

-0.5

0

0.5

1

sin 2 , 4, 40n

nu p p NN

sin 2 , 11, 40n

nu p p NN

,4 , 4ˆ( ) p pu p

,11 , 11ˆ( ) p pu p

-20 -10 0 10 20

-0.4

-0.2

0

0.2

0.4

-20 -10 0 10 20

-0.4

-0.2

0

0.2

0.4

-20 -10 0 10 20

-0.4

-0.2

0

0.2

0.4

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:

0Here, /k m -1 -0.5 0.5 1

0.5

1

1.5

2

2.5

3

/p

0/ continuum

λ = 10d, p = π/5

λ = 4d, p = π/2

λ = 2d, p = π

λ = 10/11d, p = 11π/5 (NOT PHYSICAL)

2 /p d

p

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).

-1 -0.5 0.5 1

0.5

1

1.5

2

2.5

3

/p

0/ continuum

vp

-6 -4 -2 2 4 6

0.2

0.4

0.6

0.8

1

/p

continuum

0/v v

0 0

12sin sin2 2p v p

v p

3.5 Green’s Function Methods

Dynamic response function Gn(t) is a basic structural characteristic. G describes lattice motion due to an external, unit momentum, pulse: ,0

, '

( ) ( )

1, ' , 0( ) ( ) 10, ' 0, 0

n n

n n

f t t

n n tt t dtn n t

( ) ( )n n nMu t K u f t … n-2 n-1 n n+1 n+2 …

2,0 ,0

12 2,0

( ) ( ) ( ) ( ) ( )

ˆ( ) ( ) ( , ) ( )

n n n n n n

n n n

MG t K G t t s MG s K G s

s MG s K G s G s p s M K p

LF

11 1 2

'' 0

ˆ( ) ( )

ˆˆ ˆ( , ) ( , ) ( , ) ( ) ( ) ( )

n

text

n n n nn

G t s M K p

U s p G s p F s p u t G t f d

L F

Periodic Structure: Response (Green’s) Function

Assume first neighbor interaction only: 1

' ' ,0 0 1' 1

( ) ( ) ( ), 2 ,n

n n n n nn n

Mu t K u t t K k K k

… n-2 n-1 n n+1 n+2 …

211 1 2 1 2

2 2( 1)

1( ) ( 2 42 4

nip ip

n nM kG t s M k e e s s

s s

L F L

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

20

( ) (2 )t

n nG t J d 0 2 4 6 8 10 12 14

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

2( ) (2 )n nG t J t

Displacements Velocities

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.

1' 1 1 0 0

0

( ) ( ) ( ) , ( ) ( ) ( ), ( ) ( ) ( ) ( )t

n n n n nu t G t f d U s G s F s U s G s G s U s

… -2 -1 0 1 2 …

f(t) 1 0

,0

( ) ( ) , ?( ) ( )n n

u t A u t Af t f t

1 11 0 1 0

0

( ) ( ) ( ) , ( ) ( ) ( )t

u t t u d t G s G s L

21 2

21 2( ) 4 (2 )4

t s s J tt

L

0 2 4 6 8 10 12 14

-0.2

0

0.2

0.4

0.6

22( ) (2 )t J tt

Time History Kernel (THK)

Equations for atoms n1 are no longer required

1 2 1 0

0 1 0 1 1 2 1 0

1 0 1 20 1 0 0

0

...2 0 ...

2 0 2 02 0 2 ( ) ( ) 0

...

t

u u u uu u u u u u u u

u u u u u u u t u d

1 00

( ) ( ) ( )t

u t t u d

… -2 -1 0 1 2 …

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, ua, 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

1 01 1

aa

a a

u u u

f

a–1 a…210

MD domain Coarse scale

domain

……

f

10Multiscale 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 - AuL-J Potential

FCC

12 6

( ) 4U rr r

Au - AuMorse

Potential:2 ( ) ( )( ) ( 2 )e er r r r

eU r D e e

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,mx,n

Bravais lattice

Atomic Potential and FCC Kernel Matrices

2

', ', ', , ', ', '

1,1,0 1, 1,0 1,0,1

1,0, 1 0,1,1 0,1, 1

0( )

1 1 0 1 1 0 1 0 11 1 0 , 1 1 0 , 0 0 0 ,0 0 0 0 0 0 1 0 1

1 0 1 0 0 0 0 0 00 0 0 , 0 1 1 , 0 1 11 0 1 0 1 1 0 1

|n n m m l ln m l n m l

U

k k k

k k k

uuK

u u

K K K

K K K

20,0,0

,1

1 0 00 1 0 , 20 0 1

k k

K

2 ( ) ( )( ) ( 2 )e er r r reU r D e e

( ), ,

, ,

(cos cos )cos 2 sin sin sin sin( , , ) 4 sin sin (cos cos )cos 2 sin sin

sin sin sin sin (cos cos )cos 2

i pn qm pln m l

n m l

q r p p q p rp q r e k p q p r q q r

p r q r p q r

K K

1 1( , ) ( , , )l r lp q p q r G K

F

Morse potential

K-matrices

Fourier transform in space

Inverse Fourier transform for r (evaluated numerically for all p,q and l):

z,l

y,mx,n

Atomic Potential and FCC Kernel Matrices

-10 -8 -6 -4 -2 0 2 4 6 8 10

10-3

10-2

10-1

100

m=0 m=2 m=4 m=6 m=8

n,m(1

,1)

n

n,m , element (1,1)

1

0( )1 1

0

( , ) ( , )( , ) ( , ) ( , )

( , ) ( , )aa

aa

p q p qp q p q p q

p q p q

G GΦ G G

G G

( ) 1 1 ( ) ( ) ( ), , ,( , )a a a a

n m p n q m n m n mp q Φ Φ Θ Ξ

F F

redundant block, if , ,n m a u 0

( ) ( ), ,1 ', ' ', ',0 ', ' ', ',

', '

( ), ,1 ', ' ', ',0

', '

a an m n n m m n m n n m m n m a

n m

an m n n m m n m

n m

u Θ u Ξ u

u Θ u

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 domainis assembled from the parametric matrices G (a – coarse scale parameter):

Inverse Fourier transform for p and q

Final form of the boundary conditions

Method ValidationFCC gold

Karpov, Yu, et al., 2005.

( )1, , ', ' 0, ', '

', '

am n m m n n m n

m n u Θ u

0, ,m nu

, ,a m n u 0

0, ,m nu

1, ,m nu

1/4

a

Compound Interfaces

Fixed faces

Multiscale BC at five

faces

Edge assumption

(na ) (edge)u u

Problem description

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.

Fixed edge

Load

(13,0) zigzag

( ) ( )1, ' 0, ' ' , '

'

a am m m m m m a m

m u Θ u Ξ u

a = 20

l = 0l = a

Large deformation

MSBC

Qian, Karpov, et al., 2005

MSBC: Bending of Carbon NanotubesThe 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.

( ) ( )1, ' 0, ' ' , '

'

a am m m m m m a m

m u Θ u Ξ u

l = 0l = a Qian, Karpov, et al., 2005

Computational scheme

MSBC: Deformation of Graphene MonolayersThe 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 NanomembranesShown 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 MSBCWe 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