nano mechanics and materials: theory, multiscale methods and applications
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Nano Mechanics and Materials: Theory, Multiscale Methods and Applications. by Wing Kam Liu, Eduard G. Karpov, Harold S. Park. 3. Lattice Mechanics. ( n,m ). 3.1 Elements of Lattice Symmetries. - PowerPoint PPT PresentationTRANSCRIPT
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