Heat conduction in quasi-1D systems - NUS: Faculty of Science

frequentverseΠολεοδομικά Έργα

16 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

59 εμφανίσεις

1

Heat Conduction in One
-
Dimensional Systems
:
molecular dynamics and mode
-
coupling theory

Jian
-
Sheng Wang

National University of Singapore

2

Outline


Brief review of 1D heat conduction


Introducing a chain model


Nonequilibrium molecular dynamics results


Projection formulism and mode
-
coupling
theory


Conclusion

3

Fourier Law of Heat
Conduction

Fourier, Jean Baptiste Joseph,
Baron (1768


1830)

Fourier proposed the law
of heat conduction in
materials as

J

=


κ


T

where
J

is heat current
density,
κ

is thermal
conductivity, and
T

is
temperature.

4

Normal & Anomalous Heat
Transport

T
L

T
H

J

3D bulk systems obey Fourier law
(insulating crystal: Peierls’ theory of
Umklapp scattering process of
phonons; gas: kinetic theory,
κ

=

cv
l
)


In 1D systems, variety of results are
obtained and still controversial. See
S Lepri et al, Phys Rep 377 (2003) 1,
for a review.

5

Heat Conduction in

One
-
Dimensional Systems


1D harmonic chain,
k


N

(Rieder, Lebowitz &
Lieb, 1967)



k

diverges if momentum is conserved (Prosen &
Campbell, 2000)


Fermi
-
Pasta
-
Ulam model,
k


N
2/5

(Lepri et al,
1998)


Fluctuating hydrodynamics + Renormalization
group,
k


N
1/3

(Narayan & Ramaswamy 2002)

6

Approaches to Heat
Transport


Equilibrium molecular dynamics using
linear response theory (Green
-
Kubo
formula)


Nonequilibrium steady state (computer)
experiment


Laudauer formula in quantum regime

7

Ballistic Heat Transport at
Low Temperature


Laudauer formula for heat current


2
1
( ) | ( ) |
2
I n t d
   



ikx ikx
e re


'
ik x
te
scatter

8

Carbon Nanotube

Heat conductivity of
Carbon nanotubes at
T
=
300K by nonequilibrium
molecular dynamics.

From S Maruyama,
“Microscale Thermophysics
Engineering”, 7 (2003) 41.
See also G Zhang and B Li,
cond
-
mat/0403393.

9

Carbon Nanotubes

Thermal conductance
κ
A

of carbon nanotube
of length
L
,
determined from
equilibrium molecular
dynamics with Green
-
Kubo formula,
periodic boundary
conditions, Tersoff
potential. Z Yao, J
-
S
Wang, B Li, and G
-
R
Liu, cond
-
mat/0402616.

10

Fermi
-
Pasta
-
Ulam model


A Hamiltonian system with


2
1
1
2 2 3 4
(,) ( )
2
1
( ) ( ) ( ) ( )
2 3 4
N
i
i i
i
p
H p x V x x
m
V z m z a z a z a
 



 
  
 
 
     

A strictly one
-
dimensional model.

11

A Chain Model for Heat
Conduction

m

r
i

= (
x
i
,
y
i
)

Φ
i



2
2
1
1
(,)
2 2
cos( )
i
r i i
i
i
i
H K a
m
K


 
   
 
 
 


p
p r r r
T
L

T
H

Transverse degrees of freedom introduced

12

Nonequilibrium Molecular
Dynamics


Nosé
-
Hoover thermostats at the ends at
temperature
T
L

and
T
H


Compute steady
-
state heat current:
j

=(1/
N
)
S
i

d (
e
i

r
i
)/d
t
, where
e
i

is local energy
associated with particle
i


Define thermal conductance
k

by <j> =
k

(
T
H
-
T
L
)/(
Na
)

N

is number of particles,
a

is lattice spacing.

13

Nos
é
-
Hoover Dynamics

,
,
,
2
,
2
,
if
if
if
1 1
1
i L i w
i
i w w
i H i w
L H
i
i w
B L H w
i N
d
N N i N
dt
i N N
d
dt k T N m





 

   


  

 
  
 
 

 

f p
p
f
f p
p
14

Defining Microscopic Heat
Current


Let the energy density be



then
J

satisfies



A possible choice for total current is




(,) ( )
i i
i
t
 e
 

r r r
t


 

J 0
1
( )
(,)
N
i i
i
V
d
N r t dV
dt
e

 


r
j J
15

Expression of j for the
chain model









1 1 1
1
2
2 2
1
1
( ) ( ) ( ) ( 1)
( 2,1,1) ( 1,1,)
1
( ) (| | )
4
(,,) ( cos )/| |
1
(| | ) (| | ) cos( )
4 2
i i i i i i i
i i i i i i
r i i
i k j k
i
i r i i i
i i
m i i
i i i i i i
i K a
i j k K
K a a K
m


e

e 
  



       
         
  
  
 
       
 
  
j r p p G r p p G
r p H r p H p
G r n
H n n r
p
r r
r r r
,/| |
i i i i
  
n r r
16

Temperature Profile

Temperature of
i
-
th particle
computed from
k
B
T
i
=<½
m
v
i
2

>
for parameter
set E with
N
=64 (plus), 256
(dash), 1024
(line).

17

Conductance vs Size
N

Model parameters
(
K
Φ
,
T
L
,
T
H
):

Set F (1, 5, 7),
B (1, 0.2, 0.4),
E (0.3, 0.3, 0.5),
H (0, 0.3, 0.5),
J (0.05, 0.1, 0.2) ,

m
=1,
a
=2,
K
r
=1.

From J
-
S Wang &
B Li, Phys Rev
Lett
92

(2004)
074302.


ln N

slope=1/3

slope=2/5

18

Additional MD data

Parameters (
K
Φ
,
T
L
,
T
H
,
ε
),
set
L(25,1,1.5,0.2)
G(10,0.2,0.4,0)
K(0.5,1.2,2,0.4)
I(0.1,0.3,0.5,0.2)
C(0.1,0.2,0.4,0)

From J
-
S Wang
and B Li, PRE,
70
, 021204
(2004).

19

Mode
-
Coupling Theory for
Heat Conduction


Use Fourier components as basic variables


Derive equations relating the correlation functions
of the variables with the damping of the modes,
and the damping of the modes to the square of the
correlation functions


Evoke Green
-
Kubo formula to relate correlation
function with thermal conductivity

20

Basic Variables (work in
Fourier space)

2/
2/
( ),
,
, (,,,)
1,2,,
i kj N
k j
j
i kj N
k j
j
T
k k k k k k
m
Q x ja e
N
m
Q y e
N
P Q A P P Q Q
k N


 

 
 

 



21

Equation of Motion for
A



Formal solution:


,
A H H
LA L
t q p p q
    
   
    
 
(,,) ( ( ),( )) (0,,)
tL
A t p q A p t q t e A p q
 
22

Projection Operator &
Equation


Define



We have


Apply
P

and 1

P

to the equation of motion,
we get two coupled equations. Solving
them, we get



1
† †
,,
PX X A A A A


2
P P

0
( )
( ) ( ) ( )
t
t
dA t
i A t t s A s ds R
dt
     

23

Projection Method
(Zwanzig
and Mori)


Equation for dynamical correlation
function:




where
G
(
t
) is correlation matrix of normal
-
mode Canonical coordinates (
P
k
,
Q
k
).


is
related to the correlation of “random” force.


0
( ) ( ) ( ) ( )
t
G t t G d i G t
  
    

24

Definitions

1
† †
1
† †
0
1
† †
1
† † 2
(1 )
( ) ( ),(0),
( ),,
,,
,,,
(1 )
( )
( )
t
t
t
G t A t A A A
t R R A A
i A A A A
X X A A A A
R e A
dA t
A t
dt






 

 
 

P L
P P P
P L
L
L

is Liouville operator

25

Correlation function equation
and its solution (in Fourier
-
Laplace space)




Define


the equation can be solved as




in particular


0
( ) ( ) ( ) ( )
t
G t t G d i G t
  
    

0
[ ] ( )
izt
G z e G t dt




1
[ ]
( ) [ ]
G z
i z z

 
*
2 2
2
0
( ) (0)
[ ]
[ ],,
( ) [ ]
| |
k k
izt
k
k
k k
k
Q t Q
iz z
g z e dt
z iz z
Q
 


 






   
  

26

Small Oscillation Effective
Hamiltonian









2 2
2
,
(3)
,,,,
0
1
(,) ( )
2
eff k k k
k
k p q k p q k p q k p q
k p q
H P Q P Q
v Q Q Q v Q Q Q
  


 
  
 
 


,
k k
k k
H H
P Q
Q P
 
 
 
 
  
 
Equations of
motion

27

Equation of Motion of
Modes

2
'''''
'''
2
'''
'''
( )....
( )..
k k k k k k k
k k k
k k k k k
k k k
Q Q Q Q Q Q
Q Q Q Q


 
 
   
 
   
  


28

Determine Effective
Hamiltonian Model
Parameters from MD

2 2
2
,,
2 2 2
(3)
,,
2 2 2
1 1
| | ( ),
( )
2 | | | | | |
6 | | | | | |
k
k B
k p q
k p q
k p q
k p q
k p q
k p q
Q O v
k T
Q Q Q
v
Q Q Q
Q Q Q
v
Q Q Q






 
 
  


29

Mode
-
Coupling
Approximation




(
t
)

 

<
R
(t)
R
(0)>



R



Q Q




(
t
)


<
Q
(
t
)
Q
(
t
)
Q
(0)
Q
(0)>




<
Q
(
t
)
Q
(0)><
Q
(
t
)
Q
(0)>




g
(
t
)
g
(
t
)
[mean
-
field type]


30

Full Mode
-
Coupling
Equations

(3)
,,
,
2 2
( ) ( ) ( ) ( ) ( ),
( ) ( ) ( ),
[ ]
[ ],,
( ) [ ]
k p q p q p q p q
p q k p q k
k p q p q
p q k
k
k
k k
t K g t g t K g t g t
t K g t g t
iz z
g z
z iz z


 


 
   
  
 
  
 
 
 
  
 

is Fourier
-
Laplace transform of

[ ]
k
g z

( )
k
g t

31

Damping Function

1
孺

Molecular Dynamics

Mode
-
Coupling Theory

From J
-
S Wang & B Li, PRE 70, 021204 (2004).

32

Correlation Functions

Correlation
function
g
(
t
) for
the slowest
longitudinal and
transverse modes.
Black

line: mode
-
coupling,
red
dash: MD.
N
=
256.

g
(
t
)


e
-

t
cos(
ω
t
)

33

Decay or Damping Rate

Decay rate of the
mode vs mode
index
k
.
p
=
2
π
k
/(
Na
) is lattice
momentum.
N
=
1024.

Symbols are from
MD, lines from
mode
-
coupling
theory. Straight
lines have slopes
3/2 and 2,
respectively.


longitudinal

transverse

slope=2

slope=3/2

34

Mode
-
Coupling Theory in
the Continuum Limit

/
2 (3) 2
/
/
/
,2,
1
( ) ( ) ( )
2
1
( ) ( ) ( )
2
( ) ( )
a
q q
a
a
q q
a
p
t dq K g t K g t
t dq K g t g t
t p v t












  

 
 
 
 
 

 
 


35

Asymptotic Solution


The mode
-
coupling equations predict, for
large system size
N
, and small
z
:

2
2
2
[ ],
1
[ ], =
2
p
p
k
z c p p
Na
z bz p





 

If there is no transverse coupling,
Γ

=
z
(
-
1/3)
p
2

(Result of Lepri).

36

Mode
-
Coupling


z
]/
p
2

At parameter set
B.
Blue dash

:
asymptotic
analytical result,
red line

: Full
theory on
N
=1024, solid line
:
N





limit
theory



slope = 0

|| slope =

1/2

37

Green
-
Kubo Formula



2
0
*
,
2 2
,,,
,
2
2 1/(2 )
,
1
( ) (0),
,
( ) (0) | | ( ) ( ) ( )
( )
B
k
k k k k k
k
k QQ k PP k QP k
k
k
k
k
k
J t J dt
k T aN
J b Q P b i
p
J t J b g t g t g t
b
g t t

    

   


 


k




 




 
 




38

Green
-
Kubo Integrand

Parameter set B.
Red circle
:
molecular
dynamics, solid
line: mode
-
coupling theory
(
N
= 1024),
blue
line
: asymptotic
slope of 2/3.

39

k
N

with Periodic Boundary
Condition

κ

from Green
-
Kubo formula on
finite systems
with periodic
boundary
conditions, for
parameter set B
(
K
r
=1,
K
Φ
=1,
T
=0.3)

Mode
-
coupling

Molecular
dynamics

slope=1/2

40

Relation between Exponent
in
Γ

and
κ


If mode decay with
Γ≈
z
-
δ
p
2
, then


With periodic B.C. thermal conductance
κ


N
1
-
δ


With open B.C.
κ


N
1
-
1/(2
-
δ
)


Mode coupling theory gives
δ
=1/2 with
transverse motion, and
δ
=1/3 for strictly 1D
system.

41

Conclusion


Quantitative agreement between mode
-
coupling theory and molecular dynamics is
achieved


Molecular dynamics and mode
-
coupling
theory support 1/3 power
-
law divergence for
thermal conduction in 1D models with
transverse motion, 2/5 law if there are no
transverse degrees of freedom.