2371-13

Advanced Workshop on Energy Transport in Low-Dimensional Systems: Achievements and Mysteries

Bambi HU

15 - 24 October 2012

University of Houston, Department of Physics Houston, TX

U.S.A.

Asymmetric Heat Conduction and Negative Differential Thermal Resistance in Nonlinear Systems

Asymmetric Heat Conduction andNegative Differential Thermal Resistance

in Nonlinear Systems

1

�“It seems there is no problem in modern physics for which there are on record as many false starts, and as many theories which overlook some essential feature, as in the problem of the thermal conductivity of nonconducting crystals.�”

R.E. Peierls

2

Outline

I. IntroductionII. Necessary and sufficient conditions for normal

and anomalous heat conductionIII. Heat conduction in the Frenkel-Kontorova

modelIV. Asymmetric heat conduction V. Negative differential thermal resistance

3

T+ T-

4

Fourier Law (1808)

heat fluxtemperature gradient

thermal conducti

::

:

vi

y

t

j T

jT

5

Hamiltonian

2

11

1

External potentialInter-p

[ ( ) ( )]2

article potential( ) :( ) :

Ni

i i ii

i

i i

pH U x V x xm

U xV x x

6

The crux of the heat conduction problem lies in low dimensions, i.e., one and two dimensions.

7

Green-Kubo formula

)0()(1limlim102 JJ

Vd

dkT V

t

t

8

Mode coupling theory

d ( )JC t3/5t 2/5N11t

3/ 2tln N2

finite3

9

Some outstanding problems in heat conduction

1. First-principle derivation of the Fourier law from statistical mechanics.

2. Complete set of necessary and sufficient conditions for the validity of the Fourier law.

3. The exponent and the problem of universality.4. Heat conduction in two dimensions.5. Heat conduction in three dimensions.6. Thermal devices.7. Quantum heat conduction.

10

Temperature

1T Huu

1

2

1

2

1. (0, ...0, , ..., )

2. (0, ...0, , 0, ...0)

N

N

i

i

i

i

pT

ppN N

p

pTm

Nm

u

uNm

pT

N

ii

1

2

mpT i

2

11

Heat flux

)]( )([21 )(

2

)(),(

0 ),( ),(

11

2

iiiiii

i i

ii

i

xxVxxVxUm

ph

xxhtxht

txhx

txj

12

iii xxjtxj )(),(

)]()()()[(21)( 1111 iiiiiiiiiiiii

i xxFxxxxFxxxUxxxmth

Force( ) ( ) :F x V x

13

1 1

1 1 1 1

1 1

1

( ) ( ) ( )1 [( ) ( ) (

1

) ( )]2

( )

0

( )2

i i i i i i i

ii i i i i i i i

i i

i i

i

i

i i

m x U x F x x F x xh x x F x x x x F

j a x x F x

x xt

x

j j ha t

14

Density fluctuations

( , ) ( , )

( , ) ( , )

( , ) ( , ) 0

( ) i

ikx

ikx

ikxii i

i

j k t dx j x t e

h k t dx h x t e

h k t ik j k tt

hh ikx h et t

15

1( )

1 1 1

1

1

1 ( )( ) ( )

1 1( ) ( ) [1 ]2 2

1

( ) )2

2

(

i i i i i

i i

ikx ikx ikx ik x xii i i

i i i

ikx ikxii i i

i

i i i i i i i i

i

ij x

h e e e et

kh ike x x e

x x x F x

t

x x h

16

Steady state

1

1

1

1

1 1

( ) 0

)

( )

( ( )

i i

i i

i i i i

i i i

i

i

i

V q q

q x iaq F q q q F

j a q F q q

q q

17

2

1 1

( ) 0

( ) ( )

i

i i i i i i

qt

q F q q q F q q

1i ij j j

ii

J j

18

Heat baths1. Stochastic (Langevin)

iNiiiiiii qqqqFqqFqm )()()()( 1111

02 2 Bk T

: coupling between chain and heat bath

TTT

TLj ||

19

2. Deterministic (Nosé-Hoover)

1 1

22

( ) ( )1 1(

: therm ost

1)

, at response ti

0;, 0

m e

.

i i i i i i

ii SB

m q F q q F q q q

m qk T N

T TT T

: thermostat response time

20

Necessary and sufficient conditions for normal and anomalous heat conduction

21

A. Integrable models1. Linear (harmonic) models:

Dynamics described by normal modes

2. Nonlinear but integrable models:Dynamics described by solitons

• Transport is ballistic rather than diffusive.• Temperature gradient can’t be established.• diverges.• Fourier’s law is not obeyed.

22

Z. Rieder, J. L. Lebowitz, and E. Lieb, J. Math. Phys. 8, 1073 (1967).

23

B. Non-integrable models

1. Without an on-site potential, momentum is conserved.

Ex. Fermi-Pasta-Ulam (FPU) model

24

1

1

~

~

~ diverges as

Fourier s law is not obeyed'

j NdT Ndx

NN

25

2. With an on-site potential, momentum is not conserved.

Ex. Frenkel- Kontorova (FK) model

[B. Hu, B. Li, and H. Zhao (1998)]

26

1

1

~

~

~ 1 (finite)

Fourier's law is obeyed.

j NdT Ndx

27

28

The zone in the space of parameters ( , ), where for finite chains of length 640 the heat conductivity converges [(a), gray zone] and diverges [(b), white zone]. Curve 1 divides these two zones. Fo

g TN

r finite chains ( 640) finite heat conductivity is detected only above line 3.N

29

30

A. Relevant factors

INT: IntegrabilityDIS: DisorderCS: ChaosOP: On-site potentialMC: Momentum conservationPJ: Phase jumpTG: Temperature gradientHC: Heat conductionIP: Inter-particle potential

31

B. Models

HC: Harmonic chain

FK:

SHG: Sinh-Gordon

BSW: Bounded single-well

( ) 1 cosV u u

1cosh)( uuV

21( ) (1 sech )2

V u u

32

C. Remarks

(a) Ref. 2 claims that 2D disordered harmonic web can have a normal heat condition with sufficiently strong disorder. We have doubts.

(b) Ref. 7 claims that the diatomic Toda model can have a normal heat conduction. However, Ref. 8 gives a different conclusion.

(c) At low temperature, with the Nosé-Hoover heat bath.

(d) Ref. 13 claims that the hard-point model with alternating masses has a finite heat conduction. Ref. 14 and 15 give a different conclusion. Ref. 16 studies a modified version of the hard-point model, which can have a finite heat conduction in certain temperature regimes.

(e) Ref. 19 claims that the FK model can have normal and anomalous heat conductions in different parameters and temperature regimes.

(f) Disordered right triangle model and irrational triangle model.33

D. References

1. S. Lepri, R Livi and A. Politi, Phys. Reports 377, 1 (2003).2. L. Yang, Phys. Rev. Lett. 88, 4 (2002).3. S. Lepri, Europhys. J. B 18, 411 (2000).4. K. Aoki and D. Kusezov, Phys. Rev. Lett. 86, 4029 (2001).5. S. Lepri, R Livi and A. Politi, Phys. Rev. Lett. 78, 1896 (1997).6. B. Hu, B. W. Li, H. Zhao, Phys. Rev. E 61, 3828 (2000).7. E. A. Jackson and A. D. Mistriotis, J. Phys.: Condens Matter 1, 1223 (1989).8. T. Hatano, Phys. Rev. E 59, 1 (1999).9. B. W. Li, H. Zhao, and B. Hu, Phys. Rev. Lett. 86, 63 (2001).10. O. V. Gendelman and A. V. Savin, Phys. Rev. Lett. 84, 2381 (2000).11. C. Giardina, R. Livi, A. Piliti and M. Vassalli, Phys. Rev. Lett. 84, 2144

(2000).12. A. Dhar, Phys. Rev. Lett. 86, 3554 (2001).

34

13. P. L. Garrido, P. I. Hurtado and B. Nadrowski, Phys. Rev. Lett. 86, 5496 (2001).

14. A. Dhar, Phys. Rev. Lett. 88, 249401 (2002).15. G. Casati and T. Prosen, Phys. Rev. E 67, 015203 (2003).16. A. V. Savin, G. P. Tsironis and A. V. Zolotaryuk, Phys, Rev. Lett. 88,

154301 (2002).17. G. Casati, J. Ford, F. Vivaldi and W. M. Visscher, Phys. Rev. Lett. 52,

1861 (1984).18. T. Prosen and M. Robnik, J. Phys. A: Math Gen. 25, 3449 (1992).19. A. V. Savin and O. V. Gendelman, Phys. Rev. E 67, 041205 (2003).20. D. Chen, S. Aubry and G. P. Tsironis, Phys. Rev. Lett. 77, 4776 (1996).21. B. Hu, B. W. Li, and H. Zhao, Phys. Rev. E 57, 2992 (1998).22. G. P. Tsironis, A. R. Bishop, A. V. Savin and A. V. Zolotaryuk, Phys. Rev.

E 60, 6610 (1999).23. D. Alonso, R. Artuso, G. Casati and I. Guarneri, Phys. Rev. Lett. 82, 1859

(1999).24. B. W. Li, L. Wang and B. Hu, Phys. Rev. Lett. 88, 223901 (2002).

35

A᧪Normal heat conduction1. Chaos is neither a necessary nor sufficient condition2. Necessary conditions

(a) On-site potential or vanishing pressure(b) Anharmonicity(c) Non-integrability

36

B᧪Anomalous heat conductionMomentum conservation is not a necessary condition but a sufficient condition provided the pressure isnon-vanishing.

37

Exponent

1. Renormalization group (O. Narayan and S. Ramaswamy):

2. Mode coupling (S. Lepri, R. Livi, and A. Politi):

13

25

38

39

Heat conduction in the Frenkel-Kontorova model

40

Frenkel-Kontorova (FK) model

external potential

1( ) :n nW u u spring potential

H

( ) :nV un

nnn uuWuV )]()([ 1

41

Standard FK (1938)

21 1

1( ) ( )2n n n nW u u u u a

)2cos1()2(

)( 2 nn ukuV

a: natural length of spring

42

43

44

45

46

47

48

49

50

G

G

Fig. 10. Log-log of the phonon gap ( , ). The inset shows the log-log plotof the thermal conductivity ( , ). ( , ) is a decreasing function of andan increasing function of . ( , ) is an incre

aa a aa asing function of and a decreasing

function of .a

51

Asymmetric Heat Conduction

52

Two-segment FK model

A Bintk

T T

J TT

J TT

Asymmetric heat conduction: J J

53

Hamiltonian

2int / 2 1 / 2

2,2

, , 1 2

1 ( ) ,21 ( ) cos 2 .

2 2 (2 )

A B N N

A BiA B A B i i i

i

H H H k x x a

VpH k x x a xm

54

55

56

57

58

59

60

• Numerical results

• Theoretical analysis1. Temperature-dependent phonon spectra

Low temperature limit:

High temperature limit:

2. Overlap/separation of phonon bands

/ 100J J

4V V k

0 2 k

J J and

61

A B

62

A

B

63

64

65

2 4int / 2 1 / 2 int / 2 1 / 2

1 1( ) ( )2 4A B N N N NH H H k x x a x x a

66

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

kint=0.05 kint=0.6 kint=1.2

J

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

int=0 int=0.05 int=1.0

J

Fig. 5. Dependence of heat flux J on the temperature difference for T0 = 0.07. Here TL = T0*(1+ ), TR = T0*(1- ), N =100.

67

Reversal of rectification

Material A

1 1( , )N 2 2( , )N

Material B

3R

1 1 2 2 3/ /R N N R

1 2 3, 0N N N RAssume

1 2

1 2

2

68

Phenomenological explanation

0.02 0.04 0.06 0.08 0.10 0.120

1

2

3

4

5

6

7

8

A: N = 500 A: N = 1000 B: N = 500 B: N = 1000

1

2

3 4

Fig. 6. Dependence of the thermal conductivity on the temperature T for segments A and B.

JJ

J

J

,,

1122 :

1122 :

4321

21

21

21

34

34

34

69

• ExperimentC. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, Science 314, 1121 (2006).C. W. Chang, D. Okawa, H. Garcia, A. Majumdar, and A. Zettl, Phys. Rev. Lett. 101, 075903 (2008).

70

Fig. 1. A schematic description of depositing amorphous C9H16Pt (black dots) on a nanotube (lattice structure).

71

Fig. 2. The SEM image of a CNT (light gray line in center) connected to the electrodes. Scale bar, 5 mm.

72

Negative Differential Thermal Resistance (NDTR)

73

Negative differential electrical resistance

74

A᧪NDTR in homogeneous systems

75

1. With an on-site potential

(a) FK model

22

1 2( ) cos 22 2 (2 )

iFK i i i

i

p K VH x x x

76

77

78

79

80

(b) model4

4

22 4

11 ( )

2 2 4i

i i ii

pH x x x

81

82

83

84

85

86

2. Without an on-site potential: FPU model

22 4

1 11 ( ) ( )

2 2 4i

FPU i i i ii

pH x x x x

87

88

89

3. Without an on-site potential: rotator model

2

1(1 cos( ))2

irotator i i

i

pH K x x

90

0 10 20 30 400.5

1.0

1.5

2.0

2.5

3.0

3.5

j

T

N=32 N=64 N=128 N=256 N=512 N=1024

Fig. 12. Rotator model: rescaled heat flux J=Nj as a functionof ˂T for N=32, 64, 128, 256, 512, 1024. Here, T-=1 and K=2.

91

B᧪NDTR in inhomogeneous systems

92

B. Li, L. Wang, and G. Casati, Phys. Rev. Lett. 93, 184301 (2004)

93

B. Li, L. Wang, and G. Casati, Appl. Phys. Lett. 88, 143501 (2006)

94

(a)

(b)

(c)

Heat flux as a function of TL for various kint. (a) N = 25, (b) N = 150, and (c) N = 250.

95

as a function of kint for N = 25, 150, and 250.| / |J J

96

Heat flux as a function of TL for various N. (a) kint = 0.05, (b) kint = 0.3, and (c) kint = 0.5.

97

as a function of N for kint = 0.05, 0.3, and 0.5.

98

(a) Heat flux as a function of TL for various . (b) as a function of . Herekint = 0.05 and N = 25.

int int

99

Phase diagram which depicts the range of kint and N for the exhibition of NDTR.

100

Summary

101

1. We have found a reversal of the rectification effect in the two-segment FK model.

2. When the coupling of the two segments is weak, phonon band shift leads to .

3. When the coupling is strong or the chain is long enough, phonon band mixing leads to .

4. Negative differential thermal resistance (NDTR) occurs both in homogeneous and inhomogeneous systems.

5. Nonlinearity is a necessary condition but not a sufficient condition for NDTR.

6. Normal heat conduction seems to be a necessary condition for NDTR.

7. NDTR depends on the system size. The NDTR regime shrinks as the system size increases.

8. In an inhomogeneous system, NDTR also depends on the interfacial coupling constant. The NDTR regime shrinks as the interfacial coupling constant increases.

J J

J J

CollaboratorsB. Q. AiS. BuyukdagliH. K. ChanA. FillipovD. H. HeB. Q. JinY. R. KivsharB. LiH. B. LiA. V. Savin

Z. G. ShaoP. Q. TongL. WangB. S. XieL. YangP. YangA. ZeltserY. ZhangH. ZhaoW. R. Zhong

102

ReferencesI. Asymmetric Heat Conduction

1. B. Hu and L. Yang, Chaos 15, 015119 (2005). 2. B. Hu and L. Yang, Physica A 372, 272 (2006).3. B. Hu, D. He, L. Yang, and Y. Zhang, Phys. Rev. E 74, 060101 (R) (2006).4. B. Hu, D. He, L. Yang, Y. Zhang, Phys. Rev. E 74, 060201 (R) (2006).5. B. Hu, L. Yang, and Y. Zhang, Phys. Rev. Lett. 97, 124302 (2006).

II. Negative Differential Thermal Resistance1. D. He, B. Buyukdagli, and B. Hu, Phys. Rev. B 80, 104302 (2009).2. W.R. Zhong, P. Yang, B.Q. Ai, Z.G. Shao, and B. Hu, Phys. Rev. E 79, 050103 (R) (2009).3. Z. Shao, L. Yang, H.K. Chan, and B. Hu, Phys. Rev. E 79, 061119 (2009).4. D. He, B.Q. Ai, H.K. Chan, and B. Hu, Phys. Rev. E 81, 041131 (2010).5. B.Q. Ai and B. Hu, Phys. Rev. E 83, 011131 (2011).6. B.Q. Ai, W.R. Zhong, and B. Hu, Phys. Rev. E 83, 052102 (2011).7. W.R. Zhong, M.P. Zhang, B.Q. Ai, and B. Hu, Phys. Rev. E 84, 031130 (2011).