asymmetric heat conduction and negative differential...
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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
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Asymmetric Heat Conduction andNegative Differential Thermal Resistance
in Nonlinear Systems
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�“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
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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
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T+ T-
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Fourier Law (1808)
heat fluxtemperature gradient
thermal conducti
::
:
vi
y
t
j T
jT
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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
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The crux of the heat conduction problem lies in low dimensions, i.e., one and two dimensions.
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Green-Kubo formula
)0()(1limlim102 JJ
Vd
dkT V
t
t
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Mode coupling theory
d ( )JC t3/5t 2/5N11t
3/ 2tln N2
finite3
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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.
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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
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Heat flux
)]( )([21 )(
2
)(),(
0 ),( ),(
11
2
iiiiii
i i
ii
i
xxVxxVxUm
ph
xxhtxht
txhx
txj
12
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iii xxjtxj )(),(
)]()()()[(21)( 1111 iiiiiiiiiiiii
i xxFxxxxFxxxUxxxmth
Force( ) ( ) :F x V x
13
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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
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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
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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
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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
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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
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Heat baths1. Stochastic (Langevin)
iNiiiiiii qqqqFqqFqm )()()()( 1111
02 2 Bk T
: coupling between chain and heat bath
TTT
TLj ||
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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
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Necessary and sufficient conditions for normal and anomalous heat conduction
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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.
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Z. Rieder, J. L. Lebowitz, and E. Lieb, J. Math. Phys. 8, 1073 (1967).
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B. Non-integrable models
1. Without an on-site potential, momentum is conserved.
Ex. Fermi-Pasta-Ulam (FPU) model
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1
1
~
~
~ diverges as
Fourier s law is not obeyed'
j NdT Ndx
NN
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2. With an on-site potential, momentum is not conserved.
Ex. Frenkel- Kontorova (FK) model
[B. Hu, B. Li, and H. Zhao (1998)]
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1
1
~
~
~ 1 (finite)
Fourier's law is obeyed.
j NdT Ndx
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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
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A. Relevant factors
INT: IntegrabilityDIS: DisorderCS: ChaosOP: On-site potentialMC: Momentum conservationPJ: Phase jumpTG: Temperature gradientHC: Heat conductionIP: Inter-particle potential
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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
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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
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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).
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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).
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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
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B᧪Anomalous heat conductionMomentum conservation is not a necessary condition but a sufficient condition provided the pressure isnon-vanishing.
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Exponent
1. Renormalization group (O. Narayan and S. Ramaswamy):
2. Mode coupling (S. Lepri, R. Livi, and A. Politi):
13
25
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Heat conduction in the Frenkel-Kontorova model
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Frenkel-Kontorova (FK) model
external potential
1( ) :n nW u u spring potential
H
( ) :nV un
nnn uuWuV )]()([ 1
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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
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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
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Asymmetric Heat Conduction
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Two-segment FK model
A Bintk
T T
J TT
J TT
Asymmetric heat conduction: J J
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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
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• 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
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A B
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A
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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
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-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.
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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
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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
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• 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).
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Fig. 1. A schematic description of depositing amorphous C9H16Pt (black dots) on a nanotube (lattice structure).
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Fig. 2. The SEM image of a CNT (light gray line in center) connected to the electrodes. Scale bar, 5 mm.
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Negative Differential Thermal Resistance (NDTR)
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Negative differential electrical resistance
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A᧪NDTR in homogeneous systems
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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
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(b) model4
4
22 4
11 ( )
2 2 4i
i i ii
pH x x x
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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
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3. Without an on-site potential: rotator model
2
1(1 cos( ))2
irotator i i
i
pH K x x
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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.
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B᧪NDTR in inhomogeneous systems
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B. Li, L. Wang, and G. Casati, Phys. Rev. Lett. 93, 184301 (2004)
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B. Li, L. Wang, and G. Casati, Appl. Phys. Lett. 88, 143501 (2006)
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(a)
(b)
(c)
Heat flux as a function of TL for various kint. (a) N = 25, (b) N = 150, and (c) N = 250.
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as a function of kint for N = 25, 150, and 250.| / |J J
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Heat flux as a function of TL for various N. (a) kint = 0.05, (b) kint = 0.3, and (c) kint = 0.5.
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as a function of N for kint = 0.05, 0.3, and 0.5.
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(a) Heat flux as a function of TL for various . (b) as a function of . Herekint = 0.05 and N = 25.
int int
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Phase diagram which depicts the range of kint and N for the exhibition of NDTR.
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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
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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
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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).