View
9
Download
0
Category
Preview:
Citation preview
SCATTERING APPROACH FOR MOMENTUM AND HEAT TRANSFER
In collaboration with:R.L. Jaffe, M. Kardar, M. Krüger, M. Maghrebi, J. Rahi, A. Shpunt (MIT),
G. Bimonte (Napoli),N. Graham (Middlebury),
U. Mohideen, R. Zandi, E. Noruzifar (UC Riverside),
Thorsten EmigCNRS & Université Paris Sud
Nanoscale Radiative Heat Transfer, Les Houches, 16/05/2013
OUTLINE• Equilibrium fluctuations:
• interactions between macroscopic bodies
• influenced by shape, material properties, and temperature
• correlated !
• Non-equilibrium fluctuations:
• bodies at different temperatures
• novel interaction effects
• heat radiation and transfer
• Outlook, new directions
HEAT RADIATION AND TRANSFER
• Breaking the law, at the nanoscale [MITnews, July 29, 2009]
• Planck’s law is modified for small objects and short separations
• Probing Planck’s Law with Incandescent Light Emission from a Single Carbon Nanotube [Y. Fan, S.B. Singer, R. Bergstrom, & B.C. Regan, Phys. Rev. Lett.102, 187402 (2009)]
• Probing Planck's Law for an Object Thinner than the Thermal Wavelength [C. Wuttke and A. Rauschenbeutel, arXiv:1209.0536 [quant-ph]]
SURFACE PHONON POLARITONS MEDIATED ENERGY TRANSFER BETWEEN NANOSCALE GAPS
• Beyond Stefan-Boltzmann law
• Understand heat transfer in nano-systems
• Near field effects can give huge enhancement of transfer(tunneling of evanescent waves)
S. Shen, A. Narayanaswamy, & G. Chen, Nano Lett. 9, 2909 (2009)
EQUILIBRIUM QED: INTERACTIONS
• Start from path integral for free energy of electromagnetic fieldat inverse temperature and imaginary frequency
Z =⇧
�
⌃DADA� exp
⇤��
⌃dxE�(⇥,x)
�H0(⇥) +
1⇥2
V(⇥,x)⇥
E(⇥,x)⌅
F (�) = � 1�
log Z(�)
• Free photons:
• Interaction:
1/� ⇥ = ic�
V(⇥,x) = I ⇥2 (�(ic⇥,x)� 1) + �⇥�
1µ(ic⇥,x)
� 1⇥
�⇥
H0(�) = I +1�2
����
• In terms of current fluctuations:
�J(x�)
⇥G0(⇥,x,x⇥) + V�1(⇥,x)�(3)(x� x⇥)Z ⇠
ZDJDJ
⇤|obj
exp
��
Zdxdx0
J
⇤(x)
⇣
THE T-OPERATOR AND ITS DECOMPOSITIONG0(⇥,x,x⇥) + V�1(⇥,x)�(3)(x� x⇥)T�1 =
is inverse T-operator: induced source = T-operator × applied field”
E =⇥c
2⇥
� ⇥
0d� log det(MM�1
⇥ )
M =
0
@(T1)�1 U12 U13 · · ·
U21 (T2)�1 U23 · · ·· · · · · · · · · · · ·
1
A M�11 =
0
@T1 0 0 · · ·0 T2 0 · · ·
· · · · · · · · · · · ·
1
A
1
2
U↵� couples induced sources on different objects
INTERMEZZO: OPERATOR FORMALISM• Simple electrostatic example: Two metallic objects at fixed potentials
• Find potential U and surfacecharges from T-operators
• Conditions:
• Surface charges:
• Potential:
T1
T2
�↵ G0
T↵ =1
2(S↵ � 1)
U = G2�1
U = U0 +G1�2
U = G0(�1 + �2)
G↵ = G0 �G0T↵G0
U = U0 U = 0
�1 = G�10 (1�G0T1G0T2)
�1U0 = G�10
1X
n=1
(G0T1G0T2)nU0
�2 = �T2
1X
n=0
(G0T1G0T2)nU0
U = (1�G0T2)1X
n=0
(G0T1G0T2)nU0
multiple “scattering” & multipole expansion
CASIMIR POTENTIAL• The Casimir energy can be expressed as
with
• Diagonal: Scattering amplitudes (T-matrix elements) of individual objects. They describe shape and material properties.
• Off-diagonal: Translation matrices. They are universal (depend only on dimension of space and type of field) and describe relative position of objects, i.e., geometry.
• For two objects:
E =⇥c
2⇥
� ⇥
0d� log det(MM�1
⇥ )
E2 =⇥c
2⇥
� �
0d� ln det(1� N12)
M =
0
@(T1)�1 U12 U13 · · ·U21 (T2)�1 U23 · · ·· · · · · · · · · · · ·
1
A M�11 =
0
@T1 0 0 · · ·0 T2 0 · · ·· · · · · · · · · · · ·
1
A
N12 = T1U12T2U21
INTERPRETATION FOR COMPACT OBJECTS
• 2 Objects: Expansion in number (2p) of scatterings
• For each scattering consider l partial waves
• Expand scattering amplitude in
• For - Series of Casimir energy up to order
one needs only finite p, l
• Scale-free objects (cone, wedge, …): low order expansion accurate
1/L
E2 =⇥c
2⇥
� �
0d� ln det(1� N12) =
⇥c
2⇥
� �
0d�Tr log(1� N12)
� p l7,8 1 1 6
9,10 1 2 16
11,12 1 3 30
13,14 2 4 48
15,16 2 5 70
dim(N)R ⇠ R/L
⇠ (R/L)⌘
= � ~c2⇡
Z 1
0dTr
✓N12 +
1
2N2
12 +1
3N3
12 + . . .+1
pNp
12 + . . .
◆
with N12 = T1U12T2U21 ⇠ e�2L
SOME EXAMPLES
EQUILIBRIUM INTERACTIONS BETWEEN...
SHARP SHAPED METALS
T depends on R
T independent of
M. F. Maghrebi, S. J.Rahi, T. Emig, N. Graham, R. L. Jaffe, M. Kardar, PNAS 108, 6867 (2011).
PLATE - WEDGEMultiple scattering expansion is highly accurate: analytical resultsat all distances
PLATE - CONE
E = � ln 4� 116⇡
~c
d
1| ln ✓0/2| + O(✓2
0)
0 0
T=300º KT=80º K
T=0º K d = 1µm
EM
DN
�
F ⇠ �~c16⇡| ln ✓0
2 |
ln 4� 1
d2� 2
3�2T
ln2d
�T+0.810
�2T
+ · · ·�
STABLE EQUILIBRIUM?• Earnshaw’s theorem: A charged body cannot
be held in stable equilibrium by electrostatic forces from other charged bodies.
• Extension to fluctuation-induced forces?
• Start from scattering formulation (T-operators):
• Move object A by d with the “rest” of objects (R) fixed.
• Object A is unstable ( ) if
• Stability not possible for (i) (ii)on the imaginary frequency axis (where always )
E =
~c
2⇡
Z 1
0d tr log T�1T1 =
~c
2⇡
Z 1
0d tr log(I� TAGTRG)
r2d E
��d=0
0 sign(TA)sign(TR) � 0
✏J/✏M > 1, µJ/µM 1 (positive TJ)
✏J/✏M < 1, µJ/µM � 1 (negative TJ)✏J > 1
✏J , µJ
S. J. Rahi, M. Kardar, T. Emig, Phys. Rev. Lett. 105, 070404 (2010).
NON-EQUILIBRIUM QED• Objects at different temperatures (local equilibrium)
• Environment can have different temperature
• Modification of equilibrium force ?
• Radiation and transfer of heat ?
T↵
Tenv
Tenv
T1
T2
T3
T4
Parallel plates: M. Antezza, L.P. Pitaevskii, S. Stringari, V.B. Svetovoy, Phys. Rev. A 77, 022901 (2008).General shapes: M. Krüger, T. Emig, G. Bimonte and M. Kardar, EPL 95 21002 (2011), M. Antezza et al. (2011).
M. Krüger, T. Emig and M. Kardar, PRL 106, 210404 (2011).
FLUCTUATION-DISSIPATION THEOREM
• Equilibrium field correlations:
• Three contributions: ‣ zero point fluctuations: ‣ thermal currents in object :‣ environment fluctuations
Ceq(T ) = hE(!; r)E⇤(!; r0)ieq = [aT (!) + a0(!)]c2
!2ImG(!; r, r0) = C0 +
X
↵
Csc↵ (T ) + Cenv(T )
↵
C0 = a0(!)c2
!2ImG
Csc↵ (T ) = aT (!)GIm"↵G⇤
Cenv(T ) = �aT (!)c2
!2GImG�1
0 G⇤
aT (!) ⌘ !4~(4⇡)
2
c4(exp[~!/kBT ]� 1)
�1 a0(!) ⌘ !4~(4⇡)2
2c4
↵
Cneq(Tenv, {T↵}) = C0 +X
↵
Csc↵ (T↵) + Cenv(Tenv) = Ceq(Tenv) +
X
↵
[Csc↵ (T↵)� Csc
↵ (Tenv)]
C↵(T↵) ⌘ aT↵(!)G↵Im"↵G⇤↵
Csc↵ (T↵) = O↵,� C↵(T↵) O†
↵,� , with
O↵,� = (1�G0T�)1
1�G0T↵G0T�
• Non-equilibrium correlations: change temperatures
• Scattering theory: radiation of object :scattered at all other objects:
HEAT RADIATION OF SINGLE OBJECT
• Poynting vector:
• Heat emitted by object :
• Use to get general result
• Since involves only propagating waves, in matrix notation:
↵
H↵ = Re
I
⌃↵
S · n↵ = �Z 1
�1
d!
2⇡
Z
V↵
d3r hE(r) · J⇤(r)i
S(r) =c
4⇡
Zd!
2⇡hE(r)⇥B⇤(r)i
H↵ =
2~⇡
Z 1
0d!
!
exp(~!/kBT )� 1
Tr {Im[G0] Im[T]� Im[G0]T Im[G0]T⇤}
E = 4⇡i!
c2G0 J
Im[G0]
H↵ =~2⇡
Zd!
!
e~!
kBT � 1Trpr
⇥I � SS†⇤ � 0
HEAT TRANSFER BETWEEN TWO BODIES
• Two bodies, at T1 and at T2 , in cold environment. Total heat transferred from 1 to 2:
where is radiation of 1, partly absorbed by 2.
• We get for transfer rate
with
• Since J is symmetric (trace is cyclic), one gets
Htot
= H(2)
1
(T1
)�H(1)
2
(T2
)
H(2)1 (T1)
H(2)1 (T1) =
2~⇡
Z 1
0d!
!
e~!
kBT1 � 1J(T1,T2)
J(T1,T2) = Tr
⇢[Im[T2]� T⇤
2Im[G0]T2]1
1�G0T1G0T2G0 [Im[T1]� T1Im[G0]T⇤
1]G⇤0
1
1� T⇤2G⇤
0T⇤1G⇤
0
�� 0
Htot
=2~⇡
Z 1
0
d!!
1
e~!
kBT1 � 1� 1
e~!
kBT2 � 1
!J(T
1
,T2
)
TOTAL ABSORBED HEAT
• Total heat absorbed by one object (1) in the presence of a second object (2) and the environment:
• Here we have included the heat emitted by object 1 (in the presence of object 2) which is negative,
and the radiation absorbed from the environment.
• Important for heating or cooling rate.
H(1)1 = �2~
⇡
Z 1
0d!
!
e~!
kBT1 � 1ImTr
⇢(1 +G0T2)
1
1�G0T1G0T2G0 [Im[T1]� T1Im[G0]T⇤
1]1
1�G⇤0T⇤
2G⇤0T⇤
1
�
H(1)(T1, T2, Tenv) = H(1)2 (T1) +H
(1)1 (T2) +H(1)
env(Tenv)
=X
↵=1,2
H(1)↵ (T↵)�H(1)
↵ (Tenv)
NON-EQUILIBRIUM FORCE
• Maxwell stress tensor:
• Total force on object 2 due to other objects and environment:
�ab(r) =
Zd!
16⇡3
⌧EaE
⇤b +BaB
⇤b � 1
2
�|E|2 + |B|2
��ab
�
2 211F2
1 F22⌃2 ⌃2
n2 n2
F2 = Re
I
⌃2
� · n2 =
Z 1
�1
d!
2⇡
1
!
Z
V2
d3r ImhrE(r) · J⇤i = F2,eq(Tenv) +X
�
⇥F2
�(T�)� F2�(Tenv)
⇤
F(2)1 =
2~⇡
Z 1
0d!
1
e~!
kBT1 � 1<Tr
⇢r(1 +G0T2)
1
1�G0T1G0T2G0 [=[T1]� T1=[G0]T⇤
1]G⇤0
1
1� T⇤2G⇤
0T⇤1G⇤
0
T⇤2
�
F(2)2 =
2~⇡
Z 1
0d!
1
e~!
kBT2 � 1<Tr
⇢r(1 +G0T1)
1
1�G0T2G0T1G0 [=[T2]� T2=[G0]T⇤
2]1
1�G⇤0T⇤
1G⇤0T⇤
2
�
EQUILIBRIUM VS. NON-EQUILIBRIUM
• All quantities expressed as traces over product of free Green’s function and T-operators of individual bodies.
• Equilibrium:
• Computations on imaginary frequency axis
• Non-equilibrium:
• Traces are non-analytic function of frequency, computations on real frequency axis
• Quantities sensitive to details of dielectric function: resonances
• Much richer phenomenology
NON-EQUILIBRIUM EFFECTS FOR...• Heat radiation:
• Heat transfer:
• Forces
A. P. McCauley, M. T. H. Reid, M. Krüger and S. G. Johnson, Phys. Rev. B 85, 165104 (2012).
M. Krüger, T. Emig and M. Kardar, PRL 106, 210404 (2011)
M. Krüger, T. Emig, G. Bimonte and M. Kardar, EPL 95 21002 (2011)
HEAT RADIATION• Stefan-Boltzmann law for an ideal black body with surface area A:
• Sphere and cylinder (SiO2) at T=300K:
10-2
10-1
1
10-1 1 10 102
Rad
iate
d H
eat [�
T4 A
]
R [µm]
Class.
� R �t�
spherecylindercylind. ||cylind. 2 R
2 R
plate
H = �T 4A � =⇡2k4
B
60~3c2
~ volume~ surface
polarized radiation
Polarization exp. observed: Y. Öhman, Nature 192, 254 (1961);G. Bimonte et al., New J. Phys. 11, 033014 (2009).
HEAT TRANSFER• Heat transfer rate from plate to sphere (SiO2, R=5µm)
0.5
0.55
0.6
0.65
0.7
0.5 1 1.5 2 2.5 3 3.5 4 4.5
Hea
t Tra
nsfe
r Hs [�
T4 2
�R2 ]
d / R
d = �full solutionone reflection
0.8 1
1.2 1.4 1.6 1.8
10-2 10-1 100
PTA
Rat
iod / R
10-2
10-1
100
10-1 100 101 102
Hs (
d = �
)
R [µm]
� R
Class.
2 Rd
Tenv = 0K
T = 300K
T = 0K
• Increased heat transfer at small d due to tunneling of evanesc. waves.• At small d proximity transfer approximation (PTA) is valid:• Volume-to-surface crossover around R ⇡ �T
Hs ⇠ 1/d
TWO SPHERES AT DIFFERENT TEMPERATURES
FORCE BETWEEN TWO SPHERES (SIO2)• Dipole approximation, one reflection: assume radius
• Force on sphere 2: attraction (solid lines) and repulsion (dashed lines)
10-3
10-2
10-1
1
4 5 6 7 8 9 10 15 20
F [1
0-18 N
]
d [µm]
� d-2
T1=0 K,T2=0 KT1=300 K, T2=300 K
T1=0 K, T2=300 KT1=300 K, T2=0 K
10-3
10-2
10-1
1
4 5 6 7 8 9 10 15 20
F [1
0-18 N
]
d [µm]
� d-2
T1=300 K,T2=300 KT1=0 K, T2=0 K
T1=0 K, T2=300 KT1=300 K, T2=0 K
R⌧ d,�T =~c
KBT
Tenv = 0K Tenv = 300K
• Oscillations from due to interference of reflected and non-reflected radiation. Set by material resonances.
• Stable equilibrium positions.• Self-propelled pairs: equal acceleration in the same direction.
F↵↵
CASIMIR LEVITATION
• Non-equilibrium situation:
• Hot microsphere levitates above a cold dielectric plate
• If sphere cools down (including heat transfer) it will fall down
OUTLOOK / NEW DIRECTIONS Radiation/Transfer: effect of shape? e.g. non-parallel cylinders, disorder (roughness)?
Fluctuation of forces / radiation / transfer? distribution functions? Related to friction (Einstein relation): Quantum friction?
Relation to random matrices?
Dynamic effects: Radiation due to motion
...
Recommended