Hyperbolic Heat Equation

1-D BTE 0x

f ff fv

t x

0 0f ff T

x x T x

assume

Integrate over velocity space assuming that is an averaged relaxation time

2 00x x x x

fTv fd v d v f d v fd

t x T

201

3

fv d

T

(4.63a) q fv d

(4.64)

20 ,x x

fv d

T

x xq fv d

x

x x

q Tq

t x

20x x x

f fv v v f f

t x

or

or xx x

q Tq

t x

In 3-D q

( , )( , ) ( , )

q r tq r t T r t

t

Maxwell-Cattaneo equation

Energy equation without convection

( , )( , ) ( , )p

T r tc q r t q r t

t

q

qq T

t

2

qq q Tt

p

Tc q q

t t

2

2p

T qc q

t t t

2

q q q

pcq Tq T

t t

2qp

Tq c q T

t t

Cattaneo equation

22

2q q q

pp

cT q T qc T

t t t

2q q2

2

p pc cT T q qT

t t t

2q q2

2

1T T q qT

t t t

Without internal heat generation2

22 2tw

1 1,

T TT

v t t

Damped wave equation

twq

v

Decay in amplitude: q

expt

21

3 v gc v For an insulator:

21

3 gv tw3gvv

hyperbolic equation

speed of thermal wave

Hyperbolic heat equation : valid strictly when tp >

semi-infinite solid under a constant heat flux at the surface

propagation speed : vtw

pulse wavefront: x1 = vtwt1, x2 = vtwt2

short pulse long pulse

In the case of a short pulse, temperature pulse propagates and its height decays by dissipating its energy to the medium as it travels.

Entropyu

T s qt

Local energy balance

Local entropy balance gen

s qs

t T

Entropy production rate

gen q2 2

1 1 qs q T q q

T T t

Hyperbolic heat equation sometimes predicts a negative entropy generation. (Energy transferred from lower-temp. region to higher-temp. region)

Negative entropy generation is not a violation of the 2nd law of thermodynamicsBecause the concept of “temperature” in the hyperbolic heat equation cannot be interpreted in the conventional sense due to the lack of local thermal equilibrium

Dual-Phase-Lag Model

Finite buildup time after a temperature gradient is imposed on the specimen for the onset of a heat flow: on the order of the relaxation time Lag between temperature rise and heat flow

In analogy with the stress-strain relationships of viscoelastic materials with instantaneous elasticity

( , ) ( ) ( , )t

q r t K t t T r t dt

( ) ( ),K When

( , ) ( , )q r t T r t

q/

q

( )K e

When

→ Cattaneo eq.

electrons → lattice phonons → temperature rise

By assuming that p/1

0q

( ) ( )K e

10

q q

( , ) exp ( , )t t t

q r t T T r t dt

2

q 2T2

1,

T TT T

t t t

q q 0 ,q

q T Tt t

0 1

0T q

retardation time

0 : effective conductivity (heat diffusion)

1 : elastic conductivity (hyperbolic heat)

( , )( , ) ( , )p

T r tc q r t q r t

t

dual-phase-lag model

q T( , ) ( , ),q r t T r t

q q

( , )( , ) ( , )

q r tq r t q r t

t

T T( , ) ( , ) ( , )T r t T r t T r tt

q T

( , )( , ) ( , ) ( , )

q r tq r t T r t T r t

t t

First order approximation

q q 0

qq T T

t t

q 0 T

T q

T q, : intrinsic thermal properties of the bulk material

T q The requirement may cause .1 0

1T

q q

( , ) exp ( , ) ( , )t t t

q r t T r t T r t dtt

The heat flux depends not only on the history of the temperature gradient but also on the history of the time derivative of the temperature gradient.

→ Cattaneo eq. When

T 0

When

T q → Fourier law

Solid-fluid heat exchanger

immersed long thin solid rods in a fluid inside a sealed pipe insulated from the outside

rod diameter d, number of rods N

inner diameter of pipe D

total surface area per unit lengthtotal cross-sectional area of the rodstotal cross-sectional area of the fluid

2c / 4A N d

2 2 2

2 2f c4 4 4 4

D D N dA A D Nd

average convection heat transfer coefficient h P N d

ff s f

TC G T T

t

f ff

c c

, C AhP

G CA A

2

s ss s s f2

T TC G T T

t x

2f

f 2 0T

x

s f assume

2 2 2qs s s s

T2 2 2

1T T T T

x t x t t

s sfT q T T

s f s f

, , CC

C C G C C

Due to the initial temperature difference between the rod and the fluid , a local equilibrium is not established at any x inside the pipe until after a sufficiently long time.

Two-Temperature Model

electrons → electron-phonon interaction → phononsUnder the assumption that the electron and phonon systems are each at their own local equilibrium, but not in mutual equilibrium (valid when t > )

ss e s

TC G T T

t

e : electron, s : phononC : volumetric heat capacity = cp

G : electron-phonon coupling constant

ee e e s a

TC T G T T q

t

Assume the the lattice temperature is near or above the Debye temperature so that electron-electron scattering and electron-defects scattering are insignificant compared with electron-phonon scattering. 2 2

e Be e

F2

n kC T

2

B,e

F2v

k Tc R

(5.25) esT

2 2e B

e3

n k T

m

(5.55b)

2 2e B

ee3

n kT

m

eq

es

TT

eq : thermal conductivity at

e sT T2 2 2 2 2 2

eq eqe B e B e Beq s e e

e s e s e

, , 3 3 3

n k n k n kT T T

m T m T m

2 2e e a

s6

m n vG

T

or 24e a B

eq18

n v kG

1/ 3

aD a

B

3

4

nhv

k

(5.10)1/ 3

B Da

a

4

3

kv

h n

ee e e s a

TC T G T T q

t

ss e s

TC G T T

t

2

q2 2 a a e eTe T e 2

1q q T TT T

t t t t

2

q2 2 a s ss T s 2

1q T TT T

t t t

s e eT q T T

e s e s

, , C C C

C C G C C G

q : not the same as the relaxation time due to collisionThermalization time: thermal time constant for the electron system to reach an equilibrium with the phonon system

q T: 30 ~ 40 fs, : 0.5 ~ 0.8 ps, : 60 ~ 90 ps

Heat Conduction Across Layered Structures

Equation of Phonon Radiative Transfer (EPRT)

00

scatt ( , )

f ff f fv S

t r t T

S0: electron-phonon scattering

Phonon BTE in 1-D system

0x

f ff fv

t x

*BE

B

1

exp 1

f f

k T

*

x

f f f fv

t x

or

phonon Intensity: summation over the three phonon polarizations

g

1( , , ) ( )

4 P

I x t v f D

phonon intensity under non-equilibrium distribution function: energy transfer rate in a direction from a unit area, per unit frequency, per unit solid angle

( ) ( ) ( )x xq x v f x D d

g gcosxv v v

+ forward direction ( > 0)- forward direction ( < 0)

T1

T2

q

I

I

x

*

x

f f f fv

t x

*

g g

( )1

( , )

I I I T I

v t x v T

Equation of Phonon Radiative Transfer (EPRT)

*

4

1( )

4

I Ia I a I I d

c t x

Radiative Transfer Equation (RTE)

optical thickness of the medium ,L a L

neglecting scattering 0

*1 I Ia I I

c t x

acoustical thickness of the mediumg

L

L

v

g

1 1

v

corresponds toa

Equilibrium intensity, Bose-Einstein statistics

B B

2 3g*/ 3 /3 2

p

( , )e 1 8 e 12

k T k TP P

v k dkI T

d v

g

1( , , ) ( )

4 P

I x t v f D

B

2*

/ 2

1, ( )

2e 1k T

k dkf D

d

(5.33)

22

2p p p

,p k k kv v v

g

dv

dk

analogy to blackbody intensity

B B B

33 3

/ / /2 2 2 2

2 2 / 22( )

e 1 e 1 2 e 1b h k T k T k T

hI

c c c

integrating over all frequency → total intensity for all 3 phonon modes

3 4B

SB 3 2a40

k

v

: Phonon Stefan-Boltzmann constantva: average phase velocity of the 2 transverse and 1 longitudinal phonon modes

SB

44 4 3* * SBB

3 3 20 0a

3( ) ( , )

8 e 1x

Tk T x dxI T I T d

v

2

4 341 1

4/50 0 0 02

2 2

e 11b b xC T

C C T x dxI I d d T

Ce

2 2

1 0 0 2 0 B 0 B2 , / 2 /C hc c C hc k c k

2 55 2 401 B

44 3 22 00 B

2 22

15 6015 2 /

cC k

C cc k

At temperature higher than Debye temperature

high frequency limit because the shortest wavelength of the lattice wave should be on the order of atomic distances, the lattice constants

mD

B

,h

k

D m m

mB B

1h

xT k T k T

m3

* * m B3 20

p

( ) ( , )8

kI T I T d T

v

energy flux at phonon equilibrium: I* invariant with direction

* * *ˆ ˆ( ) ( ) cos ( )q I T nd I T d I T

particle flux

4N

nvJ (4.12a)

energy carried by a phonon = energy flux / particle fluxenergy density= energy carried by a phonon X number of phonons per unit volume

At low temperature

duC

dT

**

g g

( ) 4( ) ( )

/ 4

I Tu T n I T

nv v

* 4SB

g g

4 4( ) ( )u T I T T

v v

3C T

At high temperature3

* m B3 2

g g p

4 4( ) ( )

8

ku T I T T

v v v

Dulong-Petit law: 3vc R

0 0dq

qdx

radiative equilibrium

*14

dqI G

dx

2 1

4 0 0 1

ˆ( ) ( , ) 2G x I x d I d I d

1*

1

4 2GI I d

m m 1*

0 0 1

1 14 2I d I d d

total quantities

1*

1

1

2I I d

or

2 1

4 0 0 1

ˆ ˆ cos sin 2q I nd I d d I d

T1

T2

q

I

I

x

Solution to EPRT

steady state*

g

( ( ))

( , )

I I T x I

x v T

Two-flux method in planar structures

*

, 0 1I I I

x

when

*

, 1 0I I I

x

when

*1 1 1(0, ) ( ) 1 (0, )I I T I

boundary conditions: gray medium, diffuse and gray walls

*2 2 2( , ) ( ) 1 ( , )I L I T I L

Solutions to EPRT

*

0( , ) (0, )exp ( )exp

xx x dI x I I

*( , ) ( , )exp ( )expL

x

L x x dI x I L I

Heat Flux (spectral)

1

0

* *2 20

2 (0, )exp ( , )exp

2 ( ) 2 ( )x L

x

x L xq I I L d

x d x dI E I E

1 2

0( ) expn

n

xE x d

2 1

4 4 0 0 1

ˆ ˆ cos cos 2q I nd I d I d I d

*3 20

( ) 2 (0) ( ) ( ( )) ( )q I E I T E d

*

3 22 ( ) ( ) ( ( )) ( )L

L LI E I T E d

For diffuse surface

3 3

* *2 20

2 (0) 2 ( )

2 ( ) 2 ( )x L

x

x L xq I E I L E

x d x dI E I E

total heat flux

1

12xq q d I d d

Heat flux at very low temperature

4 4SB 1 SB 2

1 2

4

3 1 1 1 1 41

2 2 3

x

T Tq

L Kn

Heat flux between blackbodies with small T difference

beff4

13

x

T Tq

KnL L

Thermal conductivity

3SB

b

16

3

T

1

02 (0, )exp ( , )exp

x L xq I I L d

* *2 20

2 ( ) 2 ( )x L

x

x d x dI E I E

acoustically thick limit

** *( ) ( ) ( )

dII x I x

dx

x

z

Let*

204 ( )

Iq zE z dz

x

1 2 D ,T T when

3SB16

3x

T dTq

dx

3

b SB

16

3T

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Effective Conductivity of the Film

3 effb SB

b

16 143 1

3

TKn

KnL

eff

b

Kn

Thermal Resistance Network

1bT

2aT 2bT

3aT3bT

xq

HT

LT1 2 3

H L

totx

T Tq

R

H1R1R

12R2R

23R3R

3LR

effective thermal conductivity of the whole layered structuretot

efftot

L

R

Internal thermal resistance with Fourier’s law

ii

i

LR

Thermal resistance at the interface

4 1 1 4 1 1

3 2 3 2ji

iji ij j ji

R

:ij transmissivity

Thermal resistance at the boundaries

31H1 3L

1 1 3 3

4 1 1 4 1 1,

3 2 3 2R R

H1R1R

12R2R

23R3R

3LR

Thermal Boundary Resistance (TBR)

TH

11 22 33TL

Acoustic Mismatch Modelspecular reflection of phononssimilar to geometric opticsno scattering or diffusion

Diffuse Mismatch Modeldiffusive reflection of phononsno information (except for energy) retained after a scattering

Thermal Contact Resistancedue to incomplete contact between two materialsthermal resistance between two bodies, usually with very rough surfaces, rms > 0.5 m Thermal Boundary Resistance

due to the difference in acoustic properties of adjacent materials

1T 2T

22

11

1 1, v 2 2, v

Acoustic Mismatch Model (AMM)For longitudinal phonon modes with linear dispersion in a Debye crystal (equilibrium, isotropic, perfect contact)

g

1( , , ) ( )

4 P

I x t v f D

1 1 1 1

1( , , ) , ( )

4 lI x t v f T D

1 2 ,1 2 12 1cosq q d I d d

m 2 / 2

1 2 1 1 1 12 1 1 1 10 0 0

1( , ) ( )cos sin

4 lq v f T D d d d

1 2 1 2 1 2

12 21 2

1 1 2 2 2 1

4 cos cos

cos cosl l

l l

v v

v v

transmission coefficient

1 2

1 2

sin sin

l lv v

Snell’s

law

Let 12 1

12

1

cos

cos

d

d

: hemispherical transmission

2 / 2 / 2

12 1 1 1 1 12 1 1 10 0 0

1cos sin 2 cos sind d d

m 2 / 2

1 2 1 1 1 12 1 1 1 10 0 0

1( , ) ( )cos sin

4 lq v f T D d d d

m 2 / 2

1 1 1 12 1 1 1 10 0 0

1( , ) ( ) cos sin

4 lv f T D d d d

m / 2

1 1 1 12 1 1 10 0

1( , ) ( ) cos sin

2 lv f T D d d

m m 312 121 1 1 1 1 120 0

1

1( , ) ( ) ( , )

4 4l ll

v f T D d v f T dv

1 2

1 2

sin sin

l lv v

Snell’s law critical angle: C

1 22

1

sin

sinl

l

vv

C21 21

2 2 22 202 2

2 cos sinl l

dv v

C / 22 221 2 12 1 121 2 1 12 2 20 0

1 2 1 1 1

cos cos2 sin 2 sin

sin sinl l l

d dv v v

m 3121 2 1 1 12 0

1

1( , )

4 ll

q v f T dv

1 2 2 1xq q q

m 3 3121 1 1 2 2 22 0

1

1( , ) ( , ) ( )

4 l ll

v f T v f T D dv

B

2

/ 2

1, ( )

2e 1k T

k dkf D

d

(5.33)

22

2p p p

,p k k kv v v

g

dv

dk

2

g g 2

1 1( , ) ( ) ( , )

4 4 2

k dkv f T D v f T

d

B

2 3

3 2 /3 2p p

1( , )

8 8 e 1k Tf T

v v

For longitudinal phonon modes with linear dispersion in a Debye crystal (equilibrium, isotropic, perfect contact)

p l

dv v

dk k

B

3

/3 2

1( , ) ( )

4 8 e 1l k T

l

v f T Dv

m 3 3121 1 1 2 2 22 0

1

1( , ) ( , ) ( )

4x l ll

q v f T v f T D dv

B

3 3

3 2 /3 2

1 ( , )( , ) ( )

4 8 8 e 1l k T

l l

f Tv f T D

v v

33

2

( , )( , ) ( )

2l

f Tv f T D

m 3121 1 2 22 2 0

1

1( , ) ( , )

8xl

q f T f T dv

B( / , 1 2)jx k T j or

m1 m24 3 3

4 412 B1 22 2 3 0 0

1 8 1 1

x x

x xl

k x dx x dxT T

v e e

In the low temperature limit,

2 4

4 412 B1 22 3

1 120xl

kq T T

v

For all 3 phonon modes

3 4

0 1 15x

x dx

e

m1 m24 3 3

4 412 B1 22 2 3 0 0

1 8 1 1

x x

x x xl

k x dx x dxq T T

v e e

2 2 21 1 1j l t

j

v v v

2 4

4 4 2B1 2 12 13120x j

j

kq T T v

Thermal boundary resistance

31 2 1 2

b,AMM 2 4 4 4 2B 1 2 12 1

120 1

x jj

T T T TR

q k T T v

3

2 4 22 2B 12 11 2 1 2

120 1 1

jj

k vT T T T

1 2T T T

TBR by AMM is proportional to T-3.

AMM assumes perfect specular reflection which is valid only when the characteristic wavelength of the phonons is much larger than the surface roughness. (mp >> rms)→ DMM (Diffuse Mismatch Model)

When the temperature difference is small,

2 2 31 2 1 2 4T T T T T

3 3

b,AMM 2 4 2B 12 1

30

jj

TR

k v

Dmp a

T

a: lattice constant

Diffusion Mismatch Model (DMM)In DMM, all phonons striking the interface are scattered once, and are emitted into the adjoining substances elastically ( = 0) with a probability proportional to the phonon density of states (DOS) in the respective substances.

Distribution of the emitted phonons is independent of the incident phonon, whether it is from side 1 or 2, longitudinal or transverse.

,12 12 ,21 21 12 21( , ) ( ), ( , ) ( ), ( ) ( ) 1j j

integration over the solid angle12 21( ) ( ) 1

2 212 1 21 2j j

j j

v v 12 212 2

1 2l lv v

2 22 1

12 212 2 2 21 2 1 2

, j j

j j

j j j jj j j j

v v

v v v v

at low temperature

by DMM (12) with the assumption1 2T T T

2 4

4 4 2B1 2 12 13120x j

j

kq T T v

3 2 2 31 2

b,DMM 2 4 2 2B 1 2

30 j jj j

j jj j

v v T

Rk v v

3 3

b,AMM 2 4 2B 12 1

30

jj

TR

k v

2 21 2 12,AMM

b,DM

2b,AM 2

j jj j

jj

v vR

R v

Measured Data

TBR between indium and sapphire[2]. X, normal indium; ● super conducting indium. Data B are for roughened sapphire surface. A and C are for smooth sapphire surfaces with different indium thicknesses. AMM predicts a flat line at 20.4 cm2K4/W