Micro/Nanoscale Heat Transfer:
Interfacial Effects Dominate the
Heat Transfer
1
Xing Zhang, Zengyuan Guo
Tsinghua University, China
Proceedings of the ASME 2012 3rd Micro/Nanoscale Heat &
Mass Transfer International Conference
BACKGROUND
2 Technology nodes for Intel nanotransistors.
Nanotechnology has been described as
a new industrial revolution
M. Chu, et al. Annu. Rev. Mater. Res. 2009. 39:203
3
BACKGROUND
Size continuously
decreases, power
density increases,
micro/nanoscale
heat transfer
becomes critical.
What is the
dominant factor in
micro/nanosclae
heat transfer?
Interfacial effects http://www.slideshare.net/gigaom/gn2010-
main-slidesfinallivevent-2
OUTLINE
4
1. Heat conduction
2. Convective heat transfer
3. Thermal radiation
4. Conclusions
1.1 Thermal conductivity of nanofilms
1. HEAT CONDUCTION
5
Single crystal silicon nanofilms (MD)
Both in-plane and out-of-plane thermal conductivities
dramatically reduced.
Exhibit anisotropic properties.
Surface limitations of the phonon transport
Bulk: 148 W m-1 K
-1
X. L. Feng, Doctoral Thesis, 2001
1.1 Thermal conductivity of nanofilms
6
Gold films
Dramatically reduce when the film is thinner than
approximately 500 nm.
Also exhibit anisotropic properties.
Bulk: 317 W m-1 K
-1
In-plane
Out-of-plane
J. P. Bourgoin, et al., J. Appl. Phys., 108, 073520, 2010
1.1 Thermal conductivity of nanofilms
7
Out-of-plane thermal conductivity
Pump-probe method
Pump laser is externally modulated and heats the sample
Probe beam detects the transient thermoreflectance
change of the sample.
J. P. Bourgoin, et al., J. Appl. Phys., 108, 073520, 2010
1.1 Thermal conductivity of nanofilms
8
Out-of-plane thermal conductivity
Varies linearly with the film thickness in layers thinner
than 500 nm.
Surface scattering of the electrons dominates the
thermal conductivity.
Gold films
Bulk: 315 W m-1 K-1
Aluminum films
Bulk: 236 W m-1 K-1
J. P. Bourgoin, et al., J. Appl. Phys., 108, 073520, 2010
Electrical
resistance
Electrical conductivity
Temperature
Thermal conductivity
I & U
Heat power
One-dimensional steady-state electrical
heating method
9
In-plane thermal conductivity
X. Zhang, et al., Appl. Phys. Lett., 86:171912, 2005.
10
The thermal conductivity decreases dramatically.
The temperature dependence is evidently different.
In-plane thermal conductivity
Gold films
Charge carrier electron metal
metallic nanofilm
Widemann-Franz law
Heat carrier
Is Wiedemann-Franz law still valid?
In-plane thermal conductivity
11
0 .b
b
L constT
Violation of the W-F law
The normal electrical conductivities do not equal
to thermal conductivities;
The electrical conductivity drop is considerably
greater than the thermal conductivity; 12
In-plane thermal conductivity
X. Zhang, et al., Chin. Phys. Lett., 25, 3360, 2008
TEM micrographs of the present nanofilms
Nanostructure of the MNFs
13
In-plane thermal conductivity
They are fine-grained(polycrystalline)
14
In-plane thermal conductivity
How to consider the effects of GB?
1. The electrons passing through the GB are considered to
have the same ability to tran. charge and to tran. heat.
2. Those reflected electrons can deliver energy to the
phonons on the GB, even though they have no
contribution to charge transport.
X. Zhang, et al., Chin. Phys. Lett., 25, 3360, 2008
In-plane thermal conductivity
Electron scattering relaxation model
background tb
grain boundary tg
surface ts
impurity ti
Combined
relaxation
time te
High purity(99.98%)
In plane-less importance ignore
Matthiessen’s rule (MR)
15
X. Zhang, et al., Chin. Phys. Lett., 25, 3360, 2008
e b g s i
1 1 1 1 1
t t t t t
Electrons being reflected
Electrons passing through
R
T* = 1-R
Relaxation time for films
Thermal conductivity
16
Electron scattering relaxation model
e1 b
1 1
t t
e2 b g
1 1 1
t t t
*
f e1 e2
1 1 1T R
t t t
f f
*
b b
1
1
l
l T
17
In-plane thermal conductivity
Data vs. model (Platinum films)
Measure films of different thickness at different
temperature
Electron relaxation model match well with the
experimental data
W. G. Ma, et al., Chin. Phys. B, 18, 2035, 2009
18
Modified W-F law
Surface Scattering:
Grain boundary scattering:
Electrical conductivity
Mayadas & Shatzkes theory (1969)
Fuchs & Sondheimer theory (1938)
2 3f
b
3 11 3 3 ln 1
2
Widemann-Franz law
Modified Widemann-Franz law
Bulk & monocrystalline: R=0 =0 Lf=L
0
19
Modified W-F law X. Zhang, et al., Chin. Phys. Lett., 25, 3360, 2008
0b
b
LT
f bf bf
f b0 f b
L
T TL
f
f
f
LT
2 3 *0
1
1 3 2 3 3 ln 1 1 1
fL
L T
20
Data vs. modified W-F law (Platinum
films)
Modified W-F law match well with the experimental
data
Interfacial effects dominate the thermal and
electrical transport
Modified W-F law
W. G. Ma, et al., Chin. Phys. B, 18, 2035, 2009
21
1.2 General conduction law
According to Einstein’s famous mass-energy
relation, Guo developed Thermomass theory.
Thermomass is defined as the equivalent mass
of thermal energy.
ED0
: thermal energy c: light speed Mh: thermomass
Heat has a dual nature of energy and mass.
Heat conduction can be analyzed with the
first principle.
Z. Y. Guo, et al, Acta Phys. Sin., 56, 3306, 2007.
2 0
2 D
h
EME M
cc
22
1.2 General conduction law
Kinematic similarity between the thermomass
and gas, both follow Newtonian mechanics.
The thermomass density and drift velocity
heat flux q
T1
T2
(T1>T
2)
gas flow
P1
P2
(P1>P
2)
Z. Y. Guo, et al, Acta Phys. Sin., 56, 3306, 2007.
2
Vh
C T
c
h
VC T
Qu
hh h 0
t
u
hh h h h h hp
t
uu u f
23
1.2 General conduction law
The general heat conduction equation can
be obtained:
The pressure of thermomass can be derived
from Debye state equation:
Resistance Driving
force
Spatial
inertia
Temporal
inertia
2
h h 2
V
V
C Tp C T
c
22 0h h vTM TM TMu C T Tt x
t t t
q q
u q
24
1.2 General conduction law
Resistance Driving force Spatial inertia Temporal inertia
Ignoring spatial inertia:
Ignoring temporal inertia:
Ignoring temporal & spatial inertia:
Fourier’s law
CV model
Nanoscale
22 0h h vTM TM TMu C T Tt x
t t t
q q
u q
0T q
0TM Tt
t
q
q
22 0h h vTM TMu C T Tx
t t
q
u q
25
How to prove?
Ignoring temporal inertia: Nanoscale
Furthermore, can be expressed as:
Spatial thermomass inertia
Steady non-Fourier heat conduction phenomena
occur at ultra-high heat flux conditions at very low
temperatures.
Thermomass is very small (8.4×10-9 kg in
1 kg gold at 300 K)
22 0h h vTM TMu C T Tx
t t
q
u q
2
2 3 3 3
1 05
2
I
h B
q dTq
dxn k T
26
How to prove?
Gold nanofilms (9.51 μm× 292 nm×76 nm)
Helium cryostat (3.2 K)
Large current density (3.24x1014
A m-2)&
large heat flux (2.2x1010
W m-2)
Due to the thermomass
inertia, the temperature
profile based on the
general heat conduction
law is higher than on
Fourier’s law. Ambient temperature: 50 K
Current: 7.19 mA
Evidence I (Influence of heat flux)
ΔT = TExperiment
– TFourier’s law
increases as the
heat flux increases.
Experimental results
Evidence II
(Influence of ambient temperature):
T0 ΔT
max
Experimental results
29
1.3 Interfacial thermal resistance
Interfacial thermal resistance between a solid
and superfluid helium was first detected by
Kapitza in 1941
Acoustic Mismatch Model (AMM)
Khalatnikov (1952)
Diffuse Mismatch Model (DMM)
Swartz and Pohl (1989)
interface
reflected
1
l , t1 , or t2
incidentl
t1
t2
transmitted
l
t1
t2
2
critical
angle
interface
reflected
1
l , t1 , or t2
incidentl
t1
t2
transmitted
l
t1
t2
2
critical
angle
Specular Diffuse
L. Shi, http://www.phys.ttu.edu
30
1.3 Interfacial thermal resistance
Yamane obtained the ITR by linear fitting the
thermal resistance of SiO2 films with different
thicknesses.
The ITR about 2x10-8
m2 K W
-1 (~30 nm SiO
2 film)
As the size is reduced, the relative importance of the
ITR will increase in thermal conduction.
T. Yamane, et al., J. Appl. Phys., 91(12): 9772, 2002.
A
A
B
B
A
A
B
B
Surface II
Surface I
31
1.4 Thermal contact resistance
Contact spot density
dominates the
temperature
distribution at the
interfaces when the
total contact area and
the loading pressure
are the same.
TCR decreases with
increasing contact
spot density for a
constant contact
pressure and average
surface roughness.
X. Zhang, et al., Int. J. Thermophys., 27, 880, 2006.
2. CONVECTIVE HEAT TRANSFER
32
The physical mechanisms for size effects can
be classified into two classes.
First: the continuum assumption still holds and
the interfaces only affect the macro parameters
Second: the interface affects not only the macro
parameters but also the micro parameters (MFP,
relaxation time, etc.). The continuum assumption
breaks down and Newton’s viscosity law and Fourier’s
heat conduction law are no longer valid.
Z. Y. Guo and Z. X. Li, Int. J. Heat Mass Transfer, 46, 149, 2003
2.1 Flow compressibility of gas in a microtube
2. CONVECTIVE HEAT TRANSFER
33
Flow rate measurement unit High-pressure nitrogen tank
Test microtube
Pressure transmitter
Thermocouple
Potentiometer Precise multi-voltmeter
Experiments for gas flow in microtube
Z. Y. Guo and Z. X. Li, Int. J. Heat Mass Transfer, 46, 149, 2003.
2.1 Flow compressibility of gas in a
microtube
34
Pressure drop is large along the flow direction.
Inlet Outlet p
Z. Y. Guo and Z. X. Li, Int. J. Heat Mass Transfer, 46, 149, 2003.
2.1 Flow compressibility of gas in a
microtube
35
Large pressure drop makes the Mach number
change significantly
Incompressible flow
parabolic
Compressibility
effect
Actual flow
Velocity profile departs from parabolic for laminar
flow in a circular tube
Z. Y. Guo and Z. X. Li, Int. J. Heat Mass Transfer, 46, 149, 2003.
2.1 Flow compressibility of gas in a microtube
36
The product of the friction coefficient and the
Reynolds number is no longer constant, but is
function of the Mach number
Effect of flow
compressibility
Z. Y. Guo and Z. X. Li, Int. J. Heat Mass Transfer, 46, 149, 2003.
2
2
16
Re 1.5 0.6
1
6
6
4Re 1.1
f
M
M MC
2.1 Flow compressibility of gas in a microtube
37
Heat transfer is markedly enhanced
Incompressible flow
parabolic
Compressibility
effect
Actual flow
The compressibility induces more flattened velocity
profiles
large velocity and temperature gradients near the
wall as the flow compressibility increases.
Z. Y. Guo and Z. X. Li, Int. J. Heat Mass Transfer, 46, 149, 2003.
2.2 Relative importance of the viscous
force over the inertial force
38
In microscale natural convectiion, the viscous
force dominate over the inertial force.
Inertial force ~ L2
Viscous force ~ L
Natural convection around small sized
2D vertical plate
Z. Y. Guo and Z. X. Li, Int. J. Heat Mass Transfer, 46, 149, 2003.
2.2 Relative importance of the viscous
force over the inertial force
39
Momentum equation
Inertial force Buoyancy force Viscous force
Natural convection around normal-sized object:
Driving force - Buoyancy force
Resistance force Inertial force (dominant)
Viscous force
2 U U U g T
40
Natural convection around normal-
sized 2D vertical plate
Buoyancy force Inertial force ~
~
~
Ratio of inertial to viscous force
Ratio of convection to conduction
U U g T
U
41
Small sized 2D vertical plate
Ratio of inertial to viscous force
Ratio of convection to conduction
Driving force - Buoyancy force
Resistance force Inertial force
Viscous force (dominant)
Buoyancy force Viscous force ~
~
~
2 U g T
U2g Tl
Different from normal-sized one
2.2 Relative importance of the viscous
force over the inertial force
42
Two-dimensional square cavity
2D natural convection in a square cavity for
Rayleigh numbers from 102 to 10
8
Boundary conditions:
Z. Y. Guo and Z. X. Li, Int. J. Heat Mass Transfer, 46, 149, 2003.
43
Two-dimensional square cavity
Flow fields for free convection:
Ra =102
Ra =106 Streamline
Z. Y. Guo and Z. X. Li, Int. J. Heat Mass Transfer, 46, 149, 2003.
44
Two-dimensional square cavity
The viscous, inertial and buoyancy forces:
Ra =102 Ra =10
6
Relative importance of the viscous force to
the inertial force markedly increases with
decreasing Ra
Z. Y. Guo and Z. X. Li, Int. J. Heat Mass Transfer, 46, 149, 2003.
45
Two-dimensional square cavity
Ratio of the inertial force to viscous force
decreases and the viscous force becomes
dominant with decreasing Ra
Z. Y. Guo and Z. X. Li, Int. J. Heat Mass Transfer, 46, 149, 2003.
46
Two-dimensional square cavity
Ra>106, same as the conventional case, Nu~Ra
0.33
103≤Ra ≤10
6, inertial force can be ignored, Nu~Ra
0.28
Ra<103, the natural convection is very weak and
heat conduction is dominant, Nu=1.
Z. Y. Guo and Z. X. Li, Int. J. Heat Mass Transfer, 46, 149, 2003.
47
2.3 Importance of roughness
Flow rate
measurement unit
High-pressure nitrogen tank
Test microtube
High-pressure liquid tank
Pressure regulator Precise multi-voltmeter
Experiments for liquid flow in microtube
Z. X. Li, et al., Microscale Thermophys. Eng., 7, 253 (2003)
48
2.3 Importance of roughness
Roughness of the tube
Glass tube Silicon tube Stainless steel tube
Glass tube and silicon tube are smooth
The relative roughness of stainless steel tube is
about 3.3~3.9％
Z. X. Li, et al., Microscale Thermophys. Eng., 7, 253 (2003)
49
2.3 Importance of roughness
Frictional factor in smooth microtube
f·Re~64 is almost the same as that in macrotubes
Flow transition from laminar to turbulent,
Re~2000-2300.
Incompressible flow as in macrotube.
Z. X. Li, et al., Microscale Thermophys. Eng., 7, 253 (2003)
50
2.3 Importance of roughness
Frictional factor in rough microtube
f·Re is 15-37% higher than the theoretical value
(relative roughness is less than 5%)
Early transition from laminar to turbulent,
Re~1800. Z. X. Li, et al., Microscale Thermophys. Eng., 7, 253 (2003)
3.1 Coherence
3. THERMAL RADIATION
51
20-μm-thick plane-
parallel Si film
X. G. Liang, Chin. Phys. Lett., 23, 1219 (2006)
Fourier Transformed
Infrared spectrometer
s-polarized p-polarized
Angular transmissivity
Wavelength is 5 μm
Transmission is directional with a number of distinct lobes
s- and p-polarized radiation present different transmission
spectra ---different reflection coefficients at the interfaces.
3.1 Coherence
52
Spectral normal transmissivity of the 20-μm-
thick plane-parallel Si film
X. G. Liang and M. H.
Han, Chin. Phys. Lett.,
23, 1219 (2006)
The normal transmissivity of the plane-parallel Si film
has a distinctly fluctuant pattern.
Varies more frequently for shorter wavelength and
more regularly for the longer
3.2 Evanescent
53
Evanescent wave:
- nearfield standing wave,
- extends about 1/2 ,
- decays exponentially with the distance
Evanescent field
x
E
3.2 Evanescent
54
600 K
300 K
d
Conduction regime: > 300 nm
Combined regime: ~ 50 nm
Radiation regime: <20 nm, 95% radiative heat flux
M. H. Han, et al., Sensors & Actuators A, 120, 397 (2005)
Simulation
4.CONCLUSIONS
55
As the size is reduced, the ratio of the surface
area to the volume increases, so the relative
importance of the interfacial effects increases.
The physical mechanisms for size effects have
been classified into two classes.
First: the continuum assumption still holds
and the interfaces only affect the macro
parameters
4.CONCLUSIONS
56
Second: the interface affects not only the
macro parameters but also the micro
parameters (MFP, relaxation time, etc.). The
continuum assumption breaks down and
Newton’s viscosity law and Fourier’s heat
conduction law are no longer valid.
The major characteristic of micro/nanoscale
heat transfer is that interfacial effects
dominate the heat transfer.
ACKNOWLEDGMENTS
Thank Professors Z. X. Li and X. G. Liang at
Tsinghua University for their contributions.
This work was supported by the National
Natural Science Foundation of China (Grant
Nos 50730006, 50976053 and 51136001).
57
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