Thermal conductance of solid-solid and solid-liquid interfaces

David G. Cahill,

Zhenbin Ge, Ho-Ki Lyeo, Xuan Zheng, Paul Braun

Frederick Seitz Materials Research Lab and Department of Materials Science

University of Illinois, Urbana

Interfaces are critical at the nanoscale

• Low thermal conductivity in nanostructured materials

– improved thermoelectric energy conversion

– improved thermal barriers

• High thermal conductivity composites and suspensions

50 nm

Interfaces are critical at the nanoscale

• High power density devices

– solid state lighting– high speed

electronics– nanoscale sensors

Micrograph of tunneling magnetoresistive sensorfor 120 GB drives, M. Kautzky (Seagate)

Interface thermal conductance

• Thermal conductance (per unit area) G is a property of an interface

• Thermal conductivity is a property of the continuum

Interface thermal conductance (2001)

• Observations (2001) span a very limited range

– Al/sapphire Pb/diamond– no data for hard/soft

• lattice dynamics (LD) theory by Stoner and Maris (1993)

• Diffuse mismatch (DMM) theory by Swartz and Pohl (1987)

Acoustic and diffuse mismatch theory

• Acoustic mismatch (AMM)

– perfect interface: average transmission coefficient <t> given by differences in acoustic impedance, Z=v

– lattice dynamics (LD) incorporates microscopics

• Diffuse mismatch (DMM)

– disordered interface: <t> given by differences in densities of vibrational states

• Predicted large range of G not observed (2001)

• For similar materials, scattering decreases G

• For dissimilar materials, scattering increases G

2005: Factor of 60 range at room temperature

nanotube/alkane

W/Al2O3

Au/surfactant/water

PMMA/Al2O3

Modulated pump-probe apparatus

f=10 MHz

rf lock-in

psec acoustics andtime-domain thermoreflectance

• Optical constants and reflectivity depend on strain and temperature

• Strain echoes give acoustic properties or film thickness

• Thermoreflectance gives thermal properties

Modulated pump-probe

Bonello et al. (1998)

• four times scales:

– pulse duration, 0.3 ps – pulse spacing, 12.5 ns – modulation period, 100 ns– time-delay, t

t

Analytical model for modulated time-domain thermoreflectance

• frequency domain solution for heat flow in cylindrical coordinates using gaussian beams.

• G(k) given by iterative solution (transfer matrix)

• In-phase and out-of-phase signals by series of sum and difference over sidebands

Iterative solution for layered geometries

Two basic types of experiments on solid samples

• thermal conductivity of bulk samples and thermal conductance of interfaces

• thermal conductivity of thin films

thermal conductivity map of cross-section of thermal barrier coating, with J.-C. Zhao (GE)

Flexible, convenient, and accurate technique...

• ...with 3 micron resolution

Interfaces between highly dissimilar materials

• high temperature limit of the radiation limit

max

max

D

: vibrational cutoff frequency of material A

( 1.8 THz for Bi, 2.23 THz for Pb)

v : Debye velocity of material B

3max23

b

D

kG

v

DOS

Material A (Pb, Bi)

Material B (diamond, BeO)

R. J. Stoner and H. J. Maris, Phys.Rev.B 48, 22, 16373 (1993)

Thermoreflectance data for Bi and Pb interfaces

t (ns)

0.05 0.2 0.5 2 50.1 1

-Vin

/Vou

t

2

5

1

10

Bi/H/Diamond

Pb/H/Diamond

Bi/H/Si(111)

Room temperature thermal conductance

• Pb and Bi show similar behavior. Electron-phonon coupling is not an important channel.

• Weak dependence on Debye velocity of the substrate.

• Pb/diamond 50% smaller than Stoner and Maris but still far too large for a purely elastic process.

Temperature dependence of the conductance

• Excess conductance has a linear temperature dependence (not observed by Stoner and Maris).

• Suggests inelastic (3-phonon?) channel for heat transport

Application: can we use interfaces to beat the minimum thermal conductivity?

• If the small thermal conductance of Bi/diamond could be reproduced in a multi-layered film, then placing interfaces every 10 nm would give an incredibly low thermal conductivity of 0.1 W/m-K (factor of 2 smaller than a polymer).

W/Al2O3 nanolaminates

• room temperature data

• sputtered in pure Ar

• atomic-layer deposition at 177 and 300 °C, S. George (U. Colorado)

• G = 220 MW m-2 K-1

50 nm

Unexpected advance: Thermal conductivity imaging

• At t=100 ps,

– in-phase signal is mostly determined by the heat capacity of the Al film

– out-of-phase signal is mostly determined by the effusivity (C)1/2 of the substrate

ZrO2:Y thermal barrier

• after 500 thermal cycles (1 h)

25 °C —>1135 °C—>25 °C

ZrO2:Y thermal barrier

• after 500 thermal cycles (1 h)

25 °C —>1135 °C—>25 °C

Solid-liquid interfaces: Two approaches

• Transient absorption measurements of nanoparticles and nanotubes in liquid suspensions.

– Measure the thermal relaxation time of a suddenly heat particle. If the particle is small enough, then we have sensitivity to the interface

– limited to interfaces that give good stability of the suspension

• Thin planar Al and Au films. Same as before but heat flows both directions: into the fluid and into the solid substrate.

Transient absorption

• Optical absorption depends on temperature of the nanotube

• Cooling rate gives interface conductance

G = 12 MW m-2 K-1

• MD suggests channel is low frequency squeezing and bending modes strongly coupled to the fluid.

Application: Critical aspect ratio for a fiber composite

• Isotropic fiber composite with high conductivity fibers (and infinite interface conductance)

• But this conductivity if obtained only if the aspect ratio of the fiber is high

HS(CH2)11

OOH

4

G = 2.0108

G = ∞ (a)

EG4

t (ps)

∆αl

(pp

m)

t (ps)

∆αl

(pp

m)

O

NH

C

SH O

OH

G = 1.8108

G = ∞ (b)

Tiopronin

Hydrophilic metal nanoparticles: 4 nm diameter Au:Pd nanoparticles in water

• transient absorption data

22 nm diameter Au:Pd nanoparticles in water; CTAB surfactant

t (ps)

G = 1.8 108

G = ∞

∆αl

(pp

m)

Nanoparticle summary

G ~ 200 MW m-2 K-1

S

O

OH

S

O

OH

S

O

OH

SS

SS

HNO

OHO

S

HNO

OHO

N

7

Br

N

7

Br

N

7

Br

N

7

Br

N

7

Br

N

7

Br

In water In Toluene

G ~ 15 MW m-2 K-1

Application: Critical particle radius for nanocomposite

• Interface conductance and thermal conductivity of the fluid determine a critical particle radius

• For particles in water, rc = 3 nm.

• For high thermal conductivity particles, dilute limit of effective medium theory

rc = G

r >> rc

r << rc

Thermoreflectance of solid-liquid interfaces

hydrophobic

37 MW/m2-K

hydrophilic

150 MW/m2-K

no water

Conclusions

• Much to learn about transport of heat across interfaces but we now have good tools.

• Pb/diamond, Bi/diamond interfaces show a temperature dependent conductance far above the radiation limit. What is the correct description of this inelastic channel?

• Can circumvent the “minimum thermal conductivity” with high densities of interfaces.

• Conductance of hydrophilic nanoparticle/surfactant/water interfaces is essentially independent of the surfactant layer.

• Heat transfer is reduced by a factor of 4 at hydrophobic interfaces with water.