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arXiv:1103.2389v1 [physics.optics] 11 Mar 2011

Near-field radiative heat transfer between macroscopic planar surfaces

Richard Ottens,1V. Quetschke,2Stacy Wise,1, ∗A.A. Alemi,1, †R.

Lundock,1, ‡G. Mueller,1D.H. Reitze,1D.B. Tanner,1and B.F. Whiting1

1Department of Physics, University of Florida, P.O. Box 118440, Gainesville, FL 32611-8440, USA

2Department of Physics and Astronomy, University of Texas at Brownsville, 80 Fort Brown, Brownsville, TX 78520, USA

(Dated: March 15, 2011)

Near-field radiative heat transfer allows heat to propagate across a small vacuum gap in quantities

that are several orders of magnitude greater then the heat transfer by far-field, blackbody radia-

tion. Although heat transfer via near-field effects has been discussed for many years, experimental

verification of this theory has been very limited. We have measured the heat transfer between two

macroscopic sapphire plates, finding an increase in agreement with expectations from theory. These

experiments, conducted near 300 K, have measured the heat transfer as a function of separation

over mm to µm and as a function of temperature differences between 2.5 and 30 K. The experiments

demonstrate that evanescence can be put to work to transfer heat from an object without actually

touching it.

PACS numbers: 44.40.+a,78.20.Ci

Humans knew of radiative heat transfer at least as

early as the discovery of fire, and physicists have investi-

gated this process for centuries, culminating in the black-

body theory of Planck and the birth of the quantum the-

ory. Planck’s equation for black-body radiation contains

only the temperature and some fundamental constants.

When actual materials are involved, their emissivities en-

ter the discussion, but little else. For example, the heat

transfer per unit area between two semi-infinite planes

is set by their temperatures and integrated emissivities

but does not depend on their separation or other geo-

metrical quantities. When the two planes approach each

other closely the situation changes.

regime, each material interacts with exponentially de-

caying evanescent electromagnetic fields generated in and

existing outside the other material; these fields can drive

currents and generate heat.[1–3] This near-field radiative

heat transfer can be several orders of magnitude greater

than far-field blackbody radiation.

In this near-field

Much like the Casimir and van der Waals force, near-

field heat transfer deals with fluctuations that only exist

over small distances. The first in-depth theory for near-

field heat transfer between planar surfaces was derived by

Polder and Van Hove,[2] building on the work of Rytov[1].

There have been several other theoretical approaches,

and in general the theory seems complete, except per-

haps at distances comparable to atomic dimensions.[4]

Although heat transfer via near-field effects has been

discussed for many years, experimental verification of the

theory for heat transfer between two planar surfaces has

been limited. Hargreaves[5] has presented room tem-

perature observations for two Cr surfaces at distances

as small as 1 µm. Domoto et al.[6] reported results at

cryogenic temperatures but for relatively large (50 µm)

separations, where near-field effects were barely observ-

able. Neither study compared experiment to theory. A

comparison at a fixed spacing has been put forward, but

the plates were separated by polyethylene spacers, so the

distance could not be varied.[7] There have also been sev-

eral recent results using a sphere-plane geometry.[4, 7–9]

There are engineering reasons for this approach, as, un-

like a parallel plane geometry, a sphere-plane geometry

needs no angular alignment. Using scanning probe and

micro-machine technologies, these experiments cover a

wide distance range.

In this paper, we report a set of measurements of heat

transfer between two planar surfaces and present a de-

tailed comparison to theory. In addition to its intrin-

sic interest, this work was motivated by possible ap-

plications in cooling objects without actually touching

them, such as the mirrors of a future laser interferometer

gravitational-wave detector.[10] This application would

require the parallel plane geometry in order to get large

areas of close approach of the two objects and thus sig-

nificant heat transfer.

The near-field heat-transfer process can be thought

of as a form of frustrated total internal reflection.

Evanescent waves, exponentially-decaying electromag-

netic fields, exist outside, but near to, the surface of

a material medium at temperature T. These decaying

fields are a consequence of travelling waves inside the

medium experiencing total internal reflection; this phe-

nomenon occurs when there is no valid solution to Snell’s

Law, nisinθi= ntsinθt, where niand ntare the indices

of refraction and θiand θtthe angle between wave vec-

tor and surface normal on the two sides of the interface.

Although there is no energy transmission across the inter-

face, an electric field exists on the far side. Furthermore,

if another medium is brought near this exponentially de-

caying field some of the energy from the incident beam

will propagate across the gap and into the new mate-

rial. If one medium is hotter than the other, this photon

tunneling will lead to heat transfer from hot to cold.

Polder and Van Hove[2] considered two half spaces of

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identical material but different temperatures. The com-

puted heat transfer coefficient W comes from the tem-

perature derivative of the z component of the Poynting

vector (Sz) from each medium, which is evaluated just in-

side surface of the other medium. W is the temperature

derivative of this difference as can be seen in Eq. 1.

W =lim

T1−T2→0|Sz(d+) − Sz(0−)

T1− T2

| =∂Sz(d+)

∂T1

(1)

It is convenient to break W into two components

W = Wsin+ Wexp.(2)

Wsinis the part where the wave number kx of the field

is in the range 0 < kx< ω/c, where the field is propa-

gating, and which gives the ordinary (Stefan-Boltzmann)

far-field radiation. Wexpis the component where the field

is exponentially decaying away from the surface; it is the

larger contributor to the near-field limit:

Wexp=

?∞

0

dω

?∞

ω/c

dkx

kx

4π2(Texp

?

+ Texp

⊥

)∂[

?ω

e?ω/kT−1]

∂T

(3)

where the energy transmission coefficients are

Texp

P

=

1 − cos(2χP)

cosh[2κ(d − δP)] − cos(2χP),(4)

with P = ? or ⊥ for the two polarizations and χ?and

χ⊥being the phase shifts on reflection

χ?= arg[(−iκǫ + kz)(−iκǫ − kz)∗], (5)

χ⊥= arg[(−iκ + kz)(−iκ − kz)∗], (6)

with ǫ the material dielectric function. δ?and δ⊥can be

derived from

e2κδ?=

????

iκǫ + kz

iκǫ − kz

????

????

2

, (7)

e2κδ⊥=

????

iκ + kz

iκ − kz

2

, (8)

and kz and κ are the z component of k for the medium

and the vacuum, respectively

kz=

?

(ǫ − 1)ω2/c2− κ2, (9)

κ = ikzv=

?

k2

x− ω2/c2.(10)

Note that kx> ω/c, so that kzv is imaginary above the

surface and the energy transmission coefficients Texp

Texp

⊥

are not derived from the energy reflection coeffi-

cients R?and R⊥.[2] We have used this theory to calcu-

late the heat transfer coefficient between sapphire plates,

using Barker’s sapphire dielectric function.[11]

?

and

FIG. 1. Heat transfer coefficient vs. distance for z-cut sap-

phire. The temperatures of the two media are Thot = 310

K and Tcold = 300 K, respectively. The violet curve shows

W?

sin+ W⊥

shows W?

terms dominate at close distances. The blue curve is the total

heat transfer coefficient W.

sinwhich dominate at far distances. The red curve

exp. The green curve shows W⊥

exp. The evanescent

Figure 1 shows the prediction of the model for z-

cut sapphire; the temperatures of the two media are

Thot= 310 K and Tcold= 300 K. The Wsin term dom-

inates at large separation and is nearly constant with

separation. The Wexpterm dominates in the near-field

regime, so that the total heat transfer changes from be-

ing independent of separation at large distances towards

Wexp∝ 1/d2at short distances. The turning point be-

tween Wsinand Wexpoccurs when the separation is ap-

proximately equal to the peak wavelength of the black-

body curve of the hot half space. Wien’s displacement

law predicts that λmax≈ 9 µm when the temperature is

310 K.

A sketch of our apparatus is shown in Fig. 2. It is de-

signed around two 50×50×5 mm3sapphire plates. These

have a specified flatness of λ/8 @ 633 nm per inch on the

largest surfaces and are cut such that the c axis is per-

pendicular to these surfaces (z-cut). Sapphire was used

because it has good thermal conductivity. It is also a

candidate for the test masses of future gravitational-wave

detectors.

One of the plates, henceforth called the cold plate, is

attached to the thermal bath that is the vacuum cham-

ber. The other plate, the hot plate, is thermally isolated

from the bath by a Macor spacer attached to the back

side of the hot plate. The hot plate also has a heater

wound on a copper ring which is itself attached to the

back of the plate. The heater current and voltage, after

correcting for lead resistance, give the power required to

maintain a given temperature difference between the hot

and cold plates. Both plates have a Si-diode thermome-

ter fastened to their backs to read the temperature (and

for the hot plate to control it). Both plates have all four

corners coated with an approximately 200 nm thick layer

of sputtered copper. These coatings have areas about

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FIG. 2. Experimental apparatus. Stepper motors allow ad-

justment of the spacing, tip, and tilt (read capacitively) of two

sapphire plates. The temperature of the hot plate is controlled

by a feedback circuit, and the power required to maintain a

temperature difference gives the heat transfer from the hot

plate to the cold plate and to the thermal bath.

1 mm2and serve as capacitor plates that are read by

four 24-bit capacitance-to-digital converter circuits[12] to

measure the separation and angular misalignments of the

plates. The metal film is wrapped around to the sides of

the sapphire to allow electrical contact to the electrodes.

The cold plate is glued to a copper disk, which in turn

is attached to the experimental structure. The Macor

spacer on the back of the hot plate is attached to a modi-

fied kinematic mirror mount which allows for z-axis linear

movement and tip and tilt angular adjustment by turn-

ing the three adjustment screws in the back. Three step-

per motors turn screws on the kinematic mount via gear

reduction boxes; each motor step translates to a linear

movement of the hot plate by 35 nm. The components

are held together by an “L” shaped backbone (not shown)

to give rigidity. The assembly is located in a UHV cham-

ber, with a base pressure below 2 × 10−7Torr, making

gas conduction negligible. Signals to the stepper motors,

capacitance readouts, temperature readouts, and current

and voltages to the heater are all controlled and/or read

by a LabVIEW computer program.

Each pair of capacitor plates is calibrated by taking ca-

pacitance readings as the plates are driven together one

step of the stepper motors at a time. A fit is made to

C = ǫ0a/d + Cstraywhere ǫ0is the dielectric constant of

the vacuum, a is the capacitor area, and Cstrayis a par-

allel contribution independent of separation. The data

fits the equation above very well, with R2values greater

than 0.999. The fitted value of a equals the metalized

area within our knowledge of this area; Cstray≈ 0.4 pF.

The average capacitance gives the distance while the in-

dividual readings are used to correct the alignment by

sending steps to the motors controlling tip and tilt.

To measure the heat transfer coefficient we compute

FIG. 3. Heat transfer coefficient vs. distance. The curves

are each offset vertically by 2 W/m2·K from the one below.

The points are the data, with error bars, determined from the

scatter in the heat transfer measurements and the uncertainty

in the distance calibration. The solid lines are the theoretical

predictions for flat plates while the dashed lines are the theo-

retical predictions for slightly curved plates (see text). Each

measured curve has a reproducible addendum due to other

heat leaks which are not included in the model and which has

been subtracted from the data. The temperatures are (top to

bottom): Thot = 327.0 K, Tcold = 308.0 K; Thot = 322.0

K, Tcold = 307.0 K; Thot = 317.0 K, Tcold = 305.8 K;

Thot= 312.0 K, Tcold= 305.2 K.

W = P/[A(Thot− Tcold)], where P is the power dissi-

pated in the heater, A is the plate surface area, and Thot

and Tcoldare the temperatures of the hot and cold plates,

respectively. The data are a sum of parallel heat path-

ways, including thermal conduction through the Macor

spacer and the other parts of the hot-plate holder, radi-

ation to the thermal bath, and the contributions of the

near- and far-field radiation between the two plates. We

can observe the near-field effect because it is the only

one of these that will change with plate separation. The

thermal conduction and radiation-to-the-bath paths just

add a constant offset to the radiative heat transfer that

the model predicts.

Data for the heat transfer coefficient versus distance

were collected for four temperature differences.

Fig. 3.) Each one shows that near-field heat transfer ex-

ists and that the data and model agree reasonably well.

Each run covers a separation range of about 2-100 µm.

The only freedom in the fit is an offset to the model. All

our measurements have an offset of 0.0435±0.0004W/K,

completely consistent with thermal conduction through

(See

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FIG. 4. Near-field heat flux vs. distance for multiple temper-

ature differences and multiple runs. (One run is shown with

open symbols and other runs with filled symbols.) The inset

shows the near-field heat flux vs. temperature difference for

several specific separation distances.

the Macor spacer and the rest of the support for the hot

plate, in parallel with radiation from the rear surface of

the plate. The chamber pressure for these runs ranged

between 5 × 10−7Torr and 2 × 10−7Torr. The vac-

uum chamber is held at constant temperature of 30.0◦C

(303.2 K). The hot plate is brought close to the cold

plate step by step. Each datum is an average of a set of

500 values, in both heat transfer coefficient and distance,

taken after the system has reached thermal equilibrium.

It takes 30 to 45 minutes after moving to a new position

to reach thermal equilibrium. We calculate the average

and its standard deviation and plot these in Fig. 3.

Theoretical curves match the experimental data well.

However, although the agreement is within experimen-

tal errors, there does appear to be a systematic offset:

the theory predicts a slightly lower heat transfer coeffi-

cient at each separation than we measure. Alternatively,

the plates could be slightly closer than measured by the

capacitive readout. We believe that the latter explana-

tion is correct. Simulations of the heat transfer between

two convex plates (shown as dashed lines) eliminate the

systematic error when the radius of curvature in each

plate is ∼1 km,corresponding to deviations from flatness

of 500 nm. Subsequently, we measured the plate cur-

vatures using optical flats and green mercury light; the

Newton’s rings interference pattern gave a central dis-

placement of 170±30 nm with respect to the perimeter,

a little smaller but of the same order of magnitude as the

simulation gave.

Figure 4 shows the heat flux caused by near-field ef-

fects. Both far-field heat transfer and the offset due to

heat leaks to the thermal bath have been subtracted.

For each temperature difference, data are shown for sev-

eral distinct data runs; these agree very well. The inset

shows the dependence on the temperature difference of

the plates. For distance values where the near-field ef-

fects dominate, the heat flux is linear in the temperature

difference. Both data and model were fitted to

φ(∆T,d) = G(d)(∆T)α(d)

(11)

where φ(∆T,d) is the total near-field heat flux, G(d)

is a multiplicative factor, and α(d) is the exponent for

∆T. Each curve follows a power law in ∆T, with the

exponent varying from 0.70 at small distances to 0.91 at

larger distances. The differential heat transfer, Eq. 1, has

α(d) = 1; the finite temperature differences used in the

experiment bring in higher-order terms.

In summary we have measured near-field heat trans-

fer across a small gap for a parallel-plane geometry. The

data agree quite well with the theory of Polder and Van

Hove[2]. The experiments demonstrate that significant

amounts of heat can be transferred via evanescent radi-

ation in the near-field regime.

Our research is supported by the National Science

Foundation through Grant No. PHY-0855313. A.A.A.

was supported by the University of Florida Physics REU

Site through NSF grant DMR-0552726.

∗Current address:

mospheric Science, Dalhousie University, Halifax, NS

B3H3J5, Canada

†Current address: Department of Physics, Cornell Univer-

sity, Ithaca, NY 14853-2501, USA

‡Current address: Astronomical Institute, Tohoku Uni-

versity, Aoba, Sendai 980-8578, Japan

[1] S. Rytov, Theory of Electric Fluctuations and Thermal

Radiation (Academy of Sciences Press, Moscow, 1953).

Translation in AFCRC-TR-59-162 (1959).

[2] D. Polder and M. Van Hove, Phys. Rev. B 4, 3303 (1971).

[3] Jackson J. Loomis and Humphrey J. Maris, Phys. Rev.

B 50, 18517 (1994).

[4] A. Kittel, W. M¨ uller-Hirsch, J. Parisi, S.A. Biehs, D.

Reddig, and M. Holthaus, Phys. Rev. Lett. 95, 224301

(2005).

[5] C.M. Hargreaves, Phys. Lett. A 30, 491 (1969).

[6] G.A. Domoto, R.F. Boehm, and C.L. Tien, J. Heat

Transfer 92, 412 (1970).

[7] A. Narayanaswamy, S. Shen, and L. Hu, Appl. Phys. A

96, 357 (2009).

[8] E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J.

Chevrier, and J. Greffet, Nature Photonics 3, 514 (2009).

[9] S. Shen, A. Narayanaswamy, and G. Chen, Nano Lett. 9,

2909 (2009).

[10] S. Wise, Sensitivity Enhancement in Future Interferomet-

ric Gravitational Wave Detectors, Ph.D. thesis, Univer-

sity of Florida, 2006. (unpublished)

[11] A. Barker Jr., Phys. Rev. 132, 1474 (1963).

[12] Analog Devices, Inc., AD7746: 24-Bit Capacitance-to-

Digital Converter with Temperature Sensor (2005).

Department of Physics and At-