Proceedings of the 15th International Heat Transfer Conference, IHTC-15 August 10-15, 2014, Kyoto, Japan
IHTC15-9188
REDUCING THERMAL RADIATION BETWEEN PARALLEL PLATES IN THE FAR-TO-NEAR FIELD TRANSITION REGIME Yoichiro Tsurimaki1,2*, Pierre-Olivier Chapuis1, Rodolphe Vaillon1, Junnosuke Okajima2, Atsuki Komiya2, Shigenao Maruyama2 1
Université de Lyon, CNRS, INSA-Lyon, UCBL, CETHIL, UMR5008, F-69621 Villeurbanne, France 2 Insutitute of Fluid Science, Tohoku University, 2-1-1, Katahira, Aoba-ku, Sendai, 980-8577, Japan
ABSTRACT The present work investigates the conditions of interference that allow decreasing the radiative heat transfer between two semi-infinite parallel plates. For this purpose, net radiative heat transfer fluxes exchanged by the plates are calculated as a function of the distance between them for various dielectric and metallic materials. A decrease of the propagative component larger than 85% is observed in the case of highly-reflective metals, while a poor reduction of few percents is observed for dielectric materials. However, the appearance of the evanescent contribution becomes significant before this reduction starts, resulting in a total flux decrease of only few percents even in the case of metals. An analysis as a function of temperature shows that the decrease of the total flux is more significant at low temperatures despite the fact that the decrease of the propagative component is larger at high temperatures. This underlines that the distance at which interferences of propagative waves appear and the distance at which evanescent waves appear are different. In the case of aluminum, the heat flux minimum and the beginning of the interferences are respectively found at T.d=900m K and T.d=4000m K. Spectral and directional analyses of propagative radiative heat fluxes are also performed and show that resonances occur at frequencies given by the waveguide theory in specific directions.
KEY WORDS: Nano/Micro scale measurement and simulation, Radiation, Thermal insulation, Interferences 1. INTRODUCTION It is now well established that the tunneling of thermally-excited evanescent electromagnetic waves induces an increase of radiative heat transfer, which can exceeds the blackbody limit valid for the far-field [1]. This tunneling happens when the distance between the hot and the cold bodies exchanging heat through thermal radiation becomes smaller than the dominant wavelength of the thermal spectrum. When the distance becomes comparable to the dominant wavelength, effects of coherence of thermal radiation may also appear because reflected propagative waves can interfere. These effects are generally overlooked and will be the focus of this paper. It has been shown that interference of thermally-excited waves in arrays of micron-size cavities [2] can explain some features of experimental far-field emission spectra. Our purpose is to observe if coherence effects are also significant for the heat exchange between two nonstructured surfaces. In the far-field regime, the radiative heat flux is independent of the distance between the bodies provided it is large enough. In the near-field, it strongly depends on the distance since evanescent waves which locally exist near surfaces become important and the radiative heat flux is larger than the radiative heat flux in the far field. In the transition regime of interest, near field effects start to emerge in addition to the effects of temporal coherence of thermal radiation. *Corresponding Author: tsurimaki @pixy.ifs.tohoku.ac.jp
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IHTC15-9188 Thermally-radiated propagative waves undergo interferences because they can be reflected by the surfaces of the bodies. As a consequence, interferences of thermally-radiated propagative waves give rise to constructive and destructive components. Thus, only specific waves with selected wavelengths can be observed between the bodies as a result of the interferences. It can be considered that radiative heat transfer through the selected waves results in the reduction of radiative heat transfer between the bodies if the contribution from evanescent waves is not so strong, in such a way that they do not hide the decrease of the propagative component. Therefore, the radiative heat flux is expected to decrease under some conditions in the transition regime, just before the enhancement by the evanescent waves. It is important to realize that these two effects, namely the increase due to tunneling of the evanescent waves and the reduction of the propagative waves due to destructive interferences, can compete for certain distance ranges between the bodies. In previous theoretical papers [3-5] addressing the case of semi-infinite parallel plates, a reduction of the propagating component of radiative heat flux caused by interferences was observed, but a significant decrease of the total heat exchange considering propagating and evanescent waves could not be observed because the evanescent contribution was overriding the drop caused by the interferences. In the present study, we search for conditions that maximize the drop while avoiding its disappearance because of the evanescent waves. The conditions depend on material properties of the bodies, temperatures of the bodies, and the distance between the bodies. For that purpose, the net radiative heat transfer flux exchanged by two semiinfinite parallel plates is calculated as a function of the distance between the plates for various dielectric and metallic materials using the expression proposed by Polder and Van Hove [3]. Spectral and directional analyses are also conducted. A radiative heat transfer coefficient between the two bodies is calculated as a function of the temperatures of the two bodies and as a function of the distance between the two bodies. As a potential application, this work could be a basis for the development of new thermal insulation systems.
2. RADIATIVE HEAT FLUX IN THE ELECTROMAGNETIC APPROACH
Fig.1 Two semi-infinite parallel plates. We consider two semi-infinite parallel plates 1 and 3 separated by a vacuum gap of thickness d as depicted in Fig.1. The net radiative heat flux q between the two plates is composed of a propagative flux qprop and an evanescent flux qevan, and is calculated by using the following expressions [3]: (, T1 ) (, T3 ) q prop d 0 8 3
/c
0
(1 r s 2 )(1 r s 2 ) (1 r p 2 )(1 r p 2 ) 21 23 21 23 2KdK 2 s s 2ik z 2 d p p 2ik z 2 d 2 1 r 21 r 23 e 21 r 23 e 1 r
(1)
s s p p Im(r Im(r 21 ) Im(r 23 ) 21 ) Im(r 23 ) s s 2 k z2d 2 p p 2 k z2d 2 1 r r e 1 r 21 23 21 r 23 e
(2)
and q
evan
(, T1 ) (, T3 ) d 0 2 3
/c
2KdKe
2 k z2d
2
IHTC15-9188 where (, T ) /(exp( / k BT ) 1) is the mean energy of Planck oscillator, K and kz are the parallel and vertical wave vectors, respectively. kzj is a complex number and is calculated by
kzj kzj ikzj j ( / c)2 K 2 . rijs and rijp are the Fresnel reflection coefficients of s-polarized and ppolarized waves at the interface between the medium i and j respectively and are expressed as: rijs rijp
k zi k zj
(3)
k zi k zj
j k zi i k zj j k zi i k zj
(4)
where i is the dielectric function of medium i ( 2 =1 in vacuum). A radiative heat transfer coefficient is defined as the temperature derivative of the radiative heat flux and is given by:
q q prop q evan lim lim h prop hevan T 0 T T 0 T T 0 T
htotal (d , T ) lim
(5)
where T T3 T1 . In this paper, the net radiative heat flux is calculated for several metallic and dielectric materials. Their dielectric functions are calculated by using a Drude model for metals and a Lorentz-Lorenz model for dielectrics. The expressions are given as follows:
1
2p ( i )
(6)
2 2 LO i 2 2 TO i
(7)
where p is the plasma frequency, and are the damping coefficients, is a constant at high frequency, and LO , TO are the angular frequencies of the longitudinal and transverse optical phonons respectively. Materials considered in this paper are aluminum and gold for metals, silicon carbide and crystalline boron nitride for dielectrics. The parameters are p = 2.24 1016rad/s, =1.22 1014rad/s for aluminum [6], p = 1.37 1016rad/s, =4.05 1013rad/s for gold [7], =6.7, LO = 1.827 1014rad/s, TO =1.495 1014rad/s, =8.971 1011rad/s for silicon carbide [7] and =4.46, LO = 2.451 1014rad/s, TO =1.985 1014rad/s, =9.934 1011rad/s for crystalline boron nitride [8].
3. RESULTS AND DISCUSSION 3.1 Radiative heat flux in the far-to-near field Figures 2 (a) and (b) show the net radiative heat flux normalized by the flux in the far-field as a function of the distance between the two plates respectively in the case of Al-Al and SiC-SiC. The temperatures of the surfaces are 400K and 300K. In these Figures, the propagative and the evanescent components, and for each of them the s-polarized and p-polarized components, are separately shown. In the case of Al-Al, it is observed that the propagative component starts to become smaller than the far-field value at a distance around 20m, reaching a minimum around 2m. This decrease is due to interferences of thermally-radiated waves. In the case of SiC-SiC, the reduction of the propagative component is not as drastic as in the case of Al-Al, because of the lower reflectivity of dielectric
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IHTC15-9188 2
1
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Normalized radiative heat flux [-]
Normalized radiative heat flux [-]
10
2
q far field 6.7Wm
0
10
-1
10
Prop. s waves Prop. p waves Prop. total flux Total flux
-2
10
Evan. s waves Evan. p waves Evan. total flux
Evan. s waves Evan. p waves Evan. total flux
1
10
q far field 561Wm2 0
10
-1
-3
10
Prop. s waves Prop. p waves Prop. total flux Total flux
1
10 Distance d [m]
10
100
0.1
1
10
100
Distance d [m]
(b)
(a)
Fig. 2 Normalized net radiative heat flux as a function of the distance between the two plates in the case of (a) Al-Al and (b) SiC-SiC. materials. For the case of Al-Al, there is an increase of the radiative heat flux due to the evanescent waves concomitantly with the decrease of the propagative component, hiding the reduction of the propagative component. As a result, an only slight reduction of the total radiative heat flux is observed. In the case of SiC-SiC, there is no decrease of the total radiative heat flux at all. For the purpose of making a quantitative analysis of the results, fluxes are normalized by the far-field values and several parameters are introduced in the following equations: s R prop
p , prop total, prop total s , prop qmin qmin qmin qmin p total total R R R , , , prop prop , prop p , prop , prop q sfar q far q total q total field field far field far field
(8)
These values are given in Table 1 in the case of Al-Al, Au-Au, SiC-SiC, cBN-cBN and Al-cBN. Reductions of 85% and 92% are observed for the propagative component of the net radiative heat flux in the case of AlAl and Au-Au, respectively. For the net total radiative heat flux, reductions of 7% and 5% are observed for the same materials. In the case of SiC-SiC and cBN-cBN, there is no decrease of the net total radiative heat flux although it is observed that the propagative components show few percents of reduction. The case with a combination of a metallic (Al) and a dielectric (cBN) materials exhibits a behaviour which is a mixing of what is observed for metals or for dielectrics only. As a consequence, a remarkable reduction of the propagative component appears to be only possible in the case of metals. A decrease of the total flux appears also to be possible but may not be large enough to be easily measured experimentally.
s R prop
Table 1 Summary of the ratios for various combinations of materials Al-Al Au-Au SiC-SiC cBN-cBN 0.021 (1.4m) 0.015 (1.2m) 0.86 (2.5m) 0.86 (4m)
Al-cBN 0.63 (2m)
p R prop
0.18 (2m)
m)
0.95 (6m)
0.93 (4m)
0.74 (2m)
R total prop
0.15 (1.8m)
0.079 (1.8m)
0.93 (6m)
0.90 (4m)
0.72 (2m)
R total
0.93 (3m)
0.95 (3.5m)
1
1
1
3.2 Spectral analysis We will restrict our analysis to a metal, aluminum, and to a dielectric, crystalline boron nitride. Figures 3 (a) and (b) show the integration over the parallel wave vector in eq. (1) as a function of the wavelength for several distances in the case of Al-Al and cBN-cBN, respectively. The distance of 1m corresponds to the far-field configuration. In the case of Al-Al, a stair-like shape is present and it is observed
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IHTC15-9188 that sudden increases occur near the particular wavelengths for each distance, which can be expressed as follows: 2d n
(9)
where d is the distance between the two plates and n is a natural number. This equation expresses the resonance conditions given by the waveguide theory. In reality, the resonances observed in Fig. 2 (a) occur at slightly-shifted wavelengths due to the imaginary part of the Fresnel reflection coefficient. Thus, the thermally-radiated waves with the wavelengths given by eq. (9) exist between the plates as a result of the constructive interferences. When the distance is 5m, the integration over the parallel wave vector K is on the order of 102 smaller than the far-field value at the wavelengths around 10-100m as a result of the destructive interferences. This leads to the reduction of the propagative component of the radiative heat flux that was observed in Fig. 2. In the case of cBN-cBN, there is no stair-like behaviour and slight wave features are observed. It is concluded from these facts that the decrease of the propagative component of the radiative heat flux is brought by interferences of thermally-radiated waves and that materials with high reflectivity lead to a larger reduction of the propagative component.
12
13
10 d=1m d=5m d=10m d=1m
11
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9
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d=1m d=5m d=10m d=1m
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Integral over wave vector K
Integral over wave vector K
10
1
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2
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10
3
1
2
10
10
10
3
10
Wavelength [m]
Wavelength [m]
Fig. 3 Integral over the parallel wave vector K in eq. (1) as a function of the wavelength for several distances in the case of (a) Al-Al and (b) cBN-cBN.
3.3 Directional analysis In the spectral analysis, we considered the integration of the contribution of thermal radiation over all the emission angles. In this section, directional behaviours of thermally-radiated waves between the two plates are discussed. For this purpose, we consider the transmission term Zprop,j (j=s, p) for propagative waves that appears in eq. (1), written as:
Z prop, j
(1 r 21
j 2
)(1 r 23
2ik z 2 d 1 r 21 r 23 e j
j
j 2 2
)
( j=s, p )
(10)
Figure 4 shows the contour plot of the transmission term of the p-polarized component as a function of the direction of emission and as a function of the angular frequency in the case of Al-Al for d=5m. On the contour plot, we observe loci which have greater magnitudes. Note that at the vertical line where the emission angle is zero, representing the direction perpendicular to the plates, it is observed that the angular frequencies associated with the loci correspond to the angular frequencies at resonance frequencies given by eq. (9), namely the constructive interferences. Hence, in the perpendicular direction, we can depict the resonance modes in the conventional way stemming from the waveguide theory. For non-zero emission
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IHTC15-9188 angles, the loci are upward-sloping curves: this means that when an angular frequency increases, the resonant mode exists at a larger angle. To analyse this behaviour, the transmission term in the polar coordinate is plotted on the right hand side of Fig. 4. Note that in this figure, the directional behaviour is symmetric since it is assumed that the emission is independent of the azimuthal angle. When we consider a wave with an angular frequency of 2.2 1014rad/s, the first resonant mode (n=1) exists at the emission angle of around 30 degrees, and the first resonant mode exists at the emission angle of around 50 degrees for a wave with a larger angular frequency 3.0 1014rad/s. Thus, as the angular frequency increases, the constructive interferences occur at larger angles. This behaviour of the resonant waves can be explained by considering the optical-path length between the two plates. Constructive interferences occur when the phase difference after one reflection is an integer multiple of 2 . Accounting for the phase difference due to absorption , the condition of resonance is given by: 2
2d cos 2n
n=1,2,3...
(11)
The expression of the phase difference due to absorption is quite complex and depends both on the dielectric function and on the wave vector. Therefore, it is concluded from this section that only waves resulting from constructive interferences can exist between the two plates, and that they exist in given specific directions. This is a result of multi-reflections. As a consequence, to achieve strong interferences of thermally-radiated waves, it is indispensable that the materials of the two plates have a high reflectivity. This can then lead to a decrease of the radiative heat transfer.
3 1014 rad/s 2.2 1014 rad/s
n=1
n=2
Fig. 4 Contour plot of the transmission term Z prop , j in the case of Al-Al for d = 5m (left-hand side) and the transmission term in polar coordinates (right-hand side). The colour scale is logarithmic.
3.4 Temperature and distance dependence of the radiative heat transfer coefficient The radiative heat transfer defined by eq. (5) is also a function of the temperatures of the plates. Figure 5 shows the propagative component of the radiative heat transfer coefficient normalized by the one in the far-field as a function of the distance for several temperatures ranging from 0.02K to 1000K in the case of Al-Al. Note that the temperature-dependence of the dielectric function is here neglected. Reductions of the propagative
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IHTC15-9188 component of the radiative heat transfer coefficient are observed at longer distances for lower temperatures. Thus it can be said that the characteristic distance for appearance of temporal coherence is longer at a lower temperature. In addition, the reductions are significant at higher temperatures since they can benefit from the larger drop due to the destructive interferences as shown in Fig. 3 (a). Also, the propagative components show proportionality to d-1 at smaller distances after reaching their minimum.
10
4
10
3
10
2
10
1
10
0
T=0.02K T=0.1K T=1K T=10K T=100K T=300K T=1000K
d 1
h
prop
(d,T) / h
total far-field
Figure 6 shows the total radiative heat transfer coefficient normalized by the one in the far-field as a function of the distance for several temperatures from 0.02K to 1000K in the case of Al-Al. It is observed that contrarily to the propagative components considered before, the total radiative heat transfer coefficient decreases significantly for the lowest temperatures. This is due to the fact that the evanescent component appears at smaller distances for a lower temperature, and cannot hide the reduction of the propagative component in this case. This is highlighted by the change of slope from the proportionality to d-1 of the propagative component to the proportionality to d-4 of the evanescent component at the lowest temperatures. Thus, the characteristic length at which temporal coherence appears for the propagative waves and the characteristic length describing the appearance of evanescent wave are different. This is the origin of the reduction of the radiative heat flux at low temperatures.
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Distance [m]
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0
d
T=0.02K T=0.1K T=1K T=10K T=100K T=300K T=1000K
4
h
total
(d,T) / h
total far-field
Fig. 5 Propagative component of the radiative heat transfer coefficient normalized by the far-field value as a function of the distance for several temperatures in the case of Al-Al.
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-1
10
-2
10
-1
10
0
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1
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2
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Distance [m]
Fig. 6 Total radiative heat transfer coefficient normalized by the far-field value as a function of the distance for several temperatures in the case of Al-Al.
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IHTC15-9188 Figures 7 and 8 show the propagative component of the radiative heat transfer coefficient as a function of T.d for several temperatures in the case of Al-Al and SiC-SiC, respectively. In the case of Al-Al, at the temperatures lower than 10K, they are perfectly superimposed from the far-field until the propagative components starts to deviate at short distances. The minimum is found approximately at the same location T.d=700m K for all temperatures. This minimum slightly shifts towards smaller T.d values when the temperature is higher than the ambient temperature. Interestingly, the radiative heat transfer coefficients start to deviate from the ones in the far-field near T.d=4000m K, where interferences enter into play. The curves start to be no longer superimposed. Thus, it can be said that the temperature-dependent distance at which interference effects appear can be expressed by T.d=4000m K for aluminum. In the case of SiC-SiC, the minimum is found for T.d=900m K, not only at low temperatures but also at higher temperatures. However, the radiative heat transfer coefficient shows a very different behaviour when the temperature is near ambient. This is due to surface phonon polaritons: indeed, SiC is a material supporting this type of surface electromagnetic wave in the infrared domain around 10m. Most of the contribution to thermal radiation comes from this infrared region near ambient temperature. SiC is a reflective material in this wavelength region, thus interferences occur, resulting in the wavy behaviour as observed in Fig. 8. Interferences appear around T.d=3000m K in the case of SiC-SiC. 10
3
10
2
10
1
10
0
d 1 T .d 4000 0.02K 0.1K 1K 10K 100K 300K 1000K
h
prop
(d,T ) / h
far-field
(T)
T .d 700
10
-1
10
-2
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-1
10
0
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2
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3
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5
T. d [m K]
Fig. 7 Propagative component of the radiative heat transfer coefficient normalized by the far-field value as a function of T.d for several temperatures in the case of Al-Al. 2.0 1.8
h
prop
(d,T ) / h
far-field
(T)
T .d 900
1.6 1.4
T .d 3000
1.2 0.02K 0.1K 1K 10K 100K 300K 1000K
1.0 0.8 0.6 10
1
10
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10
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10
5
T. d [m K]
Fig. 8 Propagative component of the radiative heat transfer coefficient normalized by the far-field value as a function of T.d for several temperatures in the case of SiC-SiC.
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IHTC15-9188 4. CONCLUSIONS We have calculated the radiative heat flux as a function of the distance between two semi-infinite parallel plates in the far-to-near field transition regime. A reduction of more than 85% of the propagative component has been observed in the case of metals whereas only few percents of reduction have been observed in the case of dielectrics. We have also conducted spectral and directional analyses. From these analyses, it is clear that a material with high reflectivity is indispensable to obtain a significant reduction of the propagative waves by interferences and that waves can only exist in directions determined by the condition of resonance as a result of the constructive interferences. Finally, calculations of the radiative heat transfer coefficient as a function of T.d have revealed that the reduction of the total flux is significant at the lowest temperatures and that the largest reduction takes place at the a given T.d value for a wide range of temperatures. In the future, an asymptotic expression may be derived to explain this dependence for materials with slowly-varying dielectric properties.
ACKNOWLEDGMENT This research is supported by ELyT Lab (Engineering and Science Lyon – Tohoku Laboratory).
NOMENCLATURE mean energy of Planck Oscillator ( J ) damping coefficient ( rad/s ) phase difference due to absorption ( - ) p plasma frequency ( rad/s )
d distance (m) K parallel wave vector ( m-1 ) kz vertical wave vector ( m-1 ) q heat flux ( W/m2 ) r Fresnel reflection coefficient ( - ) R ratio of reduction (-) Z transmission term (-) damping coefficient ( rad/s ) dielectric function (-) constant at high frequency (-)
prop evan total LO TO
propagative evanescent total (propagative + evanescent) longitudinal optical phonon transversal optical phonon
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