Journal of Quantitative Spectroscopy & Radiative Transfer 187 (2017) 310–321
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Coherent regime and far-to-near-field transition for radiative heat transfer Yoichiro Tsurimaki a,b, Pierre-Olivier Chapuis b, Junnosuke Okajima a, Atsuki Komiya a, Shigenao Maruyama a, Rodolphe Vaillon b,c,n a
Institute of Fluid Science, Tohoku University, 2-1-1, Katahira, Aoba-ku, Sendai 980-8577, Japan Univ Lyon, CNRS, INSA-Lyon, Université Claude Bernard Lyon 1, CETHIL UMR5008, F-69621, Villeurbanne, France W.W. Clyde Visiting Chair, Radiative Energy Transfer Laboratory, Department of Mechanical Engineering, University of Utah, Salt Lake City, UT 84112, USA
b c
a r t i c l e in f o
abstract
Article history: Received 10 May 2016 Received in revised form 13 July 2016 Accepted 9 August 2016 Available online 28 September 2016
Radiative heat transfer between two semi-infinite parallel media is analyzed in the transition zone between the near-field and the classical macroscopic, i.e. incoherent farfield, regimes of thermal radiation, first for model gray materials and then for real metallic (Al) and dielectric (SiC) materials. The presence of a minimum in the flux-distance curve is observed for the propagative component of the radiative heat transfer coefficient, and in some cases for the total coefficient, i.e. the sum of the propagative and evanescent components. At best this reduction can reach 15% below the far-field limit in the case of aluminum. The far-to-near-field regime taking place for the distance range between the near-field and the classical macroscopic regime involves a coherent far-field regime. One of its limits can be practically defined by the distance at which the incoherent far-field regime breaks down. This separation distance below which the standard theory of incoherent thermal radiation cannot be applied anymore is found to be larger than the usual estimate based on Wien's law and varies as a function of temperature. The aforementioned effects are due to coherence, which is present despite the broadband spectral nature of thermal radiation, and has a stronger impact for reflective materials. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Far field Near field Coherent regime Interferences
1. Introduction Radiative heat flux between two bodies at different temperatures depends on the distance separating the bodies [1]. At distances much larger than the characteristic wavelength of thermal radiation, generally estimated as being the Wien's wavelength of thermal radiation (10 μm at room temperature), the usual laws of incoherent thermal radiation – use of the specific intensity, Stefan-Boltzmann's and Wien's laws – hold. This is the incoherent farfield regime of thermal radiation, where the phases of waves can be neglected. At distances smaller than the n
Corresponding author.
http://dx.doi.org/10.1016/j.jqsrt.2016.08.006 0022-4073/& 2016 Elsevier Ltd. All rights reserved.
characteristic wavelength of thermal radiation, the contribution of evanescent waves appears. When the distance between the bodies becomes very small, the heat flux due to the evanescent waves is dominant and can exceed the far-field radiative heat flux by several orders of magnitude [2–5]. This is the near-field regime of thermal radiation. A lot of attention was paid to the strong enhancement of the net radiative heat flux beyond the blackbody configuration. It gave rise to the advent of near-field thermal radiation as a new branch of radiative heat transfer (see two recent reviews and references therein [6,7]). In addition, it had often been estimated that the broadband spectrum of thermal radiation does not allow observing sharp interference features. The possibility of interference behaviors for thermal radiation emission in
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the far field is experimentally known since the early 2000s, when Maruyama et al. [8] observed discrete thermal emission peaks from a microcavity array. The wavelengths of electromagnetic modes calculated by the cavity resonator model agreed with the dominant peaks. Among other surface structuring possibilities, early geometries involving gratings [9] and Fabry–Pérot layers [10] were shown to exhibit sharp peaks associated to coherence. Is it also possible to observe coherent effects in thermal radiation transfer between two flat surfaces? The transition zone between the near-field and the classical macroscopic (incoherent far-field) regimes of thermal radiation is the focus of the present article and termed far-to-nearfield transition regime. For monochromatic radiation, interference of multiply-reflected waves in the vacuum gap between two parallel plane media can lead to a decrease of the net radiative heat flux. The present article shows that a coherent far-field contribution to broadband thermal radiation (see Fig. 1(a)) can be observed in the total net radiative heat flux. A decrease of the net radiative heat flux in comparison to the far-field value, due to interferences, is possible even in this case of broadband thermal radiation. In the transition regime, near-field effects start to appear in addition to the reduction of the propagative contribution. Therefore, the total net radiative heat flux between two bodies decreases due to interferences unless the contribution of evanescent waves hides the reduction. To sum up, the interval of distances where the transition between the near-field and incoherent far-field regimes takes place involves a competition between the propagative and evanescent waves. Even though a reduction of the propagative component of radiative heat flux caused by interferences was noticeable in some simulation results in the case of two semi-infinite parallel media, including in the early work of Polder and Van Hove [2,11–13], a significant decrease of the total net radiative heat flux considering both propagative and evanescent waves could not be observed because the contribution of evanescent waves was overriding the drop caused by interferences. Let us mention that we realized recently [14] that Narayanaswamy and Mayo [15] were
311
also working on a similar problem. In their paper [15], they focus on the radiative energy transfer due to propagating waves between two metallic half-spaces and the emphasis is on deriving useful analytical results. In this paper, we will investigate in details the physics ruling the far-to-near-field transition and especially the coherent propagative regime of thermal radiation in the canonical case of two semi-infinite parallel media (Fig. 1 (b)). In the second section, we will briefly recall the wellknown main elements of the theory used to calculate the total net radiative heat flux and the radiative heat transfer coefficient between the surfaces, as a function of material properties which are described by the dielectric constant (ε), the temperature of the media (T), and the distance (d) separating them. In the third section, the case of two gray (frequency-independent dielectric constant) identical materials will be considered for an improved understanding of the underlying physics. The propagative component of the radiative heat transfer coefficient between the surfaces will be analyzed in four typical cases depending on the relative strengths of Re[ε] and Im[ε], and on the sign of Re[ε]. Spectral and directional analyses will be conducted to describe the interference effects taking place in the cavity separating the surfaces. The dependence on temperature of the propagative and evanescent components of the radiative heat transfer coefficient will be examined as a function of the product of temperature and distance (Td). This will allow determining characteristic distances which are typical for (see Fig. 1(a)): (i) the transition between the incoherent far-field regime and the coherent far-field regime (dincoh-coh), (ii) the distance at which evanescent and propagative waves contribute equally to the total net radiative heat flux (dpropevan), and (iii) the distance at which the total net radiative heat flux is minimum (dflux-min). In Section 4, the analysis will be applied to real metallic and dielectric materials. In particular, asymptotic expressions of the propagative component of the radiative heat transfer coefficient at small and large distances will be derived for metals. Finally, it will be discussed how the temperature dependence of the dielectric function affects the present results.
Fig. 1. (a) Schematic of the transition from the far-field incoherent to the near-field regimes. (b) Two semi-infinite parallel media at temperatures T1 and T3 with complex dielectric constants ε1 and ε3, separated by a vacuum layer of thickness d (ε2 ¼1).
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2. Theoretical background Throughout the article, we consider two semi-infinite parallel media (1 and 3) separated by a vacuum gap of thickness d (medium 2), as depicted in Fig. 1(b). Materials are assumed to be non-magnetic, isotropic, homogeneous, local, and in local thermodynamic equilibrium. We underline that we do not expect significant changes by considering the media as nonlocal as the distance between the media will stay significant throughout this work [16]. In the considered configuration, the total net radiative heat flux (qtotal) between the media is the sum of contributions from propagative waves (qprop) and from evanescent waves (qevan). The radiative heat flux can be calculated by using the well-known expressions available since Polder and Van Hove [2]: Z 1 X Θðω; T 1 Þ Θðω; T 3 Þ dω qprop ¼ j ¼ s;p 8π 3 0 2 j 2 Z ω=c 1 r j23 1 r 21 2π K dK ; ð1Þ 1 r j r j e2ik0z2 d 2 0 21 23 qevan ¼ Z
Z
X
1
ω=c
j ¼ s;p
1 0
2π K dK e
Θðω; T 1 Þ Θðω; T 3 Þ dω 2π 3 2k″z2 d
Im r j21 Im r j23 ; 1 r j r j e 2k″z2 d 2 21 23
ð2Þ
where Θðω; TÞ ¼ ℏω=ðexpðℏω=kB TÞ 1Þ is the mean energy of a Planck oscillator (h ¼ 2π ℏ and kB being the Planck and Boltzmann constants, respectively), ω is the angular frequency, K and kz are the parallel and vertical wavevectors, respectively. kzi, the z-component of the q wavevector in ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 medium i, is given by kzi ¼ kzi þ i k″zi ¼ εi ðω=cÞ2 K 2 , where c is the speed of light in vacuum. εi is the dielectric function of the medium i (ε2 ¼1 in vacuum). r sij and r pij are the Fresnel reflection coefficients of s-polarized and ppolarized waves at the interface between medium i and medium j respectively, and are given by: r sij ¼
kzi kzj ; kzi þ kzj
ð3Þ
r pij ¼
εj kzi εi kzj : εj kzi þ εi kzj
ð4Þ
As customary, a radiative heat transfer coefficient (hereafter RHTC) is defined as the temperature derivative of the radiative heat flux and can be split into the propagative and evanescent components as follows: total
h
ðd; T; εÞ ¼ lim
ΔT-0
qtotal prop evan ¼h ðd; T; εÞ þh ðd; T; εÞ ΔT
ð5Þ
In the following, we will investigate the coherent propagative contribution to thermal radiation and the far-tonear-field transition of thermal radiation using the RHTC, which depends solely on temperature T, the dielectric function ε of the materials 1 and 3, here assumed to be identical, and the distance d between the surfaces.
Fig. 2. Propagative component of the RHTC normalized by its far-field value as a function of Td for four constant dielectric functions.
3. Radiative heat transfer coefficient between semiinfinite parallel media made of similar gray materials In this section, semi-infinite parallel media made of identical gray (frequency-independent dielectric constant ε) materials are considered for the sake of understanding the basic physics involved. The RHTC is calculated for typical cases depending on the relative strengths of Re[ε] and Im[ε], and on the sign of Re[ε]. For the same magnitude of Re[ε] (100) but different signs, we consider a small (0.1) or a large (105) magnitude of Im[ε]. 3.1. Propagative component of the RHTC When the integrals over ω and K in Eqs. (1)–(2) are transformed respectively into integrals over dimensionless variables x ¼ ℏω=kB T and y ¼ K d, the bound of the integrals (ω=c in Eqs. (1)–(2)) over y becomes kB Td=cℏ. As a result, Fig. 2 shows the propagative component of the RHTC between two identical gray materials as a function of the product of temperature and distance between the surfaces. The propagative component of the RHTC is normalized by its value in the far field, given in Table 1 for the cases considered. When the temperature-distance product value becomes smaller than 104 μm K, the propagative component of the RHTC starts to deviate from its far-field value: a decrease can be observed in all of the four cases. Then the RHTC reaches a minimum at a certain Td value and increases as Td becomes smaller. Moreover, at very small temperature-distance product values, the RHTC levels off towards a value that depends significantly on ε. For instance, in the case of ε ¼ 7100þ i105, the RHTC reaches almost a hundred times the far-field value, while the RHTC in the case of ε ¼ 100þ i0.1 levels off to a value a hundred times smaller than the corresponding farfield value. The magnitude of the reduction is especially significant in the case ε ¼ 100þi0.1, where the minimum of the RHTC is more than a thousand times smaller than that in the far field. In the case ε ¼ 7100þi105, the two RHTCs are superimposed for all the Td values and a reduction of 70% is observed at around Td ¼750 μm K. For the material with
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Table 1 Propagative component of the RHTC in the far field for the four cases considered.
hfarfield (W m 2 K 1) prop
ε¼ 100þ i 0.1
ε¼ 100þ i 105
ε¼ 100þ i 0.1
ε¼ 100þ i 105
1.43
3.60 10 2
7.21 10 4
3.60 10 2
ε ¼100 þi0.1, a reduction of 32% is observed. In all of these four cases, the contribution of p-polarized waves is larger than that of s-polarized waves. Thermally-radiated propagative waves emitted by the flat surfaces experience multiple reflections between the two surfaces. When the distance is in the far-field regime, radiation is usually considered incoherent and the radiative heat exchange is described by the intensities, StefanBoltzmann's law and geometrical optics. We checked numerically that the asymptotic limit for d-1 in Eq. (1) provides the same value as with using the usual formulas involving intensities [1] with a relative difference of around 0.01%. As shown by Mulet et al. [5], Eq. (1) allows recovering the standard far-field incoherent formula. As the distance becomes smaller, the coherent regime holds, where both constructive and destructive interferences of the thermally-radiated waves take place as a result of multiple reflections. Constructive interferences of the thermally-radiated propagative waves intensify radiative heat transfer between the two surfaces and destructive interferences decrease the number of waves that are allowed to exist between the surfaces and thus the heat flux becomes smaller. The relative dominance of constructive or destructive interferences is determined by the simultaneous interplay of three parameters: the finesse of Fabry–Pérot resonances, which is related to the losses in the media, the number of waves that satisfy the Fabry– Pérot resonance for a certain distance separating the surfaces, and the wavelengths contributing to thermal radiation, which derive from the temperatures of the media. In fact, the reduction observed in Fig. 2 occurs because the contribution of destructive interferences surpasses that of constructive interferences. Effects of constructive interferences can also be observed in some cases. In the case ε ¼ 100 þi0.1, an enhancement of a few percent of the propagative RHTC over its far-field value takes place when the distance between the surfaces is decreased (it cannot be seen in Fig. 2 due to the logarithmic scale of the ordinate axis), just before the reduction due to destructive interferences starts. 3.2. Spectral and directional distributions Fig. 3 shows the spectral distribution of the propagative component of the RHTC at 300 K in the cases of ε ¼100 þi105 and ε ¼100 þi0.1. It indicates that the spectral distribution differs from that in the far field when the distance between the plates is in the coherent regime and that it changes with the distance. In the case ε ¼100 þi105, the propagative component of the spectral RHTC can be larger than the one in the far field at some angular frequencies while it is smaller at some other frequencies. The sharp increase of the propagative component of the spectral RHTC defines the boundary between these two
regions. In the angular frequency regions where the propagative component of the spectral RHTC is larger than its far-field value, constructive interferences are dominating. It is the opposite in the frequency regions where the propagative component of the spectral RHTC is smaller than its far-field value: destructive interferences are dominating. The angular frequencies at which the sharp increases take place can be found by using the following equation: kz2 d þ argðr 21 or 23 Þ ¼ m π ; m ¼ 1; 2; 3; …
ð6Þ
where the natural number m is the order of the interfering mode. For instance, the increase at ω ¼1.8 1014 rad/s for d ¼5 μm in the case ε ¼100 þi105 corresponds to constructive interferences of the first mode (m ¼1) in Eq. (6). For waves with high frequencies, the amount of modes is so large that the peaks of the transmission coefficient merge, making it similar to the transmission coefficient in the far field. This explains why the propagative components of the spectral RHTC at 2 μm and 5 μm have almost the same contributions as the ones in the far field at high frequencies. In opposite, for waves with low frequencies, the contribution of the propagative RHTC for the distances 2 μm and 5 μm is larger than the one in the far field. To discuss further the behavior of thermally-radiated propagative waves between the two plates, Fig. 4(a) shows the contour plot of the transmission coefficient (the last factor in Eq. (1), having a fractional form) as a function of angular frequency and emission angle in the case ε ¼100 þi105. The transmission coefficient is shown in logarithmic scale. Loci of larger magnitudes – Fabry–Pérot resonances – are observed. These lines are a result of the constructive interferences and correspond to a transmission coefficient near unity. In contrast, other areas correspond to destructive interferences and the thermallyradiated waves are not allowed to exist. Especially for the vertical emission angle, the angular frequencies at which constructive interferences occur are estimated with ωm ¼ m π c=d. For non-zero emission angles, as predicted by Eq. (6), the loci are upward-sloping curves (see Fig. 3). Indeed, Eq. (6) predicts that when kz is decreasing (K increasing, which means an emission angle becoming larger), the frequency where the constructive interference is maximum is growing for a prescribed distance d at a given order m. In-between the loci of maxima lay the regions where destructive interferences take place. The upward-sloping curves are observed for materials with low reflectivity as well, but the contrast between constructive and destructive interferences is less clear because reflectivities are low (not shown). We also observe in Fig. 4(a) that the transmission coefficients are large at low angular frequencies for a large range of angles. This explains why in Fig. 3 the RHTCs in
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the micrometer range are larger than those in the far-field at low angular frequencies. The transmission coefficient at angular frequency ω ¼2 1014 rad s 1 is shown in polar coordinates in Fig. 4 (b) for two cases of identical absolute value of the real part: ε ¼ 100þi105 and ε ¼100 þi0.1. At this angular frequency, the waves exist around the emission angles 26° and near 90° as a result of constructive interferences, while for other angles they vanish due to destructive interferences. When we compare the two examples depicted in Fig. 4(b), we observe sharper peaks for ε ¼ 100þ105 compared to the case ε ¼100 þi0.1. Since
Fig. 3. Spectral distribution of RHTC at 300 K for several distances between the surfaces in the case of gray materials: ε¼ 100þ i105 and ε ¼100þi0.1.
the reflectivity is larger for the former case than the latter one, Fabry–Pérot resonances occur with fewer losses, explaining why peaks are sharper for the most reflective material. 3.3. Dependence on temperature We now investigate the impact of the temperature dependence of the Bose–Einstein factor. Fig. 5 shows the propagative and evanescent components of the RHTC between two gray materials whose dielectric functions are
Fig. 5. Normalized propagative and evanescent components of the RHTC between two gray materials with ε ¼ 100 þi105 as a function of the product of temperature and distance between the surfaces for several temperatures.
Fig. 4. (a) Contour plot of the transmission coefficient as a function of angular frequency and emission angle in the case ε ¼100þi105 for d¼ 5 μm (logarithmic scale). (b) p-polarized transmission coefficient in polar coordinates for two gray materials ε¼ 100þ i105 and ε¼ 100þ i0.1 at ω ¼2 1014 rad s 1 and at the separation distance d ¼ 5 μm.
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ε ¼ 100 þi105, for several temperatures ranging from 0.1 K to 1000 K. The RHTCs are normalized by the far-field value and plotted as a function of the product of temperature and distance between the surfaces. Both the propagative and evanescent components of the RHTC are superimposed for all temperatures: it is confirmed that the RHTC can be expressed as a function of solely the product T d and material properties when the dielectric function is constant (hprop (evan) ¼hprop (evan)(Td, ε)). To investigate the regime transition from the far field to the near field, we define four characteristic distances: dincoh-coh, dflux-min, dprop-min, and dprop-evan. Firstly, dincoh-coh is the distance at which the incoherent radiation frame becomes inappropriate, defined as the distance at which the variation of the propagative component of RHTC due to interferences is 5% compared to the one in the far field: prop far field far field ðdincohcoh Þ h ðTÞ ¼ 0:05 h ðTÞ. Even h though this criterion has no physical basis, it states a practical limit below which the far-field incoherent thermal radiation theory cannot be used anymore. Secondly, dprop-min is the distance at which the net propagative heat flux reaches a minimum, defined as prop prop h ðdpropmin ; TÞ ¼ min h ðd; TÞ . In a similar way, dflux-min is the distance at which net total radiative heat flux reaches a minimum, defined as n o total total ðdfluxmin ; TÞ ¼ min h ðd; TÞ . The second of these h minima is the one that could be observed in practice. As for the transition between the far-field regime and the near-field regime, it can be characterized by the distance dprop-evan at which the contribution of the propagative component is equal to that of the evanescent component, prop evan i.e. h ðdpropevan ; TÞ ¼ h ðdpropevan ; TÞ. It will be interesting to see if this distance differs strongly from Wien's wavelength or not. It is shown in Fig. 5 that the three distances dincoh-coh, dprop-min, and dprop-evan are found at specific Td values independent of temperature. This means that the four points are solely a function of the product Td when the dielectric function is constant. 3.4. Effects of Re[ε] and Im[ε] on the radiative heat transfer coefficient We now investigate the impacts of the values of the real and imaginary parts of the dielectric function on the reduction of radiative heat transfer in the coherent regime. Note that various behaviors that are achievable in practice are weighted averages of the following investigated cases. The following results could therefore also be helpful as guidelines for future designs. To assess the impact of the imaginary part of the dielectric function, we analyze the RHTC between two identical gray media having the same magnitude of the real part of the dielectric function, but opposite signs, i.e. Re[ε] ¼ 7100, while varying the ratio of the imaginary to the real parts of the dielectric function. Similarly, we set the imaginary part to Im[ε]¼100 and vary Re[ε] to investigate the impact of the real part of the dielectric function.
315
We first analyze the impact of the imaginary part of the dielectric function on the RHTC. The propagative and evanescent components of the RHTC normalized by its farfield value are plotted in Fig. 6, as a function of Td for materials of positive real part of the dielectric function. The ratio Im[ε]/Re[ε] is changed from 0.1 to 10. A larger reduction of the propagative component of the RHTC is observed when the ratio Im[ε]/Re[ε] becomes larger. The minima of the propagative component are found at almost the same Td value, slightly decreasing with increasing Im [ε]/Re[ε]. For ratios smaller than 0.1, the propagative components of the RHTC are superimposed for all Td values (not shown), which indicates that the imaginary part of the dielectric function determines the behavior of the propagative component of RHTC. It is interesting to observe that the magnitude of the ratio contributes to the evanescent component of the RHTC in an opposite way before and after the distance dprop-evan: when the distance is smaller than dprop-evan, the dielectric function with the smaller ratio contributes greatly to the evanescent component of the RHTC (although the impact is not significant). However, at distances larger than dprop-evan, the dielectric function with the greater ratio contributes more significantly to the evanescent component of the RHTC. This is due to s-polarized evanescent waves which contribute notably at larger distances. For ratios smaller than 0.1, the evanescent components of the RHTC are also superimposed for all Td values (not shown). We now consider a negative real part of the dielectric function (see Fig. 7). In this case, the RHTC is much smaller than in the case of a positive real part (see Table 1). The reduction of the propagative component of the RHTC becomes significant as the ratio becomes smaller. Unlike the previous case, the minima are not found at the same Td value but at smaller Td values as the ratio becomes smaller. In addition, the dielectric functions with large ratios now contribute to the evanescent components of the RHTC at distances smaller than dprop-evan. The dielectric functions with small ratios lead to large evanescent component of the RHTC at distances larger than dprop-evan. Even though the reduction of the propagative component of the RHTC is significant for ratios smaller than 1, the contribution from s-polarized evanescent waves is very important at larger Td values, ruining the effect of the reduction of the propagative component in the total RHTC. We now analyze the impact of the real part of the dielectric function. Fig. 8 shows the components of the RHTC for a fixed imaginary part of the dielectric function (100) while changing the real part from the ratio of 10 2– 10. The reduction of the propagative component becomes smaller as the real part of the dielectric function becomes larger. In addition, the increase of the real part leads to an enhancement of the evanescent component with a large slope. It is interesting to observe that dprop-evan appears to be the same for all cases. Fig. 9 shows the case of a negative real part of the dielectric function. A larger real part of the dielectric function results in a larger reduction of the propagative component of the RHTC. In contrast, the real part of the dielectric function does not play an important role in the evanescent component of the RHTC. This shows that in cases of positive and negative real parts
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Fig. 6. Impact of the imaginary part of the dielectric function on the RHTC in the case of a positive real part of the dielectric function.
Fig. 9. Impact of the real part of the dielectric function on the RHTC in the case of negative real parts of the dielectric function.
minimum may not appear in the total coefficient due to the much stronger contribution of evanescent waves.
4. Radiative heat transfer between two semi-infinite parallel media made of actual materials
Fig. 7. Impact of the imaginary part of the dielectric function on the RHTC in the case of a negative real part of the dielectric function.
In the previous section, we have considered gray materials as paradigmatic examples. In this section, we discuss the radiative heat exchange between two semiinfinite parallel metallic and dielectric plates. We consider aluminum (Al) as representative for metals and silicon carbide (SiC) for dielectrics. The dielectric function of metallic materials is approximated by the Drude model ε ¼ 1 ω2p =ðωðω þiνÞÞ, where ωp and ν are the plasma frequency and the damping coefficient, respectively. The dielectric function of dielectric materials is given by the Lorentz model ε ¼ ε1 ðω2 ω2LO þ i γωÞ=ðω2 ω2TO þ i γωÞ, where ωLO , ωTO are the resonant frequencies of longitudinal and transverse optical phonons, respectively, γ is the damping coefficient, and ε1 is the dielectric constant at high frequencies. The parameters considered are ωp ¼2.241016 rad/s, ν ¼1.221014 rad/s for aluminum [16], and ε1 ¼6.7, ωLO ¼1.8271014 rad/s, ωTO ¼1.495 1014 rad/s, γ ¼8.9711011 rad/s for silicon carbide [17]. In this section, these parameters are assumed not to depend on temperature. As a result, the variations with temperature are solely due to an interplay between the frequencydependence of the dielectric function and the Bose–Einstein factor. 4.1. Aluminum
Fig. 8. Impacts of the real part of the dielectric function on the RHTC in the case of positive real parts of the dielectric function.
of the dielectric function, a significant reduction of the propagative component of the RHTC can be obtained when the ratio becomes larger and smaller, respectively. We also find that dprop-min depends on the ratio. However, the
The normalized propagative component of the RHTC between two Al media is represented in Fig. 10 as a function of the product of temperature and distance between the plates for several temperatures. The reduction of the propagative component of the RHTC at 300 K is 85% and becomes even larger at higher temperatures. The minima are found at almost the same Td value around Td¼ 680 μm K at low temperatures, and at slightly smaller Td values at high temperatures. At low temperatures, the
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317
Fig. 10. Propagative radiative heat transfer coefficient normalized by its far-field value as a function of the product of temperature and distance in the case of two Al media for a temperature-independent dielectric function.
real part of the dielectric function of Al is getting close to a constant value, 1 ω2p =ν2 . The imaginary part is larger than the real part and is inversely proportional to the angular frequency. Note that the p-polarized propagative waves provide the dominant contribution at both low and high temperatures in the case of two Al media. Thus, the propagative component of the RHTC between two Al surfaces at low temperature is similar to the case Im[ε]/Re [ε] ¼10 in Fig. 7. At high temperatures where the contributing angular frequencies are around 1015 rad s 1, the real part is on the order of 100 to 10, and the ratio Im [ε]/Re[ε] is much smaller than unity. Thus, the reduction of the propagative component of the RHTC becomes significant, as could be observed in Fig. 7. At very small Td regions, the propagative component of the RHTC levels off to a certain value which depends on temperature. At high temperatures, the RHTC at very small distances is several times larger than its far-field value while it is on the order of thousand times the far-field value at low temperature. Asymptotic expressions at these small Td regions can be obtained by making calculations at the limit d-0 in Eq. (1): ! Z 15σ T 3 1 x4 ex Re½ε prop dx h ¼ 1 þ ε 2π 4 0 ðex 1Þ2 ( 2σ T 3 ; ω oo ν ; ð7Þ 4σ T 3 ; ω 44 ν where x ¼ ℏω=kB T. While deriving this equation, it can be observed that s- and p- polarizations have the same contribution at very small distances. Note that the integral form of the asymptotic equation in Eq. (7) is valid for any material at any temperature, while the most right-hand side of Eq. (7) can be applied only to materials whose properties are described by the Drude model. Therefore, the propagative component of the RHTC at a given temperature converges at small distances toward a value independent of the kind of metal. Moreover, an asymptotic expression of the RHTC in the far field can also be
Fig. 11. In the case of two Al media: (a) Plot of characteristic Td values that show the breakdown of the incoherent thermal radiation theory, the minimum of the total and propagative components of the RHTC, and the transition from the far field to the near field. (b) Plot of the propagative, evanescent and total components of the RHTC at 300 K and location of the characteristic distances (dprop-min ¼2.2 μm, dflux-min ¼ 3.8 μm, dpropevan ¼4.6 μm, and dincoh-coh ¼29.8 μm).
analytically derived. Narayanaswamy and Mayo [15] derived asymptotic expressions of the propagative component of the RHTC between two metallic semi-infinite media by counting the number of waves that exist between them at a certain angle by approximating the transmission coefficient of a single wave with the Lorentz distribution. Following the same procedure and considering the limit d-1, the asymptotic expressions in the far field are given as follows: 8 > >
1
1
4983 k2B ν2 pffiffi 1 7 4 2 ℏ2 π 2 ωp
3 > ν : 16 ; 3 ωp σ T
σ T 2 ; ω oo ν 7
ω 44 ν
:
ð8Þ
We observed that the difference with another asymptotic expression by Polder and Van Hove [2] is of the order of 2%. It is striking that the low-frequency approximation departs from the well-known T3 behavior of the RHTC associated to the case of two blackbodies (hBB ¼4σT3).
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We now turn to the analysis of the four characteristic distances in the case of two Al plates: the one characteristic of the far-to-near-field transition, the one at which classical thermal radiation breaks down, and the minima for the net total and net propagative components of the RHTC. Temperature-characteristic distance products are given in Fig. 11(a) for temperatures comprised between 1 and 1000 K. The characteristic distances can be observed directly on the plot in Fig. 11(b) of the propagative, evanescent and total components of the RHTCs at 300 K. At low temperatures, the first two distances are respectively found at constant Td values (Td¼1110 μm K and 9500 μm K, respectively), while the minima of the total and propagative components are found at around 970 and 680 μm K, respectively. It is worth noticing that the coherent regime appears at distances two to three times larger than those expected using Wien's law (Tλ ¼2898 μm K). This means that the classical theory of blackbody radiation that assumes incoherent radiation cannot be applied once the product Td of the system becomes smaller than around 9000 μm K, i.e. 30 μm at room temperature. The transition regime becomes narrower at high temperature and therefore it is expected that the reduction of the total net radiative heat flux is less significant in this temperature range. This is due to the variations of the dielectric function as a function of frequency. We observe that the critical Td values for the four parameters do not shift in the same directions as a function of temperature. The trend is that the near-field regime starts at larger Td values when the temperature increases. Similar results are obtained for other metals such as silver, copper and gold [18]. In the case of aluminum, the values of the minima can be found in Fig. 12, which shows the contour plot of the total component of the RHTC between two Al surfaces as a function of temperature and distance. The red and green lines show the beginning and the end of the transition regime, respectively. The blue line shows the distances at which the total radiative heat flux reaches the minimum. The reduction of the total component of the RHTC is the largest at the lowest temperatures and is 15% at maximum. Even though the propagative component exhibits a much greater reduction (Fig. 10), evanescent waves contribute non-negligibly to the radiative heat transfer at the transition regime and hide the reduction in the total RHTC. As a result, the reduction of the total component cannot exceed 15%. As expected from the discussion above, the transition regime becomes narrow at larger temperature and indeed the reduction is less important as temperature rises. As a matter of fact, the reduction at ambient temperature is 10%. 4.2. Silicon carbide We now analyze the case of dielectrics. Fig. 13 shows the propagative component of the RHTC between two SiC plates normalized by the far-field value as a function of the product of temperature and distance for several temperatures. At low temperature, the propagative RHTC exhibits the same profile independent of temperature for all Td values. The real part of the dielectric function of SiC
can be approximated as ε1 ðωLO =ωTO Þ2 , and the imaginary 2 2 part ε1 ðωLO =ωTO Þ2 γωðωTO ωLO Þ is small compared to the real part. Thus, the behavior of the RHTC is similar to the case Im[ε]/Re[ε]¼10 2 of Fig. 8. The same discussion can be applied to high temperatures where the dielectric function can be considered constant with real part ε ffi ε1 . At ambient temperature, the propagative RHTC presents a wavy behavior in the transition regime because of polaritons which make SiC reflectivity large in this temperature region. The result is due to counteracting effects and depends on the spectral weighting of the different contributions to the dielectric function. Similarly to Fig. 11, characteristic distances are plotted in Fig. 14. At low temperature, the limit of the incoherent thermal radiation theory can be found for Td equal to around 1700 μm K, which is smaller than the criterion given by Wien's law. As discussed previously, the minima of the propagative RHTC are found for Td equal to around 900 μm K at both high and low temperatures. However, these characteristic distances are found at much larger Td values near ambient temperature because polaritons, which have a dominant contribution in this temperature range, make the material reflective. At room temperature, it is found that Tdincoh-coh 2200 μm K. The end of the transition regime is found for Td equal to around 805 μm K from low temperatures to temperatures up to around 50 K. In the case of dielectric materials, the end of the transition regime is found at smaller Td values than the Td values of the minima. Nevertheless, we underline that the total component of the RHTC between two SiC media does not show any reduction due to interferences (not shown here). The key result of this section is therefore that dincoh-coh is smaller (7.3 μm at 300 K in Fig. 14(b)) than the gross estimate obtained with Wien's law for (10 μm at 300 K) in the case of dielectrics.
5. Effects of temperature dependence of the dielectric function In this section, the impact of the temperature dependence of the dielectric function on the RHTC is discussed in the case of metallic materials. Note that we do not discuss the case of dielectrics. The temperature dependence of the plasma frequency and the damping coefficient was investigated for aluminum and gold [19,20]. The temperature dependence of the plasma frequency is weak (we consider it anyway), but the damping coefficient monotonically increases with temperature. The effect on the far-field limit can be important depending on the temperature range considered and was investigated in [20]. Close to room temperature, the effect can be observed but stays weak. As an example, Fig. 15 shows the normalized propagative and evanescent components of the RHTC between two aluminum media at 198 K as a function of Td, calculated with temperature dependent and independent dielectric functions. Note that the nonnormalized values of the RHTC differ, but as we focus here on the impact of the variation due to the temperature dependence, we keep the same normalization as previously. From the far field to Td values where the
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low angular frequency, comparable to or smaller than the damping coefficient. Thus, the temperature dependence of
Fig. 12. Plot of the total component of the RHTC as a function of temperature and distance in the case of two Al media for temperatureindependent dielectric function. Red, green and blue lines show the distances characteristic of the transition regime dprop-evan and dincoh-coh, and the minimum of the total component dflux-min. The contour plot may appear locally blurred due to the mesh discretization (already computerresource intensive).
Fig. 13. Propagative radiative heat transfer coefficient normalized by its far-field value as a function of the product of temperature and distance in the case of two SiC media.
propagative RHTC reaches its minimum, there is no difference between the two calculation results. However, for smaller Td values, there are some differences. When using the temperature-dependent dielectric function, the propagative RHTC takes smaller values than when using the temperature-independent dielectric function at low temperature (Fig. 15), and it takes larger values at high temperature (not shown). At large Td values where the propagative RHTC decreases due to destructive interferences (before reaching a minimum), most of the contributions comes from the waves with angular frequency larger than the damping coefficient. Thus the temperature dependence of the damping coefficient does not affect the RHTC. However, at small Td values where the propagative component of the RHTC increases due to contributions from low-frequency waves (after reaching a minimum), most of the contribution to the RHTC comes from the waves with
Fig. 14. In the case of two SiC media: (a) plot of characteristic Td values that show the breakdown of the incoherent thermal radiation theory, the minimum of the propagative radiative heat transfer coefficient and the end of the transition regime. (b) Plot of the propagative, evanescent and total components of the RHTC at 300 K and location of the characteristic distances (dprop-min ¼ 5.8 μm, dprop-evan ¼ 2.5 μm, and dincoh-coh ¼ 7.3 μm).
Fig. 15. Propagative and evanescent components of the RHTC between two Al media at 200 K, calculated with temperature-dependent and independent (T ¼ 300 K) dielectric functions.
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Fig. 16. Propagative radiative heat transfer coefficient normalized by its far-field value as a function of the product of temperature and distance in the case of two Al media for a temperature-dependent dielectric function.
the damping coefficient significantly affects the dielectric function, and as a consequence, the RHTC at these small Td values. We also observe that the evanescent component of the RHTC calculated with the temperature-dependent dielectric function shows little difference compared to the one calculated with the temperature independent dielectric function. When the damping coefficient is very different from that at room temperature (temperature being an order of magnitude smaller or larger), the dielectric function cannot be considered anymore as weakly dependent on temperature. Fig. 16 is similar to Fig. 10 but considers the temperature dependence of the dielectric function as in [20]. It can be seen that a strong difference to the temperature-independent case is observed for temperatures below 100 K. We stress that Eqs. (7) and (8) stay valid for temperature-dependent plasma frequency and damping coefficients. As a result, the previous discussion stays also valid but the values of the distances dincoh-coh, dpropevan, dflux-min may depart from those shown in Fig. 11.
6. Conclusions In this study, we have investigated the physics of radiative heat transfer taking place in the transition regime between the incoherent far-field and the near-field regimes, where a coherent propagative contribution to thermal radiation can be observed, for the case of two identical materials. The existence of the transition regime and the limit of the conventional theory of incoherent thermal radiation have been investigated in details. Spectral and directional analyses of the RHTC in the case of two identical gray materials have allowed understanding the reduction of the propagative component of RHTC due to interferences. In addition, analyses of the impacts of real and imaginary parts of the dielectric function have shown that a negative real part (metallic materials) is preferable to obtain a large reduction and that the imaginary part significantly affects the behaviors of both propagative and evanescent components of the RHTC. These behaviors
were exemplified by means of calculations of the RHTC between two identical Al and SiC media and showed the existence of the transition regime for actual materials. In the case of aluminum, a maximum reduction of 15% of the total net radiative heat flux can be obtained at low temperatures. Investigations have also revealed that the incoherent thermal radiation theory breaks down at much larger distances than the estimation from using Wien's law for this metal. In the present study, the two surfaces have been assumed to be perfectly smooth, thus the effects of surface roughness have not been included. It is reasonable to do so if the roughness characteristic size remains smaller than the wavelengths and the distances between the parallel surfaces. For the largest temperatures, this assumption may not be justified and some corrections could be applied [21]. Effects of oxide films that can be present on surfaces have not been considered in the present study. However, effects of the oxide film are negligible if its thickness is less than 10 nm [13], which is usual. It is well known that experiments requiring perfect parallelism are difficult to achieve in practice. As a consequence, it could be interesting to verify if the levels of reduction observed here could still be observed in the presence of unfortunate experimental tilts. We have proposed the idea of reducing radiative heat flux below its far-field value by employing interferences of thermal radiation in the case of bulk materials. It would surely be possible to increase the reduction by using layered materials. Among various possible applications, such technologies could allow developing vacuum insulation solutions with better performances.
Acknowledgments This research received support from ELyT Lab (Engineering and Science Lyon – Tohoku Laboratory).
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