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Thermal switch and thermal rectification enabled by near-field radiative heat transfer between three slabs
International Journal of Heat and Mass Transfer 82 (2015) 429–434
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Thermal switch and thermal rectification enabled by near-field radiative heat transfer between three slabs Wei Gu, Gui-Hua Tang, Wen-Quan Tao ⇑ Key Laboratory of Thermo-Fluid Science and Engineering of MOE, Xi’an Jiaotong University, Xi’an 710049, China
a r t i c l e
i n f o
Article history: Received 21 September 2014 Received in revised form 16 November 2014 Accepted 17 November 2014 Available online 9 December 2014 Keywords: Near-field Radiative heat flux Insulator–metal transition material Thermal switch Thermal rectification
a b s t r a c t In this paper, the near-field radiative heat flux of the system consisting of two SiO2 plate sources and one 50 nm thick VO2 film, a kind of insulator–metal transition material, placed between them is studied. The two sources are maintained at 400 and 300 K, respectively, and separated by a distance of 150 nm. By this configuration, the temperature of the film can be regulated by the control of the separation distance between the film and the sources via a piezoelectric motor. It is found that the net radiative heat flux of this system can be varied in a range of 7.5 103–3.2 104 W/m2 for different position of the VO2 film due to its phase transition. Hence, the functions of thermal switch and thermal rectification are able to be realized by this system. Particularly, the direction of the larger heat flux in the proposed thermal rectification can be reversed by changing the position of the VO2 film. The effects of some parameters are investigated to improve the performance of the thermal rectification. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction As the counterpart of an electric rectifier and an electric switch in electronics, a thermal rectifier is a device that allows heat to flow preferentially in one direction [1,2], and a thermal switch is a device that allows the heat flux in the ‘‘on’’ mode being obviously larger than that in the ‘‘off’’ mode [3,4]. Such devices may have enormous applications in thermal management and control for energy systems and microelectronic devices. The previous thermal rectifications were mainly limited to heat conduction and convection [1,5,6]. Recently, photonic thermal rectification based on nearfield radiative transfer [7–13] has attracted attention due to the advantage of obtaining large rectification factors over a broad temperature range. The reported photonic thermal rectifiers are all made of two terminal parts and realized by one or two of the following mechanisms: (1) an asymmetric geometric arrangement [9], (2) dissimilar materials with different temperature-dependent thermal properties [7,8,10–13]. Recently, Ben-Abdallah and Biehs [4] designed a photonic thermal transistor, and the functions of thermal switch, thermal modulation and thermal amplification were realized simultaneously. The thermal transistor is composed of three basic elements, thermal source, thermal drain, and gate, and the heat flux can be controlled by the gate made of an insulator–metal transition (IMT) material, ⇑ Corresponding author. Fax: +86 29 82669106. E-mail address: [email protected] (W. Q. Tao). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.11.058 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
whose optical property is able to be changed through a small variation of its temperature around the critical temperature Tc. In literature [4], the temperature of the gate was controlled by a certain amount of heat added to or removed from it. However, there may be two weaknesses of that way to regulate the temperature of the gate. One probably comes from the fact that two different elements are needed to add heat to or remove heat from the gate, which makes the system more complicated. The other is that there exists a region where a certain amount of heat added to the gate corresponds to three different temperatures of the gate (as shown in Fig. 3 of literature [4]), which makes it difficult to control the heat flux. In this paper, we propose a new way to control the temperature of the gate by changing the distance between the gate and the thermal source, which can be carried out, for example, with the piezoelectric motors from Attocube as used in the experiment conducted by Rousseau et al. [14], where the displacements of a silica microsphere were carried out with the piezoelectric stages by steps of 7 nm. With this improvement, a three-part system is realized simply to switch or rectify the heat flux carried by photons. 2. Configuration and calculation of the heat flux 2.1. Configuration The structure of the proposed near-field system is schematically plotted in Fig. 1. Medium 1 and medium 3 both made of
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W. Gu et al. / International Journal of Heat and Mass Transfer 82 (2015) 429–434
amorphous silicon dioxide (SiO2) are assumed to be semi-infinite and maintained at 400 and 300 K, respectively, by some thermostats. A thin layer (medium 2) made of vanadium dioxide (VO2) having thickness d is placed between them at distances d12 from medium 1 and d23 from medium 3. VO2 is one of the IMT material that undergoes a phase transition from a hightemperature metallic phase to a low-temperature insulating phase around its critical temperature (Tc 340 K) [15,16]. The thickness of the film is 50 nm, and distances d12 and d23 are related by d12 + d23 = 100 nm to introduce the near-field effect and to get large heat flux. The temperature T2 of the film is determined by thermal equilibrium with media 1 and 3, and undoubtedly T2 is a function of d12 or d23, which enables the control of heat flux by regulating d12 or d23 with the piezoelectric motors. 2.2. Calculation of heat flux The dielectric function of insulating VO2 in the bulk shape is related to the crystal orientation in the infrared wave region, while that of insulating VO2 in the film shape or VO2 in metallic phase does not exhibit anisotropy [15–17]. Here we consider the case where the optical axis of VO2 film is orthogonal to its interfaces, and its dielectric functions are obtained from [15,16], while the dielectric functions of SiO2 are taken from Ref. [18]. Fig. 2 shows the real (e0 ) and imaginary (e00 ) parts of the dielectric functions of VO2 and SiO2. It can be seen from Fig. 2 that SiO2 has two strong phonon modes at the frequencies 8.5 1013 and 2.0 1014 rad/s, and the phonon mode at lower frequency overlaps with that of insulating VO2. In Fig. 1, the heat flux across any planes parallel to the surfaces of the three bodies can be obtained by calculating the Poynting vector. Considering the three-body configuration, the theory developed by Messina and Antezza [19] recently is used to investigate the net radiative heat flux. This theory is valid for arbitrary bodies of a three-body system, i.e., for any set of temperatures and geometrical properties, and describes each bodies by means of their scattering operators. The net heat flux emitted by body 1 reads [20]:
Z 1 2 dx d k hx 2p ð2pÞ2 0 0 i Xh n12 ðxÞCj1=2 ðx;k;d12 ;d23 Þ þ n13 ðxÞCj1=3 ðx;k;d12 ;d23 Þ
U1 ¼
Z
1
j¼s;p
ð1Þ
Fig. 2. Dielectric functions of VO2 and SiO2: (a) real parts, (b) imaginary parts.
where, C1=2 and C2=3 denote the transmission probabilities of phoj j tons, defined by mode (x, k) for polarization state j = s, p, from body 1 to body 2 and from body 1 to body 3, respectively; k = (kx, ky) is the wave vector parallel to the surfaces of multilayer system. nij(x) = ni(x, Ti) nj(x, Tj) denotes the difference of Bosedistribution functions with ni/j = 1/[exp(hx/kBTi/j) 1]. 2ph is the Planck’s constant and kB is the Boltzmann’s constant. Similarly, the heat fluxes received by body 2 and body 3 are given by:
Z 1 2 dx d k hx 2p ð2pÞ2 0 0 i Xh n12 ðxÞCj1=2 ðx; k; d12 ; d23 Þ þ n32 ðxÞC3=2 j ðx; k; d12 ; d23 Þ
U2 ¼
Z
1
j¼s;p
ð2Þ Z 1 2 dx d k hx 2p ð2pÞ2 0 0 i Xh 1=3 2=3 n13 ðxÞCj ðx;k;d12 ; d23 Þ þ n23 ðxÞCj ðx; k; d12 ; d23 Þ
U3 ¼
Z
1
j¼s;p
ð3Þ
Fig. 1. Schematic of the system in the three-slab configuration.
It is easy to derive the relation U2 = U1 U3 from Eqs. (1)–(3). At steady state, the heat flux U1 emitted by body 1 equals the heat flux U3 received by body 3, and hence the net heat flux U2 received by medium 2 vanishes. All the transmission probabilities Cj are defined in terms of ðiÞ optical reflection coefficients qj ði ¼ 1; 2; 3 and j ¼ s; pÞ and
W. Gu et al. / International Journal of Heat and Mass Transfer 82 (2015) 429–434 ðiÞ
transmission coefficients sj , and their detail expressions can be ðiÞ ðiÞ obtained from Ref. [19]. The quantities qj and sj are expressed as [19]: ð1Þ qð1Þ j ðk; xÞ ¼ r j ðk; xÞ
sð1Þ j ðk; xÞ ¼ 0 ð2Þ
ð2Þ qð2Þ j ðk; xÞ ¼ r j ðk; xÞ
1 ð2Þ j ðk;
s
xÞ ¼
ð2Þ t j ðk;
h
1 e2ikz ð2Þ rj ðk;
d
i2 ð2Þ xÞ e2ikz d
ð2Þ
iðkz xÞtð2Þ j ðk; xÞe
ð4Þ
kz Þd
h i2 ð2Þ ð2Þ 1 rj ðk; xÞ e2ikz d
ð3Þ qð3Þ j ðk; xÞ ¼ r j ðk; xÞ
sð3Þ j ðk; xÞ ¼ 0 ðiÞ
In Eq. (4), the vacuum-medium Fresnel reflection (r j ) and transmission
ðiÞ (tj )
coefficients, as well as the medium-vacuum
ðiÞ ðiÞ Fresnel transmission (t j ) coefficients are included. kz and kz
denote the z component of the wave vector inside the medium i and vacuum, respectively. In addition to the isotropic media, Eqs. (1)–(4) can be easily extended to the anisotropic media by considering the anisotropic wave propagation [10,13,21–24].
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Note that the variation of T2 with d12 is monotonic, and this makes the control of T2 easy. The net heat flux transfer U of this system for steady state is shown in Fig. 4, which is equal to U1 or U3. A change of d12 from 55 to 56 nm will lead to an increase of net heat flux transfer by a factor of 3 at least. To further understand the effect of phase transition on the net radiative heat flux transfer, the distributions of U with modes (x, k) for d12 = 44 nm (VO2 in metallic phase) and d12 = 56 nm (VO2 in insulating phase) are plotted in Fig. 5. Note that k, the parallel component of wave vector, is normalized by the vacuum wave vector x/c. When the film is in its insulating phase, /1 (see Fig. 5(b)) shows an efficient coupling of mode around the resonance frequencies xspp1 8.5 1013 rad/s and xspp2 2.0 1014 rad/s of surface phonon–polaritons supported by SiO2. However, /1 in these two resonance frequencies will be weakened by the metallic VO2 film (see Fig. 5(a)). The results of Fig. 5(a) and (b) can be explained from two aspects. One comes from that the surface phonon–polaritons at frequency xspp1 8.5 1013 rad/s of SiO2 overlap with that of insulating VO2, and their phonon–phonon coupling will result in a significant enhancement of near-field radiative heat transfer [17]; the other is that metallic VO2 does not support surface polaritons and has high reflectance in the infrared frequencies, and hence the metallic VO2 will suppress the coupling of surface phonon–polaritons between the two sources of SiO2.
3. Result and discussion 3.2. Thermal switch and thermal rectification 3.1. Variation of temperature T2 and net heat flux U1 with the distance d12 Without external excitation, the system depicted in Fig. 1 reaches its steady state for which the net heat flux U2 received by medium 2 is zero. In this case, the temperature T2 is set by temperatures T1, T3 and distance d12. Figure 3 shows the variation of equilibrium temperature T2 with distance d12, which is determined by solving the equation U2 = 0. In Fig. 3, the solid line indicates the variation of T2 with the temperature dependence of the dielectric response of VO2 being taken into consideration, while the dash line and dash dot line give the results, respectively, for insulating VO2 and metallic VO2 with the assumption of temperature independence of the dielectric response. The classical effective medium theory is used to model the transition region (340.0 < T2 < 345.2 K) of IMT VO2 with the data provided by Qazilbash et al. [16]. It can be seen that the evolution shapes of T2 with d12 for insulating VO2 and metallic VO2 are very different, and hence there is an obvious jump of T2 after phase transition.
By using different physical parameters (temperatures, sizes, separation distances, etc.), several functions can be achieved by this system, and two evident ones of them are the thermal switch and thermal rectification. As has been shown in Fig. 4, the net heat flux exchange between the two sources is able to be tripled at least by changing the separation distance d12 from 55 to 56 nm. That means the proposed system can be used in two operating modes, the ‘‘off’’ mode for d12 < 55 nm and the ‘‘on’’ mode for d12 > 56 nm. A thermal rectifier is a device in which heat flux along a specific direction is quite large, while along the opposite direction is relatively small. The near-field thermal rectification was first proposed by Otey et al. [7], and after that, several different near-field thermal rectifications were designed [8–13,25]. All of the previous nearfield thermal rectifications are designed with two terminal parts on the basis of asymmetric geometric arrangement or dissimilar materials, and their work directions with the larger heat fluxes are unchangeable. For example, the SiO2–VO2 two terminal thermal rectification proposed in [13] allows the larger heat flux only
Fig. 3. Temperature T2 of the VO2 film for different d12.
Fig. 4. Net heat flux transfer for different d12.
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W. Gu et al. / International Journal of Heat and Mass Transfer 82 (2015) 429–434
Fig. 6. The net heat fluxes of forward-biased and reverse-biased for different separation distance d12.
Fig. 7. The rectification factor for different separation distance d12.
Fig. 5. Distribution of U1 with modes (x, k) in the steady state: (a) d12 = 44 nm with VO2 in metallic phase, (b) d12 = 56 nm with VO2 in insulating phase.
in the direction with VO2 in the temperature lower than Tc (i.e., VO2 in insulating phase), and this direction cannot be reversed with the fixed sources. However, this is not the situation for three-body system as shown in Fig. 1, which consists of two sources with temperature independence of dielectric response and one IMT film. This system enables the direction with the larger heat flux to be reversed by controlling the position of the film, which is one of the major contributions to this topic. In Fig. 1, the forward-biased scenario is defined for T1 = 400 K and T3 = 300 K, and the reverse-biased scenario is defined for T1 = 300 K and T3 = 400 K. The net heat fluxes of forward-biased and reverse-biased for different separate distances d12 are plotted in Fig. 6. As discussed above, the direction where the heat flux is larger can be reversed by changing the position of the VO2 film. The net heat flux in the forward-biased scenario (denoted by UF) is smaller than that in the reverse-biased scenario (denoted by UR) for d12 < 44 nm. On the other hand, UF is larger than UR for d12 > 56 nm. Hence, a forward-favored or reverse-favored thermal rectification can be realized simultaneously by the proposed
system just by setting the distance d12. It should be noted that the proposed system shows no rectification for d12 in the range of 44–56 nm. The thermal rectification factor f of this system is given in Fig. 7, which is calculated by (UF UR)/Min(UF, UR). The amplitude of the thermal rectification factor is able to reach 2.32, much higher than 0.41 obtained by Otey et al. [7]. In addition, the magnitude of the rectification factor of the proposed system can be varied in a range of 1.67–2.32 as plotted in Fig. 7. The proposed thermal rectification with modulable rectification factor is another major contribution to this topic. 3.3. Parameters investigation A thermal rectification with good performance should have high rectification factor and large heat flux. In the above part, we have investigated the thermal rectification of the three-slab system with parameters of d = 50 nm and d12 + d23 = 100 nm. However, it is foreseeable that the performance of this thermal rectification is sensitive to the parameters d, d12 and d23, and we will show how these parameters influence the performance of the thermal rectification in the following. In order to reduce the number of free parameters for analysis, we will focus our attention on the symmetric case, in which d12 equals d23, without considering phase change of VO2 film. Figure 8(a) plots the radiative heat flux as a function of film thickness d and distance d12 for VO2 in insulating phase. It can be
W. Gu et al. / International Journal of Heat and Mass Transfer 82 (2015) 429–434
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4. Conclusions We have studied the near-field radiative heat flux of a threeslab system consisting of two amorphous SiO2 sources and one IMT VO2 film between them. There is a monotonic relationship between the equilibrium temperature T2 of the film and the separation distance d12, and hence the control of T2 can be realized by modulating d12 via the piezoelectric motors. For insulating phase of VO2, its surface phonon–polaritons at frequency x = 8.5 1013 rad/s overlaps with that of SiO2, and their phonon–phonon coupling promotes the near-field heat transfer. In addition, metallic VO2 has large reflectance in the infrared frequencies and suppresses the heat transfer between the two sources. These two features result in an enhancement of net heat flux by a factor 3.32 after the VO2 film transits from metallic phase to insulator phase for the proposed system. By using different parameters (source temperatures and separation distances), thermal switch and thermal rectification are realized by the proposed system. In particular, the direction permitting the larger heat flux in the proposed thermal rectification can be reversed by changing the position of the film. Finally, it has been shown that by varying the thickness d of VO2 film and the separation distance d12 the performance of the thermal rectification can be greatly improved.
Acknowledgment This work is sponsored by the Natural Science Foundation of China (51136004).
References
Fig. 8. (a) Radiative heat flux as a function of d12 and d with VO2 in insulating phase, (b) the ratio of radiative heat flux for VO2 in insulating phase to that for VO2 in metallic phase as a function of d12 and d.
seen that the radiative heat flux is more sensitive to d12 than to d. Thus, the separation distances d12 and d23 should be small enough to get large radiative heat flux. Figure 8(b) shows the ratio of radiative heat flux for VO2 in insulating phase (marked by /i) to that for VO2 in metallic phase (marked by /m) as a function of d12 and d. As shown there, the values of /i//m are generally increased with thicker VO2 film for the same values of d12. The regions where /i//m is smaller than one and larger than six are indicated by the red dash line and black dash line respectively. It is certain that the rectification factor of the three-slab configuration is not equal to /i//m 1, because the thermal rectification is realized by asymmetric configuration of the three-slab system, i.e., d12 – d23. However, large values of /i//m will lead to large rectification factors, thus, the results given by Fig. 8(b) are helpful to design the thermal rectification. Accordingly, rectification factor can be increased by a thicker VO2 film for the case of d12 + d23 = 100 nm, and a rectification factor of 5.2 can be reached by replacing 50 nm thick VO2 film with a 200 nm thick one, whose value is close to /i//m 1 given by Fig. 8(b) for d = 200 nm and d12 = 50 nm.
[1] B. Li, L. Wang, G. Casati, Thermal diode: rectification of heat flux, Phys. Rev. Lett. 93 (18) (2004) 184301. [2] C.W. Chang, D. Okawa, A. Majumdar, A. Zettl, Solid-state thermal rectifier, Science 314 (5802) (2006) 1121–1124. [3] B. Li, L. Wang, G. Casati, Negative differential thermal resistance and thermal transistor, Appl. Phys. Lett. 88 (14) (2006) 143501. [4] P. Ben-Abdallah, S.-A. Biehs, Near-field thermal transistor, Phys. Rev. Lett. 112 (4) (2014) 044301. [5] N.A. Roberts, D.G. Walker, A review of thermal rectification observations and models in solid materials, Int. J. Therm. Sci. 50 (5) (2011) 648–662. [6] N. Yang, G. Zhang, B. Li, Carbon nanocone: a promising thermal rectifier, Appl. Phys. Lett. 93 (24) (2008) 243111. [7] C.R. Otey, W.T. Lau, S. Fan, Thermal rectification through vacuum, Phys. Rev. Lett. 104 (15) (2010) 154301. [8] S. Basu, M. Francoeur, Near-field radiative transfer based thermal rectification using doped silicon, Appl. Phys. Lett. 98 (11) (2011) 113106. [9] H. Iizuka, S. Fan, Rectification of evanescent heat transfer between dielectriccoated and uncoated silicon carbide plates, J. Appl. Phys. 112 (2) (2012) 024304. [10] P. Ben-Abdallah, S.-A. Biehs, Phase-change radiative thermal diode, Appl. Phys. Lett. 103 (19) (2013) 191907. [11] J. Huang, Q. Li, Z. Zheng, Y. Xuan, Thermal rectification based on thermochromic materials, Int. J. Heat Mass Transfer 67 (2013) 575–580. [12] L.P. Wang, Z.M. Zhang, Thermal rectification enabled by near-field radiative heat transfer between intrinsic silicon and a dissimilar material, Nanoscale Microscale Thermophys. Eng. 17 (4) (2013) 337–348. [13] Y. Yang, S. Basu, L. Wang, Radiation-based near-field thermal rectification with phase transition materials, Appl. Phys. Lett. 103 (16) (2013) 163101. [14] E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, J.J. Greffet, Radiative heat transfer at the nanoscale, Nat. Photonics 3 (9) (2009) 514–517. [15] A.S. Barker, H.W. Verleur, H.J. Guggenheim, Infrared optical properties of vanadium dioxide above and below the transition temperature, Phys. Rev. Lett. 17 (26) (1966) 1286–1289. [16] M.M. Qazilbash, M. Brehm, B.-G. Chae, P.-C. Ho, G.O. Andreev, B.-J. Kim, S.J. Yun, A.V. Balatsky, M.B. Maple, F. Keilmann, H.-T. Kim, D.N. Basov, Mott transition in VO2 revealed by infrared spectroscopy and nano-imaging, Science 318 (5857) (2007) 1750–1753. [17] P.J. van Zwol, K. Joulain, P. Ben-Abdallah, J. Chevrier, Phonon polaritons enhance near-field thermal transfer across the phase transition of VO2, Phys. Rev. B 84 (16) (2011) 161413. [18] E.D. Palik, Handbook of Optical Constants of Solids, 3, Academic Press, 1998. [19] R. Messina, M. Antezza, Three-body radiative heat transfer and Casimir– Lifshitz force out of thermal equilibrium for arbitrary bodies, Phys. Rev. A 89 (5) (2014) 052104.
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[20] P. Ben-Abdallah, K. Joulain, Fundamental limits for noncontact transfers between two bodies, Phys. Rev. B 82 (12) (2010) 121419. [21] Z.M. Zhang, Nano/microscale Heat Transfer, McGraw-Hill, New York, 2007. [22] A. Knoesen, M.G. Moharam, T.K. Gaylord, Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals, Appl. Phys. B 38 (3) (1985) 171–178. [23] S.A. Biehs, P. Ben-Abdallah, F.S.S. Rosa, K. Joulain, J.J. Greffet, Nanoscale heat flux between nanoporous materials, Opt. Express 19 (S5) (2011) A1088– A1103.
[24] H. Wang, X. Liu, L. Wang, Z. Zhang, Anisotropic optical properties of silicon nanowire arrays based on the effective medium approximation, Int. J. Therm. Sci. 65 (2013) 62–69. [25] L. Zhu, C.R. Otey, S. Fan, Ultrahigh-contrast and large-bandwidth thermal rectification in near-field electromagnetic thermal transfer between nanoparticles, Phys. Rev. B 88 (18) (2013) 184301.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Thermal switch and thermal rectification enabled by near-field radiative heat transfer between three slabs Wei Gu, Gui-Hua Tang, Wen-Quan Tao ⇑ Key Laboratory of Thermo-Fluid Science and Engineering of MOE, Xi’an Jiaotong University, Xi’an 710049, China
a r t i c l e
i n f o
Article history: Received 21 September 2014 Received in revised form 16 November 2014 Accepted 17 November 2014 Available online 9 December 2014 Keywords: Near-field Radiative heat flux Insulator–metal transition material Thermal switch Thermal rectification
a b s t r a c t In this paper, the near-field radiative heat flux of the system consisting of two SiO2 plate sources and one 50 nm thick VO2 film, a kind of insulator–metal transition material, placed between them is studied. The two sources are maintained at 400 and 300 K, respectively, and separated by a distance of 150 nm. By this configuration, the temperature of the film can be regulated by the control of the separation distance between the film and the sources via a piezoelectric motor. It is found that the net radiative heat flux of this system can be varied in a range of 7.5 103–3.2 104 W/m2 for different position of the VO2 film due to its phase transition. Hence, the functions of thermal switch and thermal rectification are able to be realized by this system. Particularly, the direction of the larger heat flux in the proposed thermal rectification can be reversed by changing the position of the VO2 film. The effects of some parameters are investigated to improve the performance of the thermal rectification. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction As the counterpart of an electric rectifier and an electric switch in electronics, a thermal rectifier is a device that allows heat to flow preferentially in one direction [1,2], and a thermal switch is a device that allows the heat flux in the ‘‘on’’ mode being obviously larger than that in the ‘‘off’’ mode [3,4]. Such devices may have enormous applications in thermal management and control for energy systems and microelectronic devices. The previous thermal rectifications were mainly limited to heat conduction and convection [1,5,6]. Recently, photonic thermal rectification based on nearfield radiative transfer [7–13] has attracted attention due to the advantage of obtaining large rectification factors over a broad temperature range. The reported photonic thermal rectifiers are all made of two terminal parts and realized by one or two of the following mechanisms: (1) an asymmetric geometric arrangement [9], (2) dissimilar materials with different temperature-dependent thermal properties [7,8,10–13]. Recently, Ben-Abdallah and Biehs [4] designed a photonic thermal transistor, and the functions of thermal switch, thermal modulation and thermal amplification were realized simultaneously. The thermal transistor is composed of three basic elements, thermal source, thermal drain, and gate, and the heat flux can be controlled by the gate made of an insulator–metal transition (IMT) material, ⇑ Corresponding author. Fax: +86 29 82669106. E-mail address: [email protected] (W. Q. Tao). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.11.058 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
whose optical property is able to be changed through a small variation of its temperature around the critical temperature Tc. In literature [4], the temperature of the gate was controlled by a certain amount of heat added to or removed from it. However, there may be two weaknesses of that way to regulate the temperature of the gate. One probably comes from the fact that two different elements are needed to add heat to or remove heat from the gate, which makes the system more complicated. The other is that there exists a region where a certain amount of heat added to the gate corresponds to three different temperatures of the gate (as shown in Fig. 3 of literature [4]), which makes it difficult to control the heat flux. In this paper, we propose a new way to control the temperature of the gate by changing the distance between the gate and the thermal source, which can be carried out, for example, with the piezoelectric motors from Attocube as used in the experiment conducted by Rousseau et al. [14], where the displacements of a silica microsphere were carried out with the piezoelectric stages by steps of 7 nm. With this improvement, a three-part system is realized simply to switch or rectify the heat flux carried by photons. 2. Configuration and calculation of the heat flux 2.1. Configuration The structure of the proposed near-field system is schematically plotted in Fig. 1. Medium 1 and medium 3 both made of
430
W. Gu et al. / International Journal of Heat and Mass Transfer 82 (2015) 429–434
amorphous silicon dioxide (SiO2) are assumed to be semi-infinite and maintained at 400 and 300 K, respectively, by some thermostats. A thin layer (medium 2) made of vanadium dioxide (VO2) having thickness d is placed between them at distances d12 from medium 1 and d23 from medium 3. VO2 is one of the IMT material that undergoes a phase transition from a hightemperature metallic phase to a low-temperature insulating phase around its critical temperature (Tc 340 K) [15,16]. The thickness of the film is 50 nm, and distances d12 and d23 are related by d12 + d23 = 100 nm to introduce the near-field effect and to get large heat flux. The temperature T2 of the film is determined by thermal equilibrium with media 1 and 3, and undoubtedly T2 is a function of d12 or d23, which enables the control of heat flux by regulating d12 or d23 with the piezoelectric motors. 2.2. Calculation of heat flux The dielectric function of insulating VO2 in the bulk shape is related to the crystal orientation in the infrared wave region, while that of insulating VO2 in the film shape or VO2 in metallic phase does not exhibit anisotropy [15–17]. Here we consider the case where the optical axis of VO2 film is orthogonal to its interfaces, and its dielectric functions are obtained from [15,16], while the dielectric functions of SiO2 are taken from Ref. [18]. Fig. 2 shows the real (e0 ) and imaginary (e00 ) parts of the dielectric functions of VO2 and SiO2. It can be seen from Fig. 2 that SiO2 has two strong phonon modes at the frequencies 8.5 1013 and 2.0 1014 rad/s, and the phonon mode at lower frequency overlaps with that of insulating VO2. In Fig. 1, the heat flux across any planes parallel to the surfaces of the three bodies can be obtained by calculating the Poynting vector. Considering the three-body configuration, the theory developed by Messina and Antezza [19] recently is used to investigate the net radiative heat flux. This theory is valid for arbitrary bodies of a three-body system, i.e., for any set of temperatures and geometrical properties, and describes each bodies by means of their scattering operators. The net heat flux emitted by body 1 reads [20]:
Z 1 2 dx d k hx 2p ð2pÞ2 0 0 i Xh n12 ðxÞCj1=2 ðx;k;d12 ;d23 Þ þ n13 ðxÞCj1=3 ðx;k;d12 ;d23 Þ
U1 ¼
Z
1
j¼s;p
ð1Þ
Fig. 2. Dielectric functions of VO2 and SiO2: (a) real parts, (b) imaginary parts.
where, C1=2 and C2=3 denote the transmission probabilities of phoj j tons, defined by mode (x, k) for polarization state j = s, p, from body 1 to body 2 and from body 1 to body 3, respectively; k = (kx, ky) is the wave vector parallel to the surfaces of multilayer system. nij(x) = ni(x, Ti) nj(x, Tj) denotes the difference of Bosedistribution functions with ni/j = 1/[exp(hx/kBTi/j) 1]. 2ph is the Planck’s constant and kB is the Boltzmann’s constant. Similarly, the heat fluxes received by body 2 and body 3 are given by:
Z 1 2 dx d k hx 2p ð2pÞ2 0 0 i Xh n12 ðxÞCj1=2 ðx; k; d12 ; d23 Þ þ n32 ðxÞC3=2 j ðx; k; d12 ; d23 Þ
U2 ¼
Z
1
j¼s;p
ð2Þ Z 1 2 dx d k hx 2p ð2pÞ2 0 0 i Xh 1=3 2=3 n13 ðxÞCj ðx;k;d12 ; d23 Þ þ n23 ðxÞCj ðx; k; d12 ; d23 Þ
U3 ¼
Z
1
j¼s;p
ð3Þ
Fig. 1. Schematic of the system in the three-slab configuration.
It is easy to derive the relation U2 = U1 U3 from Eqs. (1)–(3). At steady state, the heat flux U1 emitted by body 1 equals the heat flux U3 received by body 3, and hence the net heat flux U2 received by medium 2 vanishes. All the transmission probabilities Cj are defined in terms of ðiÞ optical reflection coefficients qj ði ¼ 1; 2; 3 and j ¼ s; pÞ and
W. Gu et al. / International Journal of Heat and Mass Transfer 82 (2015) 429–434 ðiÞ
transmission coefficients sj , and their detail expressions can be ðiÞ ðiÞ obtained from Ref. [19]. The quantities qj and sj are expressed as [19]: ð1Þ qð1Þ j ðk; xÞ ¼ r j ðk; xÞ
sð1Þ j ðk; xÞ ¼ 0 ð2Þ
ð2Þ qð2Þ j ðk; xÞ ¼ r j ðk; xÞ
1 ð2Þ j ðk;
s
xÞ ¼
ð2Þ t j ðk;
h
1 e2ikz ð2Þ rj ðk;
d
i2 ð2Þ xÞ e2ikz d
ð2Þ
iðkz xÞtð2Þ j ðk; xÞe
ð4Þ
kz Þd
h i2 ð2Þ ð2Þ 1 rj ðk; xÞ e2ikz d
ð3Þ qð3Þ j ðk; xÞ ¼ r j ðk; xÞ
sð3Þ j ðk; xÞ ¼ 0 ðiÞ
In Eq. (4), the vacuum-medium Fresnel reflection (r j ) and transmission
ðiÞ (tj )
coefficients, as well as the medium-vacuum
ðiÞ ðiÞ Fresnel transmission (t j ) coefficients are included. kz and kz
denote the z component of the wave vector inside the medium i and vacuum, respectively. In addition to the isotropic media, Eqs. (1)–(4) can be easily extended to the anisotropic media by considering the anisotropic wave propagation [10,13,21–24].
431
Note that the variation of T2 with d12 is monotonic, and this makes the control of T2 easy. The net heat flux transfer U of this system for steady state is shown in Fig. 4, which is equal to U1 or U3. A change of d12 from 55 to 56 nm will lead to an increase of net heat flux transfer by a factor of 3 at least. To further understand the effect of phase transition on the net radiative heat flux transfer, the distributions of U with modes (x, k) for d12 = 44 nm (VO2 in metallic phase) and d12 = 56 nm (VO2 in insulating phase) are plotted in Fig. 5. Note that k, the parallel component of wave vector, is normalized by the vacuum wave vector x/c. When the film is in its insulating phase, /1 (see Fig. 5(b)) shows an efficient coupling of mode around the resonance frequencies xspp1 8.5 1013 rad/s and xspp2 2.0 1014 rad/s of surface phonon–polaritons supported by SiO2. However, /1 in these two resonance frequencies will be weakened by the metallic VO2 film (see Fig. 5(a)). The results of Fig. 5(a) and (b) can be explained from two aspects. One comes from that the surface phonon–polaritons at frequency xspp1 8.5 1013 rad/s of SiO2 overlap with that of insulating VO2, and their phonon–phonon coupling will result in a significant enhancement of near-field radiative heat transfer [17]; the other is that metallic VO2 does not support surface polaritons and has high reflectance in the infrared frequencies, and hence the metallic VO2 will suppress the coupling of surface phonon–polaritons between the two sources of SiO2.
3. Result and discussion 3.2. Thermal switch and thermal rectification 3.1. Variation of temperature T2 and net heat flux U1 with the distance d12 Without external excitation, the system depicted in Fig. 1 reaches its steady state for which the net heat flux U2 received by medium 2 is zero. In this case, the temperature T2 is set by temperatures T1, T3 and distance d12. Figure 3 shows the variation of equilibrium temperature T2 with distance d12, which is determined by solving the equation U2 = 0. In Fig. 3, the solid line indicates the variation of T2 with the temperature dependence of the dielectric response of VO2 being taken into consideration, while the dash line and dash dot line give the results, respectively, for insulating VO2 and metallic VO2 with the assumption of temperature independence of the dielectric response. The classical effective medium theory is used to model the transition region (340.0 < T2 < 345.2 K) of IMT VO2 with the data provided by Qazilbash et al. [16]. It can be seen that the evolution shapes of T2 with d12 for insulating VO2 and metallic VO2 are very different, and hence there is an obvious jump of T2 after phase transition.
By using different physical parameters (temperatures, sizes, separation distances, etc.), several functions can be achieved by this system, and two evident ones of them are the thermal switch and thermal rectification. As has been shown in Fig. 4, the net heat flux exchange between the two sources is able to be tripled at least by changing the separation distance d12 from 55 to 56 nm. That means the proposed system can be used in two operating modes, the ‘‘off’’ mode for d12 < 55 nm and the ‘‘on’’ mode for d12 > 56 nm. A thermal rectifier is a device in which heat flux along a specific direction is quite large, while along the opposite direction is relatively small. The near-field thermal rectification was first proposed by Otey et al. [7], and after that, several different near-field thermal rectifications were designed [8–13,25]. All of the previous nearfield thermal rectifications are designed with two terminal parts on the basis of asymmetric geometric arrangement or dissimilar materials, and their work directions with the larger heat fluxes are unchangeable. For example, the SiO2–VO2 two terminal thermal rectification proposed in [13] allows the larger heat flux only
Fig. 3. Temperature T2 of the VO2 film for different d12.
Fig. 4. Net heat flux transfer for different d12.
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W. Gu et al. / International Journal of Heat and Mass Transfer 82 (2015) 429–434
Fig. 6. The net heat fluxes of forward-biased and reverse-biased for different separation distance d12.
Fig. 7. The rectification factor for different separation distance d12.
Fig. 5. Distribution of U1 with modes (x, k) in the steady state: (a) d12 = 44 nm with VO2 in metallic phase, (b) d12 = 56 nm with VO2 in insulating phase.
in the direction with VO2 in the temperature lower than Tc (i.e., VO2 in insulating phase), and this direction cannot be reversed with the fixed sources. However, this is not the situation for three-body system as shown in Fig. 1, which consists of two sources with temperature independence of dielectric response and one IMT film. This system enables the direction with the larger heat flux to be reversed by controlling the position of the film, which is one of the major contributions to this topic. In Fig. 1, the forward-biased scenario is defined for T1 = 400 K and T3 = 300 K, and the reverse-biased scenario is defined for T1 = 300 K and T3 = 400 K. The net heat fluxes of forward-biased and reverse-biased for different separate distances d12 are plotted in Fig. 6. As discussed above, the direction where the heat flux is larger can be reversed by changing the position of the VO2 film. The net heat flux in the forward-biased scenario (denoted by UF) is smaller than that in the reverse-biased scenario (denoted by UR) for d12 < 44 nm. On the other hand, UF is larger than UR for d12 > 56 nm. Hence, a forward-favored or reverse-favored thermal rectification can be realized simultaneously by the proposed
system just by setting the distance d12. It should be noted that the proposed system shows no rectification for d12 in the range of 44–56 nm. The thermal rectification factor f of this system is given in Fig. 7, which is calculated by (UF UR)/Min(UF, UR). The amplitude of the thermal rectification factor is able to reach 2.32, much higher than 0.41 obtained by Otey et al. [7]. In addition, the magnitude of the rectification factor of the proposed system can be varied in a range of 1.67–2.32 as plotted in Fig. 7. The proposed thermal rectification with modulable rectification factor is another major contribution to this topic. 3.3. Parameters investigation A thermal rectification with good performance should have high rectification factor and large heat flux. In the above part, we have investigated the thermal rectification of the three-slab system with parameters of d = 50 nm and d12 + d23 = 100 nm. However, it is foreseeable that the performance of this thermal rectification is sensitive to the parameters d, d12 and d23, and we will show how these parameters influence the performance of the thermal rectification in the following. In order to reduce the number of free parameters for analysis, we will focus our attention on the symmetric case, in which d12 equals d23, without considering phase change of VO2 film. Figure 8(a) plots the radiative heat flux as a function of film thickness d and distance d12 for VO2 in insulating phase. It can be
W. Gu et al. / International Journal of Heat and Mass Transfer 82 (2015) 429–434
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4. Conclusions We have studied the near-field radiative heat flux of a threeslab system consisting of two amorphous SiO2 sources and one IMT VO2 film between them. There is a monotonic relationship between the equilibrium temperature T2 of the film and the separation distance d12, and hence the control of T2 can be realized by modulating d12 via the piezoelectric motors. For insulating phase of VO2, its surface phonon–polaritons at frequency x = 8.5 1013 rad/s overlaps with that of SiO2, and their phonon–phonon coupling promotes the near-field heat transfer. In addition, metallic VO2 has large reflectance in the infrared frequencies and suppresses the heat transfer between the two sources. These two features result in an enhancement of net heat flux by a factor 3.32 after the VO2 film transits from metallic phase to insulator phase for the proposed system. By using different parameters (source temperatures and separation distances), thermal switch and thermal rectification are realized by the proposed system. In particular, the direction permitting the larger heat flux in the proposed thermal rectification can be reversed by changing the position of the film. Finally, it has been shown that by varying the thickness d of VO2 film and the separation distance d12 the performance of the thermal rectification can be greatly improved.
Acknowledgment This work is sponsored by the Natural Science Foundation of China (51136004).
References
Fig. 8. (a) Radiative heat flux as a function of d12 and d with VO2 in insulating phase, (b) the ratio of radiative heat flux for VO2 in insulating phase to that for VO2 in metallic phase as a function of d12 and d.
seen that the radiative heat flux is more sensitive to d12 than to d. Thus, the separation distances d12 and d23 should be small enough to get large radiative heat flux. Figure 8(b) shows the ratio of radiative heat flux for VO2 in insulating phase (marked by /i) to that for VO2 in metallic phase (marked by /m) as a function of d12 and d. As shown there, the values of /i//m are generally increased with thicker VO2 film for the same values of d12. The regions where /i//m is smaller than one and larger than six are indicated by the red dash line and black dash line respectively. It is certain that the rectification factor of the three-slab configuration is not equal to /i//m 1, because the thermal rectification is realized by asymmetric configuration of the three-slab system, i.e., d12 – d23. However, large values of /i//m will lead to large rectification factors, thus, the results given by Fig. 8(b) are helpful to design the thermal rectification. Accordingly, rectification factor can be increased by a thicker VO2 film for the case of d12 + d23 = 100 nm, and a rectification factor of 5.2 can be reached by replacing 50 nm thick VO2 film with a 200 nm thick one, whose value is close to /i//m 1 given by Fig. 8(b) for d = 200 nm and d12 = 50 nm.
[1] B. Li, L. Wang, G. Casati, Thermal diode: rectification of heat flux, Phys. Rev. Lett. 93 (18) (2004) 184301. [2] C.W. Chang, D. Okawa, A. Majumdar, A. Zettl, Solid-state thermal rectifier, Science 314 (5802) (2006) 1121–1124. [3] B. Li, L. Wang, G. Casati, Negative differential thermal resistance and thermal transistor, Appl. Phys. Lett. 88 (14) (2006) 143501. [4] P. Ben-Abdallah, S.-A. Biehs, Near-field thermal transistor, Phys. Rev. Lett. 112 (4) (2014) 044301. [5] N.A. Roberts, D.G. Walker, A review of thermal rectification observations and models in solid materials, Int. J. Therm. Sci. 50 (5) (2011) 648–662. [6] N. Yang, G. Zhang, B. Li, Carbon nanocone: a promising thermal rectifier, Appl. Phys. Lett. 93 (24) (2008) 243111. [7] C.R. Otey, W.T. Lau, S. Fan, Thermal rectification through vacuum, Phys. Rev. Lett. 104 (15) (2010) 154301. [8] S. Basu, M. Francoeur, Near-field radiative transfer based thermal rectification using doped silicon, Appl. Phys. Lett. 98 (11) (2011) 113106. [9] H. Iizuka, S. Fan, Rectification of evanescent heat transfer between dielectriccoated and uncoated silicon carbide plates, J. Appl. Phys. 112 (2) (2012) 024304. [10] P. Ben-Abdallah, S.-A. Biehs, Phase-change radiative thermal diode, Appl. Phys. Lett. 103 (19) (2013) 191907. [11] J. Huang, Q. Li, Z. Zheng, Y. Xuan, Thermal rectification based on thermochromic materials, Int. J. Heat Mass Transfer 67 (2013) 575–580. [12] L.P. Wang, Z.M. Zhang, Thermal rectification enabled by near-field radiative heat transfer between intrinsic silicon and a dissimilar material, Nanoscale Microscale Thermophys. Eng. 17 (4) (2013) 337–348. [13] Y. Yang, S. Basu, L. Wang, Radiation-based near-field thermal rectification with phase transition materials, Appl. Phys. Lett. 103 (16) (2013) 163101. [14] E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, J.J. Greffet, Radiative heat transfer at the nanoscale, Nat. Photonics 3 (9) (2009) 514–517. [15] A.S. Barker, H.W. Verleur, H.J. Guggenheim, Infrared optical properties of vanadium dioxide above and below the transition temperature, Phys. Rev. Lett. 17 (26) (1966) 1286–1289. [16] M.M. Qazilbash, M. Brehm, B.-G. Chae, P.-C. Ho, G.O. Andreev, B.-J. Kim, S.J. Yun, A.V. Balatsky, M.B. Maple, F. Keilmann, H.-T. Kim, D.N. Basov, Mott transition in VO2 revealed by infrared spectroscopy and nano-imaging, Science 318 (5857) (2007) 1750–1753. [17] P.J. van Zwol, K. Joulain, P. Ben-Abdallah, J. Chevrier, Phonon polaritons enhance near-field thermal transfer across the phase transition of VO2, Phys. Rev. B 84 (16) (2011) 161413. [18] E.D. Palik, Handbook of Optical Constants of Solids, 3, Academic Press, 1998. [19] R. Messina, M. Antezza, Three-body radiative heat transfer and Casimir– Lifshitz force out of thermal equilibrium for arbitrary bodies, Phys. Rev. A 89 (5) (2014) 052104.
434
W. Gu et al. / International Journal of Heat and Mass Transfer 82 (2015) 429–434
[20] P. Ben-Abdallah, K. Joulain, Fundamental limits for noncontact transfers between two bodies, Phys. Rev. B 82 (12) (2010) 121419. [21] Z.M. Zhang, Nano/microscale Heat Transfer, McGraw-Hill, New York, 2007. [22] A. Knoesen, M.G. Moharam, T.K. Gaylord, Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals, Appl. Phys. B 38 (3) (1985) 171–178. [23] S.A. Biehs, P. Ben-Abdallah, F.S.S. Rosa, K. Joulain, J.J. Greffet, Nanoscale heat flux between nanoporous materials, Opt. Express 19 (S5) (2011) A1088– A1103.
[24] H. Wang, X. Liu, L. Wang, Z. Zhang, Anisotropic optical properties of silicon nanowire arrays based on the effective medium approximation, Int. J. Therm. Sci. 65 (2013) 62–69. [25] L. Zhu, C.R. Otey, S. Fan, Ultrahigh-contrast and large-bandwidth thermal rectification in near-field electromagnetic thermal transfer between nanoparticles, Phys. Rev. B 88 (18) (2013) 184301.