Many-body effective thermal conductivity in phase-change nanoparticle chains due to near-field radiative heat transfer
Minggang Luo, Junming Zhao, Linhua Liu, Brahim Guizal, Mauro Antezza
MMany-body effective thermal conductivity in phase-changenanoparticle chains due to near-field radiative heat transfer
Minggang Luo a,b , Junming Zhao a,c, ∗ , Linhua Liu d , Brahim Guizal b , Mauro Antezza b,e, ∗ a School of Energy Science and Engineering, Harbin Institute of Technology, 92 West Street, Harbin 150001, China b Laboratoire Charles Coulomb (L2C) UMR 5221 CNRS-Université de Montpellier, F- 34095 Montpellier, France c Key Laboratory of Aerospace Thermophysics, Ministry of Industry and Information Technology, Harbin 150001,China d School of Energy and Power Engineering, Shandong University, Qingdao 266237, China e Institut Universitaire de France, 1 rue Descartes, F-75231 Paris Cedex 05, France
Abstract
In dense systems composed of numerous nanoparticles, direct simulations of near-field ra-diative heat transfer (NFRHT) require considerable computational resources. NFRHT for thesimple one-dimensional nanoparticle chains embedded in a non-absorbing host medium isinvestigated from the point of view of the continuum by means an approach combining themany-body radiative heat transfer theory and the Fourier law. Effects of the phase changeof the insulator-metal transition material (VO ), the complex many-body interaction (MBI)and the host medium relative permittivity on the characteristic effective thermal conductiv-ity (ETC) are analyzed. The ETC for VO nanoparticle chains below the transition temperaturecan be as high as 50 times of that above the transition temperature due to the phase changeeffect. The strong coupling in the insulator-phase VO nanoparticle chain accounts for itshigh ETC as compared to the low ETC for the chain at the metallic phase, where there is a mis-match between the characteristic thermal frequency and resonance frequency. The strongMBI is in favor of the ETC. For SiC nanoparticle chains, the MBI even can double the ETC ascompared to those without considering the MBI effect. For the dense chains, a strong MBIenhances the ETC due to the strong inter-particles couplings. When the chains go more andmore dilute, the MBI can be neglected safely due to negligible couplings. The host mediumrelative permittivity significantly affects the inter-particles couplings, which accounts for thepermittivity-dependent ETC for the VO nanoparticle chains. Keywords: effective thermal conductivity, near-field radiative heat transfer, many-bodyinteraction, insulator-metal phase-change material, nanoparticles
1. Introduction
Near-field radiative heat transfer (NFRHT) is currently attracting a lot of interests for itsfundamental and applicative facets [1, 2, 3, 4, 5]. In dense particulate systems, the separa- ∗ Corresponding author
Email addresses: [email protected] (Junming Zhao), [email protected] (Mauro Antezza)
Preprint submitted to Elsevier August 21, 2020 a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug ion distance between two nanoparticles is often comparable to or less than the character-istic thermal wavelength [6]. Due to the near-field effect (e.g., evanescent wave tunneling),the heat flux will exceed the Planck's blackbody limit by several orders of magnitude, whichhas been predicted thanks to the fluctuational electrodynamics theory [7, 8, 9, 10, 11, 12] andproved by recent experimental observations [13, 14, 15, 16, 17, 18].Many important progresses have been reported on direct simulations of NFRHT for sys-tems composed of nanoparticles. On the one hand, the inter-ensemble NFRHT between twonanoparticle ensembles (e.g., three-dimensional (3D) clusters of hundreds of nanoparticles[6, 19, 20] and two-dimensional (2D) nanoparticle ensembles [21, 22, 23], as well as the sim-ple ensembles composed of only a few nanoparticles [24, 25]) has been analyzed. On theother hand, the intra-ensemble NFRHT inside the nanoparticle ensemble itself has also beenreported. The fractional diffusion theory was applied to describe NFRHT along 3D plasmonicnanostructure networks, as well as one-dimensional (1D) ones, and the heat superdiffusionbehavior was found [26]. For 2D fractal structures, the effects of the structure morphology onthe collective properties were analyzed and the heat flux has no large-range character, in con-trast to non-fractal structures [27]. Thermal radiation behavior along a 1D nanoparticle chainhas been shown to be significantly affected by another nanoparticle chain in proximity due tostrong couplings [28]. Radiative heat flux along a linear chain considering an external mag-netic field was also analyzed [29]. In addtion, thermal transport behaviors along the atomicchains due to quantum effects have been reported recently [30, 31, 32]. In general, the di-rect simulation of NFRHT for dense particulate systems composed of hundreds of thousandsof nanoparticles will introduce considerable unknowns, which will be very time-consumingand will require considerable computational resources.From the point of view of continuum, the method applying the effective thermal con-ductivity (ETC) to characterize the NFRHT in dense particulate systems based on the diffu-sion assumption is really time-saving, as compared to direct simulation methods (e.g., many-body radiative heat transfer theory [24, 25], scattering matrix method [33, 34], trace formulasmethod [35, 36], thermal discrete dipole approximation method (T-DDA) [37, 38, 39], fluctu-ating surface currents approach (FSC)[40], boundary element method (BEM) [41], finite dif-ference time domain method (FDTD) [42, 43, 44] and the quasi-analytic solution [45], to namea few). The kinetic theory (KT) framework was applied to obtain the ETC for 1D nanoparticlechains [46, 47]. Recently, the limitations of KT framework to describe NFRHT was analyzedsystematically: 1) the KT framework is not suitable for materials with resonant modes outsidethe Planck's window (e.g., metal Ag) and 2) the KT framework cannot be applied directly to 2Dand 3D systems due to the lack of dispersion relations for these systems [48, 49]. Most recently,a new method based on the many-body radiative heat transfer theory and the Fourier law (MFmethod) was proposed by Tervo et al . [50] to obtain the ETC for arbitrary nanoparticle collec-tions allowing the comparison between different materials and different heat transfer modes.Due to the limitation of the KT framework describing ETC for nanoparticle ensemblesNFRHT, the investigation on ETC for materials supporting resonances outside the Planck'swindow is still missing. Furthermore, VO attracts lots of interests because of its specialinsulator-metal transition behavior around its phase transition temperature. Besides, thereis still lack of the investigation on the ETC for phase-change VO nanoparticle ensembles,especially for its metallic phase. Based on the promising phase-changing characteristics forVO , many potential applications were proposed recently: 1) near-field applications (e.g., theradiative thermal rectifier [51, 52], thermal transistor [53], conductive thermal diode [54], dy-namic radiative cooling [55] and the scalable radiative thermal logic gates [56]); 2) far-field ap-2lications (e.g., radiative thermal memristor [57] and radiative thermal rectifier [58]). Hence,it's worth analyzing the effects of the phase change on the ETC for VO nanoparticle ensem-bles, in addition to the phase-change effect on the thermal conductance reported very re-cently [21].Nanoparticles in a dense particulate system often lie in the near field of each other, whichresults in the many-body interaction (MBI) making the NFRHT mechanism more complex[24, 50, 59]. Though the complex MBI effects on the radiative heat flux and thermal conduc-tance for various nanoparticle systems (three-nanoparticle system [24, 60, 61], 1D nanoparti-cle chains [28], 2D nanoparticle ensembles [21, 23, 27], 3D nanoparticle ensembles [6, 19, 20])have been analyzed, the MBI effect on the ETC is still missing.We extract simple 1D nanoparticle chains from realistic 3D nanoparticle ensembles em-bedded in a non-absorbing host medium and focus on the thermal property (i.e., effectivethermal conductivity) describing and characterizing the NFRHT from the point of view ofcontinuum. The relative permittivity of the host medium significantly affects the inter-particlecouplings [49, 59]. Effects of the host medium relative permittivity on the ETC of closelyspaced 1D metallic nanoparticle chains have already been analyzed [46]. However, effectsof the host medium relative permittivity on the ETC for the specific case of phase-change VO nanoparticles have not yet been investigated.We address the aforementioned missing points in this paper, where the ETC for the 1Dnanoparticle chains of interest is obtained by means of the MF method. In Sec. 2, we give abrief description of the theoretical models for the MF method, as well as the formulas con-cerning the ETC for 1D nanoparticle chains. In Sec. 3, effects of the phase change of theinsulator-metal transition material (i.e., VO ), complex many-body interaction and host mediumrelative permittivity ( (cid:178) m ) on the effective thermal conductivity due to NFRHT are analyzed.The optical properties for the materials used in this work are also given in this section.
2. Models
In this section, we describe in brief the physical system (the schematic is shown in Fig. 1)and the theoretical aspects of the MF method for the ETC due to NFRHT in 1D nanoparticlechains. The nanoparticle chain is divided into two parts L and R by an imaginary plane. Figure 1: Schematic of the ordered nanoparticle chain embedded in a non-absorbing host medium with permittivity (cid:178) m . The radiative heat flux in the chain is the sum of the net heat exchange between nanoparticles from part L andnanoparticles from part R . Parts L and R are separated by the imaginary surface (dash line). The lattice spacing is h .Nanoparticle radius is a . A small linear temperature gradient d T /d x along the chain is assumed. The ETC due to near-field radiative heat transfer is defined as follows: k e f f = QA · | d T /d x | , (1)3here A is the cross section, Q is the net radiative heat flux in the chain under a small lineartemperature gradient d T /d x , which is given as follows [50]. Q = (cid:88) i ∈ L (cid:88) j ∈ R G ij ( T ) d ij (cid:175)(cid:175)(cid:175)(cid:175) d T d x (cid:175)(cid:175)(cid:175)(cid:175) , (2)where d ij is the separation distance center to center between the nanoparticle i from the part L of the chain and nanoparticle j from the part R of the chain, G ij ( T ) is the radiative thermalconductance between nanoparticles i and j, which yields [23, 26] G ij ( T ) = (cid:90) +∞ d ω π ∂ Θ ( ω , T ) ∂ T T i,j ( ω ), (3)where Θ ( ω , T ) is the mean energy of a harmonic Planck's oscillator, ω is the angular fre-quency, the parameter T i,j ( ω ) = k (cid:178) m Im( χ i E )Im( χ j E )Tr( G EE ij G EE †ij ), χ E = α E − ik π | α E | , α E is thenanoparticle polarizability, k = (cid:112) (cid:178) m ω / c is the wave vector in the host medium, (cid:178) m is the hostmedium relative permittivity, c is the speed of light in vacuum, the Green 's function G EE ij inthe many-particle system naturally includes the many-body interaction and is the element ofthe following left matrix. G EE ··· G EE N G EE G EE ( N − N G EEN G EEN ··· = G EE ··· G EE N G EE G EE N − N G EE N G EE N ··· (cid:65) − , (4)where G EE = e ikr π r (cid:104)(cid:179) + ikr − k r (cid:180) (cid:73) + − ikr − k r k r ˆ r ⊗ ˆ r (cid:105) is the free space Green's function connect-ing two nanoparticles at r i and r j , r is the magnitude of the separation vector r = r i − r j , ˆ r is the unit vector r / r , (cid:73) is the 3 × (cid:65) including many-bodyinteractions is defined as (cid:65) = (cid:73) N − k α E G EE ··· α E G EE N α E G EE α N − E G EE N − N α NE G EE N ··· α NE G EE N ( N − , (5)where (cid:73) N is the 3 N × N identity matrix. Hence, the effective thermal conductivity will berearranged as k e f f = A (cid:88) i ∈ L (cid:88) j ∈ R G ij ( T ) d ij . (6)The effective thermal conductivity k e f f can also be expressed as the frequency integral of thespectral effective thermal conductivity k ω : k e f f = (cid:82) +∞ k ω d ω . For materials (e.g., metal Ag)where the magnetic-magnetic polarized eddy-current Joule dissipation dominates the radia-tive heat transfer, rather than the electric-electric polarized displacement current dissipation,the magnetic dipole contribution to the radiative heat transfer can be taken into considera-tion in the parameter T i,j ( ω ) by the coupled electric and magnetic dipole approach [6, 21].4 . Results and discussion In this section, the optical properties of phase-change VO and polar SiC nanoparticles areintroduced. Effects of the phase change, complex many-body interaction and host mediumrelative permittivity on the ETC of the 1D nanoparticle chains due to the NFRHT are analyzedby means of the MF method. We consider particles with radius a =
25 nm forming a chainwith 250 elements in part L and as many in part R . This is large enough to reach convergentresults for all the calculations of interest considered here [49]. VO is a kind of phase-change materials, which undergoes an insulator-metal transitionaround 341 K (phase transition temperature). Below 341 K, VO is an uniaxial anisotropicinsulator, of which the dielectric function can be described by the following tensor [62]: (cid:178) (cid:107) (cid:178) ⊥
00 0 (cid:178) ⊥ , (7)where (cid:178) ⊥ and (cid:178) (cid:107) are the ordinary and extraordinary dielectric function component relative tothe optical axis, respectively. Both (cid:178) ⊥ and (cid:178) (cid:107) can be described by the Lorentz model as follows: (cid:178) ( ω ) = (cid:178) ∞ + N L (cid:88) n = S n ω n ω n − i γ n ω − ω , (8)where S n , ω n and γ n are the phonon strength, phonon frequency and damping coefficientof the n th phonon mode. N L is the number of phonon modes ( N L = (cid:178) ⊥ and N L = (cid:178) (cid:107) ). All the necessary parameters for both (cid:178) ⊥ and (cid:178) (cid:107) can be found in Ref.[62]. Above 341K, thedielectric function of the metallic-phase VO is described by the Drude model as follows [62]: (cid:178) ( ω ) = (cid:178) ∞ ω p ω − i ωγ , (9)where (cid:178) ∞ = ω p = 1.51 × rad · s − and γ =1.88 × rad · s − . In addition to the phase-change VO , the polar SiC is also used. The dielectric functions of SiC is described by theDrude-Lorentz model (cid:178) ( ω ) = (cid:178) ∞ ( ω − ω l + i γω )/( ω − ω t + i γω ) with parameters (cid:178) ∞ = 6.7, ω l = 1.827 × rad · s − , ω t = 1.495 × rad · s − , and γ = 0.9 × rad · s − [63].For an isotropic material embedded in the host medium with (cid:178) m , the polarizability can beobtained from the first order Lorenz-Mie scattering coefficient [64, 65]. α E = i π k a , (10)where a is the first order Lorenz-Mie scattering coefficient defined as a = (cid:178) / (cid:178) m j ( y )[ x j ( x )] (cid:48) − j ( x )[ y j ( y )] (cid:48) (cid:178) / (cid:178) m j ( y )[ xh (1)1 ( x )] (cid:48) − h (1)1 ( x )[ y j ( y )] (cid:48) , (11)where x = ka , y = (cid:112) (cid:178) / (cid:178) m ka , a is the nanoparticle radius, (cid:178) is the relative permittivity, j ( x ) = sin( x )/ x − cos( x )/ x and h (1)1 ( x ) = e ix (1/ i x − x ) are the first order Bessel function and spher-ical Hankel function. For 1D nanoparticle chains composed of many anistropic insulator-phase VO nanoparticles, we assume that nanoparticlesâ ˘A´Z anisotropic axes are randomly5riented. For this reason, we decide to use the well-known 1/3 − (cid:178) ⊥ and (cid:178) (cid:107) separately, and then add up the results according to the 1/3 − α E = α E ( (cid:178) (cid:107) ) + α E ( (cid:178) ⊥ ). (12)The polarizabilities for both insulator-phase and metallic-phase VO nanoparticles are shownin Fig. 2. In order to compare the resonance frequency to the characteristic thermal frequency,the spectral radiance of the blackbody at 400 K is also added in Fig. 2(b) for reference. Thecharacteristic thermal frequency mismatches with the polarizability resonance frequency ofmetallic VO nanoparticle. (a) Insulator-phase VO (b) metallic-phase VO Figure 2: Polarizability of VO nanoparticle: (a) insulator phase and (b) metallic phase. Nanoparticle radius a is 25nm. (cid:178) m =
1. For insulator VO particle, the “1/3 − (cid:178) (cid:107) and (cid:178) ⊥ [66]. The spectral radiance of the blackbody at 400 K is also added for reference. The ETC of the ordered nanoparticle chains (as shown in Fig. 1) as a function of tempera-ture T is shown in Fig. 3. Here a = 25 nm and h = 75 nm. SiC and insulator-metal phase-changeVO nanoparticles chains are embedded in a host medium with (cid:178) m =
5. The temperature T ranges from 300 K to 500 K, including the transition temperature of the VO .For the non-phase-change SiC nanoparticle chain, the ETC increases monotonically withtemperature. While for the insulator-metal phase-change VO nanoparticle chain, an obvi-ous transition of the ETC can be observed around the transition temperature of the VO . Inthe temperature range of interest, the ETC for the metallic-phase VO nanoparticle chain athigh temperature is even much lower than that of the insulator-phase VO nanoparticle chainat low temperature, which is due to the insulator-metal phase change of VO . As shown inFig. 2(b), an obvious mismatch between the resonance frequency of the metallic-phase VO nanoparticle and the characteristic thermal frequency (Planck's window), which accounts forthe low ETC. However, for the insulator-phase VO nanoparticles, the resonance frequency6 igure 3: Effective thermal conductivity k e f f of nanoparticle chains as a function of temperature. Phase-change VO and polar SiC are considered. Nanoparticle radius a = 25 nm. Lattice spacing h =75 nm. (cid:178) m = matches well with the characteristic thermal frequency as shown in Fig. 2(a), which accountsfor the high ETC.To give a quantitative description on the phase change effect, the dependence of the ETCon h / a is shown in Fig. 4. SiC (300K), insulator-phase VO (300K) and metall-phase VO (400K) nanoparticles chains in vacuum are considered. The dependence of the ratio of k ie f f (the ETC for insulator-phase VO ) to k me f f (the ETC for metallic-phase VO ) on h / a is alsoshown in Fig. 4. In general, the ratio k ie f f / k me f f is much larger than unity. The ETC for theinsulator-phase VO nanoparticle chains is much larger than that of the metallic-phase VO nanoparticle chains. The ratio k ie f f / k me f f increases to its maximum (around 50) and then de-creases with increasing h / a . The phase change effect is significant when the chain is compactand decreases when the chain goes dilute. In addition, the ETC decreases with increasing h / a . The inter-particle coupling decreases when the lattice spacing of the nanoparticle chain h increases. The decreasing coupling accounts for the decreasing ETC when increasing h / a . To evaluate the effects of the many-body interaction on the ETC, we define the followingparameter [6]: ϕ = k e f f k e f f , (13)where k e f f is the ETC evaluated with the help of the radiative thermal conductance in Eq.(3)and the Green's function including the MBI in Eq.(4), k e f f is the ETC without the MBI evalu-ated with the help of the Eq.(3) and the free space Green's function. Generally speaking, theMBI inhibits the ETC when ϕ <
1, enhances it when ϕ > ϕ ≈
1. 7 igure 4: Dependence of the ETC on h / a . SiC (300K), insulator-phase VO (300K) and metallic-phase VO (400K)nanoparticles chains in vacuum are considered. The dependence of the ratio of k ie f f (the ETC for insulator-phaseVO ) to k me f f (the ETC for metallic-phase VO ) on h / a is also added. Nanoparticle radius a =
25 nm.
We quantitatively evaluate the MBI effect on the ETC for polar SiC, metallic-phase VO and insulator-phase VO nanoparticle chains. The dependence of the parameter ϕ definedby Eq.(13) on the geometrical dimensionless parameter h / a is shown in Fig. 5. Nanoparticlesof three different sizes have been considered a = 5 nm, 25 nm and 50 nm. As shown in Fig. 5, ϕ is never less than unity, which indicates that the MBI does not inhibit the ETC for the chainscomposed of the considered materials. When h / a > ϕ starts to approach unity. The MBIdecreases with the increasing lattice spacing h . When h / a < ϕ >
1. The MBI is favorable tothe ETC. Small lattice spacing (i.e., h / a <
8) results in strong inter-particles couplings, whichresulting in a significant MBI.The maximal ϕ for SiC nanoparticle chains is around 2. The MBI can double the ETC forSiC chains. For SiC nanoparticle chains, an extremum value for ϕ was observed at h / a ≈ ϕ does not change with varying the nanoparticle size. That is to say that the MBIis independent of the nanoparticle size. ϕ for the metallic-phase VO and insulator-phaseVO nanoparticle chains is similar to each other, though ϕ for the metallic-phase VO chainsis a little bit larger than that of the insulator-phase VO chains. The dependence of total ETC on the host medium permittivity (cid:178) m for insulator-metalphase-change VO is show in Fig. 6. Nanoparticles with radius a = 25 nm and h =
75 nmare used. ETC increases with the host medium relative permittivity (cid:178) m for both insulator-phase and metallic-phase VO . The host medium relative permittivity significantly affects the8 igure 5: The dependence of the ratio ϕ of the effective thermal conductivity k e f f of nanoparticle chains with theMBI to that without MBI on h / a . Insulator-phase VO , metallic-phase VO and polar SiC are considered. Nanoparti-cles of three different sizes have been considered a = 5 nm, 25 nm and 50 nm. (cid:178) m = inter-particle coupling, which finally significantly affects the ETC. High relative permittivity isfavorable to enhancing the ETC and radiative heat transfer in the nanoparticle chain, whichis consistent with the reported results for closely spaced metallic nanoparticle chains [46]. Inaddition, at low relative permittivity (cid:178) m , the difference between the ETC for the VO nanopar-ticle chains at different phases (i.e., metallic phase and insulator phase) is much smaller thanthat at high relative permittivity, of which the insight will be analyzed in the following fromthe thermal conductivity spectrum standpoint.The dependence of the spectral effective thermal conductivity k ω on the angular frequency ω and the relative permittivity of the host medium (cid:178) m is shown in Fig. 7: (a) insulator-phaseVO (300 K) and (b) metallic-phase VO (400 K) ( a =
25 nm and h =
75 nm). For a fixed angu-lar frequency, the value of the spectral effective thermal conductivity increases significantlywith increasing (cid:178) m , which is consistent with the dependence of total ETC on (cid:178) m , as shownin Fig. 6. Increasing the relative permittivity is in favor of enhancing the radiative effectivethermal conductivity. For metallic-phase VO nanoparticle chains, the frequency peak of thespectral effective thermal conductivity corresponds to the characteristic thermal frequency,as shown in Fig. 7(b). However, for insulator-phase VO nanoparticle chains, besides the peakof the spectral effective thermal conductivity corresponding to the characteristic thermal fre-quency, there are several secondary peaks, as shown in Fig. 7(a). It's worthwhile to mentionthat the peak of the spectral effective thermal conductivity shows a red-shift behavior with theincrease of relative permittivity (cid:178) m for both insulator-phase and metallic-phase VO , as canbe seen from Fig. 7.From the formulas for the ETC (i.e., Eq. (6) combined with the Eq. (3)), the polarizabil-ity for the nanoparticle plays a significant role in determining the ETC for the nanoparticlechains. To understand the insight of the red-shift behavior of the peaks with increasing therelative permittivity (cid:178) m and different spectral behaviors of the effective thermal conductivity9 igure 6: The dependence of ETC on the host medium permittivity (cid:178) m for insulator-metal phase-change VO .Insulator-phase VO (300 K) and metallic-phase VO (400 K) are considered. Nanoparticle radius a = 25 nm. h = (b) metallic-phase VO Figure 7: Dependence of the spectral effective thermal conductivity k ω on the ω and relative permittivity of the hostmedium (cid:178) m : (a) insulator phase (300 K) and (b) metallic phase (400 K). Nanoparticle radius a is 25 nm. h =
75 nm. and metallic-phase VO nanoparticle chains, the polarizability ofthe VO nanoparticles embedded in the host medium with several different relative permittiv-ities (cid:178) m is given in Fig. 8 (a) for insulator-phase VO nanoparticles and (b) for metallic-phaseVO nanoparticles with the following parameters. Nanoparticle radius a = 25 nm. Relativepermittivity (cid:178) m = 1, 3, 5 and 7, respectively. The increasing directions of the (cid:178) m , as well as theangular frequency corresponding to the main peaks, are also added for reference. (a) Insulator-phase VO (b) metallic-phase VO Figure 8: Polarizability of VO nanoparticle at several different relative permittivities (cid:178) m : (a) insulator phase and (b)metallic phase. Nanoparticle radius a is 25 nm. Relative permittivity (cid:178) m = 1, 3, 5 and 7, respectively. The increasingdirections of the (cid:178) m , as well as the angular frequency corresponding to the main peaks, are added for reference. As shown in Fig. 8, the peak of the polarizability shows an obvious red-shift behavior forboth the insulator-phase VO and metallic-phase VO nanoparticles, which accounts for thered-shift behaviors of the peaks for the spectral effective thermal conductivity as shown inFig. 7. This red-shift behavior of the peaks for the polarizability with increasing (cid:178) m results inthe increasing match between the peak frequency of the polarizability and the characteristicthermal frequency (Planck's window, as shown in Fig. 2(b)), which accounts for the increas-ing total ETC with increasing (cid:178) m as shown in Fig. 6. For the metallic-phase VO nanoparticlechain, we also can observe a competition between the following two processes: (1) the de-creasing peak value of the metallic-phase VO nanoparticle polarizability with increasing (cid:178) m and (2) the increasing match degree between the peak frequency of the polarizability and thecharacteristic thermal frequency (Planck's window) with increasing (cid:178) m . As shown in Fig. 6,the total ETC increases with increasing the (cid:178) m , therefore, the match degree between the peakfrequency of the polarizability and the characteristic thermal frequency (Planck's window) isthe influencing factor prior to the exact value of the polarizability peak.
4. Conclusion
Near-field radiative heat transfer for 1D nanoparticle chains embedded in a non-absorbinghost medium is investigated from the point view of the continuum by means of the MF methodcombining the many-body radiative heat transfer theory and the Fourier law together. Effects11f the phase change of materials, complex many-body interaction and host medium relativepermittivity on the effective thermal conductivity ETC are analyzed.The value of the ETC for VO nanoparticle chains below the transition temperature canreach around 50 times that above the transition temperature due to the phase change effect.The strong coupling in the insulator-phase VO nanoparticle chain accounts for its high ETCas compared to the low ETC for the metallic-phase VO nanoparticle chain, where there isa mismatch between the characteristic thermal frequency and the polarizability resonancefrequency.Strong MBI is in favor of the ETC. For dense chains (the ratio of the lattice spacing tonanoparticle radius h / a < h / a > nanoparticle chains. The red-shiftbehavior of the peaks for the polarizability with increasing (cid:178) m results in the increasing de-gree of match between the peak frequency of the polarizability and the characteristic thermalfrequency (Planck's window), which accounts for increasing the total ETC with increasing (cid:178) m . Acknowledgements
The support of this work by the National Natural Science Foundation of China (No. 51976045)is gratefully acknowledged. M.A. acknowledges support from the Institute Universitaire deFrance, Paris, France (UE). M.G.L. also thanks for the support from China Scholarship Coun-cil (No.201906120208).
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