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Z. Naturforsch. 2017; 72(2)a: 129–134 Eva Yazmin Santiago and Raul Esquivel-Sirvent* Near Field Heat Transfer between Random Composite Materials: Applications and Limitations DOI 10.1515/zna-2016-0368 Received September 26, 2016; accepted November 7, 2016; previously published online December 19, 2016 Abstract: We present a theoretical study of the limits and bounds of using effective medium approximations in the calculation of the near field radiative heat transfer between a composite system made of Au nanoparticles in a SiC host and an homogeneous SiC slab. The effective dielectric function of the composite slab is calculated using three different approximations: Maxwell-Garnett, Bruggeman, and Looyenga’s. In addition, we considered an empirical fit to the effective dielectric function by Grundquist and Hunderi. We show that the calculated value of the heat flux in the near field is dependent on the model, and the difference in the effective dielectric function is larger around the plasmonic response of the Au nanoparticles. This, in turn, accounts for the difference in the near field radiative heat flux. For all values of filling fractions, the Looyenga approximation gives a lower bound for the heat flux. Keywords: Composites; Fluctuations; Heat Transfer. PACS numbers: 73.20Mf; 42.25.Kb; 44.40+a; 42.50.Lcx. 1 Introduction Composite material engineering has become an important technique to control the physical properties of materials from the macro- to the mesoscale [1–4]. In the static response or in the long wavelength limit for elastic [5] or electromagnetic waves [6], the Lamé constants or the dielectric function of composites can be calculated using effective medium approximations (EMAs). In this article, we are interested in the effective dielectric function of a composite defined as a homogeneous *Corresponding author: Raul Esquivel-Sirvent, Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, Ciudad Universitaria, D. F. 01000, México, E-mail: [email protected] Eva Yazmin Santiago: Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, Ciudad Universitaria, D. F. 01000, México matrix or host with a dispersive dielectric function εh, and inclusions that occupy a volume fraction fi with a dielectric function εi and their shape is defined by the depolarisation factor Li. For simplicity, the frequency dependence of the dielectric functions is assumed. The problem of EMA is to find an effective dielectric function [7] of the form ε = F (εh , εi , fi , Li ) with the constraint 1 = fh + ∑ i fi , where fh is the volume fraction occupied by the host and F is a function of the parameters of the system. Examples of EMA include the Maxwell-Garnett approximation, the Bruggeman approximation [8] that is symmetric with respect to the exchange of host and inclusions, thus making it more appropriate for random porous media [9]. EMAs are not unique and there are at least a dozen different models or recipes described in the literature to obtain the effective dielectric function. In an extensive review, Prasad and Prasad [10] compared these models for the same composite material showing different results depending on the model, even for dilute systems. In the static limit, where there is no dissipation, any effective medium model is bounded above and below by the Hashim-Strickman variational [11] bounds that impose a maximum and minimum value for the effective dielectric function. The variations in the effective models have also been explored in the context of the Casimir effect where Lifshitz model imposes the condition that the effective dielectric function has to satisfy Kramers-Kronig relations [12]. Experimentally, Grundquist and Hunderi [13, 14] measured the optical response of Ag-SiO2 cermet films as well as films made by depositing Au particles on substrate. In both cases, classical models such as MaxwellGarnett or Bruggeman do not fit the experimental data and a modification to the Maxwell-Garnett model was proposed. However, there are cases where the theoretical description of the optical properties using EMA and experiments match, such as in porous Si whose measured optical response, is described accurately by the Bruggeman model [9] or the calculation of the extinction coefficient of nested nanoparticles can be described by Maxwell-Garnett model [15]. Near field heat transfer can be tuned using composite materials. The simplest composite materials are periodic layered media that are described accurately by EMA [4] and the near field radiative heat transfer [16]. For more Unauthenticated Download Date | 8/10/17 11:54 AM 130 E. Y. Santiago and R. Esquivel-Sirvent: Near Field Heat Transfer between Random Composite Materials complex structures, Ben Abdallah [17] considered the near field radiative heat transfer between two SiC slabs separated by vacuum, each one with a periodic array of cylindrical pores. As the volume fraction occupied by the pores changes, the near field heat transfer increases due to different mechanisms related to the allowed modes and additional surface waves present in the system. Another uniaxial system, also described by an effective dielectric function, are arrays of silicon nanowires also showing an increase in the heat transfer after a critical volume fraction of nanowires is reached [18]. Another system where the percolation transition in the near field heat transfer was calculated [19] between a composite sphere made of a host of polystyrene with metallic inclusions and a SiC half-space. The changes in the radiative heat transfer close to the insulator-metal transition are shown and are described by two EMA models, Bruggeman and Lagarkov-Sarychev. In a plane-plane radiative heat transfer calculation, the EMA was used to describe the insulating-conducting transition of a thin Au film as a function of its thickness [20]. In this article, we study the near field radiative heat transfer between a SiC slab and a composite slab made of an homogeneous host and spherical inclusions made of Au nanoparticles. We calculate the spectral heat function and the total radiative heat transfer for this system using different EMAs, showing the validity of the different approaches. 2 Theory 2.1 E ffective Dielectric Functions In this article, we consider, without loss of generality, composites made of an homogeneous matrix with spherical inclusions made of Au nanoparticles of diameter 10 nm, in a host material of SiC as shown in Figure 1a. The approximations we consider in this article for the effective dielectric function are the Maxwell-Garnett approximation (εMG ), Bruggemans (εB ), and LooyengaLifshitz [21, 22] (εL ) that are obtained from the following equations εMG − εh ε + 2ε = MG h ε −ε f i h , εi + 2εh (1) ε − ε ε − ε f B i + (1 − f ) h B = 0 ε ε + 2 B εh + 2εB i (2) d a b Figure 1: (a) Schematics of the SiC slab (host) with spherical Au inclusions of diameter D. The host has a dielectric function εh and the inclusions εi. (b) Parallel slab configuration of the composite at temperature T1 and a homogeneous SiC half-space at temperature T2. and εL 3 = f εi3 + (1 − f )εh3 . (3) In the Mawell-Garnett approximation the term on the right-hand side of (1) is proportional to the polarisability of the inclusion. Bruggeman’s approximation yields a second-order equation for εB for a fixed value of the filling fraction f. The physically meaningful root is such that Im(εB ) > 0. Finally, Looyengas-Lifshitz approximations belong to the class of general power law models ε α = f εiα + (1 − f )εhα , where the exponent α depends on the shape of the inclusions. When α = 1, the inclusions have no depolarisation and we obtain the arithmetic mean of the dielectric functions [23]. The dielectric function of Au is taken from the experimental values of Johnson and Christie [24] corrected for finite size effects [25], this is ε( ω)i = εexp ( ω) + ω2p ω2 + i ωγ − ω2p ω2 + i ω( γ + γ ′ ) , (4) εexp(ω) being the complex experimental dielectric function, ωp = 1.36 × 1016 rad/s, the plasma frequency, γ = 4.05 × 1013 rad/s, is the bulk damping, vf = 1.39 × 106 m/s the Fermi velocity of Au and γ′ = vF/D, where D is the diameter of the Au nanosphere. The effect of the finite size of the particle is shown in Figure 2, where we present the real part (2a) of the dielectric function with and without finite size correction, and the imaginary part also with and without correction (2b). Thus, finite size effects have to be considered in the calculation of the effective dielectric function of the composite. Unauthenticated Download Date | 8/10/17 11:54 AM E. Y. Santiago and R. Esquivel-Sirvent: Near Field Heat Transfer between Random Composite Materials 131 a 50 Re ( −50 −100 500 1000 1500 eff) 2000 f = 0.05 eff) 10 8 6 4 30 f = 0.01 7.5 20 10 0 500 1000 λ (nm) 1500 7.0 6.5 6.0 200 2000 Figure 2: Real part (a) and imaginary part (b) of the dielectric function of Au (bulk) and of a Au nanoparticle of diameter D = 10 nm. The finite size effects corrections are more relevant in the imaginary part. 3 Near Field Radiative Heat Transfer The near field radiative heat transfer (NFHT) is calculated between a SiC half-space at temperature T2 = 1 K and the eff) Im ( eff) with ε∞ = 6.7, ωL = 1.827 × 1014 rad/s, ωT = 1.495 × 1014 rad/s, and γ = 0.9 × 1012 rad/s. With the dielectric function of the host and the Au inclusions, we calculate the effective dielectric function using the three different models as a function of wavelength and for different values of the filling fraction. This is shown in Figure 3 for the real part of the effective dielectric function and in Figure 4 for the imaginary part. The filling fractions considered are f = 0.01, 0.05, 0.1, 0.15. For small filling fractions, Bruggeman and Maxwell-Garnett models give the same results at large wavelengths. The largest difference occurs at around a wavelength of λ = λp = 620 nm that corresponds to the plasmonic resonance of the Au nanoparticles. For λ > λp, Looyenga’s model shows the largest variation as compared with the other approximations. Im ( (5) 800 1000 1200 Wavelength (nm) 1400 1600 1800 f = 0.15 MG B L f = 0.1 15 12 9 6 3 0 f = 0.05 8 eff) 600 25 20 15 10 5 0 6 Im ( , 4 2 0 eff) ωT2 − ω2 − i γω Im ( ε( ω)h = ε∞ ω2L − ω2 − i γω 400 Figure 3: Real part of the effective dielectric function for the three models as a function of wavelength. Each panel corresponds to different filling fractions as indicated. At λp = 620 nm, the plamonic response of the spherical Au nanoparticles is evident. The dielectric function of SiC εh is given by a Lorentz type oscillator: 4 0 Re ( 40 0 f = 0.1 12 60 50 Im ( (ω)) 8 eff) b 0 12 Re ( −150 −200 MG B L 16 Re ( Re ( (ω)) eff) D = 10 nm Bulk 0 f = 0.15 20 15 10 5 0 –5 1.5 1.2 0.9 0.6 0.3 0.0 200 f = 0.01 400 600 800 1000 1200 Wavelength (nm) 1400 1600 1800 Figure 4: Same as in Figure 3 but for the imaginary part of the effective dielectric function. composite slab at a temperature T1 = 300 K, both slabs separated by a distance d. In the case of the composite slab, the nanospheres are randomly distributed and the Unauthenticated Download Date | 8/10/17 11:54 AM 132 E. Y. Santiago and R. Esquivel-Sirvent: Near Field Heat Transfer between Random Composite Materials effective dielectric function is isotropic. Using the known results of fluctuating electrodynamics [26, 27], the heat flux Q is calculated as Q= 1 θ( ω, T1 ) − θ( ω, T2 ) ∑ ∫ q τ p ,s ( ω, q, d )dq d ω, (6) 4 π2 ∫ p ,s where θ(ω, T) = ħω/(exp(ħω/kBT) − 1) is Planck’s distribution function, q is the parallel component to the slabs of the wave vector, and the perpendicular component is k0 = ω2 / c 2 − q 2 . For either p or s polarised waves, the transmission coefficient is 1 2 2 2 12 2 (1− | rp ,s | )(1− | rp ,s | )/ | Dp ,s | , if q < ω/c τ p ,s ( ω, q, L) = ( −2 k0 d ) 1 2 / | Dp12,s |2 . if q > ω/c 4 Im(rp ,s ) Im(rp ,s )e (7) The terms rp1,s and rp2,s are the Fresnel coefficients for the slabs at temperature T1 and T2, respectively. The Fabry-Perot type term in the denominator is D12p ,s =| 1 − rp1,srp2,s exp(2ik0d )|. a 105 103 104 103 102 102 Q Q b 105 SiC MG B L 104 101 101 100 100 f = 0.1 f = 0.01 10– 1 10– 9 10– 8 10– 6 10– 5 10– 1 – 9 10 104 104 103 103 102 102 10– 9 10– 7 d (nm) 10– 6 10– 5 10– 6 10– 5 101 101 10– 1 10– 8 d 105 Q Q 10– 7 d (nm) c 105 100 Setting the temperatures at T1 = 300 K and T2 = 1 K, we calculate the heat flux as a function of the separation d for the different effective medium models at various filling fractions of the nanospheres. In Figure 1b, we show the configuration of the system. In Figure 5, we present the values of the heat flux, normalised to the far field value Qbb = σ (T14 − T24 ), where σ = 5.670 × 10 − 8 Wm − 2 K − 4. As a reference, the value of Q for two homogeneous SiC slabs is also plotted. For a filling fraction of f = 0.010 = 1 %, we see that there is no significant difference between the different effective medium models, except for Looyengas that shows a smaller value of Q for all separations. As the filling fraction increases to f = 0.1 and f = 0.15, we observe differences on the predicted value of Q depending on the model. Looyenga’s approximation gives a lower bound for the total near field radiative heat transfer for all values of the filling fractions. At a filling fraction of f = 0.33, the Bruggeman model predicts a percolation transition, that corresponds to the close packing of the spheres. At this filling fraction, we see the largest differences in the predicted values of Q. 100 f = 0.15 10– 8 10– 7 d (nm) 10– 6 10– 5 10– 1 10– 9 f = 0.33 10– 8 10– 7 d (nm) Figure 5: Near field radiative heat flux normalised to the black body ideal case Qbb = σ (T14 − T24 ) as a function of slab separations. The panels correspond to different values of the filling fraction, as indicated (a) f = 0.01, (b) f = 0.1, (c) f = 0.15 and (d) f = 0.33. As a reference, the solid line is the NFHT between two homogeneous SiC slabs. Unauthenticated Download Date | 8/10/17 11:54 AM E. Y. Santiago and R. Esquivel-Sirvent: Near Field Heat Transfer between Random Composite Materials 133 For a system with a small filling fraction, MaxwellGarnett does not match the experimental measurements of the optical properties for the systems studied by Grundquist and Hunderi. Neither does the Bruggeman approximation. Grundquist introduced an ad-hoc correction [13, 14] to the Maxwell-Garnett model. The effective dielectric function in this case is 2 4 + f α , εG = εh 4 − f α (8) where α=3 For comparison, equation (1) as we εi − εh . εi + 2εh can (9) write Maxwell-Garnett a 105 Maxwell−Garnett 104 Q 10 Hunderi 3 102 101 100 f = 0.10 10–1 10–9 10–8 10–7 10–6 10–5 b 104 Q 103 1+2f α . 1− f α (10) For two different filling fractions, we show in Figure 6 the total heat flow obtained using Maxwell-Garnett (8) of the Grundquist-Hunderi (10) EMAs. At a filling fraction of f = 0.1, the value of Q differs for both models for all separations considered. However, for f = 0.15, the difference becomes smaller. This is because the larger the amount of Au in the sample, the effective dielectric function becomes more metallic and the dielectric function of Au begins to be more prevalent in the effective response. 4 Conclusions In this article, we calculate the near field heat transfer between a composite medium and an homogeneous slab using different models for the effective dielectric function of the composite. The real and imaginary parts of the effective dielectric function have different values depending on the model used. This in turn yields also different values for the total heat transfer Q, even for small filling fractions. We also show that ad-hoc models fitted to experimental optical data, such as Grandquist-Hunderi approximation, differ from the values of Maxwell-Garnett model, even at small filling fractions. Furthermore, in the case of metallic inclusions, the different models differ around the plasmonic resonance of the inclusions. The sensitivity of both the effective dielectric function and the radiative heat transfer shows that any prediction based on effective medium models gives at best an estimate of the heat transfer and the choice of the effective dielectric function can only be validated if experimental data of the optical properties are available. Acknowledgments: Partial support from DGAPA-UNAM project IN110916, PIIF-IFUNAM projects and CONACyT Fronteras project No. 1290. 102 101 100 References f = 0.15 10–1 10–9 εMG = εh 10 –8 –7 10 d (nm) 10 –6 10 –5 Figure 6: Comparison of the near field radiative heat flux normal4 4 ised to the black body ideal case Qbb = σ (T1 − T2 ) as a function of slab separations for the Maxwell-Garnett and the GrundquistHunderi models for filling fractions f = 0.1 (a) and f = 0.15 (b). Although the two models are very similar, there are differences in the predicted values of Q. M. Wang and N. Pan, Mat. Sci. Eng. R 63, 1 (2008). X. Zhang and Y. Wu, Sci. Rep. 5, 7892 (2015). 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