Near Field Heat Transfer between Random Composite Materials: Applications and Limitations

1 Introduction

Composite material engineering has become an important technique to control the physical properties of materials from the macro- to the mesoscale [1], [2], [3], [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 matrix or host with a dispersive dielectric function ϵ_h, and inclusions that occupy a volume fraction f_i with a dielectric function ϵ_i and their shape is defined by the depolarisation factor L_i. 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+∑ifi, where f_h 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 Maxwell-Garnett 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 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 Effective 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.

Figure 1:

(a) Schematics of the SiC slab (host) with spherical Au inclusions of diameter D. The host has a dielectric function ϵ_h and the inclusions ϵ_i. (b) Parallel slab configuration of the composite at temperature T₁ and a homogeneous SiC half-space at temperature T₂.

The approximations we consider in this article for the effective dielectric function are the Maxwell-Garnett approximation (ϵ˜MG), Bruggemans (ϵ˜B), and Looyenga-Lifshitz [21], [22] (ϵ˜L) that are obtained from the following equations

(1)(ϵ˜MG−ϵhϵ˜MG+2ϵh)=f(ϵi−ϵhϵi+2ϵh),

(2)f(ϵ˜B−ϵiϵ˜B+2ϵi)+(1−f)(ϵh−ϵ˜Bϵh+2ϵ˜B)=0

and

(3)ϵ˜L3=fϵi3+(1−f)ϵh3.

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 ϵ˜B for a fixed value of the filling fraction f. The physically meaningful root is such that Im(ϵ˜B)>0. Finally, Looyengas-Lifshitz approximations belong to the class of general power law models ϵ˜α=fϵiα+(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

(4)ϵ(ω)i=ϵexp(ω)+ωp2ω2+iωγ−ωp2ω2+iω(γ+γ′),

ϵ_exp(ω) being the complex experimental dielectric function, ω_p=1.36×10¹⁶ rad/s, the plasma frequency, γ=4.05×10¹³ rad/s, is the bulk damping, v_f=1.39×10⁶ m/s the Fermi velocity of Au and γ′=v_F/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.

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.

The dielectric function of SiC ϵ_h is given by a Lorentz type oscillator:

(5)ϵ(ω)h=ϵ∞ωL2−ω2−iγωωT2−ω2−iγω,

with ϵ_∞=6.7, ω_L=1.827×10¹⁴ rad/s, ω_T=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=λ_p=620 nm that corresponds to the plasmonic resonance of the Au nanoparticles. For λ>λ_p, Looyenga’s model shows the largest variation as compared with the other approximations.

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 λ_p=620 nm, the plamonic response of the spherical Au nanoparticles is evident.

Figure 4:

Same as in Figure 3 but for the imaginary part of the effective dielectric function.

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 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 effective dielectric function is isotropic. Using the known results of fluctuating electrodynamics [26], [27], the heat flux Q is calculated as

where θ(ω, T)=ħω/(exp(ħω/k_BT)−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/c2−q2. For either p or s polarised waves, the transmission coefficient is

The terms rp,s1 and rp,s2 are the Fresnel coefficients for the slabs at temperature T₁ and T₂, respectively. The Fabry-Perot type term in the denominator is Dp,s12=|1−rp,s1rp,s2exp(2ik0d)|.

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.

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.

For a system with a small filling fraction, Maxwell-Garnett 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

where

For comparison, we can write Maxwell-Garnett equation (1) as

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.

Figure 6:

Comparison of the near field radiative heat flux normalised to the black body ideal case Qbb=σ(T14−T24) as a function of slab separations for the Maxwell-Garnett and the Grundquist-Hunderi 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.

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.