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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
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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.
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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
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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.
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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).
A. C. Balazs, T. Emrick, and T. P. Russel, Science 314, 1107 (2006).
A. A. Krokhin, J. Arriaga, L. N. Gumen, and V. P. Drachev, Phys.
Rev. B 93, 075418 (2016).
[5] T. H. Jordan, Geop. J. Int. 207, 1343 (2015).
[6] T. C. Choy, Effective Medium Theory: Principles and Applications,
2nd Ed. Oxford Science Publications, New York, USA 2016.
[1]
[2]
[3]
[4]
Unauthenticated
Download Date | 8/10/17 11:54 AM
134 E. Y. Santiago and R. Esquivel-Sirvent: Near Field Heat Transfer between Random Composite Materials
[7] G. A. Niklasson, C. G. Grandquist, and O. Hunderi, Appl. Opt.
20, 26 (1981).
[8] A. Bittar, S. Berthier, and J. Lafait, J. Phys. 45, 623 (1984).
[9] M. Khardani, M. Bouaicha, and B. Bessais, Phys. Stat. Sol. C 4,
1986 (2007).
[10] A. Prasad and K. Prasad, Physica B 396, 132 (2007).
[11] U. Felderhoff, J. Phys. C 15, 3953 (2000).
[12] R. Esquivel-Sirvent and G. C. Schatz, Phys. Rev. A 83, 042512
(2011).
[13] C. G. Grundquist and O. Hunderi, Phys. Rev. B 16, 3513 (1977).
[14] C. G. Grundquist and O. Hunderi, Phys. Rev. B 18, 2897 (1978).
[15] R. Diaz-H.R, R. Esquivel-Sirvent, and C. Noguez, J. Phys. Chem.
C 120, 2349 (2016).
[16] X. L. Liu, T. J. Bright, and Z. M. Zhang, ASME J. Heat Transfer
136, 092703 (2014).
[17] S.-A. Biehs, P. Ben-Abdallah, F. S. S. Rosa, K. Joulain, and
J. J. Greffet, Opt. Exp. 19, A1088 (2011).
[18] S. Basu and L. Wang, Appl. Phys. Lett. 102, 053101 (2013).
[19] W. J. M. Kort-Kamp, P. I. Caneda, F. S. S. Rosa, and
F. A. ­Pinheiro, Phys. Rev. B 90, 140202R (2014).
[20] R. Esquivel-Sirvent, AIP Adv. 6, 095214 (2016).
[21] H. Looyenga, Physica 31, 401 (1965).
[22] L. Landau and E. Lifhitz, Electrodynamics of Continuous Media,
2nd Ed. Vol. 8 Pergammon Press, New York 2008.
[23] P. H. Zhou, L. J. deng, B. I. Wu, and J. A. Kong, Prog. Elect. Res.
85, 69 (2008).
[24] P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[25] K.-L.-Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, J. Phys.
Chem B 107, 668 (2003).
[26] K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J-J. Greffet,
Surf. Sci. Rep. 57, 5 (2005).
[27] B. Song, A. Fiorino, E. Meyhofer, and P. Reddy, AIP Adv. 5,
053503 (2015).
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