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Multifunctional hybrids for electromagnetic absorption
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Acta Materialia 59 (2011) 3255–3266 www.elsevier.com/locate/actamat
Multifunctional hybrids for electromagnetic absorption I. Huynen a,b, N. Quie´vy c, C. Bailly a,c,d, P. Bollen b,c,d, C. Detrembleur e, S. Eggermont b, I. Molenberg b, J.M. Thomassin e, L. Urbanczyk e, T. Pardoen a,d,⇑ a
Research Center in Micro and Nanoscopic Materials and Electronic Devices, CeRMiN, Universite´ catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium b Information and Communications Technologies, Electronics and Applied Mathematics (ICTEAM), Microwave Laboratory, Universite´ catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium c Institute of Condensed Matter and Nanosciences (IMCN), Universite´ catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium d Institute of Mechanics, Materials and Civil Engineering (iMMC), Universite´ catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium e Center for Education and Research on Macromolecules (CERM), University of Liege, Sart-Tilman B6a, 4000 Lie`ge, Belgium Received 18 November 2010; accepted 31 January 2011
Abstract Electromagnetic (EM) interferences are ubiquitous in modern technologies and impact on the reliability of electronic devices and on living cells. Shielding by EM absorption, which is preferable over reflection in certain instances, requires combining a low dielectric constant with high electrical conductivity, which are antagonist properties in the world of materials. A novel class of hybrid materials for EM absorption in the gigahertz range has been developed based on a hierarchical architecture involving a metallic honeycomb filled with a carbon nanotube-reinforced polymer foam. The waveguide characteristics of the honeycomb combined with the performance of the foam lead to unexpectedly large EM power absorption over a wide frequency range, superior to any known material. The peak absorption frequency can be tuned by varying the shape of the honeycomb unit cell. A closed form model of the EM reflection and absorption provides a tool for the optimization of the hybrid. This designed material sets the stage for a new class of sandwich panels combining high EM absorption with mass efficiency, stiffness and thermal management. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Electromagnetic interferences; Electrical properties; Carbon nanotubes; Nanocomposites; Cellular solids
1. Introduction Electromagnetic (EM) pollution is a subject of worldwide preoccupation regarding potentially harmful health issues and the proper operation of a wide range of electrical and radiofrequency (RF) instruments [1]. Electromagnetic interferences (EMI) result from electromagnetic radiations emitted, on purpose or not, by electrical circuits. Indications of potential radiative damage to living cells have recently emerged [2]. EMI can also have dramatic consequences on the functioning of critical electrical devices such ⇑ Corresponding author at: Institute of Mechanics, Materials and Civil Engineering (iMMC), Universite´ catholique de Louvain, B-1348 Louvainla-Neuve, Belgium. Tel.: +32 10 472417. E-mail address: [email protected] (T. Pardoen).
as in medical or aerospace applications. In many instances, a simple metallic foil covering the internal or external surface of the structure is sufficient to reflect the incident EM waves and to preserve the electrical integrity of the system, or to prevent EM waves to escape the system. However, in a series of applications, true absorption of the electromagnetic radiation, at least from one side of an interface, is recommended and sometimes mandatory. In several RF applications involving electronic circuits, the self-reflection of the waves inside the package affects the proper operation of the system [3]. Several stealth applications require true EM wave absorption: anechoic chambers for testing electronic devices and antennas, or ships and aircrafts for military operations. Absorption of electromagnetic waves through a slab of material requires the prevention of both reflection of the
1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.01.065
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incident wave and transmission through the medium. The absorbed power Pabs is given by Pabs = Pin Pref Pout, where Pin is the incident power, Pref is the reflected power and Pout is the total transmitted power. The absorption is quantified by the index A = Pabs/Pin, which writes (see the derivation in Appendix A) CðT 2 1Þ2 T ð1 C2 Þ2 A ¼ 1 ð1Þ 1 C2 T 2 1 C2 T 2 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi where the reflection coefficient C ¼ ð eeff 1Þ=ð eeff þ 1Þ pffiffiffiffiffiffiffi and the transmission coefficient T ¼ expðjxt eeff =co Þ involve the dielectric constant er and electrical conductivity r entering in the definition of the complex effective dielectric constant eeff = er jr/xeo, where x = 2pf, with f the frequency, co the speed of light, t the thickness and j the imaginary number. In a composite material, the dielectric constant er and electrical conductivity r are function of the properties, volume fraction and topology of the constituents. For moderate volume fractions, the dielectric constant and conductivity are obtained by the rule of mixture as er = fer1 + (1 f)er2 and r = fr1 + (1 f)r2, where eri and ri are respectively the dielectric constant and conductivity of material i, and f is the volume fraction of material 1. Logarithmic rules of mixtures have also been reported in the literature, especially for composites containing conductive inclusions [4]. The absorption index A can be calculated for a given thickness and frequency for all existing materials based on a material property database [5]. Fig. 1 is a diagram showing the dielectric constant er as a function of electrical conductivity r for a selection of materials. Isoabsorption (A) contours calculated at two frequencies, f = 1 and 10 GHz, and for a material thickness t = 10 mm are drawn on this material properties chart. The best absorbing materials combine a dielectric constant as close as possible to 1 and a moderately high conductivity around 1 S m1 (see later). No homogeneous material is found in the optimum
region based on known data. Note that the conductivity of metals (not represented in the chart) is very large, hence the norm of the reflection coefficient |C| is almost equal to 1 and the EM wave is entirely reflected. Hybrid materials naturally emerge as the solution when seemingly antagonist properties must be combined [6]. Most polymers are electrical insulators, and hence are transparent to electromagnetic radiations. Reinforcing polymers with carbon-based fillers, such as carbon black, carbon fibers and carbon nanotubes (CNT), constitutes an attractive option to reach the level of conductivity (around 1 S m1) at high frequency [7,8] required in EMI shielding applications. A small percentage of CNT raises the electrical conductivity at high frequency owing to virtual connections created by electrical capacitances existing between closely spaced nanotubes [8–10]. However, the dielectric constant is also increased by the incorporation of CNT (see the arrow towards higher electrical conductivity in Fig. 1), which has a detrimental impact on the reflection. The introduction of open space is a direct way to decrease the dielectric constant again (see the second vertical arrow in Fig. 1), proportional to the volume fraction of air, while keeping the high conductivity introduced by the nanofiller. By combining the foam strategy and the incorporation of CNT, the resulting composite involves a dielectric constant closer to that of the bulk polymer without nanotubes and a conductivity close to the optimum region [11]. In this paper, we will show both experimentally and by modeling that high EM absorption can be achieved in the gigahertz range by using a multiscale hybrid strategy involving the waveguide characteristic of a metallic honeycomb structure and the absorption properties of a CNTfilled polymer foam. In addition, a closed form model for the EM absorption will allow, after validation, different performances indices to be defined, based on which the hybrid performances can be optimized in terms of EM absorption, lightness and thermal management.
Fig. 1. Chart of electrical properties of materials: dielectric constant plotted against electrical conductivity (using CES EduPack 2009 software from Granta) and resulting isoabsorption (A) contours.
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The paper starts with a description of the processing of the hybrid material, followed by the EM characterizations. The analytical EM model used to predict the absorption of the hybrid is derived in the next section, followed by the validation against the experimental results and against numerical simulations. The last section proposes an analysis of the performances of the new hybrid regarding lightness and thermal conductivity, with a comparison to the other possible competitors.
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(ISCO pump) for 24 h in order to ensure equilibrium of saturation. The pressure was then released within 30 s. The saturated PU-based sheets were quickly transferred between two hot plates at 120 °C for 2 min to induce foaming. The foams were finally quenched in an ice/water bath. The volume and weight of the samples were measured and the density of the PU–2 wt.% CNT foam was 0.38 g cm3. The foam structure was observed by scanning electron microscopy (JEOL JSM 840-A). The average pore size was 5 lm.
2. Processing of the hybrid 2.3. Hybrid preparation The hierarchical structure of the hybrid material, starting from the CNT-filled polymer composite at the nanometer scale, the foam level at the micrometer scale and the filled honeycomb at the millimeter scale, is shown in Fig. 2. The steps involved in the processing of the hybrid are described in this section. 2.1. Nanocomposite preparation Thermoplastic polyurethane (PU; DesmopanÒ 2590A, Bayer) was melt blended with 1 or 2 wt.% of multiwalled CNT (Nanocyl 7000, 90%) in a counter-rotating twin-screw static mixer (BrabenderÒ) at 180 °C for 5 min at 60 rpm. The nanocomposite was then pressed for 5 min in a hot press at 180 °C in order to obtain ready-to-foam 4-mmthick sheets. 2.2. Nanocomposite foaming The sheets were placed in a vessel at 60 °C and pressurized with CO2 (99.5%, Air Liquide, Belgium) at 300 bars
A 6-mm-thick honeycomb (Liming HoneycombComposites Co., Ltd., China), made of 100-lm-thick aluminum sheets forming 6-mm-sided hexagons, was mechanically inserted in the nanocomposite foam after heating at 220 °C in an oven. The honeycomb was slowly penetrated into the foam under compression within performed at a constant rate of 10 mm min1 using an universal mechanical testing machine. The system involves many degrees of freedom with several parameters that can easily be varied: dimensions of the honeycomb, nature of the polymer matrice, cell size of the foam and volume fraction of CNT. 3. Electromagnetic characterization The samples were characterized using a Vector Network Analyzer (VNA) Model Wiltron (Anritsu) 360, which allows simultaneous measurement of both the reflected and the transmitted power at its two ports over a given frequency range. It measures the scattering parameters (or
Composite Polymer
α
CNT
Honeycomb
2-10cm
Air
10-100µm
0.1-1µm
Fig. 2. Hierarchical structure of the hybrid material starting from the CNT-filled polymer composite at the nanometer scale, the foam level at the micrometer scale and the filled honeycomb at the millimeter scale – schematic drawings and micrographs.
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S-parameters), phase and magnitude that are characteristic of the device connected between its two ports: |S11|2 corresponds to the power reflected back at port 1 while |S12|2 is related to the power transmitted from port 1 to port 2, through the device. After eliminating the influence of the feeding probes on the S-parameters by an adequate calibration, the ratio of the power absorbed by the sample (Pabs) to the incident power (Pin) under test can thus be calculated as Pabs/Pin = 1 |S11|2 |S12|2. Fig. 3 shows (a) the picture of an hybrid sample inserted into the measurement set-up, (b) the absorption index A per unit thickness of sample and (c) the variation of the measured reflected power 10log 10(Pref) as a function of frequency. The multicomposite is a 2 wt.% CNT-reinforced PU foam filling a honeycomb with cell size X of 6 mm and a thickness t of 5 mm (see Fig. 2). The results obtained with the honeycomb (solid line) are compared to those obtained with a CNT-filled polymer foam alone (dashed line). The absorption is improved with the honeycomb, involving a specific peak of absorption at 6.6 GHz (see Fig. 3b). 4. Electromagnetic modeling 4.1. Closed form model When the composite material is inserted into the honeycomb structure, propagation inside each cell of the honeycomb is affected by the metallic walls of the cell. The problem is similar to that of a metallic waveguide (see Fig. 4) filled with a material of known complex permittivity: the presence of the walls modifies the propagation constant, which becomes dependent on the width a and height b of the waveguide. For rectangular waveguides or cells, the canonical expression for the complex propagation constant is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 x2o x2 mp2 np2 c ¼ j eeff ¼ j e eff 2 a b c2o co rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp2 np2 ¼ c2c þ þ ð2Þ a b When the size of the waveguide increases (a and b ! 1), the propagation constant tends to cc, defined in Appendix A for an unbounded slab. For rectangular cells of finite size, expression (2) implies the existence of a cutoff frequency fo that depends on the size of the cell and on the index pair (m, n) of the mode propagating inside the waveguide. r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi xo co m 2 n 2 fo ¼ ¼ pffiffiffiffiffiffiffi þ ð3Þ 2a 2b eeff 2p Above the cut-off frequency (f > fo), c = jb is purely imaginary, meaning that the propagation of the waves takes place along the through-thickness z-axis, with transmission factor ejbz accounting for the phase shift (or equivalently a delay of propagation in time domain). Below
the cut-off (f < fo), c = a is purely real, meaning that the wave is attenuated with factor eaz, implying no propagation. By identification of Eq. (2) with the expression of cc given in Appendix A, the effective permittivity of the waveguide structure filled with the composite material is given by 2 cco mp 2 np2 co 2 D eeffw ¼ ¼ eeff þ a b jx x 2 x ¼ er 1 o2 jr=xeo ð4Þ x Eq. (4) reveals that the equivalent dielectric constant seen by the wave passing through the waveguide structure (=Rðeeffw Þ) varies with frequency, being negative below the cut-off frequency and positive above it, and always remains lower than the dielectric constant er of the filling composite. In contrast, the imaginary part is not modified with respect to the composite alone, meaning that the conductivity seen by the wave is not affected by the waveguide. Hence, it is theoretically predicted that the power reflected by the waveguide structure filled with composite will, above the cut-off, always be lower than the power reflected by the same composite without honeycomb, meaning that the absorption will be superior in the presence of the waveguide by virtue of the power balance. Canonical expressions for the propagation constant exist only for rectangular or circular geometries. Using perturbation techniques, the cut-off frequency of a waveguide of arbitrary cross-section geometry is calculated from the variation DS of the surface of the cross-section with respect to the surface S of a rectangular one. Considering the hexagonal cell with edge size X (see Fig. 2, right) and expressing X as a function of angle a and dimensions a and b of the rectangular cross-section fitting the hexagon, the cut-off frequency foh of a hexagonal cell is given as DS cos a 2 2 2 ð5Þ ¼ for 1 þ foh ¼ for 1 Sr 1 þ 2 cos a with for given by expression (3) for a rectangular waveguide. For a regular hexagon (a = p/3), relation (5) reduces to 5 foh2 ¼ for2 4
ð6Þ
Next, it can be shown (see [12]) that expressions (2) and (4), which are valid for a rectangular cell, remain valid for the hexagonal cell, provided that expressions (5) and (6) are used for the cut-off frequency fo. Final expressions for the complex propagation constant, noted ch, and the corresponding effective permittivity, noted effh, associated to waves propagating through the hybrid material reduce to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi x2 cos a mp 2 5 np2 ch ¼ j eeff 2 1 þ ð7Þ 1 þ 2 cos a a 4 b co
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(a) Reflected Power Pref
Absorption Index per unit thickness from measurements 16
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x 10
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Fig. 3. (a) Measurement configuration for hybrid sample; (b) absorption index in per cent per unit thickness measured with (solid) and without (dashed) honeycomb with edge size X = 6 mm and thickness t = 5 mm; (c) corresponding reflected power measured with (solid) and without (dashed) honeycomb; (d) corresponding absorption index, calculated from analytical formulation (1); (e) effective dielectric constant calculated from analytical formula (4) or (8) with a = 60°, er = 5, X = 6 mm, m = 1 and n = 0.
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a=x+2u=(1+2cos α) X u
b/2 =X sin α
X
α
Fig. 4. Topology of cross-section for rectangular (left) and hexagonal (right) cells of metallic lattice inserted into a composite slab.
eeffh
¼ er 1 þ jr=xeo
cos a 1 þ 2 cos a
mp 2 np2 co 2 þ a b x ð8Þ
with a = X(1 + 2cos a) and b = 2Xsin a. For the dominant mode (i.e. the mode having the lowest cut-off frequency), m = 1 and n = 0. For a regular hexagonal cell (a = p/3), the expressions simplify into: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 5 p2 5 p2 ch ¼ j eeff 2 ¼ c2c þ ð9Þ 4 a co 4 a 5 p2 co 2 jr=xeo ð10Þ eeffh ¼ er 4 a x with a = 2X and b = 1.772X. Expressions derived in Appendix A for the absorption remain fully valid for the case of the honeycomb filling the composite, provided that cc is replaced by ch when calculating expressions (A.1)– (A.9). Eq. (10) shows that inserting the honeycomb into the composite further reduces the real part of the effective dielectric constant, by a factor depending on geometry and frequency. Fig. 3d and e shows the predictions of the model using the parameters of the system tested experimentally. Above the cut-off frequency (where Re(effh) = 0), predicted around 6.3 GHz in Fig. 3e, the real part of the effective dielectric constant for the hybrid (with honeycomb, solid line) remains lower than for the composite foam alone (dashed line), implying that the reflected power is reduced, as confirmed by the measurement in Fig. 3c. The predictions obtained for A using Eq. (1) with eeff replaced by the new expression (10) are also given in Fig. 3d, showing a good agreement. 4.2. Numerical model and validation of the closed form model The analytical model was also validated by comparing the predicted absorption index with respect to numerical simulations performed with COMSOL software, which solves Maxwell’s equations via the finite element method. Two kinds of simulations were performed: one series with only the slab of the composite conductive material and another series considering a regular honeycomb structure inserted in the composite slab, as represented in Fig. 5a
Fig. 5. Geometry used in Comsol Software for validating analytical model. (a) Slab of composite material of thickness t, dielectric constant er and conductivity r, sandwiched between two thin air layers modeling the surrounding environment. (b) Same configuration with metallic honeycomb lattice inserted in the composite slab. Proper boundary conditions are imposed on exterior boundaries in order to ensure propagation of a transverse electromagnetic wave with incidence normal to the air– composite interface and periodicity of the honeycomb lattice.
and b, respectively. The following parameters were varied: the conductivity r of the composite slab, the thickness t of the slab and the size X of the honeycomb. A summary of the validations is presented in Fig. 6. Each figure shows, for r = 0.5 and 1 S m1 respectively, the absorption index A for (a) t = 5 mm and X = 1 mm without and with a honeycomb, (b) t = 5 mm and X = 2 mm without and with a honeycomb and (c) t = 10 mm and X = 0.9 or 2 mm. The agreement between the analytical predictions and results of the numerical simulations is conspicuous, demonstrating again the enhancement of the EM absorption resulting from the presence of the honeycomb. 4.3. Tailoring of the absorption range The size of the honeycomb can be tailored in order to improve the absorption in a range of frequencies. As an example, Fig. 7 shows the absorption index (1) calculated with the analytical model for a cell size X = 0.9 mm. The frequency of maximum absorption is shifted towards 60 GHz as compared to Fig. 3. Furthermore, active tun-
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Absorption index 90
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(c) Fig. 6. Absorption index, in per cent, respectively calculated from analytical (line) formulation and simulated using the commercial FEM solver Comsol (symbol). The parameters of the composite slab are: (a) thickness t = 5 mm, dielectric constant er = 3, conductivity r = 0.5 S m1, (b) thickness t = 5 mm, dielectric constant er = 3, conductivity r = 1 S m1 and (c) thickness t = 10 mm, dielectric constant er = 3, conductivity r = 0.5 S m1. Two sizes of honeycomb cells are considered: X = 0.9 mm (dashed) and 2 mm (dash-dotted), while the third pair of curves holds for a composite slab without honeycomb.
ing/filtering is possible by changing the morphology of the material through an external deformation. Fig. 7 illustrates the effect of a compression of the regular honeycomb lattice inducing a deformation of angle a from 60° (regular honeycomb, Fig. 2) to 18°. These calculations were also validated by full-wave finite element method (FEM) simulations (results now shown). The maximum absorption frequency is shifted towards lower frequencies, by more than 15 GHz. 5. Multifunctional optimization of the hybrid Compared to a simple polymer foam, the incorporation into a honeycomb introduces additional degrees of freedom towards the design of multifunctional sandwich panels (see
[13]), with a potential to optimize density, stiffness and thermal insulation. In order to show the potential of the new hybrid, two material selection procedures involving an EM absorption constraint and an objective towards optimizing another material property are now described, following the rationale developed by Ashby [14]. A first important design constraint typical of transportation or human body protection (e.g. a helmet) is lightness. Assuming that the thickness of a panel is a free constant, the performance index M1 for a light EM absorption panel is given by M1 = 1/(qt), where q and t are the relative density and thickness respectively. M1 can be calculated from Eq. (1) at several frequencies corresponding to an imposed level of absorption A set
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Absorption index 90
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50 40 30 X=0.9 mm, α=18°
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Fig. 7. Absorption index, in per cent, calculated from the analytical formulation derived in Appendix A. The parameters of the composite slab are: thickness t = 5 mm, dielectric constant er = 3, conductivity r = 1 S m1. Two shapes of honeycomb cells are considered: regular (a = 60°) and deformed (a = 18°), both with edge size X = 0.9 mm. The dashed curve holds for composite slab without honeycomb.
equal to 0.9. Fig. 8a provides the value of M1 based on data available for the most relevant materials extracted from the chart in Fig. 1 (leather, plaster of Paris and SiC foam), for a 2 wt.% CNT-reinforced PU foam (nanocomposite foam) and for the two corresponding hybrids having different honeycomb dimensions (hybrid 1 with X = 4.7 mm and hybrid 2 with X = 1.58 mm). The SiC foam has a low performance index (M1 = 2) for all the frequencies and an absorption level of 0.9 is never attained with plaster of Paris due to reflection, hence M1 = 0. For these reasons, these material do not show up in Fig. 8a. Leather could theoretically compete with the nanocomposite foam at 10 GHz. Above this frequency, the latter is significantly better. The hybrids developed in this work perform extremely well when a high absorption is desired (A = 0.9), not only because of high EM absorption, which is better than any other material, but owing also to the lightness of the foam/honeycomb association. The performance can be tuned for a desired frequency range by the honeycomb cell dimensions (e.g. 10–20 GHz for hybrid 1 and 30–50 GHz for hybrid 2). An even lighter hybrid could be obtained with a metallized polymer-based honeycomb. Now, in most applications, face sheets will be used on one or both sides of the hybrid for protection and mechanical strength in bending, and will have to be accounted for in the procedure. A similar optimization procedure can be worked out for EM absorption at minimum thickness. It should be noted that the corresponding thicknesses (between 5 and 15 mm) relative to the hybrid materials are realistic. The second procedure aims at finding the optimum material for an application involving EM absorption and thermal dissipation. This is typical of an electric or electronic device from which heat must be evacuated. The per-
Fig. 8. (a) Performance index M1 of materials and hybrids for an absorption level A = 0.9 at five frequencies. Nanocomposite foam: er = 3– 3.5, r = 0.7–1.2 S m1, q = 0.3; hybrid 1: X = 4. 7 mm, q = 0.39; hybrid 2: X = 1.58 mm, q = 0.44; (b) performance index M2 of materials and hybrids with minimal thickness for an absorption level A = 0.9 at five frequencies. Nanocomposite foam: k = 0.067 W m1 K1, hybrid 1: k = 4.13 W m1 K1, hybrid 2: k = 11.56 W m1 K1.
formance index M2 for a panel with maximum thermal conductivity at a given absorption level is given by M2 = k/t, where k is the thermal conductivity of the hybrid. The thermal conductivity can be predicted from a simple multi-step homogenization theory starting from the CNT-reinforced polymer up to the hybrid (see Appendix B). Experimental measurements (see Appendix B) confirmed that the presence of nanotubes leads to only a very limited enhancement of the thermal conductivity of the polymer [15,16]. Hence, a lower bound homogenization model has to be used [14,17]. The thermal conductivity of the hybrid essentially comes from the metallic honeycomb. Fig. 8b shows, again, that the new hybrids perform very well. Leather and SiC foam present acceptable performances, though much lower than hybrid 1 over the entire frequency range and lower than hybrid 2 for frequencies of 30 GHz and above. In order to keep good thermal conductivity, only one metallic face sheet can be used (on the outer surface, to avoid reflection) if one considers a sandwich panel. On the other hand, in applications requiring good thermal insulation, the metallic honeycomb can be replaced by a polymer honeycomb with metallized surfaces.
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Other properties, such as the bending stiffness and strength of the new hybrid material can be optimized by proper selection of the face sheet materials. Additional degrees of freedom are provided by using graded foam densities and more complex cell shapes. The results presented in this work show that unique EM absorption can be combined in the form of stiff, light panels with thermal management coming from the choice of the honeycomb, opening many new avenues to applications requiring EM absorption in housing, electronics and health protection, as well as EM active filtering by changing dimensions. 6. Conclusions A new class of hybrid materials with high EM absorption capabilities was fabricated by inserting a CNT-filled polymer foam into a metallic honeycomb. The waveguide characteristics of the honeycomb combined to the conductive nanocomposite foam reduces the real part of the effective dielectric constant of the hybrid, resulting in very high EM absorption in the gigahertz range, superior to any known material. A closed form model, validated towards FE simulations, has been used to design the hybrid material with desired EM absorption levels in a specific frequency range opening to active filtering applications. Moreover, this designed material offers extremely high performances when also considering lightness, thermal management, bending stiffness and strength. Acknowledgements The authors are grateful to the Wallonia Region DG06 for supporting this research through a Winnomat II project “Multimasec”. The support of S. Benyoub for the thermal measurements and of D. Spote for electromagnetic measurements as well as the provision of the Al honeycombs by the company Liming Honeycomb Composites are gratefully acknowledged. I.H. and C.D. are research director and senior research associate for the National Funds for Scientific Research (F.R.S.-FNRS), respectively. J.M.T. is a logistics collaborator for the National Funds for Scientific Research (F.R.S.-FNRS). The authors declare no competing financial interests. Appendix A. General expression of the absorbed power in a composite slab Propagation of electromagnetic waves through a slab of composite conductive material with a thickness t and a permittivity (or complex dielectric constant) er jr/xeo is described via a transmission line formalism associated to electric (E) and magnetic (H) field components [18]. Assuming normal incidence of the wave with respect to the air–material interface, the field components have the following z-dependence in each of the three areas (see Fig. A.1):
Fig. A.1. Topology of the composite slab.
Input air region z < 0 : E ¼ V þa1 eca z þ V a1 eca z H ¼ Y a ðV þa1 e
ca z
þ V a1 e Þ
Composite 0 < z < t : E ¼ V þc e H ¼ Y c ðV þc ecc z þ V c ecc z Þ
ðA:1Þ cc z
Output air region z > t : E ¼ V þa2 e H ¼ Y a V þa2 e
and
ca z
ca ðztÞ
þ V c e
cc z
and ðA:2Þ
ca ðztÞ
and ðA:3Þ
with V the voltage amplitude of the waves and, for air, the complex propagation constant ca = jx/co, the angular pulsation x = 2pf, the frequency f, the velocity of light co and the wave admittance Ya = ca/jxlo, and, for the composite, pffiffiffiffiffiffiffi the complex propagation constant cc = j eeff x/co and the wave admittance Yc = cc/jxlo. Forward terms with a subscript “+” represent a wave propagating along positive zaxis, while terms with a subscript “” hold for a wave propagating in the reverse direction. This transmission line formalism based on complex phasors ensures that all successive reflections occurring in the time domain at input (z = 0) and output (z = t) air–slab interfaces are accounted for. Imposing the continuity of E and H fields on the output air–slab interface (at z = t) yields V þc ecc t þ V c ecc t ¼ V þa2 Y c ðV þc e
cc t
cc t
and
V c e Þ ¼ Y a V þa2
ðA:4Þ
After elimination of V+a2 and rearranging, the ratio of the reflected to the incident wave component in the composite material is related to the interfacial reflection coefficient C as pffiffiffiffiffiffiffi eeff 1 D Yc Ya ðA:5Þ V c =V þc ¼ Ce2cc t with C ¼ ¼ pffiffiffiffiffiffiffi eeff þ 1 Yc þ Ya Similarly applying the continuity conditions on E and H fields at the input interface z = 0, rearranging and using expression (A.5) for the ratio Vc/V+c, the ratio of the reflected to the incident wave in the first air area is obtained as V a1 Cðe2cc t 1Þ ¼ V þa1 1 C2 e2cc t
ðA:6Þ
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Finally, the ratio of the transmitted wave V+a2 in region 3 to the incident wave V+a1 in region 1 at z = 0 is obtained by recombination of the continuity equations and expressions (A.5) and (A.6) as
level to be considered in the thermal conductivity homogenization is shown in Fig. B.1a and corresponds to the polymer–CNT composite. Upper and lower bounds for the thermal conductivity write
V þa2 ecc t ð1 C2 Þ ¼ V þa1 1 C2 e2cc t
ðB:1Þ kupperbound ¼ fCNT kCNT þ ð1 fCNT Þkmatrix kCNT þ 2kmatrix 2f CNT ðkmatrix kCNT Þ klowerbound ¼ kmatrix kCNT þ 2kmatrix þ fCNT ðkmatrix kCNT Þ
ðA:7Þ
The power absorbed in the composite slab is obtained from the power balance between the power respectively incident in the air region 1 (Pin), reflected at the input air–composite interface (Pref) and transmitted into region 3 (Pt). These powers are defined from classical transmission line theory as the cross products of the electric and magnetic fields at the input (z = 0) and at the output (z = t) air–composite interfaces, respectively, as: P abs ¼ P in P ref P t 1 1 1 ¼ Y ca jV þa1 j2 Y ca jV a1 j2 Y ca jV þa2 j2 ðA:8Þ 2 2 2 The absorption index A is then defined as the absorbed power Pabs normalized by the incoming power Pin, and D with the definition T ¼ ecc t introduced in expressions (A.6) and (A.7), the final relationship writes V a1 2 V þa2 2 CðT 2 1Þ2 T ð1 C2 Þ2 A¼1 ¼1 V þa1 V þa1 1 C2 T 2 1 C2 T 2 ðA:9Þ Appendix B. Thermal characterization and modeling The thermal conductivity of the hybrids can be estimated based on simple homogenization rules. The first
1µm
ðB:2Þ where fCNT is the volume fraction of CNT, kCNT is the thermal conductivity of the CNT and kmatrix is the thermal conductivity of the polymer matrix [14]. Several studies in the literature have shown that the thermal transfer between matrix and CNT is usually poor due to the so-called Kaptiza resistance [19,16]. Hence, the lower bound is considered as the most realistic. The second level in the homogenization is shown in Fig. B.1b. The thermal conductivity of a foam can be computed using [17] " 3 # qfoam 2 qfoam 1 qfoam kfoam ¼ þ2 kgaz kmat þ 1 2 3 qmat qmat qmat ðB:3Þ where kmat is the thermal conductivity of the material in the cell walls (here, of the composite as predicted by Eq. (B.1) or (B.2)), kgaz is the thermal conductivity of the gaz entrapped in the foam (usually air), qfoam is the density of the foam and qmat is the density of the material in the cell walls (here the polymer–CNT composite). The third homogenization level shown in Fig. B.1c corresponds to the hybrid. The thermal conductivity of a
50µm
Composite, λcomp Density, ρcomp, ρfoam
Matrix, λmatrix
CNT, CNT, λλCNT, vol. fraction f CNT
Air, λair
(a)
(b) 2mm
1mm
Bi-foam ffoam, λfoam foam Foam Thickness t, thermal cond. λhybrid hybrid λhoneyc
Honeycomb
Plate Thickness tplate plate plate, thermal cond. λplate
(c)
(d)
Fig. B.1. (a) Level 1 of thermal model – polymer/CNT composite, (b) level 2 of the thermal model – foam made of air and composite matrix, (c) level 3 of the thermal model – hybrid made of honeycomb filled with CNT-reinforced polymer foam and (d) level 4 of the thermal model – multimaterial panel (with one face sheet).
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Thermal conductivity [W/(m.K)]
7 6
3 mm 2 mm
5 4 3
nanocomposite nanocomposite foam hybrid panel
2 1 0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
CNT volume fraction Fig. B.2. Prediction of the thermal conductivity of the multimaterial panel.
honeycomb filled by another material (here, by a foam) is given by khybrid ¼ ð1 ffoam Þkhoneyc þ ffoam kfoam
ðB:4Þ
where ffoam is the volume fraction of foam in the system and khoneyc is the thermal conductivity of the honeycomb material The fourth level of homogenization, shown in Fig. B.1d, corresponds to the multimaterial panel (here with one face sheet). The thermal conductivity of the panel is given by 1 tplate =ðt þ tplate Þ ðt=ðt þ tplate ÞÞ k¼ þ ðB:5Þ kplate khybrid where kplate is the thermal conductivity of the face sheet material, t is the thickness of the hybrid and tplate is the thickness of the face sheet. For Al honeycomb and Al face sheet, the model chain predicts, using the following set of typical parameter values: kal = 100 W m1 K1, kair = 0.025 W m1 K1, kpoly = between 0.1 and 0.5 W m1 K1, kCNT = 3000 W m1 K1, qfoam = 0.3, qcomp = 1.2, and fCNT = 0.025, that the thermal conductivity of the panel is dominated by the thermal conductivity of the honeycomb and face sheet (see Fig. B.2). In other words, the foam material essentially behaves as a thermally insulating material, and the thermal conductivity can only be significantly enhanced by adding another thermally conductive material. This result is a direct outcome of the assumption that the thermal conductivity of the polymer/CNT composite is given by the lower Table B.1 Comparison between measurements and predictions of thermal conductivity on polymer foam, CNT nanocomposite and CNT nanocomposite foam. Samples
k Measured (W m1 K1)
k Predicted (m1 K1)
PU PU + 2 wt.% CNT PU foam (q = 0.35) PU + 2 wt.% CNT foam (q = 0.46)
0.23 0.28 0.05 0.08
0.23 0.247 0.057 0.075
bound (B.2), in agreement with results in the literature [20,21], systematically showing very small effects of incorporation of CNT on the thermal conductivity. In order to test this conclusion and the validity of the model, the thermal conductivity of a PU polymer, a PU foam, a PU–CNT composite and a PU–CNT composite foam has been characterized and compared to the predictions in Table B.1. The measured thermal conductivity for PU agrees with literature data [5]. Introducing 2 wt.% CNT increases the thermal conductivity but only by 20%, close to the lower bound (B.2). The thermal conductivity of the foam is predicted well by the model (B.3). The more significant increase in thermal conductivity for the CNTreinforced PU foam results mainly from the change in density (the presence of CNT increases the viscosity of the system and, for the same processing conditions, decreases the foamability). References [1] White DRJ. Electromagnetic interference and compatibility. Germantown, Maryland: Don White Consultants, Inc.; 1976. [2] Adang D, Remacle C, Vander Vorst A. IEEE Trans Microw Theory Tech 2009;57:2488. [3] Chen HH, Chung SS. IEEE Trans Microw Theory Tech 1998;46:2124. [4] Peng ZH, Peng JC, Peng YF, Wang JY. Chin Sci Bull 2008;58:3497. [5] CES EduPack. Software from Granta Design Ltd, Cambridge; 2009. [6] Ashby MF, Bre´chet Y. Acta Mater 2003;51:5801. [7] White DRJ. A handbook on electromagnetic shielding materials and performances. Germantown, Maryland: Don White Consultants, Inc.; 1975. [8] Thomassin JM, Lou X, Pagnoulle C, Saib A, Bednarz L, Huynen I, et al. J Phys Chem C 2007;11:11186. [9] Saib A, Bednarz L, Daussin R, Bailly C, Lou X, Thomassin JM, et al. IEEE Trans Microw Theory Tech 2006;54:2745. [10] Huynen I, Bednarz L, Thomassin JM, Pagnoulle C, Jerome R, Detrembleur C. In: Proceedings of the 38th European microwave conference, October 28–30, Amsterdam. London, UK: # Horizon House Publications; 2008. p. 5–8. [11] Thomassin JM, Pagnoulle C, Bednarz L, Huynen I, Jerome R, Detrembleur C. J Mater Chem 2008;18:792. [12] Harrington RF. Time-harmonic electromagnetic fields. Piscataway, NJ: IEEE Press; 2001.
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[13] He Y, Gong R. Europhys Lett 2009;85:58003; Park KY, Lee SE, Kim CG, Han JH. Compos Sci Technol 2006;66:576. [14] Ashby MF. Materials selection in mechanical design. 3rd ed. Oxford: Elsevier; 2005. [15] Huxtable ST, Cahill DG, Shenogin S, Xue L, Ozisik R, Barone P, et al. Nat Mater 2003;2:731. [16] Clancy T, Gates T. Polymer 2006;47:5990. [17] Gibson LJ, Ashby MF. Cellular solids. 2nd ed. Cambridge: Cambridge University Press; 1997.
[18] Ulaby FT. Fundamentals of applied electromagnetics. 5th ed. Upper Saddle River, NJ: Pearson Prentice Hall; 2006. [19] Seidel GD, Lagoudas DC. J Appl Mech 2008;75:041025/1. [20] Kim HS, Chae YS, Park BH, Yoon JS, Kang M, Jin HJ. Curr Appl Phys 2008;8:803. [21] Yuen M, Ma CCM, Chiang CL, Chang JA, Huang SW, Chen SC, et al. Composites Part A 2007;38:2527.
Acta Materialia 59 (2011) 3255–3266 www.elsevier.com/locate/actamat
Multifunctional hybrids for electromagnetic absorption I. Huynen a,b, N. Quie´vy c, C. Bailly a,c,d, P. Bollen b,c,d, C. Detrembleur e, S. Eggermont b, I. Molenberg b, J.M. Thomassin e, L. Urbanczyk e, T. Pardoen a,d,⇑ a
Research Center in Micro and Nanoscopic Materials and Electronic Devices, CeRMiN, Universite´ catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium b Information and Communications Technologies, Electronics and Applied Mathematics (ICTEAM), Microwave Laboratory, Universite´ catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium c Institute of Condensed Matter and Nanosciences (IMCN), Universite´ catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium d Institute of Mechanics, Materials and Civil Engineering (iMMC), Universite´ catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium e Center for Education and Research on Macromolecules (CERM), University of Liege, Sart-Tilman B6a, 4000 Lie`ge, Belgium Received 18 November 2010; accepted 31 January 2011
Abstract Electromagnetic (EM) interferences are ubiquitous in modern technologies and impact on the reliability of electronic devices and on living cells. Shielding by EM absorption, which is preferable over reflection in certain instances, requires combining a low dielectric constant with high electrical conductivity, which are antagonist properties in the world of materials. A novel class of hybrid materials for EM absorption in the gigahertz range has been developed based on a hierarchical architecture involving a metallic honeycomb filled with a carbon nanotube-reinforced polymer foam. The waveguide characteristics of the honeycomb combined with the performance of the foam lead to unexpectedly large EM power absorption over a wide frequency range, superior to any known material. The peak absorption frequency can be tuned by varying the shape of the honeycomb unit cell. A closed form model of the EM reflection and absorption provides a tool for the optimization of the hybrid. This designed material sets the stage for a new class of sandwich panels combining high EM absorption with mass efficiency, stiffness and thermal management. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Electromagnetic interferences; Electrical properties; Carbon nanotubes; Nanocomposites; Cellular solids
1. Introduction Electromagnetic (EM) pollution is a subject of worldwide preoccupation regarding potentially harmful health issues and the proper operation of a wide range of electrical and radiofrequency (RF) instruments [1]. Electromagnetic interferences (EMI) result from electromagnetic radiations emitted, on purpose or not, by electrical circuits. Indications of potential radiative damage to living cells have recently emerged [2]. EMI can also have dramatic consequences on the functioning of critical electrical devices such ⇑ Corresponding author at: Institute of Mechanics, Materials and Civil Engineering (iMMC), Universite´ catholique de Louvain, B-1348 Louvainla-Neuve, Belgium. Tel.: +32 10 472417. E-mail address: [email protected] (T. Pardoen).
as in medical or aerospace applications. In many instances, a simple metallic foil covering the internal or external surface of the structure is sufficient to reflect the incident EM waves and to preserve the electrical integrity of the system, or to prevent EM waves to escape the system. However, in a series of applications, true absorption of the electromagnetic radiation, at least from one side of an interface, is recommended and sometimes mandatory. In several RF applications involving electronic circuits, the self-reflection of the waves inside the package affects the proper operation of the system [3]. Several stealth applications require true EM wave absorption: anechoic chambers for testing electronic devices and antennas, or ships and aircrafts for military operations. Absorption of electromagnetic waves through a slab of material requires the prevention of both reflection of the
1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.01.065
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incident wave and transmission through the medium. The absorbed power Pabs is given by Pabs = Pin Pref Pout, where Pin is the incident power, Pref is the reflected power and Pout is the total transmitted power. The absorption is quantified by the index A = Pabs/Pin, which writes (see the derivation in Appendix A) CðT 2 1Þ2 T ð1 C2 Þ2 A ¼ 1 ð1Þ 1 C2 T 2 1 C2 T 2 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi where the reflection coefficient C ¼ ð eeff 1Þ=ð eeff þ 1Þ pffiffiffiffiffiffiffi and the transmission coefficient T ¼ expðjxt eeff =co Þ involve the dielectric constant er and electrical conductivity r entering in the definition of the complex effective dielectric constant eeff = er jr/xeo, where x = 2pf, with f the frequency, co the speed of light, t the thickness and j the imaginary number. In a composite material, the dielectric constant er and electrical conductivity r are function of the properties, volume fraction and topology of the constituents. For moderate volume fractions, the dielectric constant and conductivity are obtained by the rule of mixture as er = fer1 + (1 f)er2 and r = fr1 + (1 f)r2, where eri and ri are respectively the dielectric constant and conductivity of material i, and f is the volume fraction of material 1. Logarithmic rules of mixtures have also been reported in the literature, especially for composites containing conductive inclusions [4]. The absorption index A can be calculated for a given thickness and frequency for all existing materials based on a material property database [5]. Fig. 1 is a diagram showing the dielectric constant er as a function of electrical conductivity r for a selection of materials. Isoabsorption (A) contours calculated at two frequencies, f = 1 and 10 GHz, and for a material thickness t = 10 mm are drawn on this material properties chart. The best absorbing materials combine a dielectric constant as close as possible to 1 and a moderately high conductivity around 1 S m1 (see later). No homogeneous material is found in the optimum
region based on known data. Note that the conductivity of metals (not represented in the chart) is very large, hence the norm of the reflection coefficient |C| is almost equal to 1 and the EM wave is entirely reflected. Hybrid materials naturally emerge as the solution when seemingly antagonist properties must be combined [6]. Most polymers are electrical insulators, and hence are transparent to electromagnetic radiations. Reinforcing polymers with carbon-based fillers, such as carbon black, carbon fibers and carbon nanotubes (CNT), constitutes an attractive option to reach the level of conductivity (around 1 S m1) at high frequency [7,8] required in EMI shielding applications. A small percentage of CNT raises the electrical conductivity at high frequency owing to virtual connections created by electrical capacitances existing between closely spaced nanotubes [8–10]. However, the dielectric constant is also increased by the incorporation of CNT (see the arrow towards higher electrical conductivity in Fig. 1), which has a detrimental impact on the reflection. The introduction of open space is a direct way to decrease the dielectric constant again (see the second vertical arrow in Fig. 1), proportional to the volume fraction of air, while keeping the high conductivity introduced by the nanofiller. By combining the foam strategy and the incorporation of CNT, the resulting composite involves a dielectric constant closer to that of the bulk polymer without nanotubes and a conductivity close to the optimum region [11]. In this paper, we will show both experimentally and by modeling that high EM absorption can be achieved in the gigahertz range by using a multiscale hybrid strategy involving the waveguide characteristic of a metallic honeycomb structure and the absorption properties of a CNTfilled polymer foam. In addition, a closed form model for the EM absorption will allow, after validation, different performances indices to be defined, based on which the hybrid performances can be optimized in terms of EM absorption, lightness and thermal management.
Fig. 1. Chart of electrical properties of materials: dielectric constant plotted against electrical conductivity (using CES EduPack 2009 software from Granta) and resulting isoabsorption (A) contours.
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The paper starts with a description of the processing of the hybrid material, followed by the EM characterizations. The analytical EM model used to predict the absorption of the hybrid is derived in the next section, followed by the validation against the experimental results and against numerical simulations. The last section proposes an analysis of the performances of the new hybrid regarding lightness and thermal conductivity, with a comparison to the other possible competitors.
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(ISCO pump) for 24 h in order to ensure equilibrium of saturation. The pressure was then released within 30 s. The saturated PU-based sheets were quickly transferred between two hot plates at 120 °C for 2 min to induce foaming. The foams were finally quenched in an ice/water bath. The volume and weight of the samples were measured and the density of the PU–2 wt.% CNT foam was 0.38 g cm3. The foam structure was observed by scanning electron microscopy (JEOL JSM 840-A). The average pore size was 5 lm.
2. Processing of the hybrid 2.3. Hybrid preparation The hierarchical structure of the hybrid material, starting from the CNT-filled polymer composite at the nanometer scale, the foam level at the micrometer scale and the filled honeycomb at the millimeter scale, is shown in Fig. 2. The steps involved in the processing of the hybrid are described in this section. 2.1. Nanocomposite preparation Thermoplastic polyurethane (PU; DesmopanÒ 2590A, Bayer) was melt blended with 1 or 2 wt.% of multiwalled CNT (Nanocyl 7000, 90%) in a counter-rotating twin-screw static mixer (BrabenderÒ) at 180 °C for 5 min at 60 rpm. The nanocomposite was then pressed for 5 min in a hot press at 180 °C in order to obtain ready-to-foam 4-mmthick sheets. 2.2. Nanocomposite foaming The sheets were placed in a vessel at 60 °C and pressurized with CO2 (99.5%, Air Liquide, Belgium) at 300 bars
A 6-mm-thick honeycomb (Liming HoneycombComposites Co., Ltd., China), made of 100-lm-thick aluminum sheets forming 6-mm-sided hexagons, was mechanically inserted in the nanocomposite foam after heating at 220 °C in an oven. The honeycomb was slowly penetrated into the foam under compression within performed at a constant rate of 10 mm min1 using an universal mechanical testing machine. The system involves many degrees of freedom with several parameters that can easily be varied: dimensions of the honeycomb, nature of the polymer matrice, cell size of the foam and volume fraction of CNT. 3. Electromagnetic characterization The samples were characterized using a Vector Network Analyzer (VNA) Model Wiltron (Anritsu) 360, which allows simultaneous measurement of both the reflected and the transmitted power at its two ports over a given frequency range. It measures the scattering parameters (or
Composite Polymer
α
CNT
Honeycomb
2-10cm
Air
10-100µm
0.1-1µm
Fig. 2. Hierarchical structure of the hybrid material starting from the CNT-filled polymer composite at the nanometer scale, the foam level at the micrometer scale and the filled honeycomb at the millimeter scale – schematic drawings and micrographs.
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S-parameters), phase and magnitude that are characteristic of the device connected between its two ports: |S11|2 corresponds to the power reflected back at port 1 while |S12|2 is related to the power transmitted from port 1 to port 2, through the device. After eliminating the influence of the feeding probes on the S-parameters by an adequate calibration, the ratio of the power absorbed by the sample (Pabs) to the incident power (Pin) under test can thus be calculated as Pabs/Pin = 1 |S11|2 |S12|2. Fig. 3 shows (a) the picture of an hybrid sample inserted into the measurement set-up, (b) the absorption index A per unit thickness of sample and (c) the variation of the measured reflected power 10log 10(Pref) as a function of frequency. The multicomposite is a 2 wt.% CNT-reinforced PU foam filling a honeycomb with cell size X of 6 mm and a thickness t of 5 mm (see Fig. 2). The results obtained with the honeycomb (solid line) are compared to those obtained with a CNT-filled polymer foam alone (dashed line). The absorption is improved with the honeycomb, involving a specific peak of absorption at 6.6 GHz (see Fig. 3b). 4. Electromagnetic modeling 4.1. Closed form model When the composite material is inserted into the honeycomb structure, propagation inside each cell of the honeycomb is affected by the metallic walls of the cell. The problem is similar to that of a metallic waveguide (see Fig. 4) filled with a material of known complex permittivity: the presence of the walls modifies the propagation constant, which becomes dependent on the width a and height b of the waveguide. For rectangular waveguides or cells, the canonical expression for the complex propagation constant is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 x2o x2 mp2 np2 c ¼ j eeff ¼ j e eff 2 a b c2o co rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp2 np2 ¼ c2c þ þ ð2Þ a b When the size of the waveguide increases (a and b ! 1), the propagation constant tends to cc, defined in Appendix A for an unbounded slab. For rectangular cells of finite size, expression (2) implies the existence of a cutoff frequency fo that depends on the size of the cell and on the index pair (m, n) of the mode propagating inside the waveguide. r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi xo co m 2 n 2 fo ¼ ¼ pffiffiffiffiffiffiffi þ ð3Þ 2a 2b eeff 2p Above the cut-off frequency (f > fo), c = jb is purely imaginary, meaning that the propagation of the waves takes place along the through-thickness z-axis, with transmission factor ejbz accounting for the phase shift (or equivalently a delay of propagation in time domain). Below
the cut-off (f < fo), c = a is purely real, meaning that the wave is attenuated with factor eaz, implying no propagation. By identification of Eq. (2) with the expression of cc given in Appendix A, the effective permittivity of the waveguide structure filled with the composite material is given by 2 cco mp 2 np2 co 2 D eeffw ¼ ¼ eeff þ a b jx x 2 x ¼ er 1 o2 jr=xeo ð4Þ x Eq. (4) reveals that the equivalent dielectric constant seen by the wave passing through the waveguide structure (=Rðeeffw Þ) varies with frequency, being negative below the cut-off frequency and positive above it, and always remains lower than the dielectric constant er of the filling composite. In contrast, the imaginary part is not modified with respect to the composite alone, meaning that the conductivity seen by the wave is not affected by the waveguide. Hence, it is theoretically predicted that the power reflected by the waveguide structure filled with composite will, above the cut-off, always be lower than the power reflected by the same composite without honeycomb, meaning that the absorption will be superior in the presence of the waveguide by virtue of the power balance. Canonical expressions for the propagation constant exist only for rectangular or circular geometries. Using perturbation techniques, the cut-off frequency of a waveguide of arbitrary cross-section geometry is calculated from the variation DS of the surface of the cross-section with respect to the surface S of a rectangular one. Considering the hexagonal cell with edge size X (see Fig. 2, right) and expressing X as a function of angle a and dimensions a and b of the rectangular cross-section fitting the hexagon, the cut-off frequency foh of a hexagonal cell is given as DS cos a 2 2 2 ð5Þ ¼ for 1 þ foh ¼ for 1 Sr 1 þ 2 cos a with for given by expression (3) for a rectangular waveguide. For a regular hexagon (a = p/3), relation (5) reduces to 5 foh2 ¼ for2 4
ð6Þ
Next, it can be shown (see [12]) that expressions (2) and (4), which are valid for a rectangular cell, remain valid for the hexagonal cell, provided that expressions (5) and (6) are used for the cut-off frequency fo. Final expressions for the complex propagation constant, noted ch, and the corresponding effective permittivity, noted effh, associated to waves propagating through the hybrid material reduce to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi x2 cos a mp 2 5 np2 ch ¼ j eeff 2 1 þ ð7Þ 1 þ 2 cos a a 4 b co
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(a) Reflected Power Pref
Absorption Index per unit thickness from measurements 16
-1 -2 -3 -4
12
-5
db
Percent (%)/mm
14
10
-6 -7
8
-8 -9
6
-10 4 0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
-11 0.4
0.6
0.8
10
Frequency in Hz
1
1.2
1.4
1.6
(b)
1.8 10
Frequency in Hz
x 10
x 10
(c)
Absorption index per unit thickness 14
8 X= 6 mm no Honeycomb
13
6
12 4 11 2
ℜ (εeff )
%/mm
10 9 8
X= 6 mm no Honeycomb
0 -2
7 -4 6 -6
5 4 0.4
0.6
0.8
1
1.2
Frequency in Hz
(d)
1.4
1.6
1.8 10
x 10
-8 0.4
0.6
0.8
1
1.2
Frequency in Hz
1.4
1.6
1.8 10
x 10
(e)
Fig. 3. (a) Measurement configuration for hybrid sample; (b) absorption index in per cent per unit thickness measured with (solid) and without (dashed) honeycomb with edge size X = 6 mm and thickness t = 5 mm; (c) corresponding reflected power measured with (solid) and without (dashed) honeycomb; (d) corresponding absorption index, calculated from analytical formulation (1); (e) effective dielectric constant calculated from analytical formula (4) or (8) with a = 60°, er = 5, X = 6 mm, m = 1 and n = 0.
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a=x+2u=(1+2cos α) X u
b/2 =X sin α
X
α
Fig. 4. Topology of cross-section for rectangular (left) and hexagonal (right) cells of metallic lattice inserted into a composite slab.
eeffh
¼ er 1 þ jr=xeo
cos a 1 þ 2 cos a
mp 2 np2 co 2 þ a b x ð8Þ
with a = X(1 + 2cos a) and b = 2Xsin a. For the dominant mode (i.e. the mode having the lowest cut-off frequency), m = 1 and n = 0. For a regular hexagonal cell (a = p/3), the expressions simplify into: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 5 p2 5 p2 ch ¼ j eeff 2 ¼ c2c þ ð9Þ 4 a co 4 a 5 p2 co 2 jr=xeo ð10Þ eeffh ¼ er 4 a x with a = 2X and b = 1.772X. Expressions derived in Appendix A for the absorption remain fully valid for the case of the honeycomb filling the composite, provided that cc is replaced by ch when calculating expressions (A.1)– (A.9). Eq. (10) shows that inserting the honeycomb into the composite further reduces the real part of the effective dielectric constant, by a factor depending on geometry and frequency. Fig. 3d and e shows the predictions of the model using the parameters of the system tested experimentally. Above the cut-off frequency (where Re(effh) = 0), predicted around 6.3 GHz in Fig. 3e, the real part of the effective dielectric constant for the hybrid (with honeycomb, solid line) remains lower than for the composite foam alone (dashed line), implying that the reflected power is reduced, as confirmed by the measurement in Fig. 3c. The predictions obtained for A using Eq. (1) with eeff replaced by the new expression (10) are also given in Fig. 3d, showing a good agreement. 4.2. Numerical model and validation of the closed form model The analytical model was also validated by comparing the predicted absorption index with respect to numerical simulations performed with COMSOL software, which solves Maxwell’s equations via the finite element method. Two kinds of simulations were performed: one series with only the slab of the composite conductive material and another series considering a regular honeycomb structure inserted in the composite slab, as represented in Fig. 5a
Fig. 5. Geometry used in Comsol Software for validating analytical model. (a) Slab of composite material of thickness t, dielectric constant er and conductivity r, sandwiched between two thin air layers modeling the surrounding environment. (b) Same configuration with metallic honeycomb lattice inserted in the composite slab. Proper boundary conditions are imposed on exterior boundaries in order to ensure propagation of a transverse electromagnetic wave with incidence normal to the air– composite interface and periodicity of the honeycomb lattice.
and b, respectively. The following parameters were varied: the conductivity r of the composite slab, the thickness t of the slab and the size X of the honeycomb. A summary of the validations is presented in Fig. 6. Each figure shows, for r = 0.5 and 1 S m1 respectively, the absorption index A for (a) t = 5 mm and X = 1 mm without and with a honeycomb, (b) t = 5 mm and X = 2 mm without and with a honeycomb and (c) t = 10 mm and X = 0.9 or 2 mm. The agreement between the analytical predictions and results of the numerical simulations is conspicuous, demonstrating again the enhancement of the EM absorption resulting from the presence of the honeycomb. 4.3. Tailoring of the absorption range The size of the honeycomb can be tailored in order to improve the absorption in a range of frequencies. As an example, Fig. 7 shows the absorption index (1) calculated with the analytical model for a cell size X = 0.9 mm. The frequency of maximum absorption is shifted towards 60 GHz as compared to Fig. 3. Furthermore, active tun-
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80
80
70
70
60
60
50
50
%
%
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30
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Comsol, no Honeycomb Analytic, non Honeycomb Analytic, X=2 mm Analytic, X=0.9 mm
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Analytic, X=2 mm
Comsol, X=0.9 mm Comsol, X=2 mm
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Frequency in Hz
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8
9
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x 10
(b) Absorption index 100 90 80 70
%
60 50 40 30
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Comsol, X=0.9 mm Analytic, X=0.9 mm
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Analytic, X=2 mm Comsol, X=2 mm
0 0
1
2
3
4
5
6
Frequency in Hz
7
8
9
10 10
x 10
(c) Fig. 6. Absorption index, in per cent, respectively calculated from analytical (line) formulation and simulated using the commercial FEM solver Comsol (symbol). The parameters of the composite slab are: (a) thickness t = 5 mm, dielectric constant er = 3, conductivity r = 0.5 S m1, (b) thickness t = 5 mm, dielectric constant er = 3, conductivity r = 1 S m1 and (c) thickness t = 10 mm, dielectric constant er = 3, conductivity r = 0.5 S m1. Two sizes of honeycomb cells are considered: X = 0.9 mm (dashed) and 2 mm (dash-dotted), while the third pair of curves holds for a composite slab without honeycomb.
ing/filtering is possible by changing the morphology of the material through an external deformation. Fig. 7 illustrates the effect of a compression of the regular honeycomb lattice inducing a deformation of angle a from 60° (regular honeycomb, Fig. 2) to 18°. These calculations were also validated by full-wave finite element method (FEM) simulations (results now shown). The maximum absorption frequency is shifted towards lower frequencies, by more than 15 GHz. 5. Multifunctional optimization of the hybrid Compared to a simple polymer foam, the incorporation into a honeycomb introduces additional degrees of freedom towards the design of multifunctional sandwich panels (see
[13]), with a potential to optimize density, stiffness and thermal insulation. In order to show the potential of the new hybrid, two material selection procedures involving an EM absorption constraint and an objective towards optimizing another material property are now described, following the rationale developed by Ashby [14]. A first important design constraint typical of transportation or human body protection (e.g. a helmet) is lightness. Assuming that the thickness of a panel is a free constant, the performance index M1 for a light EM absorption panel is given by M1 = 1/(qt), where q and t are the relative density and thickness respectively. M1 can be calculated from Eq. (1) at several frequencies corresponding to an imposed level of absorption A set
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80 70 60
%
50 40 30 X=0.9 mm, α=18°
20
X=0.9 mm, α=60° no honeycomb
10 0 0
1
2
3
4
5
6
Frequency in Hz
7
8
9
10 10
x 10
Fig. 7. Absorption index, in per cent, calculated from the analytical formulation derived in Appendix A. The parameters of the composite slab are: thickness t = 5 mm, dielectric constant er = 3, conductivity r = 1 S m1. Two shapes of honeycomb cells are considered: regular (a = 60°) and deformed (a = 18°), both with edge size X = 0.9 mm. The dashed curve holds for composite slab without honeycomb.
equal to 0.9. Fig. 8a provides the value of M1 based on data available for the most relevant materials extracted from the chart in Fig. 1 (leather, plaster of Paris and SiC foam), for a 2 wt.% CNT-reinforced PU foam (nanocomposite foam) and for the two corresponding hybrids having different honeycomb dimensions (hybrid 1 with X = 4.7 mm and hybrid 2 with X = 1.58 mm). The SiC foam has a low performance index (M1 = 2) for all the frequencies and an absorption level of 0.9 is never attained with plaster of Paris due to reflection, hence M1 = 0. For these reasons, these material do not show up in Fig. 8a. Leather could theoretically compete with the nanocomposite foam at 10 GHz. Above this frequency, the latter is significantly better. The hybrids developed in this work perform extremely well when a high absorption is desired (A = 0.9), not only because of high EM absorption, which is better than any other material, but owing also to the lightness of the foam/honeycomb association. The performance can be tuned for a desired frequency range by the honeycomb cell dimensions (e.g. 10–20 GHz for hybrid 1 and 30–50 GHz for hybrid 2). An even lighter hybrid could be obtained with a metallized polymer-based honeycomb. Now, in most applications, face sheets will be used on one or both sides of the hybrid for protection and mechanical strength in bending, and will have to be accounted for in the procedure. A similar optimization procedure can be worked out for EM absorption at minimum thickness. It should be noted that the corresponding thicknesses (between 5 and 15 mm) relative to the hybrid materials are realistic. The second procedure aims at finding the optimum material for an application involving EM absorption and thermal dissipation. This is typical of an electric or electronic device from which heat must be evacuated. The per-
Fig. 8. (a) Performance index M1 of materials and hybrids for an absorption level A = 0.9 at five frequencies. Nanocomposite foam: er = 3– 3.5, r = 0.7–1.2 S m1, q = 0.3; hybrid 1: X = 4. 7 mm, q = 0.39; hybrid 2: X = 1.58 mm, q = 0.44; (b) performance index M2 of materials and hybrids with minimal thickness for an absorption level A = 0.9 at five frequencies. Nanocomposite foam: k = 0.067 W m1 K1, hybrid 1: k = 4.13 W m1 K1, hybrid 2: k = 11.56 W m1 K1.
formance index M2 for a panel with maximum thermal conductivity at a given absorption level is given by M2 = k/t, where k is the thermal conductivity of the hybrid. The thermal conductivity can be predicted from a simple multi-step homogenization theory starting from the CNT-reinforced polymer up to the hybrid (see Appendix B). Experimental measurements (see Appendix B) confirmed that the presence of nanotubes leads to only a very limited enhancement of the thermal conductivity of the polymer [15,16]. Hence, a lower bound homogenization model has to be used [14,17]. The thermal conductivity of the hybrid essentially comes from the metallic honeycomb. Fig. 8b shows, again, that the new hybrids perform very well. Leather and SiC foam present acceptable performances, though much lower than hybrid 1 over the entire frequency range and lower than hybrid 2 for frequencies of 30 GHz and above. In order to keep good thermal conductivity, only one metallic face sheet can be used (on the outer surface, to avoid reflection) if one considers a sandwich panel. On the other hand, in applications requiring good thermal insulation, the metallic honeycomb can be replaced by a polymer honeycomb with metallized surfaces.
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Other properties, such as the bending stiffness and strength of the new hybrid material can be optimized by proper selection of the face sheet materials. Additional degrees of freedom are provided by using graded foam densities and more complex cell shapes. The results presented in this work show that unique EM absorption can be combined in the form of stiff, light panels with thermal management coming from the choice of the honeycomb, opening many new avenues to applications requiring EM absorption in housing, electronics and health protection, as well as EM active filtering by changing dimensions. 6. Conclusions A new class of hybrid materials with high EM absorption capabilities was fabricated by inserting a CNT-filled polymer foam into a metallic honeycomb. The waveguide characteristics of the honeycomb combined to the conductive nanocomposite foam reduces the real part of the effective dielectric constant of the hybrid, resulting in very high EM absorption in the gigahertz range, superior to any known material. A closed form model, validated towards FE simulations, has been used to design the hybrid material with desired EM absorption levels in a specific frequency range opening to active filtering applications. Moreover, this designed material offers extremely high performances when also considering lightness, thermal management, bending stiffness and strength. Acknowledgements The authors are grateful to the Wallonia Region DG06 for supporting this research through a Winnomat II project “Multimasec”. The support of S. Benyoub for the thermal measurements and of D. Spote for electromagnetic measurements as well as the provision of the Al honeycombs by the company Liming Honeycomb Composites are gratefully acknowledged. I.H. and C.D. are research director and senior research associate for the National Funds for Scientific Research (F.R.S.-FNRS), respectively. J.M.T. is a logistics collaborator for the National Funds for Scientific Research (F.R.S.-FNRS). The authors declare no competing financial interests. Appendix A. General expression of the absorbed power in a composite slab Propagation of electromagnetic waves through a slab of composite conductive material with a thickness t and a permittivity (or complex dielectric constant) er jr/xeo is described via a transmission line formalism associated to electric (E) and magnetic (H) field components [18]. Assuming normal incidence of the wave with respect to the air–material interface, the field components have the following z-dependence in each of the three areas (see Fig. A.1):
Fig. A.1. Topology of the composite slab.
Input air region z < 0 : E ¼ V þa1 eca z þ V a1 eca z H ¼ Y a ðV þa1 e
ca z
þ V a1 e Þ
Composite 0 < z < t : E ¼ V þc e H ¼ Y c ðV þc ecc z þ V c ecc z Þ
ðA:1Þ cc z
Output air region z > t : E ¼ V þa2 e H ¼ Y a V þa2 e
and
ca z
ca ðztÞ
þ V c e
cc z
and ðA:2Þ
ca ðztÞ
and ðA:3Þ
with V the voltage amplitude of the waves and, for air, the complex propagation constant ca = jx/co, the angular pulsation x = 2pf, the frequency f, the velocity of light co and the wave admittance Ya = ca/jxlo, and, for the composite, pffiffiffiffiffiffiffi the complex propagation constant cc = j eeff x/co and the wave admittance Yc = cc/jxlo. Forward terms with a subscript “+” represent a wave propagating along positive zaxis, while terms with a subscript “” hold for a wave propagating in the reverse direction. This transmission line formalism based on complex phasors ensures that all successive reflections occurring in the time domain at input (z = 0) and output (z = t) air–slab interfaces are accounted for. Imposing the continuity of E and H fields on the output air–slab interface (at z = t) yields V þc ecc t þ V c ecc t ¼ V þa2 Y c ðV þc e
cc t
cc t
and
V c e Þ ¼ Y a V þa2
ðA:4Þ
After elimination of V+a2 and rearranging, the ratio of the reflected to the incident wave component in the composite material is related to the interfacial reflection coefficient C as pffiffiffiffiffiffiffi eeff 1 D Yc Ya ðA:5Þ V c =V þc ¼ Ce2cc t with C ¼ ¼ pffiffiffiffiffiffiffi eeff þ 1 Yc þ Ya Similarly applying the continuity conditions on E and H fields at the input interface z = 0, rearranging and using expression (A.5) for the ratio Vc/V+c, the ratio of the reflected to the incident wave in the first air area is obtained as V a1 Cðe2cc t 1Þ ¼ V þa1 1 C2 e2cc t
ðA:6Þ
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Finally, the ratio of the transmitted wave V+a2 in region 3 to the incident wave V+a1 in region 1 at z = 0 is obtained by recombination of the continuity equations and expressions (A.5) and (A.6) as
level to be considered in the thermal conductivity homogenization is shown in Fig. B.1a and corresponds to the polymer–CNT composite. Upper and lower bounds for the thermal conductivity write
V þa2 ecc t ð1 C2 Þ ¼ V þa1 1 C2 e2cc t
ðB:1Þ kupperbound ¼ fCNT kCNT þ ð1 fCNT Þkmatrix kCNT þ 2kmatrix 2f CNT ðkmatrix kCNT Þ klowerbound ¼ kmatrix kCNT þ 2kmatrix þ fCNT ðkmatrix kCNT Þ
ðA:7Þ
The power absorbed in the composite slab is obtained from the power balance between the power respectively incident in the air region 1 (Pin), reflected at the input air–composite interface (Pref) and transmitted into region 3 (Pt). These powers are defined from classical transmission line theory as the cross products of the electric and magnetic fields at the input (z = 0) and at the output (z = t) air–composite interfaces, respectively, as: P abs ¼ P in P ref P t 1 1 1 ¼ Y ca jV þa1 j2 Y ca jV a1 j2 Y ca jV þa2 j2 ðA:8Þ 2 2 2 The absorption index A is then defined as the absorbed power Pabs normalized by the incoming power Pin, and D with the definition T ¼ ecc t introduced in expressions (A.6) and (A.7), the final relationship writes V a1 2 V þa2 2 CðT 2 1Þ2 T ð1 C2 Þ2 A¼1 ¼1 V þa1 V þa1 1 C2 T 2 1 C2 T 2 ðA:9Þ Appendix B. Thermal characterization and modeling The thermal conductivity of the hybrids can be estimated based on simple homogenization rules. The first
1µm
ðB:2Þ where fCNT is the volume fraction of CNT, kCNT is the thermal conductivity of the CNT and kmatrix is the thermal conductivity of the polymer matrix [14]. Several studies in the literature have shown that the thermal transfer between matrix and CNT is usually poor due to the so-called Kaptiza resistance [19,16]. Hence, the lower bound is considered as the most realistic. The second level in the homogenization is shown in Fig. B.1b. The thermal conductivity of a foam can be computed using [17] " 3 # qfoam 2 qfoam 1 qfoam kfoam ¼ þ2 kgaz kmat þ 1 2 3 qmat qmat qmat ðB:3Þ where kmat is the thermal conductivity of the material in the cell walls (here, of the composite as predicted by Eq. (B.1) or (B.2)), kgaz is the thermal conductivity of the gaz entrapped in the foam (usually air), qfoam is the density of the foam and qmat is the density of the material in the cell walls (here the polymer–CNT composite). The third homogenization level shown in Fig. B.1c corresponds to the hybrid. The thermal conductivity of a
50µm
Composite, λcomp Density, ρcomp, ρfoam
Matrix, λmatrix
CNT, CNT, λλCNT, vol. fraction f CNT
Air, λair
(a)
(b) 2mm
1mm
Bi-foam ffoam, λfoam foam Foam Thickness t, thermal cond. λhybrid hybrid λhoneyc
Honeycomb
Plate Thickness tplate plate plate, thermal cond. λplate
(c)
(d)
Fig. B.1. (a) Level 1 of thermal model – polymer/CNT composite, (b) level 2 of the thermal model – foam made of air and composite matrix, (c) level 3 of the thermal model – hybrid made of honeycomb filled with CNT-reinforced polymer foam and (d) level 4 of the thermal model – multimaterial panel (with one face sheet).
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Thermal conductivity [W/(m.K)]
7 6
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5 4 3
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2 1 0
0
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CNT volume fraction Fig. B.2. Prediction of the thermal conductivity of the multimaterial panel.
honeycomb filled by another material (here, by a foam) is given by khybrid ¼ ð1 ffoam Þkhoneyc þ ffoam kfoam
ðB:4Þ
where ffoam is the volume fraction of foam in the system and khoneyc is the thermal conductivity of the honeycomb material The fourth level of homogenization, shown in Fig. B.1d, corresponds to the multimaterial panel (here with one face sheet). The thermal conductivity of the panel is given by 1 tplate =ðt þ tplate Þ ðt=ðt þ tplate ÞÞ k¼ þ ðB:5Þ kplate khybrid where kplate is the thermal conductivity of the face sheet material, t is the thickness of the hybrid and tplate is the thickness of the face sheet. For Al honeycomb and Al face sheet, the model chain predicts, using the following set of typical parameter values: kal = 100 W m1 K1, kair = 0.025 W m1 K1, kpoly = between 0.1 and 0.5 W m1 K1, kCNT = 3000 W m1 K1, qfoam = 0.3, qcomp = 1.2, and fCNT = 0.025, that the thermal conductivity of the panel is dominated by the thermal conductivity of the honeycomb and face sheet (see Fig. B.2). In other words, the foam material essentially behaves as a thermally insulating material, and the thermal conductivity can only be significantly enhanced by adding another thermally conductive material. This result is a direct outcome of the assumption that the thermal conductivity of the polymer/CNT composite is given by the lower Table B.1 Comparison between measurements and predictions of thermal conductivity on polymer foam, CNT nanocomposite and CNT nanocomposite foam. Samples
k Measured (W m1 K1)
k Predicted (m1 K1)
PU PU + 2 wt.% CNT PU foam (q = 0.35) PU + 2 wt.% CNT foam (q = 0.46)
0.23 0.28 0.05 0.08
0.23 0.247 0.057 0.075
bound (B.2), in agreement with results in the literature [20,21], systematically showing very small effects of incorporation of CNT on the thermal conductivity. In order to test this conclusion and the validity of the model, the thermal conductivity of a PU polymer, a PU foam, a PU–CNT composite and a PU–CNT composite foam has been characterized and compared to the predictions in Table B.1. The measured thermal conductivity for PU agrees with literature data [5]. Introducing 2 wt.% CNT increases the thermal conductivity but only by 20%, close to the lower bound (B.2). The thermal conductivity of the foam is predicted well by the model (B.3). The more significant increase in thermal conductivity for the CNTreinforced PU foam results mainly from the change in density (the presence of CNT increases the viscosity of the system and, for the same processing conditions, decreases the foamability). References [1] White DRJ. Electromagnetic interference and compatibility. Germantown, Maryland: Don White Consultants, Inc.; 1976. [2] Adang D, Remacle C, Vander Vorst A. IEEE Trans Microw Theory Tech 2009;57:2488. [3] Chen HH, Chung SS. IEEE Trans Microw Theory Tech 1998;46:2124. [4] Peng ZH, Peng JC, Peng YF, Wang JY. Chin Sci Bull 2008;58:3497. [5] CES EduPack. Software from Granta Design Ltd, Cambridge; 2009. [6] Ashby MF, Bre´chet Y. Acta Mater 2003;51:5801. [7] White DRJ. A handbook on electromagnetic shielding materials and performances. Germantown, Maryland: Don White Consultants, Inc.; 1975. [8] Thomassin JM, Lou X, Pagnoulle C, Saib A, Bednarz L, Huynen I, et al. J Phys Chem C 2007;11:11186. [9] Saib A, Bednarz L, Daussin R, Bailly C, Lou X, Thomassin JM, et al. IEEE Trans Microw Theory Tech 2006;54:2745. [10] Huynen I, Bednarz L, Thomassin JM, Pagnoulle C, Jerome R, Detrembleur C. In: Proceedings of the 38th European microwave conference, October 28–30, Amsterdam. London, UK: # Horizon House Publications; 2008. p. 5–8. [11] Thomassin JM, Pagnoulle C, Bednarz L, Huynen I, Jerome R, Detrembleur C. J Mater Chem 2008;18:792. [12] Harrington RF. Time-harmonic electromagnetic fields. Piscataway, NJ: IEEE Press; 2001.
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[13] He Y, Gong R. Europhys Lett 2009;85:58003; Park KY, Lee SE, Kim CG, Han JH. Compos Sci Technol 2006;66:576. [14] Ashby MF. Materials selection in mechanical design. 3rd ed. Oxford: Elsevier; 2005. [15] Huxtable ST, Cahill DG, Shenogin S, Xue L, Ozisik R, Barone P, et al. Nat Mater 2003;2:731. [16] Clancy T, Gates T. Polymer 2006;47:5990. [17] Gibson LJ, Ashby MF. Cellular solids. 2nd ed. Cambridge: Cambridge University Press; 1997.
[18] Ulaby FT. Fundamentals of applied electromagnetics. 5th ed. Upper Saddle River, NJ: Pearson Prentice Hall; 2006. [19] Seidel GD, Lagoudas DC. J Appl Mech 2008;75:041025/1. [20] Kim HS, Chae YS, Park BH, Yoon JS, Kang M, Jin HJ. Curr Appl Phys 2008;8:803. [21] Yuen M, Ma CCM, Chiang CL, Chang JA, Huang SW, Chen SC, et al. Composites Part A 2007;38:2527.