Electromagnetic Absorption Properties of Carbon ... - IEEE Xplore

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 54, NO. 1, FEBRUARY 2012

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Electromagnetic Absorption Properties of Carbon Nanotube Nanocomposite Foam Filling Honeycomb Waveguide Structures Nicolas Quiévy, Pierre Bollen, Jean-Michel Thomassin, Christophe Detrembleur, Thomas Pardoen, Christian Bailly, and Isabelle Huynen, Senior Member, IEEE

Abstract—Carbon nanotube reinforced polymer foams filling a metallic honeycomb were processed and characterized for the production of hybrid materials with high electromagnetic absorption potential. Electromagnetic modeling and experimental characterization of the hybrids proved that the honeycomb, acting as a hexagonal waveguide, improves the absorption properties in the gigahertz range above the cutoff frequency. The electromagnetic absorption can be tuned by changing the hybrid material properties. The required levels of electrical conductivity are attained owing to the dispersion of low amounts (1–2 wt%) of carbon nanotubes inside the polymer matrix. The combination of the foam and honeycomb architecture contributes to decrease the real part of the relative effective permittivity Re{εr,e ff }. Varying the cell shape of the honeycomb changes the frequency range for high absorption. An analytical model for the absorption has been developed, showing good agreement with the experimental results. Index Terms—Carbon nanotubes (CNTs), electromagnetic wave absorption, nanocomposites, polymer foams.

I. INTRODUCTION UE to the generalization of electronic devices in most human technologies and the growth of wireless communications and radar detection, electromagnetic (EM) compatibility

D

Manuscript received April 30, 2011; revised September 20, 2011 and December 2, 2011; accepted December 3, 2011. Date of publication January 11, 2012; date of current version February 17, 2012. This work was supported by the Wallonia Region DG06 through a Winnomat II Project “Multimasec,” by Fonds pour la formation a` la Recherche dans l’Industrie et dans l’Agriculture” and by the National Fund for Scientific Research, Brussels, Belgium. N. Quiévy and C. Bailly are with the Institute of Condensed Matter and Nanosciences, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium (e-mail: [email protected]; [email protected]). P. Bollen is with the Institute of Mechanics, Materials and Civil Engineering, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium (e-mail: [email protected]). J.-M. Thomassin and C. Detrembleur are with the Center for Education and Research on Macromolecules, University of Liege, 4000 Liège, Belgium (e-mail: [email protected]; [email protected]). T. Pardoen is with the Research Center in Micro and Nanoscopic Materials and Electronic Devices, the Research Center in Architectured and Composite Materials, and the Institute of Mechanics, Materials and Civil Engineering, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium (e-mail: [email protected]). I. Huynen is with the Information and Communications Technologies, Electronics and Applied Mathematics, Microwave Engineering and Applied Electromagnetics Group, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium, and also with the National Fund For Scientific Research, 1000 Brussels, Belgium (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEMC.2011.2179928

has become essential for the proper operation of electronic devices in their environment. Electromagnetic interference (EMI) shielding prevents the propagation of new emissions outside the devices and the penetration of potential external radiations. Shielding can be achieved by reflecting the incident EM waves. However, for stealth applications related to radar technology, the absorption of EMI can be required, leading to the concept of radar absorbing materials (RAMs). Salisbury screens strongly decrease the reflection of an incident EM wave, but are efficient only for a single frequency [1]. Thick absorbing materials with specific design are used in anechoic chambers in order to avoid the reflection of EM waves. Composites or nanocomposites made from conductive particles filled polymers have emerged as a solution for EM absorption while keeping the material thickness acceptable. Ferrite particles fillers provide high-performance microwave absorption [2]–[6], but the required amount of filler is large and the absorption is limited to a narrow frequency range for a given material thickness. Conducting polymers like polyaniline (PANI) and polypyrrole (PPy) are also used to absorb EM waves [7], [8]. Recent work on the dispersion of metallic nanowires in polystyrene showed that low volume fractions of Cu nanowires improve the EMI shielding capacity of the nanocomposites [9], [10], with promising application for EM wave absorption. Carbon-based fillers such as carbon nanofibers and carbon nanotubes (CNTs) dispersed in polymer matrices are becoming more and more attractive for EM absorption [11]–[15]. Thanks to the formation of a percolating network, a relatively low amount of CNT, typically smaller than 1–2%, is sufficient to raise the electrical conductivity of the insulating polymer in the range of 1 S/m [16]. This phenomenon is enhanced by orders of magnitude at high frequencies thanks to the virtual connections created by electrical capacitances existing between closely spaced nanotubes [17]–[19]. However, this effect goes along with the increase of the relative permittivity of the material promoting EM wave reflections, which is undesirable for RAMs. Decreasing the permittivity of materials can be achieved by foaming. This kind of strategy has already been developed by our group for CNT filled polymers [17]–[20]. The hybrid solution developed in this paper involves a CNTreinforced nanocomposite foam filling a metallic honeycomb lattice, as illustrated in Fig. 1 [20]. Above the cutoff frequency of the honeycomb waveguide [21], higher EM absorption levels are reached by the hybrid compared to the conductive foam alone. This multimaterial and multiscale strategy, starting at the

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Fig. 1. Multimaterial and multiscale strategy developed to reach high EM absorption levels in the gigahertz range starting from (a) CNT dispersed in a polymer matrix which is (b) foamed and (c) inserted in a metallic honeycomb.

nanoscale with the CNTs, combines the high electrical conductivity and low permittivity of the nanocomposite foam and the waveguide characteristics of the honeycomb lattice. The required antagonist properties for high EM absorption (without using magnetic material), i.e., the lowest permittivity and a high electrical conductivity, are obtained in the gigahertz range. The objective of this paper is to describe the optimization of the absorption performance, which is a function of the CNT content, foaming, and waveguide shape and size, with the help of an analytical model developed for the hybrid material. Furthermore, the analysis is validated by comparison to experimental data obtained on hybrids involving chemically foamed nanocomposites. II. OPTIMIZATION OF EM ABSORPTION IN MULTIHIERARCHICHAL NANOCOMPOSITE MATERIAL A. Analytical Formulation of EM Absorption Designing materials with high absorption properties is achieved by reducing both reflection and transmission of EM waves. The fraction of EM power absorbed by a slab of material, denoted hereby absorption index A, is defined by the ratio between absorbed power Pabs and incident power Pin , expressed as Pabs Pref Pout =1− − Pin Pin Pin 2 Γ T2 − 1 T 1 − Γ2 2 =1− − . 1 − Γ2 T 2 1 − Γ2 T 2

A=

(1)

Expression (1) shows that A can be rewritten, after some basic algebra, in terms of wave and material parameters involving the reflection coefficient at input interface of the slab √ εr,eff − 1 (2) Γ= √ εr,eff + 1 and the transmission coefficient through the slab √ −jωt εr,eff T = exp c0

(3)

where ω = 2πf, f is the frequency, t is the material thickness, c0 is the speed of light, while the relative effective permittivity

Fig. 2. Optimum material parameters to reach an absorption index A = 0.9 between 0.1 and 100 GHz. tm in is the minimum thickness, σ is the electrical conductivity in S/m, and the relative permittivity ε is equal to 1 in each case.

εr,eff of the slab involved in (2) and (3) is written as εr,eff = εr −

jσ ωε0

(4)

with εr is the relative permittivity, ε0 is the permittivity of vacuum, and σ is the electrical conductivity of the material. Definition (1) implies that the absorption index A is improved when reflected power Pref and transmitted power Pout are decreased with respect to Pin . For a monolayer single material, the ideal combination of electrical properties (εr and σ) required to reach A = 0.9 in the gigahertz range while minimizing the thickness is shown in Fig. 2. First of all, the relative permittivity must be equal to 1. Then, keeping a high absorption index at lower frequencies requires increasing the thickness and decreasing the conductivity of the material in an exponential way as shown in Fig. 2. For a reasonable thickness, no single material owns this particular set of electrical properties to reach A = 0.9. A multimaterial strategy based on a multiscale structure shown in Fig. 1 meets these requirements. As it will be shown in Section II-B, the dispersion of CNT shown in Fig. 1(a) (CNT appear as bright dots) increases both the conductivity and the permittivity of the insulating polymer matrix for a low amount of filler. Foaming the nanocomposite, as displayed in Fig. 1(b), reduces the relative permittivity following the rule of mixture [22] εr,foam = f εr,air +(1−f )εr,nano com p osite =

εr,foam − jσfoam ωεo

= f εr,air + (1 − f )εr,nano com p osite − (1 − f )jσnancom p osite /ωεo

(5)

where f is the air volume fraction, εr, air = 1, and εr, nano com p osite is the relative permittivity of the nanocomposite involving its conductivity, using the formalism (4). However, foaming does not permit the attainment of a relative permittivity equal to one. The incorporation of the honeycomb structure in the nanocomposite foam shown in Fig. 1(c) acts as a hexagonal waveguide with, as a consequence, that the resulting

´ QUIEVY et al.: EM ABSORPTION PROPERTIES OF CN NANOCOMPOSITE FOAM FILLING HONEYCOMB WAVEGUIDE STRUCTURES

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multihierarchical material exhibits a decrease of the real part of its relative effective permittivity (Re{εr,eff }) near the cutoff frequency of the waveguide. As detailed in the Appendix, the expression of the effective permittivity can be obtained from classical waveguide theory as cos α εeffh = εr,foam − 1 + 1 + 2 cos α

mπ 2 nπ 2 co 2 jσfoam × + − (6) a b ω ωεo where a = (1 + 2 cos α)X and b = 2X sin α are the dimensions of the rectangular waveguide section enclosing the hexagon, while the cell size X and the angle α set the hexagonal shape (see Fig. 1) and m and n are the propagation modes. In the following, α will be set to 60◦ and only the transverse electric TE10 (m = 1 and n = 0) mode will be considered. Equation (6) implies that the effective permittivity is calculated based on a rectangular waveguide of dimensions a and b, with a cutoff frequency modified using a perturbation formula relying on the relative reduction of waveguide cross section [23]. Perturbation calculation for a hexagon yields the corrective factor cos α/(1 + 2 cos α), leading to an increase of the cutoff frequency due to the cross-section reduction from rectangle to enclosed hexagon. Equation (6) reveals that the equivalent dielectric constant seen by the wave passing through the waveguide structure (=(εeffw )) varies with frequency, being negative below the cutoff frequency, and positive above, and remains always lower than the dielectric constant εr of the filling composite. Considering that the hybrid structure {nanocomposite foam + honeycomb} is surrounded by air with a permittivity equal to 1, we can predict that the power reflected by the hybrid structure should, above the cutoff, always be lower than power reflected by the same composite without honeycomb, meaning that the absorption will be superior in presence of waveguide by virtue of the power balance (1). These predictions will be confirmed in the next sections by simulations (see Figs. 3 and 4) and experiments (see Figs. 9 and 10). On the contrary, (6) also predicts that the imaginary part of the permittivity is not modified with respect to the composite foam alone, meaning that the conductivity seen by the wave is not affected by the waveguide. The negative value of the dielectric constant below the cutoff results from the derivation of equivalent permittivity from the complex propagation constant of the equivalent waveguide, as detailed in the Appendix. It also means that, in lossless media, the equivalent wave impedance μ/ε is imaginary, and only reactive power is able to penetrate in the waveguide, with, however, strong attenuation and local storage of energy. Nevertheless, waveguides operating below cutoff with associated negative equivalent permittivity can, for example, be combined with periodic resonant magnetic loads [24], for the demonstration of left-handed (or backward) propagation with negative effective refraction index resulting from permittivity and permeability simultaneously negative. Similarly, structures with permittivity (or permeability) close to zero are of great interest for designing compact circuits and low-profile antennas exploit-

Fig. 3. (a) Influence of the waveguide cell size X (4.76, 6.35, and 9.525 mm) on Re{εr, e ff } for the honeycomb filled with air (εr = 1) (solid lines) or with material characterized by σ = 1 S/m and εr = 3 (dashed lines). (b) Influence of εr of the filling material on the cutoff frequency for different cell sizes X .

ing zeroth-order resonators [25]. They support standing waves having zero spatial variation, owing to the cancellation of the effective permittivity (or permeability), hence of the propagation constant. B. Parametric Study Depending on the hexagonal cell shape and on the relative permittivity εr of the filling material, the real part of the structured composite permittivity Re{εr,eff } is equal to 0 at the cutoff frequency, as shown by the dashed and solid lines in Fig. 3(a) for the filled and unfilled hexagonal waveguides, respectively. Above the cutoff frequency, the EM waves can propagate inside the waveguide. If the filling material has EM attenuation properties, i.e., is lossy, the corresponding absorption index A will be maximized at the frequency where Re{εr,eff } is equal to 1. The frequency range over which Re{εr,eff } is negative is fixed by the cutoff phenomenon. The cutoff frequency is a function of the cell shape and of the permittivity εr of the filling material, as shown in Fig. 3(b). Decreasing the permittivity will, on one hand, increase the cutoff frequency for a given cell size X and, on the other hand, increase the absorption index A. Adapting the cell size to the relative permittivity of the filling foam is thus essential for reaching high absorption levels in a desired frequency range.

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Fig. 5. Measurement setup using a VNA. (a) Microstrip line method. (b) Coax-to-waveguide transitions launching for the microwave signal.

Fig. 4. (a) Absorption index of a hexagonal waveguide, a nanocomposite foam, and the corresponding hybrid. T = 30 mm. (b) Influence of the honeycomb cell size X on the absorption index in the gigahertz range for a 15-mm-thick hybrid filled with a nanocomposite foam having σ = 0.7 S/m and εr = 2.

Above the cutoff, a synergy is predicted between the hexagonal waveguide propagating the EM waves and the filling nanocomposite foam attenuating these waves as displayed in Fig. 4(a). The hybrid provides higher absorption indexes in a selected frequency range by varying the cell size X as shown in Fig. 4(b). The performance of the hybrid can be optimized by playing with material thickness, conductivity of the foam, i.e., by controlling the CNT content and dispersion, honeycomb cell size X, and foam structure. The simulations of Fig. 4 use (1)–(3) with (4) replaced either by (5) for foams or (6) for hybrids. III. EXPERIMENTAL VALIDATION A. Nanocomposites and Hybrids Preparation Multiwalled carbon nanotubes (CNT) (Nanocyl NC 7000, 90%) and chemical foaming agent (CFA) (Hydrocerol HK40B, Clariant) were dispersed in thermoplastic polyurethane (TPU Desmopan 2590 A, Bayer) or polycarbonate (PC Makrolon 2805, Bayer) using a corotating twin-screw minicompounder (DSM Xplore Microcompounder 15 cm3 ) with a bypass allowing continuous recycling of the material at the head of the mixing chamber. Processing conditions were the following: the temperature was set at 180 ◦ C for TPU and 270 ◦ C for PC. The

compounds were mixed for 3 min at 100 r/min. 7-mm thick hybrids were prepared using a mould introduced in a Fontijne press working at 15 tons. The nanocomposite powder was poured in the honeycomb cells inserted in the mould. The amount of powder was calculated to reach a foam density of 0.55 g/cm3 each time. In situ foaming was realized above the decomposition temperature of the foaming agent (>230 ◦ C) for 2 min. Honeycombs (CRIII-1/8-5052-.002-8.1 and CRIII-3/8-5052-.004-5.4, R , Hexcel), made of 50- or 100-μm-thick aluminum HexWeb (5052) sheets forming 1.58- or 4.76-mm-sided hexagons, respectively, were used. The chemical foaming method used here differs from the one used in earlier studies based on a CO2 supercritical technique [21]. Chemical foaming is easier but is expected to generate larger cell sizes. B. Measurement Setup The measurements were performed using a vector network analyzer (VNA) Model Wiltron (Anritsu) 360, which allows covering the 40 MHz–40 GHz frequency range of the four scattering parameters for any device connected between its two ports. Two different configurations were considered for the results proposed in this paper. 2-mm thick solid plates of nanocomposites were characterized using the microstrip line method [see Fig. 5(a)] presented in [19], in order to extract the dielectric constant εr and conductivity σ from the S-parameters, as presented in Fig. 6. The measurement method used for the characterization of the thick hybrid with honeycomb (t = 7 mm) uses coaxialto-waveguide transitions, also called in the literature flanged rectangular waveguides. As reported in [26], this technique is widely applied in the industry for testing materials and substances that are difficult to be machined or held to fit accurately the inner dimensions or shape of a waveguide or cavity. The configuration can be used either in transmission or reflection mode [27]. The transmission mode, where the slab is inserted between two flanged waveguides, enables a straightforward


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sorb power once having penetrated. The S-parameters are also converted to the corresponding ABCD or chain matrix T of the hybrid slab, using classical conversion formulas from microwave circuit theory. The diagonal term T11 resulting from de-embedded four Sij parameters is expressed as T11 =

1 + S11 − S22 + S11 S22 − S21 S12 = cosh γh t (7) 2S21

where γ h is the propagation constant in the hybrid, and t is its thickness. Inversion of (7) knowing thickness t yields γ h , from which the dielectric constant and conductivity shown in Fig. 9 are extracted using (A1) of the Appendix. Referring to the power balance associated with S-parameters definition, |S11 |2 corresponds to the power reflected back at port 1, while |S21 |2 is related to the power transmitted from port 1 to port 2, through the device. The absorption index defined in (1) can be rewritten as resulting from the power balance of the de-embedded S-parameters of the hybrid as A = Pabs /Pin = 1 − |S11 |2 − |S21 |2 .

Fig. 6. (a) Electrical conductivity of PC-CNT nanocomposites filled with 0.1, 0.5, 1, and 2 wt% of CNT. (b) Corresponding real part of the relative permittivity.

extraction of both permittivity and permeability from Sparameters referenced to the two air–slab interfaces, using, for example, the Nicolson Ross Weir technique [28], [29]. This latter configuration has been further developed and adapted in this paper to the particular topology and EM behavior of the hybrid samples to handle. To this end, a particular test fixture was built. It allows a precise line-of-sight alignment of the respective openings of the two waveguides, together with a good contact between sample and flanges, without requiring using clamps or screws throughout the test hybrid sample and waveguide flanges [see Fig. 5(b)]. Each flanged waveguide, with its transition to coaxial connector, is clamped at its bottom on a block sliding on a guiding rail. The sample is positioned between the flanges. The contact between the flanges and the samples is ensured by sliding the blocks and tuning the distance with the screw. Clamping the flanges on the upper side compared to the used setup shown in Fig. 5(b) was checked and no change in the results was observed, confirming the good contact between the flanges and the sample. A first calibration measurement is performed with the two flanged waveguides in contact (“thru” configuration). These measured Sthru -parameters are used to move the reference planes of S-parameters measured with the sample inserted between flanges from the coaxial connector of the flanged waveguide to the flange-sample interfaces. The obtained de-embedded S-parameters are first used for computing the absorption index (8), characterizing both the matching of the hybrid at the air–sample interface, other said the ability of incoming power to enter into the slab, and its capability to ab-

(8)

This definition of the absorption index, because of its normalization to incoming power, truly characterizes the performances of a microwave absorber, i.e., a material showing simultaneously low reflection at the input interface between air and hybrid and high absorption inside the hybrid: a value of A close to 1 (or 100%) implies that both S11 associated with input reflection and mismatch and S21 associated with residual power outgoing from the hybrid, are minimized. It is, thus, an alternate formulation, compared to reflectivity, for characterizing the performances of an absorber. C. EM Characterization of Hybrids Low amounts of CNT raise the electrical conductivity of the polymer matrix as displayed in Fig. 6(a) for PC-CNT nanocomposites. However, it simultaneously produces an increase of the relative permittivity of the nanocomposite as shown in Fig. 6(b). Foaming was performed to decrease the permittivity while keeping acceptable levels of conductivity around 1 S/m in order to obtain high absorption index A. The microstructure of the processed foams obtained using the CFA is shown in Fig. 7. Scanning electron microscope (SEM) micrographs show that different structures were obtained by varying the CFA content. It is observed that increasing the amount of foaming agent [see Fig. 7(a)–(d)] increases the pore size of the foam due to a coalescence phenomenon. The resulting foam structure with a lower cell density is more heterogeneous for the higher CFA contents. This could negatively impact the electrical properties of the material due to a less efficient dispersion of CNT. The absorption index of the PC-CNT 1 wt% foams processed with different amounts of CFA is shown in Fig. 8. The lower the cell density, the lower the absorption ability. The waveguide effect on Re{εr,eff } is highlighted in Fig. 9(a) and (b) for honeycomb filled with PC-CNT 1 wt% (using 10 wt% of CFA) and TPU-CNT 1 wt% (using 7.5 wt% of CFA) foams, respectively. Below the cutoff frequency, Re{εr,eff } is negative for both hybrids, while it remains almost constant and positive

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Fig. 7. SEM micrographs of PC-CNT 1wt% foams processed using (a) 5, (b) 7.5, (c) 10, and (d) 15 wt% of CFA. Scale bar 500 μm.

Fig. 9. Measurements (solid lines) and predictions (dashed lines) of the real part of εr , e ff with frequency for 7-mm-thick (a) PC-CNT 1 wt% foams with and without honeycomb (X = 4.76 mm) and (b) TPU-CNT 1wt% foams with and without honeycomb (X = 4.76 mm). Inset: Measured conductivities of the samples.

Fig. 8. Variation of the absorption index A as a function of the frequency for 7-mm-thick PC-CNT 1 wt% foams obtained using 7.5, 10, and 15 wt% of CFA.

over the whole frequency range for the nanocomposite foams alone. The conductivity is not significantly modified by the presence of the metallic honeycomb. The cutoff frequencies were measured at 11.4 and 10.4 GHz for the TPU- and PCbased hybrids, respectively. It can be seen in Fig. 9 that the predictions obtained with the help of the analytical model match the measurements, validating (6). In each case, the geometrical parameters were X = 0 or 4.76 mm and α = 60◦ . The measured permittivity at 40 GHz represents here the relative permittivity of the foam, and a linear fit was used for the measured conductivity over the studied frequency range. The absorption indexes of the foams and hybrids extracted from the measured S-parameters are shown in Fig. 10 for the PC-CNT 1 wt% (using 15% of CFA) system. Below 10 GHz, the foam has better absorption indexes than both hybrids due to the cutoff, i.e., negative values of Re{εr,eff }. However from 10 to 35 GHz, the hybrid with X = 4.76 mm performs better than the nanocomposite foam. Above 35 GHz, the absorption index is almost the one of the nanocomposite foam. The cutoff of the other hybrid with X = 1.58 mm limits the absorption be-

Fig. 10. Variation of the measured (solid lines) and predicted (dashed lines) absorption indexes as a function of the frequency for 7-mm-thick PC-CNT 1 wt% foam and the corresponding hybrids with X = 4.76 and 1.58 mm.

low 35 GHz. At high frequencies, the absorption indexes of the hybrid and the nanocomposite foam become similar. Some differences between measurements and predictions were observed. Predictions were made using a linear fit of σ over the frequency range and by picking εr ,foam at 40 GHz. Locally, variations in σ are measured changing the absorption levels. The true value of εr ,foam should be a little bit higher because of the remaining influence of the honeycomb waveguide at 40 GHz (see example in Fig. 3). The foam microstructure and the nonperfect


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acceptable cell size X and material thickness t. Investigations are now made to decrease this limit. The particular hierarchical organization of the studied hybrid materials offers interesting perspectives for multifunctional structures. In particular, high EM absorption could be combined with lightness and stiffness if a foam-filled honeycomb was sandwiched between rigid dielectric plates. Good thermal conductivity across the structure is also achieved owing to the presence of a metallic honeycomb. APPENDIX Fig. 11. Variation of the absorption index as a function of the frequency for 7-mm-thick honeycomb (X = 4.76 mm) filled with PC-CNT foams containing 1, 1.25, and 1.5 wt% of CNT.

hexagonal shape of the metallic honeycomb could involve some oscillations and shift of the cutoff in the data compared to the predictions. However, a good trend exhibiting the influence of the waveguide structure is shown. Designing the hybrids for higher EM absorption levels requires, thus, selecting the right foam properties (εr and σ) for a given honeycomb cell size. EM absorption properties of hybrids filled with PC foams (using 5 wt% of CFA) containing different amounts of CNT were measured. The absorption indexes shown in Fig. 11 are similar over the whole frequency range for increasing CNT content. Filling PC with only 1 wt% of CNT raises the conductivity of the 7-mm-thick hybrid around 1–2 S/m which is already an upper limit for achieving high EM absorption in the considered frequency range for this material. In summary, experimental characterization and analytical prediction of CNT nanocomposite foams filling metallic honeycombs have proved that the hybrid structures exhibit Re{εr,eff } equal to 1 at a well-defined frequency. Dispersion of CNT in the polymer matrix raised the conductivity σ in the range of 1 S/m for only 1 wt%. This efficient combination of electrical properties yields high levels of absorption index A above the cutoff imposed by the hexagonal waveguide for a 7-mm-thick material.

IV. CONCLUSION EM absorption in the gigahertz frequency range was achieved by combining the waveguide properties of a metallic honeycomb structure filled with a TPU or PC foam reinforced with CNT. The nanocomposite filling foam is able to attenuate the incident EM waves for low content of CNT dispersed inside the polymer matrix, because the exceptional aspect ratio of these nanostructures promotes the formation of a conductive network. Above the cutoff frequency set by the hexagonal waveguide, the absorption index of the hybrid is higher than for the nanocomposite foam alone. Experimental results validate the developed analytical model. Tuning and optimizing the performance of the hybrid is now possible thanks to the model. However, reaching high absorption index below 5 GHz remains difficult while keeping

When the composite foam 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 waveguide of rectangular cross section with metallic walls (see inset in Fig. 1) filled with a material of known complex permittivity εr ,foam . The presence of the walls modifies the propagation constant with respect to pure TEM propagation in free space: it 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 √ ω 2 − ωo2 γ = jω εeffw /co = j εr,foam c2o ω 2 mπ 2 nπ 2 = j εr,foam 2 − − (A1) co a b where εeffw is the equivalent or effective permittivity associated with an equivalent TEM propagation. Expression (A1) implies the existence of a cutoff frequency fo depending on the size of the cell and on the index pair (m, n) of the mode propagating inside the waveguide

m 2 n 2 co ωo fo = =√ + . (A2) 2π εr,foam 2a 2b Above the cutoff frequency (f > fo ), γ = jβ is purely imaginary, meaning that the propagation of the waves takes place along zaxis with transmission factor e−j β z accounting for phase shift (or equivalently a delay of propagation in time domain). Below the cutoff (f < fo ), γ = α is purely real, meaning that the wave is attenuated with factor e−α z , implying no propagation. Combining (A1) with (5) of the foam permittivity, the effective permittivity of the waveguide can be rewritten as ω2 jσfoam εeffw = εr,foam 1 − o2 − . (A3) ω ωεo The positive or negative value of the real part of the effective permittivity is, thus, strongly related to the behavior of the propagation constant, being purely real below cutoff, and imaginary above cutoff. Canonical expressions for the propagation constant exist only for rectangular or circular geometries. Using perturbation techniques, the cutoff frequency of a waveguide of arbitrary crosssection geometry is calculated from the variation ΔS of the surface of the cross section with respect to the surface S of a rectangular one. The change in cutoff frequency resulting from

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an inward perturbation of the waveguide wall expresses as, according to Harrington [23] (ε|E|2 − μ|H|2 )ds Δfo ΔS . (A4) ≈ Δ S =− 2 + μ|H|2 )ds fo S (ε|E| S The left-hand side of this expression assumes that the actual EM fields are approximated by corresponding electric field E and magnetic field H existing in the waveguide in absence of perturbation. It is valid when the perturbation is smooth enough, which is the case here: the lateral walls of the rectangular waveguide are progressively modified. Since the electric field is close to zero in the vicinity of the lateral walls for the dominant TE10 mode of interest here, while the longitudinal magnetic field is maximum, the change in cutoff frequency is derived as righthand side by using the analogy with the formulation available for cavity resonators [23], for the case of a perturbation occurring in a area where magnetic field is maximum. The validity of this analogy can be verified by analytical calculations. Considering the hexagonal cell with edge size X (see inset in Fig. 1) and expressing X as a function of angle α and dimensions a and b of the rectangular cross section fitting the hexagon, the cutoff frequency foh of an hexagonal cell is given as cos α 5 foh = fo 1 + = fo , for α = π/3 (A5) 1 + 2 cos α 4 taking into account the fact that ΔS is negative owing to crosssection reduction, and that α = π/3 for a regular hexagon. Next, according to [23], (A1) and (A3) established for a rectangular cell remain valid for the hexagonal cell, provided that (A5) is used for the cutoff frequency fo . Final expressions for the complex propagation constant, noted γ h , and corresponding effective permittivity, noted εffh , associated with waves propagating through the hybrid material reduce to γh = j

mπ 2 nπ 2 ω2 cos α εr,foam 2 − 1 + + co 1 + 2 cos α a b

εeffh = εr,foam ×

− 1+

mπ 2 a

+

(A6)

cos α 1 + 2 cos α nπ 2 c 2 o

b

ω

−

jσfoam ωεo

(A7)

with a = X(1 + 2 cosα) and b = 2Xsinα. For the dominant mode (i.e., the mode having the lowest cutoff frequency), m = 1 and n = 0. For a regular hexagonal cell (α = π/3), (A7) simplifies into 5 π 2 co 2 jσr,foam − . (A8) εeffh = εr,foam − 4 a ω ωεo ACKNOWLEDGMENT The authors would like to thank L. Monnereau, S. Eggermont, and I. Molenberg for sample preparation and fruitful discussions. The support of D. Spote for electromagnetic measure-

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Nicolas Quiévy received the Civil Engineer degree in chemistry and the Ph.D. degree in sciences of the Engineer from the Université catholique de Louvain (UCL), Louvain-la-Neuve, Belgium, in 2005 and 2010, respectively. In 2005, he joined the Bio and Soft Matter Research Division, Institute of Condensed Matter and Nanosciences, UCL, where he is currently a Postdoctoral Researcher. He has a particular interest in the structure–property relationships of polymer– nanoparticle systems and in the design of advanced materials with multifunctional properties.

Pierre Bollen received the degree of Civil Engineer in chemistry and material sciences from the Université catholique de Louvain (UCL), Louvainla-Neuve, Belgium, in 2010, where he is currently working toward the Ph.D. degree from the Institute of Mechanics, Materials and Civil Engineering. His Ph.D. study is on multiscale nanocomposite for electromagnetic interference shielding with a focus on the process, mechanical performance, and electromagnetic performance.

Jean-Michel Thomassin was born in Verviers, Belgium, in 1979. He received the M.S. degree in chemistry and the Ph.D. degree in sciences from the University of Liège (ULg), Liège, Belgium, in 2001 and 2005, respectively. He is currently with the Center for Education and Research on Macromolecules, ULg as a “Logistics Collaborator” (F.R.S.-FNRS). He is mainly involved with the development of electromagnetic interference shielding materials. His main research activities also include the development of new polymeric membranes for fuel-cell applications and rheological characterization.

51

Christophe Detrembleur was born in Verviers, Belgium, in 1974. He studied chemistry at the University of Liège, Liège, Belgium, where he received the Ph.D. degree from the Center for Education and Research on Macromolecules (CERM), in 2001, under the supervision of Prof. R. Jérôme. His major research topic was the search for new regulators for the controlled radical polymerization of (meth)acrylic monomers. He was an Invited Researcher at IBM, Almaden Research Center, San Jose, CA, under the supervision of Dr. J. Hedrick for three months in 1998. In May 2001, he joined the Research Center of Bayer AG, Leverkusen, Germany, where he was involved on materials synthesis and polymer processing. In January 2003, he moved to the polyurethane research division at Bayer, where he was involved in the development of new high-performance UV coatings. In October 2003, he became a permanent Research Associate at CERM under the auspices of the National Fund for Scientific Research (F.R.S.FNRS). In October 2008, he was promoted Senior Research Associate by the F.R.S.-FNRS and heads a research team at CERM. His main research interests are in the field of new controlled radical polymerization techniques, preparation of new multifunctional polymeric materials, and contribution of macromolecular engineering to the nanotechnology. Thomas Pardoen received the Degree in engineering in 1994 and the Ph.D. degree in 1998 from the Université catholique de Louvain (UCL), Louvainla-Neuve, Belgium. He was a Postdoctoral Researcher at Harvard University from 1998 to 2000 before joining UCL in 2000 as a Faculty Member. His research interests span the area of nano-, micro-, and macromechanics of materials and systems, with an emphasis on multiscale modeling and experimental investigation of deformation and fracture phenomena. The main applications concern advanced metallic alloys, microelectromechanical systems and nanoelectromechanical systems, thin films, multifunctional architectured materials, polymer-based composites, adhesive and molecular bonding, fracture mechanics, welding, and forming. He is currently a Full Professor at UCL, Vice President of the Institute of Mechanics, Materials and Civil Engineering and member of the Research Center in Micro and Nanoscopic Materials and Electronic Devices and of the Research Center in Architectured and Composite Materials at UCL. He has published approximately 100 papers in international peer reviewed journals. Dr. Pardoen received the Grand Prize Alcan from the French Academy of Sciences in 2011. Christian Bailly received the Ph.D. degree in applied sciences from the Université catholique de Louvain (UCL), Louvain-la-Neuve, Belgium, in 1993. He is now a Full Professor with the School of Engineering, Université catholique de Louvain (UCL), Louvain-la-Neuve, Belgium. His research activities with the Bio and Soft Matter Research Division, Institute of Condensed Matter and Nanosciences, UCL focus on the structure–property relationships of polymer-based systems, including blends and nanocomposites. His recent emphasis is on the quantitative prediction of rheological properties from structural knowledge. Isabelle Huynen (S’90–A’95–M’96–SM’06) received the Electrical Engineer and Ph.D. degrees in applied sciences from the Université catholique de Louvain (UCL), Louvain-la-Neuve, Belgium, in 1989 and 1994, respectively. In 1989, she joined the Microwave Laboratory, UCL. She is currently the Research Director with the National Fund for Scientific Research, Brussels, Belgium, and a Part-Time Professor at UCL. She is the author or coauthor of one book and more than 250 publications in journals and conference proceedings. She holds three patents. She has particular interest in the development of devices based on nanoscaled materials and topologies for applications at microwave, millimeter-wave, and optical wavelengths.