Hollow carbon spheres in microwaves : Bio inspired absorbing coating

The electromagnetic response of a heterostructure based on a monolayer of hollow glassy carbon spheres packed in 2D was experimentally surveyed with respect to its response to microwaves, namely, the Ka-band (26–37 GHz) frequency range. Such an ordered monolayer of spheres mimics the well-known “moth-eye”-like coating structures, which are widely used for designing anti-reflective surfaces, and was modelled with the long-wave approximation. Based on the experimental and modelling results, we demonstrate that carbon hollow spheres may be used for building an extremely lightweight, almost perfectly absorbing, coating for Ka-band applications.

The electromagnetic response of a heterostructure based on a monolayer of hollow glassy carbon spheres packed in 2D was experimentally surveyed with respect to its response to microwaves, namely, the Ka-band (26-37 GHz) frequency range.Such an ordered monolayer of spheres mimics the well-known "moth-eye"-like coating structures, which are widely used for designing anti-reflective surfaces, and was modelled with the long-wave approximation.Based on the experimental and modelling results, we demonstrate that carbon hollow spheres may be used for building an extremely lightweight, almost perfectly absorbing, coating for Ka-band applications.V C 2016 AIP Publishing LLC.
[http://dx.doi.org/10.1063/1.4938537]2][3][4][5][6][7] Unlike most random carbon foams, for which it is still challenging to accurately control geometric parameters, hollow carbon spheres can be more easily tailored through a synthesis method based on well-calibrated sacrificial templates.This advantage turns such spheres into very interesting candidates for using them in electromagnetic applications.
In particular, hollow spheres (HS) made of glasslike carbon with uniform diameter can be potentially used for producing ordered periodic structures.In this work, the electromagnetic properties of a system formed by a single layer of carbon HS packed in 2D were investigated.][10] In the present case, however, the HS monolayer is an electrically conductive material with losses; thus, due to these attractive properties, it may be used for EM absorption applications.With this aim in view, the long-wave approximation, used for modelling structures presenting a spatial dispersion of refractive index, 9 was adapted to the description of the HS monolayer response in microwaves.For calculating reflection R, transmission T, and absorption A coefficients of the HS monolayer, the model of wave distribution in multi-layered medium, widely used in optics, 11 was applied.
Carbon HS samples were prepared by a template method based on two kinds of sacrificial spherical polymer beads, having two different diameters.As both present a nearly zero carbon yield when directly pyrolysed at 900 C under inert atmosphere, hydrothermal conditions were used for depositing a sucrose-derived hydrochar at their surface.The resultant spheres were next converted into glasslike carbon through an additional subsequently pyrolysis step at 900 C in nitrogen flow.During such heating, the initial material of the spheres was largely destroyed so that both kinds of HS can be considered as purely made from the same carbon derived from sucrose.In the following, the carbon hollow spheres are referred to as HS-A and HS-B, the former having a lower diameter than the latter (see Table I).
Scanning electron microscopy images of the two kinds of HS discussed here are presented in Fig. 1.Pictures showing damaged spheres were deliberately chosen to demonstrate their hollow character, but the vast majority of carbon HS was neither broken nor cracked.
The dimensions of several tens of carbon HS were measured by electron microscopy, using full particles for average diameter determination.For further calculations, we assume that carbon spheres consist of glassy carbon with a density q ¼ 1:55 g/cm 3 and a static conductivity r ¼ 20 000 S/m. [12][13][14] For microwave characterisation, the spheres were assembled in monolayers.The average shell thickness of the spheres was deduced from the known values of density of glassy carbon, weight and number of spheres in the monolayer, and average sphere diameter.All these physical parameters of spheres and monolayers are collected in Table I.
The electromagnetic response of a monolayer based on each kind of carbon hollow spheres was investigated in the Ka-band (26-37 GHz).All measurements were performed in a 7.2 Â 3.4 mm waveguide system.In a typical experiment, the spheres were initially packed to form a close-compact layer which was then placed inside the waveguide between two 1 mm-thick layers of transparent to microwave Styrofoam V R .
Due to the unavoidable, slight distribution of HS sizes and due to some defects in the packing of such lightweight, fragile objects, the as-obtained 2D ordering was closer to a square-shaped lattice.

APPLIED PHYSICS LETTERS 108, 013701 (2016)
This article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: responses of the samples were obtained as ratios of transmitted to input (S 21 ) and reflected to input (S 11 ) signals.
The simplified geometry, assumed for investigating the monolayers based on two separate sizes of carbon hollow spheres, is the one presented in Fig. 2. The wavelength of the radiation in the Ka-band, 1 cm at 30 GHz, is much larger than the HS diameter.][10] This approximation allows introducing a homogenisation procedure to calculate the spatial dispersion of the effective refractive index of the monolayer formed by the 2D packed lattice of spheres (see below).Therefore, the investigated system can be considered as a stack of thin layers of constant refractive index so that the characteristic matrix of the corresponding assembly of thin films can be calculated and directly converted into reflection R ¼ S 2 11 and transmission T ¼ S 2 21 coefficients.The equations below were written using SI units and assume an exp½ixt À ikz harmonic time convention.
Let us consider the cross-section of a HS monolayer along the yz-plane at the coordinate x ¼ x 0 , where jx 0 j < r o , where r o is the outer sphere radius, as shown in Fig. 2(b).This cross-section consists of ring-like carbon regions and air regions.The homogenisation procedure in this case means that the non-conductive air regions can be averaged with highly conductive carbon regions according to their relative surface fractions.The symmetry of the system allows considering just one unit cell which is a square of side s ¼ 2r o .For calculating the surface fractions of carbon and air regions parallel to the yz-plane at the coordinate x 0 , let us consider a cross-section of a hollow sphere along the xz-plane, as presented in Fig. 2(b).From this figure, it can be easily calculated that the glassy carbon ring surface area is Þ, whereas that of air is 4r 2 o À Sðx 0 Þ along the yz-plane at the coordinate x 0 .The dependence of Sðx 0 Þ on x 0 vanishes when jx 0 j < r i , where r i is the inner radius of HS (in the center region terms with x 0 cancel out and Sðx 0 Þ ¼ Sðr i Þ).Averaging inside the unit cell directly leads to the following equation for the refractive index: where and where n s ðÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 À ir 2pe 0 q is the refractive index of carbon, i being the imaginary unit, r ¼ 20 000 S/m the static conductivity of glassy carbon, the frequency, e 0 ¼ 8:85 Â 10 À12 F/m is the permittivity of vacuum, and n 0 ¼ 1 the refractive index of air.Thus, the EM response of a monolayer of carbon HS in the long-wave approximation is considered as equivalent to that of a layer of bulk material of thickness s ¼ 2r o and with a spatial dispersion of refractive index which obeys Equation (1).
Fig. 3 shows such spatial dispersions (1) for bulk and hollow spheres.The void inside the spheres leads to a constant refractive index in the central region of the HS monolayer.
The maximum values of the refractive index of bulk (index b) and hollow (index h) spheres are, respectively n b ðÞ ¼ pn s ðÞ=4 À n 0 ð1 À p=4Þ; (3) It is worth noticing that, within the long wave approximation, the type of ordering of HS in the monolayer does not significantly affect the EM absorption properties, unlike, for example, changes of HS shell thickness.
For calculating the EM response of the HS monolayer, the model of wave distribution in a multi-layered medium 11 was used.Let us consider a layer of bulk material with the  spatial dispersion obeying Eq. ( 1).For calculating its R and T coefficients, the layer was first divided into N parts.Using Maxwell equations, it is possible to obtain the direct solutions for electric and magnetic fields at the boundaries of each thin layer.Written in matrix notations, they have the following form 11,15 where E 1 , H 1 and E 2 , H 2 are electric and magnetic field in the left and right side of a layer of thickness s 0 , q are the wave vectors in the layer of refractive index n(x) and in air (refractive index n 0 ¼ 1) inside the waveguide of width a ¼ 7.2 mm, respectively; k being the wavelength.The first term in the right part of Eq. ( 5) is known as the characteristic matrix of the layer.
In the case of a multi-layered structure, the characteristic matrix of the whole layer is the product of all single layer matrices.The R, T coefficients of the N-layered subsystem inside the waveguide can be calculated as where In Fig. 4 are presented the measured and modelled Kaband spectra of two different monolayers made from HS-A and HS-B with diameters D and shell thicknesses dr.A good agreement can be observed between the experiment and modelling for the reflection coefficient R. The possible reason for the difference between modelled and measured reflection coefficients in the high-frequency part in the spectra of Fig. 4 is the unevenness of shell thickness dr in the real spheres forming the monolayer.The experimental values of the transmission coefficient T are low, as expected, but considerably higher than those predicted by the model.This might be due to a bad electrical contact between the HS monolayer and the walls of the waveguide. 1,5The small gap between the waveguide and the sample leads to the penetration of electromagnetic radiation and therefore to an increase of transmission coefficient.
The model proposed above was applied to estimate how some changes in the carbon HS parameters (i.e., radius r o and shell thickness dr) would affect the electromagnetic response of the corresponding monolayer.First, the HS radius was fixed, whereas the thickness of the shell dr was changed.The resultant spectra of R and T coefficients (not shown) are closely behaved as those presented in Fig. 4. R and T increased and decreased when dr increased, respectively.It is especially interesting to look at the absorption spectra A ¼ 1 À R À T. For example, the calculated results for monolayers based on hollow spheres of constant radius r o ¼ 0:67 mm (corresponding to radius of HS-A) and variable shell thicknesses dr are presented in Fig. 5.
Fig. 5 shows that the absorption coefficient is the highest when the shell thickness dr ranges from 4 to 8 lm.From these results, it can be concluded that there is an optimal thickness leading to a maximal absorption of the monolayer formed by HS of radius r o ¼ 0:67 mm.When the shell FIG. 3. Spatial dispersion of the homogenised refractive index described by Eq. ( 1).The values of maxima n b and n h were calculated from Eqs. ( 3) and (4).thickness was fixed at dr ¼ 4 lm while changing the radius of the hollow spheres in the monolayer, the results led to the absorption spectra shown in Fig. 6.The analysis of Fig. 6 shows that an increase of sphere radius leads to a monotonic increase of absorption coefficient A. This is because the increase of sphere radius leads to an increase of the total thickness of the system and summary amount of carbon in the monolayer.It can be concluded from these data that, for EM absorption applications, spheres with larger diameters are to be preferred.
The frequency dependencies presented in Figs. 5 and 6 have generally the same appearance.For a clearer and more detailed analysis of these data, the frequency was fixed at 30 GHz, and both r o and dr were varied, see Fig. 7.
Based on the results given in Fig. 7, it can be seen that the absorption coefficient presents a maximum for all varied values of r o when dr is in range 5-10 lm.The maximum of absorption is especially explicit for smaller values of r o .To explain the origin of this maximum, let us consider Eq. ( 4), where r 2 o À r 2 i % 2r o dr, so that dr directly controls the real and imaginary parts of the refractive index of the carbon HS monolayer.A similar maximum was observed in thin film optics.It is indeed shown in Ref. 15 that, for each single homogeneous layer, there are existing relation between Im(n) and Re(n) for maximum absorption at fixed layer thickness.Generally, on one hand, the wall thickness dr controls the averaged conductivity and hence the energy losses in the monolayer of spheres.On the other hand, high values of averaged conductivity lead to increase of reflection and decrease of absorption coefficients.
Additionally, it can be seen from Fig. 7 that A increased with r o .Therefore, for EM absorption applications, the most preferable hollow spheres are those having large radii r o .Our model predicts that, for high values of r o , it is possible to achieve almost perfect microwave absorption with A > 95%.It is also worth mentioning, however, that high values of A are obtained when r o starts to be comparable or larger than the wavelength of the initial radiation.In this case, the validity of the long-wave approximation used here should be experimentally proved by free-space measurements or more exact Mie scattering-based models 16 should be used.
The graphs corresponding to Fig. 7 calculated at 26 and 37 GHz have the same general form, but the maximum in the plane section r o ¼ const is more explicit and narrow at 37 GHz, and broader and flattened at 26 GHz.Therefore, at higher frequencies, the electromagnetic response of the monolayer is more sensitive to changes of dr.
In summary, a model for describing the electromagnetic properties in Ka-band of monolayers based on a 2D packing of hollow carbon spheres was presented.The results showed that, for EM applications requiring high absorption, the most preferable hollow spheres are those of larger radii r o .Additionally, it was estimated that, for each value of HS radius, there is an optimum shell thickness dr such that the absorption coefficient of the monolayer is the highest.The present work therefore pointed out that "moth-eye"-like 2D ordered structures based on hollow conducting spheres are promising systems for being used in microwave radiation absorption applications.

FIG. 2 .
FIG. 2. (a) Geometry of a carbon hollow spheres monolayer, and (b) crosssection of a single sphere of outer and inner radii r o and r i , respectively, along the xz-plane.

FIG. 4 .FIG. 5 .
FIG.4.Measured and modelled Ka-band spectra of hollow spheres with different diameter D and wall thicknesses dr.

TABLE I .
Parameters of carbon hollow spheres (HS) and monolayers (ML) based on them.
FIG. 1. SEM images of carbon hollow spheres: (a) A-series and (b) B-series, shell edge (c) of a broken sphere.
This work was supported in part by FP7-PEOPLE-2013-IRSES-610875 NAmiceMC, FP7 Twinning Grant Inconet EaP_004.FIG. 6. Calculated absorption spectra of monolayers based on carbon hollow spheres of variable radii r o and constant shell thickness dr ¼ 4 lm.FIG. 7. Calculated absorption spectra of monolayers based on carbon hollow spheres at 30 GHz for various radii r o and shell thicknesses dr.