RESEARCH OUTPUTS / RÉSULTATS DE RECHERCHE How effectively do carbon nanotube inclusions contribute to theelectromagnetic performance of a composite material? Estimation criteria from microwave and terahertz measurements

Screening eﬀect in ﬁnite-length carbon nanotubes (CNT) and their agglomerates hinders signiﬁcantly the electromagnetic interaction in composite materials. Screening eﬀect is strong in the microwave range, and it decreases with increasing frequency resulting in a strong frequency dependence of the eﬀective conductivity of the composite. Since screening eﬀect is rather small in the terahertz range, the eﬀective conductivity in this range is determined di-rectly by the intrinsic conductivity of the inclusions. The ratio of the microwave to terahertz eﬀective conductivities was proposed as a parameter to estimate how eﬀectively carbon nanotube inclusions contribute to the electromagnetic performance of composite materials in the microwave range. CNT ﬁlm was considered as a material where maximal possible interaction of the CNTs with EM ﬁeld occurs. Single-walled CNT ﬁlms and CNT-based composite materials, as well as hybrid ﬁlm comprising mixtures of WS 2 nanotubes and CNTs were fabricated and measured in the microwave and terahertz ranges. The electromagnetic ﬁeld interaction with the inclusions has been estimated for all the samples fabricated. monotonic multiwalled carbon nanocomposite on temperature ranging 373 to 873

quency dependence of the effective conductivity of the composite. Since screening effect is rather small in the terahertz range, the effective conductivity in this range is determined directly by the intrinsic conductivity of the inclusions. The ratio of the microwave to terahertz effective conductivities was proposed as a parameter to estimate how effectively carbon nanotube inclusions contribute to the electromagnetic performance of composite materials in the microwave range. CNT film was considered as a material where maximal possible interaction of the CNTs with EM field occurs. Single-walled CNT films and CNT-based composite materials, as well as hybrid film comprising mixtures of WS 2 nanotubes and CNTs were fabricated and measured in the microwave and terahertz ranges. The electromagnetic field interaction with the inclusions has been estimated for all the samples fabricated.

Introduction
High conductivity and very high aspect ratio of carbon nanotubes (CNT) cause their effective interaction with electromagnetic (EM) radiations [1,2,3,4,5,6,7,8]. Many interesting effects in CNTs have been recently predicted and demonstrated, including: propagation of slowed-down surface waves [8], localized plasmon (antenna) resonance in the terahertz range [9,4], screening effect in microwave and radio-frequency ranges [5,10], strong near-field enhancement [5,11], and Purcell effect [12]. CNTs have been proposed as elements for different integrated circuits and electromagnetic devices, such as transmission lines [8,7], nanoantennas [13,14,15], interconnects [16,17], terahertz detectors and emitters [18]. Moreover, CNTs have real perspective applications as inclusions in composite materials for different electromagnetic applications [11,19,20,21]. A large variety of CNT-based composites have been produced and investigated during the past two decades [22,19]. They aimed at obtaining the maximal electrical conductivity of the composite material with minimal concentration of inclusions. This can be achieved by a suitable choice of the CNT geometry [10] and homogeneous distribution of the CNTs in the host matrix [23,24].
Many efforts have been made to achieve a homogeneous dispersion of CNTs in the host matrix [22,25,26,27,28]. To that end, different strategies have been applied, including melt mixing [23], ultrasonic treatment, CNT cutting [29], functionalization of CNTs [28,22,25,27], or additional inclusions preventing CNT aggregation [30]. Though all these methods help to improve CNT dispersion, additional side effects may partly prevent the achievement of the most effective interaction of the CNT inclusions with an electromagnetic field. For instance, functionalization of the CNTs may increase intertube contact resistance [26,28]; ultrasonic and chemical treatment may lead to the CNT cutting [31]. Different techniques are used to estimate the morphology of the prepared composites, among which terahertz spectroscopy [32,33], rheological measurements [34], microscopic observations [23,24], X-rays diffraction tests [35], and dc-electric conductivity measurements [23,24].
Electromagnetic shielding performance of a composite slab is usually evaluated by its shielding effectiveness (SE) that depends on the effective permittivity of both the sample and surrounding media and on the thickness of the sample as well. SE parameters are determined by interference effects in the slab and impedance mismatch between the surrounding media 2 and the sample.
Currently, there is no criteria which can assess how effectively CNT inclusions contribute to the electromagnetic performances of a composite material, and how far the electrical conductivity of the fabricated composite is from maximal possible conductivity at the given CNT volume fraction. The major cause hindering electromagnetic interaction is a screening effect in the individual inclusions or their aggregates. Screening effect is maximal in a wide range, from radiofrequency to microwave, and is minimal in the terahertz range [5,10].
By comparing the conductivity of a CNT film and a CNT-based composite material in the terahertz and microwave (or static) ranges, we formulate a criterion for the evaluation of the quality of the CNT-based material with respect to its electromagnetic performance. We also propose to compare the conductivity of CNT composites with that of CNT films, where the EM response of CNTs is as large as possible.

Theoretical background
In this section we present theoretically a criterion that characterizes how effectively CNTinclusions interact with the EM field. Though the consideration is carried out for the case of single-walled CNTs, it remains true for composite comprising multi-walled CNTs and bundles of single-walled CNTs.
The conductivity spectra of CNT composite materials have a broad terahertz peak [1,2,3,4] due to a finite-length effect [15]. The peak frequency f p increases with decreasing CNT length [9]. For further consideration we assume that the CNTs are long enough, so that f p < ν/2π, where ν = τ −1 ; τ is the electron relaxation time. This condition is satisfied for CNTs with length L ≥ 1 µm and τ < 50 fs; it provides non-resonant interaction of CNT with the electromagnetic wave below 3 THz. Calculations are performed here for undoped metallic CNTs. Since doped semiconducting tubes have conductivity comparable to that of metallic CNTs [36], equivalent results would be obtained with doped semiconducting CNTs.

Individual carbon nanotube
In the terahertz and subterahertz ranges, the surface conductivity of the metallic singlewalled CNT is determined mainly by intraband electron transitions and described by the Drude formula [8]: where υ F is the Fermi velocity, υ F 10 6 m/c; R is the CNT radius; ω = 2πf is the angular frequency, f is the frequency, e is the charge of electron; is the reduced Planck constant.
As an example, the conductivity spectrum for zigzag (12,0) CNT is shown in Fig. 1(a).
One can divide the spectrum in two regions: (i) the low frequency region, f < ν/4π, where the conductivity is mainly real and equal to the static conductivity; (ii) the higher frequency region, f > ν/4π, where the conductivity is complex and frequency dependent.  Let an isolated single-walled CNT of length L and radius R, and aligned parallel to the z axis of a Cartesian coordinate system, occupies the region z ∈ (0, L). The CNT is exposed to an incident field with z-component E z = E 0 z exp(−iωt). The scattered EM field is induced by the axial surface current excited in the CNT. The surface current density j(z) depends both on the tube conductivity σ cn and the axial component of the total electric field E tot z on the CNT surface: where E tot z = E 0 z + E sc z , E sc z is the axial component of the field scattered by the CNT. Let us introduce the effective surface conductivity of the CNT Following [10], one can identify three different regimes of CNT interaction with the In the quasi-static regime, due to the polarization of finite-length CNT, the scattering In the dynamic regime, the screening effect is absent, i.e., E tot z ≈ E 0 z , and consequently σ (eff) cn ≈ σ cn . The same is true for the intermediate regime at high electron relaxation rate, i.e. at the condition of f p < ν/2π.

Carbon nanotube based composite
Let us now consider a composite material where the CNT inclusions are randomly dispersed and randomly oriented in the dielectric matrix with the relative effective permittivity ε h . For simplicity, we assume that tubes are isolated from each other and there is no electromagnetic interaction between them. We use adapted the Waterman-Truell formula to estimate the effective relative permittivity of this composite material [4] ε where ε 0 = 8.85 × 10 −12 F m −1 ; the function N m (L) describes the number density of the CNTs of type m with radius R m and length L. The factor 1/3 in equation (4) is due to the random orientations of the CNTs; and is the axial polarizability calculated using the integral equation approach [37]. We shall describe the contribution from the inclusions to the total conductivity of the composite material by the effective conductivity Let us note that the frequency dependence of σ eff in the range (0,f p ) is explained in some papers by Drude-Smith model [38,39]. However, application of this model is not justified, as it cannot explain a variation of the frequency dependence σ eff (f ) with the CNT concentration in the network (see Sec. 3). Fig. 1(b,c) shows the spectra of Re(σ eff ), Re(ε eff ), and Im(ε eff ) calculated for composites comprising identical metallic single-walled CNTs at different, but consistent within each sample, CNT lengths L ∈ {1, 5, 50}µm and the same nanotube volume fraction F = 2πR 2 L n, where n is a number density of the tubes of length L and radius R. The spectrum of Re(σ eff ) for the longest tubes (L = 50 µm) follows the Drude law, whereas the spectra for shorter tubes with L = 5 µm and L = 1 µm have a terahertz peak at f p = 0.9 and 3 THz, respectively.
Let us note that for all composites in Fig. 1(b), the value of Re(σ eff ) is approximately the same at the peak frequency f p and approximately equal to that found for "infinitely long" tubes (L = 50 µm). This is due to the absence of the screening effect at frequency f p in the Indeed, after substitution of (2) into (5), (5) into (4), and (4) into (6), taking into account that E tot z ≈ E 0 z , for composite with identical CNTs at frequency f p , we obtain 6 One can conclude from (7) and Fig. 1 These conclusions give us a criterion to estimate the quality of the prepared composite material. Let us introduce the following ratio as a parameter showing the contribution of the inclusions to the electromagnetic performance of the prepared composite. f 0 in (8) is a frequency at which the screening effect within inclusions or aggregates of inclusions can be neglected. For the composites considered in Fig. 1(b,c), the screening effect is small in the range f 0 ∈ (f s , f p ), where f s is a frequency at which the equality Im(ε eff − ε h )/Re(ε eff − ε h ) = 1 is true; f s is indicated by vertical arrows in Fig. 1(c).
For hypothetical composite material comprising "infinitely-long" homogeneously distributed CNTs, from (8), one gets g = 1, while for realistic composite, it is expected that g < 1. The higher the value g is, the more effectively the inclusions contribute to the electromagnetic response of the composite.
Realistic CNT-based composites are complex systems where electron transport is caused by intrinsic conductivity of the CNTs and intertube electron tunneling [40,41]. Moreover, the electromagnetic interaction between nanoparticles takes place [42]. All these microscopic effects determine the charge and the total field distribution along the CNTs. According to eq.
(2), the total field is responsible for the current induced in the CNT network. If the total field is less than the incident field, one can talk about screening effect. Thus, microscopic effects influence the effective conductivity via the total field and consequently via the screening effect.
Let us notice that the intrinsic conductivity of the CNTs can change when they form 7 bundles. This occurs through additional electron scattering by phonons in the adjacent tubes of the bundle. Similar phenomena have been reported for multilayer graphene structures [43].
However, this effect does not modify the frequency dependence of the tube conductivity in the range f ν. As shown in recent experiments [44,45] (see also Fig. 3(a)), the effective conductivity of films containing long-length bundled single-walled CNTs has weak frequency dependence below 1 THz. Therefore, we attribute the strong frequency dependence of the observed value of Re(σ eff ) to screening effect in the inclusions and their agglomerates.
It is expected that the parameter g depends on the volume fraction F of the CNTs. At low F , microwave effective conductivity of the composite material is determined mainly by the electromagnetic response of individual tubes, i.e. tube polarizability [46,40]. Then, the screening effect is strong and the parameter g is small. When F increases, nanotubes start to interconnect and form a highly conductive network that significantly increases the effective conductivity of the composite [46,40], which decreases the screening effect due to the intertube electron tunneling. As a result, g increases with F .
For diluted CNT composites (F 1), we propose to use a normalized parameter where g film is the parameter g for a pure film containing the very same kind of CNTs as the composite material does. A CNT film has the largest possible density of CNT network where screening effect in each tube is minimized due to charge transfers through the tunneling contacts between adjacent CNTs. Then for any composite material the following equality The parameter g 0 characterizes the dispersion of the CNTs in the composite. If the tubes are aggregated, forming dense particle, the depolarizing field within each aggregate leads to the screening effect resulting in the low value of the parameter g 0 . The composite material with g 0 = 1 has the highest possible electromagnetic performance at the given volume fraction of the inclusions. The smaller g 0 with respect to unity is, the less effectively the inclusions are involved in the electromagnetic interaction.
To calculate the parameters g and g 0 , one needs to know the frequency f 0 . Our estimations made for the composite comprising single-and multi-walled CNTs [10,47] show that for long tubes (L > 1 µm), the screening effect is rather weak at 1 THz. This is supported by the experimental data showing linear concentration dependence of the absorption at 1 THz for CNT-based composite [32]. Therefore, one can apply f 0 = 1 THz in (8) and (9).
Parenthetically, for CNT-based composites at high CNT loading, the real part of the effective conductivity remains practically constant in the range (0,30) GHz [48,38]. Therefore, the values of g and g 0 practically do not depend on the frequency in this range.
In the next sections, we present terahertz and microwave conductivity data for fabricated CNT films and CNT-based composites and evaluate the parameters g and g 0 at 30 GHz for them.

Materials, sample preparation, and measurements
In our experiment we used purified single-walled CNTs (OCSiAl Inc.) produced by CVD process with diameters of 1.8-2.2 nm, and a purity exceeding 95%.
A free standing CNT film was prepared via the vacuum filtration technique [49]. Briefly, a CNT material was dispersed by ultrasonic treatment (Ultrasonic device UZDN-2T, 44 kHz, maximum power) for 1 hour in an aqueous suspension with 1% sodium-dodecylsulfate (SDS). The suspension was centrifuged at 8000g for 20 minutes and then filtrated through a cellulose nitrate membrane filter (0.2 µm pore size) causing a CNT film to collect on the filter. This film was washed with distillate water to remove the surfactant. The filter paper was dissolved by acetone and the CNT film was transferred onto a frame with a hole of 8 mm in diameter. Film thickness was 500 nm. Typical scanning electron microscopy (SEM) image of the film is represented in Fig. 2(a).
The filtration method was also used to fabricate 700 nm thin film comprising short CNTs (L < 500 nm). To obtain short tubes, purified single-walled CNTs were cut by ultrasonic treatment in a mixture of nitric and sulfuric acids at low temperature [50].
Polystyrene composite with 1 wt.% of single-walled CNTs was prepared by forge rolling method [51]. CNT material was dispersed in toluene by mechanical stirring for an hour.
Polystyrene was mixed with CNT-toluene suspension by stirring until complete dissolution of polymer. The suspension was under ultrasonication treatment for 5 min and then stirred The obtained CNT-based polymer film was dried under a light load at room temperature for 12 h. The film thickness was 0.2 mm. Optical image of the film is represented in Fig.   2(b). The black spots with the size of less than 20 µm indicate CNT agglomerates in Fig.   2(b).
To study the influence of the tube density on the EM field interaction with the inclusions, show that thin CNTs are dispersed evenly between thick INTs (see Fig. 2(c,d)). The films were transferred to 10 µm thick PTFE substrate for terahertz and microwave measurements. The spectrum of the sample conductivity was calculated by means of the Fourier transform of the measured time-domain signal and application of the Fresnel equations for dielectric slab [53]. The microwave conductivity measurements at 30 GHz were performed by waveguide method [54,55] using a scalar network analyzer R2-408R (ELMIKA, Vilnius, Lithuania). This means, roughly speaking, that the effective medium at 30 GHz has only 64% of quasifree electrons with respect to the total number of quasi-free electrons in the CNT film.

Results and discussion
Parameter g for the CNT film depends on the type and geometry of the inclusions. It is shown in [47] that frequency dependence of the CNT film conductivity gets stronger as CNT length decreases. This tendency is demonstrated in Fig. 3(b) for thin film comprising short CNTs. From Fig. 3(b) we get g(30GHz, 1THz) = 0.33. So, the parameter g decreases as the tube length decreases. In order to show how the parameters g and g 0 vary with the CNT density, we measured the conductivity spectra of the CNT/INT films at different CNT volume fractions (see Fig.   4(a)). We roughly estimated the volume fraction as the ratio of CNT density ρ to the density of graphite ρ 0 = 2.2 g/cm 3 . Fig. 4(b) demonstrates the dependence of the parameters g(30GHz, 1THz) and g 0 (30GHz, 1THz) on the CNT volume fraction. As shown in Fig. 4(b) both values g and g 0 decrease as CNT volume fraction decreases. The less the ratio ρ/ρ 0 , the stronger CNT separation in the composite is, and the stronger the screening effect within each CNT is. It should be noticed that CNT/INT films do not contain agglomerated CNTs.
In order to show the influence of agglomeration on the parameters g and g 0 , we measured 13 the conductivity spectrum of the polystyrene composite with wt 1 wt% of the CNT inclusions (see Fig. 3(c)). As shown in Fig. 3(c), the frequency dependence of the conductivity is very strong. From the data presented in Fig. 3(c), we obtain g(30GHz, 1THz) = 0.045 and g 0 (30GHz, 1THz) = 0.071. This means that the contribution of the CNTs to the EM response of the composite material obtained is about 7% of that we got for the CNT film.
This value of g 0 is about 10 times smaller than that we obtained for the CNT/INT film at approximately the same CNT volume fraction. The reason is that the CNTs are mainly aggregated in the polymer composite, and, consequently, the screening effect within each agglomerate hinders significantly the interaction between the EM field and CNTs. Note, the strong difference in the frequency dependence of the optical density between CNT film and CNT/polyethyelene composite was observed in the subterahertz range (see Fig. 2 in [56]).
The screening effect is rather strong in finite-length multi-walled CNTs [10]. From the conductivity spectrum of the polyethylene based composite with 10 wt% of multi-walled CNTs ( Fig. 9 in [57]), we extracted g(30GHz, 300GHz) = 0.04, that demonstrates low efficiency of the interaction between the CNTs and EM field.
Though we applied g and g 0 for evaluation of CNT-based composite, these parameters can be applied to any type of composite material, where the screening effect is the main factor hindering the electromagnetic interaction.

Conclusion
We show theoretically that the effective conductivity of single-walled CNT composite at frequencies close to 1 Thz is determined mainly by the intrinsic conductivity of the CNTs with length > 1µm. The microwave effective conductivity strongly depends on the screening effect in both the individual CNTs and CNT aggregates. We propose to use a parameter g, defined as the ratio of the microwave to terahertz conductivities, to estimate how effectively carbon nanotube inclusions contribute to the electromagnetic performances of a composite material in the microwave range. Also, we propose to consider a CNT film as a reference material where the strongest possible interaction of the CNTs with the EM field occurs.
The ratio of the parameters g of the composite material to that of the CNT film gives us another performance parameter g 0 , which is helpful to know how far the EM response of the 14 composite is from the maximal possible one.
Thin films comprising long and short single-walled CNTs and diluted (1 wt.%) polymer CNT composites were fabricated. The effective conductivities of the samples were measured in the microwave (30 GHz) and terahertz (0.2-1.5 THz) ranges. The parameters g and g 0 for the prepared composite materials were found to be of several percents. The same values of g was extracted from the data taken in the literature. This indicates very low efficiency of the field interaction with CNT inclusions due to aggregation effects in the polymer composites.
The hybrid films comprising mixture of single-walled CNTs and WS 2 nanotubes with different volume fractions were fabricated and investigated in the microwave and terahertz ranges. It has been shown that both parameters g and g 0 decrease as the CNT volume fraction decreases. Opposite to polymer composite, the parameters g and g 0 remain large (g > 25%, g 0 > 75%) even at low CNT volume fraction (up to 1%) indicating strong EM interactions with CNTs within the hybrid films. This is due to the absence of the CNT agglomerates in these films.
The parameters g and g 0 can be used to estimate the quality of any other composite materials comprising conductive inclusions non-resonantly interacting with electromagnetic wave below 1 THz.