Introduction

Modern applied science and technology intensively use the possibilities of expanding the spectrum of material properties due to the peculiarities of the behavior of matter on a nanometre scale, which are manifested at the macro level. When one or more dimensions of the system are of the order of a few nanometres, the main role in the physical characteristics formation appertains to the increased contribution of the surface (compared to the volume), the quantum-dimensional effects, and under certain conditions, to the multi-particle effects can be dominated in low-dimensional systems. Another way to significantly changing the physical properties of matter is the creation of non-equilibrium states in systems, as well as the synthesis of heterogeneous systems, for example, composite materials that acquire properties differ from both the properties of the individual components of the composite and their averaged characteristics. The polymer–carbon nanotubes (CNTs) composites retain the advantages of the polymer matrix and additionally acquire the features of their stronger and conductive carbon filler. In such composites, several of the above-mentioned ways of radically changing the properties of materials can be used simultaneously. Thus, after adding conductive CNTs to the polytetrafluoroethylene (PTFE) dielectric matrix in an amount exceeding a certain critical value—the percolation threshold of 2.2–4.5 wt. % CNTs (O. M. Garkusha (5), Kotenok O. V. (13)), a conductive network of CNTs is formed and a dielectric–conductor transition occurs. The concrete values of the conductivity of such systems will be determined by the electronic structure and conductivity of individual CNTs, their linear dimensions and aspect ratio, the degree of ordering of the polymer molecules and filler particles, the homogeneity of the composite (including the degree of CNTs’ aggregation), the type and number of contacts of the filler particles with each other and with matrix material, the presence of defects in both components (in particular, free volumes in the polymer and Stone–Wales defects in CNTs), redistribution of charges between the components of the composite, etc.

The presence of a filler significantly changes not only the electrical properties of polymer-based composites, but also their mechanical (Vail J. R. (24), Huang An (8)), thermal (Kwon Y.-J. (14)), optical (Huang An (8)) and other properties. The large surface area of the filler (CNTs or graphene) leads to an increase in the filler–polymer interaction, which improves the transfer of load between the filler and the matrix, while contacts between filler particles provide electrical and thermal conductivity of typically non-conductive polymer materials and increase the absorption of ultra and super high frequencies electromagnetic waves with their intensity attenuation by several orders of magnitude (Lihtorovich S. P. (15), Xue G. (25), Awad et al (2019a), Huang An (8), Hao B. (6)). These properties, as well as low density, high stability in aggressive environments provide to the polymer–nanocarbon composites a promising future as protective and absorbing coatings in a wide range of wavelengths, covering at least frequency bands known as FR1 (below 7.125 GHz) of 5G wireless technology as well as frequency bands of other branches of radio electronics. Therefore, the answers to the questions regarding the possible mechanisms of excitation of the electronic subsystem in the CNTs’ conducting network under the action of external electromagnetic waves (Qi X. (19)) and dissipation of the corresponding energy, as well as their connection with the structural features of such nanocomposites, for example, with their defect structure, are a great importance from both fundamental and applied points of view.

The most versatile experimental methods of studying both the defect structure and the electronic characteristics of crystalline and non-crystalline materials and defects in them are the methods based on the study of the characteristics of γ-radiation, which arises because of the annihilation of electrons of matter with positrons appeared after nuclear transformations in corresponding source. During the process of thermalization, positrons move through the sample and attract to places with reduced positive charge density, for example, vacancy-type defects, nanopores, and interlayer spaces in multi-walled carbon nanotubes (MWCNTs). Positrons can annihilate both with bounded electrons of atoms (mainly from outer shells) and with free electrons (if any). When the vacancy-type defect has a sufficiently large size, the ortho-positronium can appear in it. Free volumes (as a dynamic type of nanopores) also trap positrons. At the same time, the positron polarizes the atoms on the inner surface of the nanopore. The probability density of finding electrons in the cavity is determined mainly by the tails of the wave functions of atoms located on the surface of the corresponding defect. Therefore, the probability of positron annihilation becomes dependent both on the electronic characteristics of the substance in which the cavity is located and on the size and number of such defects (Tao S. J. J. (20), Jean Y. C. (9)). One of the main parameters of low-density systems such as polymers and ceramics are the average sizes of free volumes and nanopores. They are easiest to determine by various types of positron spectroscopy (PS) (Awad et al (2019a), Xue G. (25), Klym et al (2016a), (b), (2021)), which can also determine some electron characteristics of these defects, namely, the probability of positron annihilation reflected corresponding local electron density. In the previous works (Tsapko E. A. (21), Nishchenko M. M. (17), Lihtorovich S. P. (15)), the authors have shown that within the framework of angular correlation of annihilation radiation (ACAR) method, it is possible to obtain parameters of the electrons distribution in MWCNTs and in their defects, as well as corresponding spatial characteristics, namely, the MWCNTs’ layer thickness, the interlayer distance, and effective radius of defects.

In the present work, the angular correlation of annihilation radiation, attenuation of electromagnetic radiation, and optical ellipsometry methods are used to investigate the influence of different amount of multi-walled carbon nanotubes in polytetrafluoroethylene on the parameters of electronic and defect structures and on absorption of ultra-high-frequency radiation in PTFE–MWCNTs composites.

Objects and methods

The investigated samples of PTFE–MWCNTs composites were obtained in Chuiko Institute of Surface Chemistry of N.A.S. of Ukraine by the method of catalytic pyrolysis of polypropylene. The structural characteristics of MWCNTs were determined by the method of transmission microscopy (TEM, JEM-100CXII). The average diameter of the MWCNTs used in the composites was 10–20 nm, the specific surface, which was determined by the desorption of argon, was 200–400 m2/g, and the bulk density varied in the range of 20–40 g/dm3.

We investigated PTFE–MWCNTs composites obtained by coprecipitation of a stabilized suspension of polytetrafluoroethylene and an aqueous suspension of MWCNTs (Garkusha O. M. (5), Kotenok O. V. (13)). Powdered composites with concentrations of 0, 5, 10, 15, and 20 wt.% MWCNTs was pressed at a temperature of 380 °C and a pressure of 5 MPa in disk-shape form with a diameter of 20 mm and a thickness of 2 mm. All measurements were carried out in the air at room temperature.

The measurements of the attenuation of electromagnetic radiation by the samples were carried out in (Lihtorovich S. P. (15)) using the automatic attenuation meter P2-52/3 in the frequency range 1.5–2.0 GHz according to the standard scheme for measuring microwave attenuation in four-pole networks using a copper standard.

The optical properties of the samples (the spectral dependences of the refractive indices <n> and absorption <k>) were obtained using angular ellipsometry according to method described in Ref. (Poperenko L. V. (18))).

The ACAR spectra for the samples were obtained using the one-dimensional long-slit spectrometer with an angular resolution of 1.07 mrad. To interpret the experimental data for investigated composites we used the classical analytical representation of the ACAR spectra as the sum of (i) parabolic part appears due to positrons annihilation with free electrons, (ii) wide Gaussian part (with dispersion from 6.0 to 9.5 mrad) caused by annihilation with electrons in Stone–Welsh defects in MWCNTs and defects in PTFE polymer chains, (iii) medium Gaussian part (with dispersion from 2.5 to 5.0 mrad) defined by annihilation with bounded electrons of carbon and fluorine atoms (mainly carbon atoms in the interlayer space of MWCNTs), and (iv) narrow Gaussian part (with dispersion from 0.2 to 1.2 mrad) determined by ‘pick off’ annihilation from the state of ortho-positronium (o-Ps) in free volumes (Tsapko E. A. (2020), Yang M. K. (26)):

$${I}^{theor}\left(\theta \right)={I}_{P}\left[\frac{{\theta }_{F}^{2}-{\theta }^{2}}{2}+{A}_{B}Tln\left(1+exp\left\{-\frac{{\theta }_{F}^{2}-{\theta }^{2}}{2{A}_{B}T}\right\}\right)\right]+\sum_{j=\mathrm{b},\mathrm{i},\mathrm{n}}\frac{{I}_{G}^{j}}{{\sigma }_{j}\sqrt{2\pi }}exp\left\{-\frac{{\theta }^{2}}{2{\sigma }_{j}^{2}}\right\},$$
(1)

where θ is the deviation from π of the angle of recession of annihilation γ-quanta, Itheor(θ) is the analytical representation of the ACAR intensity, \({I}_{P},{I}_{G}^{j}\) are the amplitude coefficients (intensities) of the parabolic and Gaussian parts of the ACAR spectra, respectively, \({\sigma }_{j}\) are the dispersions of Gaussians, \({\theta }_{F}\) is the angle corresponding to the Fermi momentum (\({P}_{F}\)), \({A}_{B}\) is the renormalized Boltzmann constant, T is the sample temperature.

Under the conditions of the long-slit geometry of the experiment, the electron dispersion law is partially averaged over the electrons’ energy and wave vector components lying in the slit plane. The averaging masks the fine structure of the energy and momentum distributions of electrons. Traditionally, the interpretation of band electrons contribution to ACAR spectrum is based on replacing the real Fermi surface by a sphere with equal volume. The effective radius kF of this sphere depends on the average band electron density n, and can also be expressed in terms of θF and certain electron effective mass m*, according to the expressions:

$$n = \frac{{k_{F}^{3} }}{{3\pi^{2} }} = \frac{{\left( {m^{*} c\theta_{F} /\hbar } \right)^{3} }}{{3\pi^{2} }},$$
(2)

where ћ is Planck’s constant, c is the speed of light. The appearance of the effective mass m* is due to the difference in the momentum distribution of real electrons inside real Fermi surface from the distribution of free electrons inside the Fermi sphere used to interpret PS experimental data. If we take the value kF, determined from the experimental value θF and corresponded to the real dispersion law, as the radius of the Fermi sphere, then the volume of that sphere will give according to Eq. (2) the overestimated values of the average free electron density n. This increasing in the volume of the reciprocal space with the states occupied by electrons is compensated by introducing the effective mass m*. According to our estimation and known values of n and θF for MWCNTs, the values of m* lie within 0.56m0 ≤ m* ≤ 0.79m0, where m0 is the rest mass of electron.

Thereby, from the ACAR spectrum, we can obtain some characteristics of free electrons subsystem. One of them is the Fermi momentum \(P_{F} = \hbar k_{F}\) related to the angle \(\theta_{F}\) according to the equation:

$$P_{F} = m^{*} c\theta_{F} .$$
(3)

The other one is the probability of positron annihilation with free electrons \({P}_{{\theta }_{F}}\), which is proportional to the corresponding electrons concentration and obtained as the ratio of the area under the parabolic part to the total area under the ACAR spectrum (after integration both the parabolic part and the full ACAR spectrum over the angle θ).

Analogously, we can define some parameters of bounded electrons subsystems. The dispersions \({\sigma }_{j}\) of wide (for electrons localized in small defects—j = b) and medium (for bounded electrons of atoms in solids–j = i) Gaussian parts determine the widths of the momentum distributions of electrons. The dispersions are inversely proportional to the widths of electrons spatial distributions and are related to the distances \({r}_{{m}_{j}}\) at which the overlap of the wave functions of the annihilating positron and electrons reaches a maximums, by the relations (j = i, b) (Tsapko E. A. (21)):

$$r_{{m_{j} }} = \sqrt{\frac{3}{2}} \frac{\hbar }{{m_{0}c\sigma_{j} }}.$$
(4)

The probabilities of annihilation of positrons with bounded electrons \({P}_{{r}_{{m}_{j}}}\) are defined as the ratios of the areas under the corresponding components (j = i, b) to the area under the full ACAR spectrum.

Through the dispersion of a narrow Gaussian σn (j = n) in the framework of the Tao model (Tao S. J. J (1972)), we can calculate the average radius R of the free volumes:

$$R = \frac{0.74049}{{\sigma_{{\text{n}}} }} - 0.1656,nm.$$
(5)

In the simplest approach, the free volume is represented as a sphere of radius R with a layer on inner surface of sphere characterized by uniform electron density and thick of ΔR. The electron density in this layer is formed by the tails of the wave functions of bounded electrons leaving the atoms into vacuum at a characteristic effective length ΔR ≈ 0.1656 nm, which is almost independent of the type of sample atoms. The probability \({P}_{R}\) of positron annihilation with electrons in free volumes depends on average electron density in these defects (and their average radius). It is also defined as the ratio of the area under the corresponding component to the area under the full ACAR spectrum of investigated PTFE–MWCNTs composites.

The existence of free-volume-type defects with sizes from 0.1 to several nanometres is necessary for the molecular dynamics of polymer chains in composites. These defects are dynamic, i.e., as a result of internal stresses, a free volume appears in the polymer chains, the chain fragments rotate or shift relative to each other, and the free volume collapses. The lifetime of free volumes is much longer (by several orders of magnitude) than the lifetime of positron in solids, so the PS method is the unique method for studying the electronic structure of the free volumes in polymer matrix (PM).

Note, that the ACAR spectrum for pure MWCNTs consists of 3 components only: (i) the wide Gaussian appeared due to annihilation of positrons in Stone–Welsh defects, (ii) the medium one appeared because of annihilation with bounded electrons of carbon atoms in interlayer space, and (iii) the parabolic contribution determined by the annihilation with free electrons (Tsapko E. A. (21)).

Results and discussion

The addition of different amounts of MWCNTs to the PTFE polymer matrix changes the properties of the composite material in different ways. Note we investigate composites with MWCNTs concentration higher than percolation threshold. The complex nature of these changes affects both the atomic and the electronic structures of the composite. Positron spectroscopy makes it possible to investigate both these aspects of changes simultaneously. At the same time, as known (Awad et al (2019b) from the infrared (IR) absorption spectra, the formation of PM–MWCNTs composite does not lead to the new absorption peaks appearing, i.e., new covalent bonds do not appear and the interaction between PM and MWCNTs remains weak (Xue G. (25)) because of hydrogen and/or Van der Waals bonds. Despite this, the composite can show unique properties that are not inherent in either the matrix or filler. As one can see below, in composites based on polytetrafluoroethylene the addition of 10 wt.% MWCNTs leads to a decrease in the probability of positron annihilation in free volumes from 4 to 2%, which is accompanied by changes of other parameters of PS about 8–29% (Fig. 1). Under such conditions the completely transparent for electromagnetic radiation (ER) pure PTFE becomes a composite with huge ER absorption at a frequency of 2 GHz (Lihtorovich S. P. (15)) (Fig. 2).

Fig. 1
figure 1

Concentration dependences of PS parameters for PTFE–MWCNTs composites (dashed lines correspond to disordered array of pure MWCNTs)

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Fig. 2
figure 2

Concentration dependence of the attenuation of electromagnetic radiation at a frequency of 2 GHz in composites PTFE–MWCNTs (Lihtorovich S. P. (2010))

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Figure 1 shows the concentration dependences of the PS parameters for the PTFE–MWCNTs composites. It can be seen from these dependences that such quantities as Fermi momentum, widths of electrons’ spatial distributions, and probabilities of annihilation of positrons with electrons of different types have significant correlations, i.e., all processes observed in defect and electron subsystems of composites are induced by filer adding and interconnected.

For most values of the MWCNTs’ concentration, the sizes R of the free volumes (Fig. 1) are almost independent of the content of MWCNTs and their values remain almost the same as in pure PTFE. However, in pure PTFE, the probability PR of positrons’ annihilation in free volumes has a biggest value in comparison with one for any studied composites. Since a small amount of filler (5 wt.% MWCNTs) should not significantly change the number of free volumes in PTFE, an essential decrease in the probability PR indicates the transfer of electrons from these defects to MWCNTs. This conclusion also correlates with changes in other ACAR parameters, which are due to the processes of annihilation mainly in MWCNTs (Nischenko M. M. (16))—about 90% of all annihilation acts. The last one can conclude due to the proximity of the values of the corresponding probabilities to their values in pure MWCNTs (see dashed lines in Fig. 1). Only a composite with a 10 wt.% MWCNTs is distinguished from others. It is characterized by 5 times free volumes sizes R increasing (in comparison with pure PTFE), which indicates a change in thermodynamic conditions for these defects formation in the matrix. However, such significant increase in size leads to an increase in the probability PR annihilation in these defects by only 27% (compared to the composite with 5 wt.% MWCNTs). This effect can be explained by the partial compensation of the influence of free volumes size increasing on PR because of reducing the density of electrons in these defects due to charge flow from them to the MWCNTs. Further increase in the MWCNTs content is accompanied by a decrease in the probability of annihilation in free volumes, and therefore electron density in these defects.

The cause of the difference between the properties of the composite and its components is the physical processes that occur on the interfaces between these components. The transfer of charges from dynamic defects in PTFE to MWCNTs is the primary cause of these processes in our case. It directly or indirectly determines all subsequent changes in the system, such as an increase in the size \({r}_{{m}_{\mathrm{b}}}\) of Stone–Welsh-type small defects (j = b) (i.e., defects mainly in MWCNTs) and significant growth of electron density in them. The latter is evidenced by a significant increase in the probability \({P}_{{m}_{\mathrm{b}}}\) of annihilation in small defects compared to pure PTFE and array of pure MWCNTs. Figure 1 also demonstrates that the growth of both the electron density in small defects (j = b) and the amount of band electrons (\(\sim {P}_{{\theta }_{F}}\)) are occurred owing to not only the redistribution of charge between free volumes and MWCNTs, but also the reducing the density of electrons, which form covalent bonds between carbon atoms in defect graphene-like layers of MWCNTs. This is manifested in a decrease of the probability \({P}_{{m}_{i}}\) of annihilation with bounded electrons (Fig. 1) in composites strongly absorbed ultra-high-frequency radiation (see Fig. 2). The size of the spatial localization areas of the bounded electrons in the MWCNTs we connect with the average value of the interlayer distance in them, which is equal to \({2r}_{{m}_{\mathrm{i}}}\) (Fig. 1). The differences between the values \({2r}_{{m}_{\mathrm{i}}}\) for pure MWCNTs and MWCNTs in composites indicate the deformed state of the MWCNTs in the PM; the primary source of which is also the processes of charges redistribution between composites’ components.

It should be noted that in the pure PTFE the main channel of positrons’ annihilation connects with bounded electrons (j = i; \({P}_{{m}_{i}}\) ≈ 62.3%), the next contributions to the ACAR spectrum of polymer associate with structural defects (j = b; \({P}_{{m}_{\mathrm{b}}}\) ≈ 33.6%) and free volumes (j = n; \({P}_{R}\) ≈ 4.1%). As we demonstrate above, after nanotubes addition to PTFE, most positrons begin to annihilate in MWCNTs. The other fundamental difference connects with an appearance of a new and significant channel of positrons’ annihilation, namely, with free electrons (\({P}_{{\theta }_{F}}\)≈ 5.3–22%), which can exist only in MWCNTs. A presence of free electrons in investigated composites provides a huge absorption of electromagnetic waves (Fig. 2) when MWCNTs’ concentration is sufficiently high.

Figure 2 shows the dependence of the attenuation of electromagnetic radiation at a frequency of 2 GHz in the PTFE–MWCNTs composites on the filler concentration. One can see that the highest absorption of electromagnetic radiation is observed in a sample with 10 wt.% MWCNTs. The composites with 15 and 20 wt.% MWCNTs also demonstrate relatively high values of absorption coefficient of electromagnetic waves, but they have lower both the PS parameters and the electron properties than one with 10 wt.% MWCNTs. As known (Garkusha O. M. (4)), at high concentrations (15–20 wt.%) of MWCNTs in polymers, a strong aggregation of nanotubes is observed; the polymer does not “wetting” the nanotubes. In this case, the volume of the interphase decreases; the interfaces, through which the charge transfer occurs, are formed by bundles of nanotubes pressed into the polymer matrix. Accordingly, in the concentration range of 15–20 wt.% MWCNTs, the values of \({P}_{{\theta }_{F}}\) in corresponding composites do not significantly differ from one for pure MWCNTs (Fig. 1). Despite this, the conductive network formed by the nanotubes is preserved. All studied samples, except for pure PTFE, had finite electrical resistance due to formation in these samples branched conductive MWCNTs’ network.

The characteristics \({\theta }_{F}\) and \({P}_{{\theta }_{F}}\) of the ACAR spectra’s parabolic component (Fig. 1), which describes the free electrons’ contribution to the positrons annihilation, can be consider in their connection with the ability of polymer composites to absorb electromagnetic radiation. Recall that the concentration dependence of the probability of annihilation in free volumes clear demonstrates the charge transferring from these volumes to the MWCNTs for all concentrations of the filler. But only for concentrations of 10–20 wt.% MWCNTs, this effect leads to an increase (in comparison with pure MWCNTs) in the probability \({P}_{{\theta }_{F}}\) of annihilation with free electrons. At the same time, the values of \({\theta }_{F}\) can be both larger (at 10 wt.%) and smaller (at 15–20 wt.% MWCNTs) than in pure MWCNTs. In addition, the absorption of ultra-high-frequency radiation remains quite large (≅ 200–410 times) (Fig. 2). In composite with 5 wt.% MWCNTs, the concentration of free electrons is small (small \({P}_{{\theta }_{F}}\)) and, in terms of transparency for ultra-high-frequency radiation, the composite is closer to the pure PTFE (only 5 times attenuation). Note that the experimental value \({\theta }_{F}\) (Fig. 1) actually determines only the maximal value (kF) of electrons’ wave vector along the kz axis in reciprocal space, i.e., the effective mass of electrons, introduced by us in Eq. (2), changes the area under the parabolic part IPtheor(θ) of ACAR spectrum (this is the first term in Eq. (1) reflected the distribution of free electrons on momentum component Pz ~ θ) due to the quantity IP in Eq. (1) only (Tsapko E. A. (21)). The distribution IPtheor(θ) itself is obtained by integrating over the energy and momentum components Px, Py of the full function of the free electrons distribution. Then, in composite with 5 wt.% MWCNTs, the large value of \({\theta }_{F}\) and the small value of \({P}_{{\theta }_{F}}\) (in comparison with pure MWCNTs) indicate that the distribution IPtheor(θ) of electrons through their momenta in such a composite should be extended along the Pz axis, but the area below it should be small, i.e., the average concentration of free electrons and, accordingly, their effective mass should be small. At large concentrations of the MWCNTs, the real distribution of electrons through their momenta is compressed by Pz (\({\theta }_{F}\) is decreased), but the area under it is increased (\({P}_{{\theta }_{F}}\) is grown). At the same time, the effective mass of band electrons and the real form of electrons distribution by Pz are approached to the rest mass of electron and parabolic distribution (as for free electrons), correspondingly. The last is possible because, according to other data of the PS, the interlayer spaces and sizes of defects in the MWCNTs change in composite (the sign of deformation is not important). Deformation and defects should cause a transformation of the Dirac cones in the dispersion law of graphene-like plane of conducting MWCNTs into ordinary paraboloids (which can be associated with almost free electrons yet) separated by energy gap located below or higher than the Fermi level. Simultaneous shrinking of the electron distribution tails (\({\theta }_{F}\) decrease) and decreasing of the average electron densities (~ \({P}_{{\theta }_{F}}\)) for MWCNTs concentrations of 15–20 wt.% compared to the composite with 10 wt.% MWCNTs can be explained due to not only the deformation of individual MWCNTs, but also their aggregation (Garkusha O. M. (4)), because the same number of free electrons, which have passed from free volumes, is now divided between all aggregated MWCNTs. During aggregation, the probability of annihilation approaches to the one in a disordered array of the pure MWCNTs, and the quality of the conducting network is deteriorated, that leads to a certain decrease in the ultra-high-frequency radiation absorption coefficient (Fig. 2).

Let us determine what other characteristics of the annihilation spectra, besides the already considered annihilation on free electrons, distinguish a weakly absorbing composite with 5 wt.% MWCNTs from strongly absorbing ones with higher concentrations of MWCNTs. From Fig. 1 it can be seen that such a characteristic is the probability \({P}_{{m}_{i}}\) of annihilation in the interlayer spaces of MWCNTs, which becomes lower for concentrations of 10–20 wt.% MWCNTs than in pure MWCNTs, while at 5 wt.% MWCNTs it becomes higher. Since, in this case, the corresponding interlayer distances \({2r}_{{m}_{\mathrm{i}}}\) do not show a similar regularity, these characteristics probably do not directly affect the absorption in the composites. However, they enhance the role of defects, in which, as already noted, the electron density and the corresponding annihilation probability \({P}_{{m}_{\mathrm{b}}}\) increase. In composites, probably, a part of the electron density from defects in MWCNTs passes into the free electrons subsystem of MWCNTs, as well as it also forms ‘tails’ of the electronic density of states in the band gap.

Note that under conditions when the Fermi level lies in the energy region where for an ideal graphene plane there are Dirac peculiarities, in deformed and defect MWCNTs with modified electron dispersion law, even small changes in the average electron density also lead to a significant change in the value of density of electronic states on the Fermi level, as well as in the corresponding kinetic characteristics (electro- and thermal conductivities) of the system. Moreover, the presence of disorder in both the structure of MWCNTs (due to defects) and the branched conducting network consisting of them can produce localized electronic states (Urbach F. (22), Hassanien A. S. (7)) near the band edge in the energy gap opening in the electron spectrum of complex system based on imperfect MWCNTs. Such states are strongly interacted with the phonon subsystem. The excitation of these states by electromagnetic wave quanta (with an energy of the order of the Urbach energy) in a MWCNTs’ conducting network probably provides the efficient absorption of ultra-high-frequency radiation in composites with 10–20 wt.% MWCNTs.

The maximum of this absorption is observed in the composite with 10 wt.% MWCNTs. A well-branched conducting network of non-aggregated MWCNTs with a high concentration of conduction electrons is formed in this material (see the maximum in the concentration dependence of the annihilation probability \({P}_{{\theta }_{F}}\) in Fig. 1). This composite also exhibits extrema of all other ACAR characteristics (the highest electron density in defects and the lowest one for bounded electrons of carbon atoms in MWCNTs, the largest radius R of the free volumes in PM, etc.), except Fermi angle \({\theta }_{F}\). The latter fact is related to the previously discussed change in the effective mass of free electrons. However, the main effect on micro-waves absorption (Fig. 2) is connected with increasing in average electron density (~ \({P}_{{\theta }_{F}}\)) approximately 1.5 times compared to other well-absorbing composites and 4.2 times compared to a weakly absorbing composite with 5% MWCNTs.

Figure 3 shows the spectral dependences of the refractive index < n > and the absorption coefficient < k > at three different incidence angles in elipsometric investigations for a composite with 10 wt.% MWCNTs. For this composite, a small peak is observed in the near-ultraviolet region on the spectral dependence of the refractive index < n > . The absorption index < k > increases monotonically over the entire investigated range of incident radiation frequencies. Such spectral dependences of the refractive indices < n > and absorption coefficient < k > are characteristic of some allotropic forms of graphite (Ermolaev G. A. (3)), where a small peak in the spectral dependence of the refractive index < n > in the near-ultraviolet region (200–400 nm) corresponds to localized sp2-hybridized states and a further monotonic increase in the refractive indices < n > and absorption indices < k > is due to transitions of collectivized π-electrons of carbon, i.e., it confirms the presence of free electrons in the composite materials and demonstrate they ability to be excited in a wide range of wavelengths. In pure PTFE, fundamentally different spectral dependences of the indices < n > and coefficients < k > are observed—low absorption and small changes (of the order of 10–1) over the entire investigated wavelength interval (Yang M. K. (2008)).

Fig. 3
figure 3

Spectral dependences of refractive index < n > and absorption coefficient < k > at three different angles of incidence for a composite with 10 wt.% MWCNTs

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The correlations between various PS parameters, as well as between them and the absorption coefficient in the investigated composites, can be seen in detail in Figs. 4, 5, 6. The dependences of the electronic structure and electromagnetic radiation absorption parameters at a frequency of 2 GHz in the PTFE–MWCNTs composites on the probability \({P}_{{\theta }_{F}}\) of annihilation of positrons with free electrons are shown in Fig. 4. If we assume a relationship between the radius of the Fermi sphere (~ \({\theta }_{F}\)) and the average electron density (~ \({P}_{{\theta }_{F}}\)) as for free electrons, then the Fermi angle \({\theta }_{F}\) should be monotonically increased from the minimum to the maximum value of the probability \({P}_{{\theta }_{F}}\) approximately along the upper dashed line in the upper left graph in Fig. 4. Deviations from this line demonstrate the essential role of the effective mass and relating changes in the electron energy spectra due to composites formation. The presented results also confirm that in the concentration range from 5 to 10 wt.% MWCNTs the growing and branching of conductive chains are continued in the composites (Garkusha O. M. (2010)).

Fig. 4
figure 4

Dependences of the PS parameters and the electromagnetic radiation absorption at a frequency of 2 GHz on the probability \({P}_{{\theta }_{F}}\) of annihilation of positrons with free electrons in the PTFE–MWCNTs composites

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Fig. 5
figure 5

Dependences of the electromagnetic radiation absorption at a frequency of 2 GHz in the PTFE–MWCNTs composites on the PS parameters \({r}_{{m}_{b}}\), \({P}_{{r}_{{m}_{b}}}\), R, and PR

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Fig. 6
figure 6

Dependences of the absorption of electromagnetic radiation at a frequency of 2 GHz in the PTFE–MWCNTs composites on the PS parameters \({r}_{{m}_{i}}\), \({P}_{{r}_{{m}_{i}}}\), \({\theta }_{F}\), and \({P}_{{\theta }_{F}}\)

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We previously note that the average radius R of the free volumes and the probability of annihilation of positrons PR in them are determined only by the parameters of the polymer matrix electronic and defect structures, while the Fermi angle \({\theta }_{F}\) and the probability of annihilation of positrons with free electrons \({P}_{{\theta }_{F}}\) are determined by analogous parameters of the MWCNTs only. From Fig. 4 we can see that the rest of the PS parameters are determined by the parameters of both the polymer matrix and MWCNTs. Since the bounded electron (parameters \({r}_{{m}_{i}}\) and \({P}_{{r}_{{m}_{i}}}\)) and electronic structure of small defects (parameters \({r}_{{m}_{b}}\) and \({P}_{{r}_{{m}_{b}}}\)) of the polymer matrix, at least outside the interphase, do not change, then the variations of corresponding quantities are also defined mainly by the MWCNTs parameters. Note the general trends of free volumes changes after the percolation threshold are in good agreement with the trend described in (Ermolaev G. A. (3), Yang M. K. (26)).

From the dependences of the absorption of electromagnetic radiation at a frequency of 2 GHz on the electronic structure parameters of composites (Figs. 5, 6) one can see that dependences of the absorption coefficient on the PS parameters described annihilation in defects and interlayer spaces are monotonically increase (Figs. 5 a, b, 6 (a)) or decrease (Fig. 6 b). The dependences of the absorption coefficient on the average radius R of free volumes and corresponding probability of annihilation in the composites have complex non-monotonic trends (Fig. 5 c, d) determined by the formation, branching, and MWCNTs aggregation processes in conducting circuits.

The electromagnetic radiation absorption at a frequency of 2 GHz monotonically increases with an increase in the density of free electrons in MWCNTs (Fig. 6 d) in the presence of branched conducting chains (infinity percolation clusters) in the composite. This effect is due to the fine structure of the density of states of carbon π-electrons in composites near the Fermi level. In the perfect semimetal MWCNT, there are a sufficiently high density of almost free π-electrons and, at the same time, a low density of states at the Fermi level, i.e., there are almost no available free states near the Fermi level, and in processes of small excitations, band electrons do not mass participate. In composites with imperfect MWCNTs, as a result of charge transport from the matrix to MWCNTs (or vice versa in other PM), the chemical potential shifts and above it the available states appear. That leads, as can be seen from the spectral dependence of the refractive indices < n > and absorption < k > (Fig. 3), to an almost linear increase in absorption in the entire investigated wavelength range. Moreover, it can be possible that for considering composites, the same high absorption may be observed in the entire infrared range of radiation.

As can be seen from Fig. 2, in the sample with 10 wt.% MWCNTs and thickness of ≈2 mm, the intensity of incident radiation at the exit from the sample decreases by a factor of 410, i.e., 99.9976% absorption is observed. Nanotubes do not reemit absorbed energy in the form of photons. In free carbon nanotubes, the electrons stand in excited state approximately 10–9 s and under transition to the ground state emit the absorbed energy as photons. In (Awad et al (2019b), for the first time, an almost linear relationship between the average size R of free volumes in the 50% (Na–CMC)/50% PAM + MWCNT composite and the energy (Urbach energy, EU), which can transfer by excited electrons to the phonon spectrum in one act of scattering on the ‘lattice’ of disordered solid, was established. Most probably, in the PTFE–MWCNTs composites, the dependences of the EU versus R will be different. For approximate estimation, we take the value EU. = 0.58 eV (Awad et al 2019b). One can see that for frequencies of electromagnetic radiation up to 1.4⋅1014 Hz (the wavelength of the incident radiation λ ≈ 2.14 μ), the free electrons have an alternative channel (not photons reradiation) to leave the excited states, namely the transfer of excitation energy to phonon spectrum of the composite. Thus, a high absorption of EW can be observed not only at the studied frequency of 2 GHz, but also in the long and medium ranges of infrared radiation. High scattering of free electrons by phonons was also noted in (Xue G. (25), Zhong J. (27)) when studying the thermal and electrical conductivity of composites filled with reduced graphene oxide.

Conclusion

As shown, the ACAR spectrum for PTFE–MWCNTs composites consists of 4 components: 3 Gaussians—a wide one (determined by the PS parameters \({r}_{{m}_{\mathrm{b}}}\) and\({P}_{{r}_{{m}_{\mathrm{b}}}}\)), caused by the annihilation of positrons mainly in Stone–Welsh defects in MWCNTs, an medium one (determined by \({r}_{{m}_{\mathrm{i}}}\) and\({P}_{{r}_{{m}_{\mathrm{i}}}}\)) caused by annihilation of positrons with bounded electrons of matrix and filler atoms, and narrow one (determined by \(R\) and\({P}_{R}\)) caused by positrons annihilation from the o-Ps state in the free volumes in PTFE, as well as the parabolic contribution (determined by \({\theta }_{F}\) and\({P}_{{\theta }_{F}}\)) appears due to the annihilation of positrons with free electrons in MWCNTs. For PTFE–MWCNTs composites, the extremes of PS parameters\({r}_{{m}_{\mathrm{i}}}\),\({P}_{{r}_{{m}_{\mathrm{i}}}}\),\({r}_{{m}_{\mathrm{b}}}\),\({P}_{{r}_{{m}_{\mathrm{b}}}}\), and \({P}_{{\theta }_{F}}\) at a filler concentration of 10 wt.% correlated with the maximum of the absorption coefficient of electromagnetic radiation at a frequency of 2 GHz. The primary reason of this effect is an increasing of free electrons density in MWCNTs, which is caused by both the charge transfer from the free volumes in PTFE matrix to MWCNTs and the changes of MWCNTs’ electron structure because of their own defects and deformation.

Using the methods of angular correlation of annihilation radiation (ACAR), attenuation of electromagnetic radiation, and optical ellipsometry, it is shown that composites of polytetrafluoroethylene–multi-walled carbon nanotubes (PTFE–MWCNTs), as opposed to completely transparent to electromagnetic radiation their PTFE matrix, after the addition of 10 wt.% MWCNTs or more demonstrate high absorption of electromagnetic radiation at a frequency of 2 GHz, i.e., 200–410 times intensity attenuation for samples with thickness of ≈2 mm. We established that in the PTFE–MWCNTs composites can be realized an alternative channel for free electrons leaving the excited states, which is connected not with photons reradiation but with energy transfer by portions of the order of Urbach energy to the phonon spectrum of disordered solid due to strong electrons’ scattering on the atomic oscillations. The estimating value of the Urbach energy for PTFE–MWCNTs composites and the presence in them branched conducting chains containing free electrons, which can be excited in a wide range of electromagnetic radiation wavelengths, give a possibility to these composite materials to have a high absorption characteristics not only in ultra-high-frequency diapason of radiation but also in the whole micro-waves and infrared diapasons.

The PTFE–MWCNTs composite with 10 wt.% MWCNTs, due to their low density, high stability in aggressive environments, and high absorption coefficient of electromagnetic radiation, is a promising material for protective and absorbing coatings in a wide range of wavelengths including frequencies used in 5G wireless technology.