The present study suggests a novel model for simulating electromagnetic characteristics of spheroidal nanofillers. The electromagnetic interference shielding efficiency of prolate and oblate ellipsoids in the X-band frequency range is studied. Different multilayered nanocomposite configurations incorporating carbon nanotubes, graphene nanoplatelets, and carbon blacks are fabricated and tested. The best performance for a specific thickness is observed for the multilayered composite with a gradual increase in the thickness and electrical conductivity of layers. The simulation results based on the proposed model are shown to be in good agreement with the experimental data. The effect of filler alignment on shielding efficiency is also studied by using the nematic order parameter. The ability of a nanocomposite to shield the incident power is found to decrease by increasing alignment especially for high volume fractions of prolate fillers. The interaction of the electromagnetic wave and the fillers is mainly affected by the polarization of the electric field; when the electric field is perpendicular to the equatorial axis of a spheroid, the interaction is significantly reduced and results in a lower shielding efficiency. Apart from the filler alignment, size polydispersity is found to have a significant effect on reflected and transmitted powers. It is demonstrated that the nanofillers with a higher aspect ratio mainly contribute to the shielding performance. The results are of interest in both shielding structures and microwave absorbing materials.

Conductive polymer nanocomposites incorporating multiwalled carbon nanotubes (CNTs), graphene nanoplatelets (GNPs), and carbon blacks (CBs) have emerged as an attractive class of materials due to their light weight, flexibility, corrosion resistivity, mechanical strength, and processing advantages over the past two decades, but the relationship between their macrobehavior and nanostructural properties is still poorly understood.1 The electrical conductivity of insulating polymers can be increased by several orders of magnitude by adding a small amount of CNTs, GNPs, or CBs. This sharp rise in conductivity results from the formation of a percolating network of electrically interconnected nanofillers through electron tunneling.2 The tunneling conductance between two nanoparticles is given as3–5 

g i j = g 0 exp ( 2 δ i j ξ ) ,
(1)

where δij is the shortest distance between two particles i and j, ξ is the tunneling length scale, and g0 is a constant proportional to the contact resistance.

Figure 1 (Multimedia view in the supplementary material) illustrates a realization of prolate and oblate ellipsoids with aspect ratios 10 and 1/10, respectively. Polycarbonate (PC) is chosen as the polymer matrix in this study for its high impact strength at temperatures between −40 °C and 100 °C.6 

FIG. 1.

Realization of fillers with volume fraction ϕ=0.1. In both videos, the fillers in green belong to the percolating cluster. Left clip: prolate ellipsoids. Right clip: oblate ellipsoids (Multimedia view in the supplementary material).

FIG. 1.

Realization of fillers with volume fraction ϕ=0.1. In both videos, the fillers in green belong to the percolating cluster. Left clip: prolate ellipsoids. Right clip: oblate ellipsoids (Multimedia view in the supplementary material).

Close modal

The superior nature of these nanocomposites makes them good candidates for different electromagnetic interference (EMI) applications such as weather monitoring, vehicle detection, air traffic control, and defense tracking in the microwave frequency range.7 Shields are designed and developed to inhibit undesirable radiation. EMI shielding in the X-band frequency range (8–12 GHz) is more important for commercial and military applications.8 Some applications, especially in military systems, demand not only having good shielding efficiency but also high absorption. To enhance the performance, a higher volume fraction of fillers can be used. However, increasing the content of fillers will lead to more reflection, which is caused by impedance mismatch between the free space and the composite. To design an efficient absorber, there are two requirements: first, a low dielectric constant (close to 1) to minimize the reflection. Second, the entering power must be attenuated over the composite thickness. This goal can be achieved for a given thickness by replacing the monolayer of nanocomposites by a multilayer with an increasing filler concentration and a similar overall thickness.

The increasing use of multilayered nanocomposites in the military and industrial applications shows a need for vigorous studies of their electromagnetic characteristics.9 These electromagnetic properties rely strongly on the anisotropic filler's shape, aspect ratio, alignment, size distribution, electrical properties, dispersion pattern, and fabrication techniques.10,11 For nanosized fillers, such as CNTs and GNPs, the homogeneous dispersion is hard to achieve due to van der Waals interactions between the nanofillers. However, microsized fillers such as carbon black have advantages of easily dispersible and low cost. Unfortunately, experimental results demonstrated that a higher weight fraction of microsized fillers is needed to reach the desired properties compared with nanofillers. Investigating the effect of the aforementioned factors on the shielding and absorbing efficiency of nanocomposites experimentally is expensive and time-consuming. Therefore, having a mathematical model can help us to examine different inputs on the shielding ability of nanocomposites.

In the previous studies on the use of nanofillers as EMI shielding materials, only small emphasis has been placed on the filler alignment and size distribution although they remain important factors. These parameters are addressed in this work.

In this work, the focus is on comparing the simulation results, for monolayer and multilayered nanocomposites, with the experimental data to validate the model and investigating the effect of different factors on the shielding ability of nanocomposites to provide an appropriate way to design a shield. This work has three objectives. First, the EMI shielding efficiency of nanocomposites is investigated experimentally in Sec. II. Second, the mathematical model to create geometry is described and finite element results for prolate ellipsoids, oblate ellipsoids, and spherical inclusions representing CNTs, GNPs, and CBs, respectively, are compared with experimental results in Sec. III. Finally, after validation of the model, the effects of increasing filler alignment and size polydispersity on shielding performance are described in Secs. IV and V, respectively.

The materials used in this study and their manufacturing process have been described in detail in Ref. 6. The composites are cut into pieces of size 10.16 mm × 22.86 mm with different thicknesses for the EMI shielding efficiency characterization in the frequency range of 8.5–12 GHz (X-band) with a WR-90 waveguide excited by the TE10 mode using a vector network analyzer. For each weight fraction, at least three samples are tested, and the average is reported in this study. Multilayered polycarbonate nanocomposites with various carbon filler and weight percentages are fabricated by compression molding. Nanocomposite specimens are compression molded at 0.1 MP and 150 °C. The pressure was sustained, while the specimens were allowed to cool down gradually to the ambient temperature (typically, 23 °C) at a typical rate of 2.5 °C/min. Scanning electron microscopy (SEM) using a Zeiss NEON 40EsB High-resolution SEM was performed for characterization of dispersed carbon fillers in polycarbonate. Figure 2 shows the SEM images of a fractured surface of two layered polycarbonate nanocomposites with CNT and CB fillers dispersed in the polycarbonate matrix, where the boundary can be found easily. The scattering parameters Sij are measured, which correspond to reflected power and transmitted power. The shielding effectiveness is obtained using the normalized transmitted power shielding efficiency (SE) =  20 log | S 21 | = 20 log | E i | | E t | , where i stands for incident and t stands for transmitted.

FIG. 2.

(a) and (b) SEM images of a fractured surface of polycarbonate nanocomposites including CB and CNT. (a) CB 6 wt. %/CNT 15 wt. % and (b) CNT 5 wt. %/CB 6 wt. %.

FIG. 2.

(a) and (b) SEM images of a fractured surface of polycarbonate nanocomposites including CB and CNT. (a) CB 6 wt. %/CNT 15 wt. % and (b) CNT 5 wt. %/CB 6 wt. %.

Close modal

Figure 3 shows the electrical percolation behavior of CNT-PC, CB-PC, and GNP-PC nanocomposites at room temperature.6 It is shown that the percolation threshold for CNT, CB, and GNP fillers occurred at ∼1 wt. %, ∼3.5 wt. %, and ∼7 wt. %, respectively, since the electrical resistivity suddenly decreases at those weight fractions. CNTs need a lower amount to reach percolation threshold, which can be attributed to their high aspect ratio. However, dispersion of fillers in the matrix has a significant effect on the percolation behavior.

FIG. 3.

Percolation behavior of polycarbonate composites incorporating CNTs, CBs, and GNPs.

FIG. 3.

Percolation behavior of polycarbonate composites incorporating CNTs, CBs, and GNPs.

Close modal

Figure 4 clearly shows that σ / ϵ f increases with nanofillers' weight fraction, where σ denotes the conductivity, ϵ is the permittivity, and f is the frequency. This increase occurs at two different rates, depending on the nanocomposite percolation threshold. The change in the rate of increase in σ / ϵ f occurs at the percolation threshold. For nanocomposites below the percolation threshold, the change in σ / ϵ f is significant. This observation confirms that achieving good SE requires not only connectivity but also good conductivity.12–14 

FIG. 4.

Measured σ / ϵ f of polycarbonate nanocomposites incorporating CNT, GNP, and CB at different weight fractions.

FIG. 4.

Measured σ / ϵ f of polycarbonate nanocomposites incorporating CNT, GNP, and CB at different weight fractions.

Close modal

One way to improve absorption is to prepare a multilayered structure in which the filler concentration is gradually increased from one layer to the next one. The low difference in the dielectric constant limits the reflectivity at the first interface. One proposed solution is using a three layered composite in which the thickness of the first layer is about one-third of the total thickness to have a smooth penetration of the incident wave, while the thin second layer aims to match the low dielectric constant of the first layer to the third layer.14 Figure 5 shows the results for three different configurations of multilayered composites. The thicknesses of first and last layers are 1 mm and 2 mm, respectively. The figure clearly shows the shielding efficiency gained by absorption in the last layer since the second layer has a very small effect on the total SE.

FIG. 5.

(a)–(c) Shielding effectiveness for three layered nanocomposites incorporating CNT and CB over the 8.5–12 GHz frequency range. (a) CB 3 wt. %-CNT 4 wt. %-CNT 15 wt. % multilayered composite for different CNT 4 wt. % thicknesses. (b) CNT 1 wt. %-CNT 5 wt. %-CNT 15 wt. % multilayered composite for different CNT 5 wt. % thicknesses. (c) GNP 2 wt. %-CNT 2 wt. %-CNT 15 wt. % multilayered composite for different CNT 2 wt. % thicknesses.

FIG. 5.

(a)–(c) Shielding effectiveness for three layered nanocomposites incorporating CNT and CB over the 8.5–12 GHz frequency range. (a) CB 3 wt. %-CNT 4 wt. %-CNT 15 wt. % multilayered composite for different CNT 4 wt. % thicknesses. (b) CNT 1 wt. %-CNT 5 wt. %-CNT 15 wt. % multilayered composite for different CNT 5 wt. % thicknesses. (c) GNP 2 wt. %-CNT 2 wt. %-CNT 15 wt. % multilayered composite for different CNT 2 wt. % thicknesses.

Close modal

Figure 6 shows the normalized transmitted power and reflected power for GNP 2 wt. %-CB 6 wt. %-CNT 15%. The thickness of the first and last layers is 3.2 mm. The figure clearly shows that S11 is more sensitive with respect to frequency, especially in a lower frequency range. By increasing the thickness of the middle layer, the reflection power changes significantly. However, the total shielding efficiency is not changing too much.

FIG. 6.

GNP 2 wt. %-CB 6 wt. %-CNT 15 wt. % multilayered composite for different CB 6 wt. % thicknesses, with solid lines and dashed lines showing S11 and S21, respectively.

FIG. 6.

GNP 2 wt. %-CB 6 wt. %-CNT 15 wt. % multilayered composite for different CB 6 wt. % thicknesses, with solid lines and dashed lines showing S11 and S21, respectively.

Close modal

Figure 7 shows S12 and S11 for monolayer and multilayered GNP composites. The thickness is adjusted to make sure that the weights of the two samples are the same. The S12 parameter indicates that both samples have almost the same shielding efficiency, whereas the reflection (S11) of the multilayered composite is less than that of the monolayer composite. The performances of GNP-PC composites are relatively poor since a large GNP content (20 wt. %, 8.5–12 GHz, and t = 1 mm) or a large thickness (8 wt. %, 8.5–12 GHz, and t = 6 mm) is necessary to achieve a shielding effectiveness of 20 dB which is adequate for most applications.15 

FIG. 7.

Scattering parameter for the GNP 2 wt. %-GNP 8 wt. %-GNP 12 wt. % multilayered composite with 2 mm-1 mm-1 mm and GNP 8 wt. % with 3 mm thicknesses.

FIG. 7.

Scattering parameter for the GNP 2 wt. %-GNP 8 wt. %-GNP 12 wt. % multilayered composite with 2 mm-1 mm-1 mm and GNP 8 wt. % with 3 mm thicknesses.

Close modal

A large number of multilayered nanocomposites incorporating CNT, GNP, and CB are fabricated and tested in this study, which are used to compare with the simulation results in Sec. III. However, it is worth mentioning that these experiments are conducted to validate our computational model. Experimentally studying the effect of different factors, size polydispersity, and filler alignment on the electromagnetic characterization of nanocomposites is expensive and time-consuming. Having a validated model facilitates effective simulation of various variables. In Secs. IV and V, the effects of filler alignment and size polydispersity on shielding efficiency are studied.

In order to understand the influence of different factors on the overall EMI shielding efficiency of nanocomposites, a theoretical background will be first given. Then, a computational model will be described in detail. Computations were done utilizing up to 256 CPU cores and large memory capabilities of the Blue Waters16 supercomputer hosted at the National Center for Supercomputing Applications (NCSA) at the University of Illinois at Urbana-Champaign.

A low reflection of waves at the interface with the surrounding air and a good absorption of power over the thickness are needed to design an efficient microwave absorber. However, these two aspects are mutually antagonist since the low reflection needs a low dielectric constant and high dissipation can only be gained by a high conductivity through adding more fillers into the matrix. Replacing a single layer of nanocomposites by a multilayer composite with the identical thickness by increasing filler's concentration can be used to resolve this issue.

Such a multilayered composite can be modeled by the chain matrix product. The global chain matrix is obtained by the matrix product of the corresponding chain matrix for each layer

[ E out H out ] = i = 1 n [ cosh β c t Z c sinh β c t Y c sinh β c t cosh β c t ] i [ E in H in ] ,
(2)

where, for each layer, t is the thickness, β c = j ω μ ( σ + j ω ϵ ) is the propagation constant, and the wave admittance Y c = β c / j ω μ .14 

The shielding results for a CNT 1 wt. %-CNT 2 wt. %-CNT 4 wt. % multilayered composite with 1 mm, 0.5 mm, and 1 mm thicknesses, respectively, and CNT 2 wt. % with 3 mm thickness are summarized in Table I. The theoretical prediction for total shielding effectiveness (and shielding by absorption) is lower than measured values, while the theoretical SER values are higher than the experimental values. The difference can be attributed to (i) the negative multireflection effect17 and (ii) ignoring the intrinsic properties and volume fraction of fillers in the theoretical prediction.12 

TABLE I.

Comparison between shielding data theoretically calculated and experimentally obtained for CNT 1 wt. %-CNT 2 wt. %-CNT 4 wt. % multilayered and CNT 2 wt. % monolayer composites at 10 (GHz).a

Total SE (dB) SER (dB)
Sample Experimental Theoretical Experimental Theoretical
CNT 1 wt. %-CNT 2 wt. %-CNT 4 wt. %  9.29  8.36  5.75  6.39 
CNT 2 wt. %  8.09  7.19  4.04  5.37 
Total SE (dB) SER (dB)
Sample Experimental Theoretical Experimental Theoretical
CNT 1 wt. %-CNT 2 wt. %-CNT 4 wt. %  9.29  8.36  5.75  6.39 
CNT 2 wt. %  8.09  7.19  4.04  5.37 
a

The samples have an equivalent CNT weight fraction with different thicknesses.

To obtain an overall picture of the nanocomposite behavior that captures a wide range of factors, the finite element modeling (FEM)18 is performed. First, the mathematical model to create the geometry is described in detail. Next, the FEM simulation is performed by solving Maxwell's equations, from which the reflection and transmission coefficients are calculated. The simulation results are then compared with measured data to validate the approach employed in this work.

The fillers are modeled as impenetrable spheroids with aspect ratio α defined as the ratio of the polar and equatorial semiaxes. The Random Sequential Addition (RAS) method19 is used to generate the realization of fillers followed by the Monte Carlo (MC) relaxation step to obtain an equilibrated configuration.20 For each filler, the position is chosen randomly within the sample size. Filler's orientation of the polar axis is chosen using the random azimuthal angle, θ, from [0,2π] and polar angle, β, from [0,π]. Fillers are added one by one until achieving the desired volume fraction, ensuring that the newly added filler does not overlap with any of the existing fillers.21 

Based on the method explained in Ref. 11 for volume fractions above the percolation threshold, after the desired volume fraction is achieved, all fillers are coated with a soft shell of thickness δ / 2 . Next, δc is found, which is the minimum δ for which a percolating cluster of interconnected fillers spanning the entire domain is formed. The highest number of fillers (volume fraction) and aspect ratio are limited by the available time and resources. These critical path based tunneling-percolation model predictions are compared with experimental data available on the electrical conductivity of CNTs and GNPs and are found to be well within the expected range.11 To simulate the shielding performance of nanocomposites, this extra soft shell is considered to compensate for the tunneling effect between nanofillers.

After creating the geometry, FEM simulation (FES) is done using high-performance computational resources with our in-house finite element code, and the FEM simulation on the composites is carried out considering a partially filled rectangular waveguide with a TE10 mode incident on each sample. A typical simulation involves solving a numerical system with two million degrees of freedom. Table II shows the property of the matrix and fillers used in this study for the FE simulations.

TABLE II.

Properties of the matrix and nanofillers including resistivity, relative permittivity, and density.

Fabrication ρ (Ω m) ϵr D (g/cc)
Polycarbonate  … 
CNT  10 7   30  2.00 
GNP  10 5   15  2.00 
CB  10 4   20  1.80 
Fabrication ρ (Ω m) ϵr D (g/cc)
Polycarbonate  … 
CNT  10 7   30  2.00 
GNP  10 5   15  2.00 
CB  10 4   20  1.80 

Figure 8 shows the simulation results in comparison to experimental data for monolayer and three layered nanocomposites. The difference can be attributed to the neglect of tunneling anisotropy, the interface layer, filler alignment, agglomeration, and uniform size-distribution in the model.22 The three latter factors are expected to have a direct influence on the shielding ability of the nanocomposite. Figure 9 shows the Smith plot for the reflection of CB 6 wt. %/PC.

FIG. 8.

(a) and (b) Comparison of simulation results for normalized reflected and transmitted powers with the experiment for monolayer and three layered nanocomposites. (a) Comparison of simulation results with the experiment for CNT 5 wt. %/PC. (b) Comparison of simulation results with the experiment for GNP 2 wt. %-GNP 8 wt. %-GNP 12 wt. %/PC.

FIG. 8.

(a) and (b) Comparison of simulation results for normalized reflected and transmitted powers with the experiment for monolayer and three layered nanocomposites. (a) Comparison of simulation results with the experiment for CNT 5 wt. %/PC. (b) Comparison of simulation results with the experiment for GNP 2 wt. %-GNP 8 wt. %-GNP 12 wt. %/PC.

Close modal
FIG. 9.

Smith plot for the reflection of the CB 6 wt. %/PC nanocomposite with 3.2 mm thickness in the frequency range of 8.5–12 GHz.

FIG. 9.

Smith plot for the reflection of the CB 6 wt. %/PC nanocomposite with 3.2 mm thickness in the frequency range of 8.5–12 GHz.

Close modal

Figure 10(a) depicts the simulation results and experimental data for GNP 2 wt. %- CNT 2 wt. %-CNT 4 wt. % multilayered nanocomposites. As can be seen, the simulation results agree well with the experimental data. Figure 10(b) shows the realization of the GNP-CNT-CNT three layered composite.

FIG. 10.

(a) and (b) Simulation of three layered nanocomposites incorporating oblate and prolate ellipsoids. (a) Simulation results with the experiment for GNP 2 wt. %-CNT 2 wt. %-CNT 4 wt. %/PC. (b) Realization of the GNP-CNT-CNT three-layered composite.

FIG. 10.

(a) and (b) Simulation of three layered nanocomposites incorporating oblate and prolate ellipsoids. (a) Simulation results with the experiment for GNP 2 wt. %-CNT 2 wt. %-CNT 4 wt. %/PC. (b) Realization of the GNP-CNT-CNT three-layered composite.

Close modal

To compare the effects of the volume fraction and thickness of layers on the performance of the multilayered structure, four different cases are considered, which are summarized in Table III. The last column is the average weight fraction for the equivalent monolayer with 4 mm thickness. Figure 11 shows that the most important factor for shielding performance is the average weight fraction, as samples C and D have the best shielding. Also, sample D is slightly better than sample C in shielding, clearly indicating that the multilayer arrangement combined with a gradual increase in thickness and weight fraction is the most efficient shielding system. The lowest reflection is for sample A as expected: it has the multilayered structure and a minimum average volume fraction. It is interesting that sample D, which has the best shielding ability, has a good absorption performance as well. Therefore, for a specific thickness and average weight fraction, multilayered composites with a gradual increase in thickness and volume fraction give the best shielding and absorption performances.

TABLE III.

Properties of samples A-D.a

Sample Configuration Thickness Average wt. %
CNT 1 wt. %-CNT 2 wt. %-CNT 4 wt. %  2 mm–1 mm–1 mm  2.00 
CNT 2 wt. %  4 mm  2.00 
CNT 1 wt. %-CNT 2 wt. %-CNT 4 wt. %  1.1 mm–0.40 mm–2.50 mm  2.98 
CNT 1 wt. %-CNT 2 wt. %-CNT 4 wt. %  0.6 mm–1.15 mm–2.25 mm  2.98 
Sample Configuration Thickness Average wt. %
CNT 1 wt. %-CNT 2 wt. %-CNT 4 wt. %  2 mm–1 mm–1 mm  2.00 
CNT 2 wt. %  4 mm  2.00 
CNT 1 wt. %-CNT 2 wt. %-CNT 4 wt. %  1.1 mm–0.40 mm–2.50 mm  2.98 
CNT 1 wt. %-CNT 2 wt. %-CNT 4 wt. %  0.6 mm–1.15 mm–2.25 mm  2.98 
a

Overall thickness of all samples is 4 mm.

FIG. 11.

Scattering parameters for cases A-D of different configurations listed in Table III, with solid lines and dashed lines showing S21 and S11, respectively.

FIG. 11.

Scattering parameters for cases A-D of different configurations listed in Table III, with solid lines and dashed lines showing S21 and S11, respectively.

Close modal

In summary, the obtained results prove the effectiveness of the multilayer structure over the monolayer shield. This gradient structure has the advantage to enhance the shielding effectiveness by absorption due to a reduced reflected power since the incident wave penetrates the multilayer at the low concentration of the first layer, and the wave is not transmitted at the last interface due to a large impedance mismatch. Thus, the maximum energy is trapped between the conductive layers with a lower outside reflection.23 

Apart from the volume fraction, the EMI shielding efficiency of nanocomposites is expected to be affected by the intrinsic electrical conductivity, dielectric constant, magnetic permeability, size polydispersity, aspect ratio, and distribution state of fillers.14 In most studies, an isotropic dispersion of filler orientations is assumed. However, due to the van der Waals interaction and shear force active within the polymer, anisotropic nanofillers including CNTs and GNPs tend to align during different processing stages.22 Very few experimental studies have been done on the effect of filler alignment on the EMI shielding. Compared with the experimental studies focusing on different fillers, no attention has been given to studying the effect of filler alignment for spheroidal fillers through simulations.

The approach used in this study is similar to the one used in Refs. 11 and 22. The nematic order parameter, S , and m are used to quantify the alignment

S = 1 2 3 cos θ 2 1 , m m min = 3 S 1 S ,
(3)

where the angle brackets represent the average over all the fillers in the sample. We can obtain filler distribution with a desired S , with S = 0 representing the isotropic distribution, while for fully aligned system, S = 1 [see Figs. 12(a) and 12(b)]. The m value for all cases in this study is m min .

FIG. 12.

(a) and (b) Realizations of oblate and prolate ellipsoids with an increasing alignment. In all figures, the fillers shown in magenta belong to the percolating cluster. Much larger systems are used in this study to generate results. The m value for all cases is equal to m min as given in Eq. (3). (a) α = 1 10 ( 2300 fillers) for an increasing alignment: S = 0 , S = 0.25 , S = 0.5 , and S = 0.9 . (b) α = 10 , ( 2500 fillers) for an increasing alignment: S = 0 , S = 0.25 , S = 0.5 , and S = 0.9 .

FIG. 12.

(a) and (b) Realizations of oblate and prolate ellipsoids with an increasing alignment. In all figures, the fillers shown in magenta belong to the percolating cluster. Much larger systems are used in this study to generate results. The m value for all cases is equal to m min as given in Eq. (3). (a) α = 1 10 ( 2300 fillers) for an increasing alignment: S = 0 , S = 0.25 , S = 0.5 , and S = 0.9 . (b) α = 10 , ( 2500 fillers) for an increasing alignment: S = 0 , S = 0.25 , S = 0.5 , and S = 0.9 .

Close modal

From Fig. 13, it is observed that for prolate ellipsoids, as the concentration of fillers increases, the efficiency of shielding increases by more than factor two in the case of the electric field oriented parallel to the equatorial axis of fillers for a high volume fraction. The main mechanism of shielding is the absorption of incident power due to high polarizability.24 When the electric field, E, is perpendicular to the CNT equatorial axis, the interaction of the electromagnetic wave and the fillers is significantly reduced. The increasing shielding efficiency dependence on the weight fraction for low concentrations changes into a decay function after ∼5 wt. %.

FIG. 13.

S21 as a function of the CNT weight fraction for isotropic and fully aligned systems.

FIG. 13.

S21 as a function of the CNT weight fraction for isotropic and fully aligned systems.

Close modal

The results for the aspect ratio α = { 10 , 1 10 } at wt % = { 4 , 12 } are shown in Fig. 14, with both cases above the percolation threshold.6 It is observed that for prolate and oblate ellipsoids, the shielding efficiency decreases as S increases. With an increasing alignment, the interconnectivity of the fillers is adversely affected, leading to a lower conductivity and shielding effectiveness. For prolate ellipsoids, the effect of filler alignment seems to be more severe than oblate fillers. However, the variation of different configurations has a more significant difference compared with prolate ellipsoids. Unfortunately, there are no experimental results available for samples considered here to compare with the simulation results. With an increasing m, the percolation threshold is observed to decrease for all cases.22 Also, the percolation threshold is more sensitive to m for highly aligned systems than isotropically oriented systems.22 It was observed that the conductivity, and hence shielding effectiveness, decreases by increasing m.22 

FIG. 14.

(a) and (b) Shielding effectiveness of prolate and oblate ellipsoids plotted as a function of S : (a) CNT 4 wt. % and (b) GNP 12 wt. %.

FIG. 14.

(a) and (b) Shielding effectiveness of prolate and oblate ellipsoids plotted as a function of S : (a) CNT 4 wt. % and (b) GNP 12 wt. %.

Close modal

It has been found that the permittivity is anisotropic24 and the dielectric loss varies for the same volume fraction with different filler orientations. For isotropically distributed fillers, the effect of anisotropic permittivity on the shielding effectiveness is minimal. However, for strongly aligned systems, anisotropic conductivity results in different dielectric permittivity values compared with the isotropic case, especially at higher volume fractions. To estimate the dielectric permittivity of nanocomposites, an anisotropic permittivity needs to be considered. Such consideration has a notable effect on the electrical conductivity of the nanocomposite.

To improve the filler dispersion, mixing processes such as sonication and shear mixing are used, which change the filler size significantly. Modifying nanofillers, for example, CNTs with HNO3 or an HNO3/H2SO4 acid mixture, can purify the nanofiller and significantly improve the dispersity in a polymer matrix. However, modifying fillers may increase the defects and shorten them.25 Therefore, the effect of filler size polydispersity needs to be investigated on the shielding efficiency of the nanocomposites.

Analytical studies have shown that the percolation threshold is extremely sensitive to polydispersity.10,26–28 Here, we analyze the effect of size polydispersity on shielding ability of randomly oriented prolate and oblate ellipsoids.

The semi-axes of the prolate and oblate ellipsoids are given as { L / 2 , D / 2 , D / 2 } and { D / 2 , D / 2 , L / 2 } , respectively. Ellipsoids following a bimodal distribution f ( x ) = p δ ( x x 1 ) + ( 1 p ) δ ( x x 2 ) are used to account for length polydispersity, where p is the number fraction, and x is L and D for prolate and oblate ellipsoids, respectively. All prolate (oblate) ellipsoids are assumed to have the same D(L), and all lengths are normalized with D(L). For CNTs, the cylinder-like shape is obtained using α 1 , while for GNPs, the plate-like shape is obtained with α 1 . In this section, an isotropic dispersion of the filler orientation is assumed.

The plots for monodisperse and polydisperse cases are shown in Fig. 15, for both prolate and oblate ellipsoids. A bidisperse system of ellipsoids [see Figs. 15(a) and 15(b)] shows that the filler size distribution has a significant effect on the shielding ability of the composite. For prolate and oblate ellipsoids, all data have the same α but have longer rod-like or wider plate-like fillers to improve the shielding ability of the composite while keeping the same weight fraction.29 The figure shows that the effect of the aspect ratio on the transmitted power is more than on the reflected power for both prolate and oblate ellipsoids.

FIG. 15.

(a) and (b) S21 and S11 as a function of monodisperse and polydisperse prolate and oblate ellipsoids, with solid lines and dashed lines showing S11 and S21, respectively. The thickness is 1 mm. (a) Normalized reflected and transmitted powers for prolate ellipsoids. (b) Normalized reflected and transmitted powers for oblate ellipsoids.

FIG. 15.

(a) and (b) S21 and S11 as a function of monodisperse and polydisperse prolate and oblate ellipsoids, with solid lines and dashed lines showing S11 and S21, respectively. The thickness is 1 mm. (a) Normalized reflected and transmitted powers for prolate ellipsoids. (b) Normalized reflected and transmitted powers for oblate ellipsoids.

Close modal

This work deals with the electromagnetic interference shielding ability of nanocomposites made of nanofillers in the X-band frequency range through the experimental study and validated numerical model. Experimental data show that σ / ϵ f increases with nanofillers' weight fraction. Depending on the nanocomposite percolation threshold, this increase occurs at two different rates. The change takes place at the percolation threshold. This observation confirms that achieving high shielding performance requires not only good conductivity but also connectivity.

One way to enhance absorption performance is using multilayered structures. Using a three layered structure in which the filler concentration is gradually increased shows that the shielding efficiency can be gained by absorption in the last layer since the second layer has a very small effect on the total shielding effectiveness. However, the reflected power changes significantly by increasing the second layer thickness.

Both experimental data and simulation results show that for a specific thickness, the multilayered composite with a gradient filler concentration and thickness has the best performance. With this multilayer structure, the maximum energy is trapped between the conductive layers with a lower outside reflection. The scattering parameters show that monolayer and multilayered composites with the same overall weight of fillers have almost the same shielding performance. However, the reflected power of multilayered composite is less than that of the monolayer composite.

The MC simulations of prolate and oblate ellipsoids and spherical fillers are performed using the soft shell around the fillers to compensate for the electron tunneling. The simulation results are in good agreement with the experimental data for monolayer and multilayered nanocomposites. The difference can be attributed to neglecting tunneling anisotropy, interface layer, filler alignment, and uniform size-distribution in the model. The two latter factors are expected to have a direct influence on the shielding ability of the nanocomposite. The performances of GNP fillers are relatively poor since a large GNP content is necessary to achieve a shielding effectiveness of 20 dB which is needed for most applications.

It is found that the theoretical prediction for total shielding effectiveness is lower than measured values, which can be attributed to the negative multireflection effect and ignoring the intrinsic properties in the theoretical prediction.

The simulation results indicate that with an increase in the alignment of prolate and oblate ellipsoids (CNT and GNP, respectively), shielding efficiency is affected significantly. The interaction of fillers and the electromagnetic wave is significantly affected by the polarization of the electromagnetic fields. In the case of the parallel oriented electric field to the equatorial axis of nanofillers, the interaction is maximum. However, the dielectric permittivity and shielding effectiveness would certainly be affected by filler alignment. To obtain more accurate electromagnetic characteristics of nanocomposites, the global filler network needs to be solved.

It is also shown that the nanofiller with a maximum aspect ratio has the main contribution to the absorbing performance. For bidisperse systems that have the same average aspect ratio, the best shielding performance is obtained for the system with a maximum aspect ratio. The aspect ratio has more effect on the transmitted power than on the reflected power.

Apart from filler alignment and polydispersity, other factors such as clustering of nanofillers, the interface layer, and crumpling are also expected to affect the shielding behavior. The analytical models would provide some important insights into the impact of such properties but are applicable, in general, for the limiting cases. Further work through modeling is in progress to better understand the electromagnetic properties of nanocomposites.

See supplementary material for the realization of oblate and prolate fillers with volume fraction ϕ = 0.1 .

We would like to thank the Private Sector Program and the Blue Waters sustained-petascale computing project at the National Center for Supercomputing Applications (NCSA), which was supported by the National Science Foundation (Award Nos. OCI-0725070 and ACI-1238993) and the state of Illinois. The authors gratefully acknowledge the use of the Taub cluster resources provided under the Computer Science and Engineering (CSE) program at the University of Illinois. This work was financially supported by the NSF Center for Novel High Voltage/Temperature Materials and Structures [NSF I/UCRC (IIP-1362146)].

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Supplementary Material