Conductive composites possessing a polymeric matrix have been developed as an auspicious class of materials yielding superior properties to metal-based materials. The electromagnetic shielding effectiveness and bandgaps of a novel interpenetrating phase composite with a polymeric matrix are studied computationally. This composite is generated from a so-called Schwarz Primitive surface, a member of the triply periodic minimal surfaces family. The shielding effectiveness of the resulting Primitive-based composite is compared with those of composites reinforced with periodically and randomly distributed spherical conductive particles. For the composites with random spherical particles, the random sequential addition method is used to generate the realizations of fillers followed by the Monte Carlo relaxation step to obtain an equilibrated configuration. The Primitive-based composite shows higher shielding effectiveness due to the interconnectivity of both phases (conductive phase and polymeric matrix) leading to a higher effective electrical conductivity. Employing a finite element analysis leads to dispersion curves, which reveal the existence of electromagnetic bandgaps at low frequencies and low volume fractions of the conductive phase, in comparison to those of other structures reported in the literature. The Primitive-based composite shows the bandgaps for transverse-electric modes, where the widths of the bandgaps vary with the volume fraction of the conductive phase.

## I. INTRODUCTION

Most polymers possess a very poor electrical conductivity.^{1,2} Effective conductivity of such composites can be increased by introducing conductive inclusions into polymeric resins. Conductive polymer-based composites become more prevalent, as they can serve as alternatives to metals due to their versatility, lightweight, lower cost, no corrosion, good mechanical properties, tailored electrical properties, and manufacturability.^{1,3} One application of conductive polymers is for the use as electromagnetic interference (EMI) shielding materials. The capability to shield depends on several factors including the volume fraction, the morphology and chemical compositions of phases, and the temperature of the composite.^{3,4}

The morphology of phases significantly affects the electrical [including the shielding effectiveness (SE)] and mechanical properties of composite materials.^{5,6} Recently, interpenetrating phase composites (IPCs) have been investigated by many researchers due to their robust physical and mechanical properties compared to those of conventional composites.^{7,8} The IPCs are composites with microstructures characterized by continuous and interconnected phases.^{9} In other words, if one of the phases is removed from the microstructure, then the remaining phase(s) will form a self-supporting open-celled foam that preserves the load bearing capability and structural integrity.^{8,9} Additionally, the interconnection and continuity of IPCs’ phases allow one to simultaneously benefit from each continuous phase leading to superior multifunctional materials.^{7,10} For example, having a continuous conductive phase leads to a higher conductivity for the composite. The microstructure of IPCs can be either periodic or random, and the resulting composite can be either isotropic or anisotropic.^{8,11}

The IPC discussed in this paper is based on the Schwarz Primitive surface shown in Fig. 1. The Schwarz Primitive surface is a member of the triply periodic minimal surfaces (TPMS) family.^{12–14} TPMS are infinitely extending, continuous, and smooth surfaces that are locally area minimizing which means any sufficiently small patch taken from the TPMS has the smallest area among all the patches created under the same boundaries. Moreover, TPMS divide the space into two intertwined regions, where each region is periodic and has a volume fraction of 50%.^{15} Also, the sum of principal curvatures at each point on the TPMS is zero; hence, the mean curvature of TPMS is zero.^{16,17} TPMS can be used to design and fabricate promising structures (IPCs or cellular materials) that can target various applications.^{15,16,18} The Primitive surface can be described using the following level-set approximation technique^{19,20}

where *C* is a constant.

The proposed IPC consists of a solid sheet attained from thickening the Primitive surface embodying the conductive phase and its complementary cube representing the polymeric matrix. Figure 2 portrays the unit cell of the Primitive interpenetrating phase composite (P-IPC) and the transition of the Primitive sheet (conductive phase) through the matrix. Abueidda *et al*.^{16,21} studied the electrical conductivity of the P-IPC experimentally and numerically and showed its high electrical conductivity compared to other structures with similar volume fractions. This high electrical conductivity induced by the Primitive structure should lead to high shielding effectiveness which is vital for commercial and military applications such as the weather monitoring, vehicle detection, air traffic control, and defense tracking. Other researchers have found that Primitive surface-based composites yield an optimum performance when two transport properties (electrical and thermal conductivities) and/or mechanical properties (shear and bulk moduli) compete with each other.^{12,22,23}

The propagation of electromagnetic (EM) waves can be manipulated or cloaked based on the choice of the geometry and materials of the structure.^{24,25} Heo and Yoo^{26} used shape optimization and the finite element method to design 2D cloaking dielectric structures that manipulate the propagation path at a specific microwave frequency. They also studied the effect of material properties on the size of the cloaking structure.

In this paper, the electrical and EMI shielding effectiveness (SE) properties of the P-IPC are studied. The SE of a material refers to the ratio of the power received with and without the presence of the material under the same incident power and is quantified in decibels (dB).^{1–3} The primary shielding mechanisms are the reflection, absorption, and multiple-bounces.^{27–29} Shielding via reflection takes place in materials holding mobile charge carriers interacting with the incident EM wave, while shielding via absorption requires a material with electrical or magnetic dipoles interacting with the incident EM wave. Hence, shielding by absorption depends on the thickness of the material. On the other hand, shielding may be resulting from a multiple-bounce mechanism which refers to the internal reflections occurring in the shielding material. Reflection is usually the dominant shielding mechanism for a homogeneous conductive material, whilst absorption represents the second important shielding mechanism.^{3,27,28} Shielding by multiple reflections can be ignored in case of a thick shield compared to its skin depth.^{30}

In addition to the shielding effectiveness, the EM bandgaps of the P-IPC are also investigated. There has been an increasing interest in studying and designing structures with the EM bandgaps. The structures with a complete EM bandgap, called photonic crystals, forbid the EM waves to propagate along any direction.^{31} The EM bandgaps have many engineering applications such as antennae, power amplifiers, and frequency selective reflection surfaces.^{32,33} Michielsen and Kole^{20} studied the photonic bandgaps of several structures and investigated the effect of structure symmetry and topology on the bandgap formation. Dolan *et al*.^{34} provided a review of the optical properties of structures inspired by the Gyroid surface, another member of the TPMS family, at different length scales. They argued that the Gyroid structure possesses complete photonic bandgaps provided that appropriate volume fractions and large enough dielectric contrasts for the phases are assigned. One of the crucial components of the bandgap analysis of a periodic structure is the employment of the Floquet-Bloch theory. When employed appropriately, it makes the finite element analysis much faster as only one unit cell is required to be meshed to complete the analysis.^{35} Mias *et al*.^{36} employed the finite element method together with the Floquet-Bloch theory to find the dispersion curves of 2D and 3D periodic structures. Also, Fietz^{37} derived an absorbing boundary condition in the frequency domain for the Bloch eigenmodes of periodic structures. The derivation of this boundary condition is based on the orthogonality relation for the Bloch eigenmodes. He tested this boundary condition numerically and showed its validity.

## II. FINITE ELEMENT ANALYSIS

The propagation of EM waves in the P-IPC and two different particulate-based composites is studied in this paper. The analyses are performed using the finite element software COMSOL Multiphysics^{38} to calculate the shielding effectiveness of the composites. The governing equation of EM waves in the frequency domain can be written as

where $\mu r$, $\u03f5r$, and $\sigma $ are the relative permeability, relative permittivity, and conductivity of the material, respectively; $E$ is the electric field vector, $\u2207\xd7$ is the curl operator, $\omega $ is the angular frequency, and *i* is the unit imaginary number; $k0$ is the wave number of free space, defined as $k0=\omega \u03f50\mu 0,$ where $\u03f50$ and $\mu 0$ are the permittivity and permeability of free space, respectively. A perfect electric conductor is applied on the four lateral sides of the waveguide, $n\xd7E=0,$ where $n$ is the normal unit vector at any point on the lateral sides. The waveguide considered here has two rectangular ports. At port 1, a wave with the transverse mode $TE01$ and incoming source power $Pi$ is sent. The scattering parameters $Sij$ (or *S*-parameters) are then measured in terms of phase and magnitude which are characteristic of the composite connected between the two ports. Here, $|S11|2=Pr/Pi$ corresponds to the ratio of the power reflected back at port 1 to the incoming source power $Pi,$ whereas $|S21|2=Pt/Pi$ is related to the normalized transmitted power from port 1 to port 2 for the incoming power $Pi$. The normalized absorbed power ($Pa$) by the composite can be calculated by

The SE is obtained by using the normalized transmitted power, and it can also be written as a sum of SER and SEA

where *i* stands for incident and *t* stands for transmitted powers. SER represents the reflection loss (shielding effectiveness by reflection) caused by reflection at the left and right interfaces, and SEA represents the absorption loss (shielding effectiveness by absorption) of the wave as it propagates through the slab.

The EM bandgaps are attained based on the use of one unit cell where Floquet periodic boundary conditions (sometimes also called Bloch periodic boundary conditions) are applied at the faces of the unit cell^{39}

where *a* and $(kx,ky,kz)$ are the side length of the cubic unit cell and components of the Floquet wave vector $kF$. The magnetic field can be found using

The eigenmodes and eigenfrequencies are obtained by scanning the wave vector in the first irreducible Brillouin zone demonstrated in Fig. 3, where the red lines illustrate the sides of the cube bounding one unit cell. Utilizing the concept of the Brillouin zone helps one to restrict one's attention to a finite zone in the reciprocal space in which there is no redundancy in the wave vector; for more details about the Brillouin zone, see, e.g., Ref. 40.

## III. RESULTS AND DISCUSSION

This section discusses the shielding effectiveness and EM bandgaps of the P-IPC. The finite element analyses are carried out using 10-node tetrahedral elements, with the accuracy of the results ensured by a mesh sensitivity study. Subsection III A discusses the shielding effectiveness of the P-IPC, while Subsection III B talks about the microwave EM bandgaps of the P-IPC.

### A. Shielding effectiveness

If an EM wave propagates from left to right in free space with a characteristic impedance $\eta 0=\mu o/\epsilon o(=377\Omega )$ into a homogeneous slab of a conductive composite with the characteristic impedance $\eta =i\omega \mu /(i\omega \epsilon \u2212\sigma )$ and thickness *t*, the first effect is the reflection at the left surface of the composite. The portion of the incident electric field that is reflected is given by the reflection coefficient for that surface, which is$R=(\eta 0\u2212\eta )/(\eta 0+\eta )$.

When the incident wave having a transverse electric field amplitude$Ei$ transmits into the composite, the portion of the wave that crosses this surface propagates within the slab and this transmitted wave has a generalized propagation constant given by $\gamma =i\omega \mu (\sigma +i\omega \epsilon )=3D\alpha \u2212i\beta $. As the wave passes through this conductive slab, its amplitude is attenuated according to the factor $e\u2212\alpha t$, and the wave accrues the phase shift. This attenuation is called the absorption loss. The first transmission through the right interface of the slab leads to a transmitted wave with fields given by $(1+|R|2)e\u2212\gamma tEi$. The infinite series of reflections and transmissions of the wave within the composite leads to outgoing waves, with the *n*-th multiple-reflected wave given by $En=(1+|R|2)e\u2212\gamma t(Re\u2212\gamma t)2nEi$. Considering all the reflected waves leads to a total transmitted wave $En=Ei(1+|R|2)e\u2212\gamma t\u2211n=0\u221e\u2061(Re\u2212\gamma t)2n$. Each successive multiply reflected wave gains a factor $(Re\u2212\gamma t)2$. This wave is greatly attenuated through the slab if *t* is much larger than the skin depth (*δ*) of the composite, and then only the first transmitted wave $(1+|R|2)e\u2212\gamma tEi$ can be considered as the total transmitted field. A partially filled rectangular waveguide (WR90) with a TE_{10} incident mode is considered as shown in Fig. 4(a). Composite specimens of standard size 22.86 mm × 10.16 mm × 2.54 mm were used for the EMI SE characterization in the frequency range of 8.5–12.4 GHz (X-band). The sample thickness is 2.54 mm. The matrix and filler properties are listed in Table I.

. | Conductivity (S/m) . | Relative permittivity . | Relative permeability . |
---|---|---|---|

Polycarbonate | $10\u221213$ | 2.9 | 0.866 |

Carbon | $108$ | 2.6 | 1 |

. | Conductivity (S/m) . | Relative permittivity . | Relative permeability . |
---|---|---|---|

Polycarbonate | $10\u221213$ | 2.9 | 0.866 |

Carbon | $108$ | 2.6 | 1 |

Figure 5(a) shows the EMI SE of P-IPC composites with volume fractions: 2, 4.5, and 10% of conductive phase. It should be mentioned that the EMI SE variation with the frequency is rather small, but it increases with the volume fraction. This increase in the EMI SE with the increase in the carbon content is due to the larger source of free electrons in the material that can interact with the incoming EM wave. For high volume fractions, the dominant mechanism is SER as can be seen in Fig. 5(b). The shielding is mostly gained by reflection since the conductivity is high, so the incident power is reflected at the first interface.

To assess the performance of the composite discussed in this study, a simulation of a composites with spherical carbon fillers with the same volume fractions is implemented. Figure 6(a) shows a realization of random spherical carbon fillers (particles, much smaller than the smallest EM wavelength) in the polycarbonate composite. The random sequential addition (RAS) method is used to generate the realization of fillers followed by the Monte Carlo (MC) relaxation step to obtain an equilibrated configuration. For each particle, the position is chosen randomly within the sample size. Fillers are added one by one until the desired volume fraction is achieved, ensuring that the newly added particle does not overlap with any of the existing fillers.

For the sake of comparison between the performance of the two composites for shielding applications, the properties of the phases in the P-IPC studied in this paper are the same as the properties of the matrix and particles for the random spherical composite. Figure 6(b) shows the shielding ability of random spherical fillers for the same volume fractions. It can be observed that the spherical fillers show a non-dispersive behavior for all volume fractions.

Table II shows the comparison between the composite with spherical carbon fillers and the P-IPC studied in this paper. It reveals that for the same volume concentration, the P-IPC has a better shielding performance, which confirms that the high conductivity is essential for good shielding. For spherical carbon fillers, the high volume fraction of particles is needed to form a percolating path which leads to a high conductivity.^{43–45} It has been shown that a high $\sigma /\omega \epsilon $ leads to a higher shielding ability.^{3,46}

Also, a periodic unit cell with a spherical inclusion (particle) in the matrix is studied; it represents a periodic composite with spherical inclusions as shown in Fig. 7(a). The objective is to compare the performance of the P-IPC composite with another periodic structure having the same volume fractions and material properties of phases, and same sample thickness. The simulation results are shown in Fig. 7(b) for 2%, 4.5%, and 10% volume fractions of fillers. Figure 7(b) and Table II indicate that the P-IPC has a much higher shielding ability compared with the periodic spherical composite. For a 10% volume fraction, periodic spherical composite exhibits around 3 dB in the X-band frequency range. However, for the P-IPC, it is around 64 dB, for the same volume fraction and thickness. As mentioned before, the percolation of the conductive phase in P-IPC leads to a higher reflection and dissipation of the incident EM power in this type of composite.

Volume fraction of the conductive phase (%) . | P-IPC (dB) . | Random spherical fillers (dB) . | Periodic spherical fillers (dB) . |
---|---|---|---|

2 | $63$ | 15.5 | 1.61 |

4.5 | $65$ | 23 | 1.94 |

10 | $69$ | 41 | 2.73 |

Volume fraction of the conductive phase (%) . | P-IPC (dB) . | Random spherical fillers (dB) . | Periodic spherical fillers (dB) . |
---|---|---|---|

2 | $63$ | 15.5 | 1.61 |

4.5 | $65$ | 23 | 1.94 |

10 | $69$ | 41 | 2.73 |

### B. Bandgap analysis

The propagation of EM waves in periodic media has been modeled by the finite element method.^{47–49} Prior works reported the formation of the EM bandgaps in materials based on the TPMS.^{50,51} Here, the transverse-electric (TE) mode bandgaps of the P-IPC are studied. The same material properties, volume fractions, and unit cell dimensions as the ones used in Subsection III A are used for the bandgap analysis. Figures 8–10 illustrate the bandgaps of the P-IPC for three different volume fractions. Several partial TE bandgaps are induced by the architecture of the Primitive surface and mismatch of the material properties of the two phases. However, two complete TE bandgaps exist for each volume fraction over a significant range of frequency. The complete TE bandgap refers to the range of frequencies in which the propagation of TE waves is forbidden in all directions of the wave vector.

The shaded regions shown in Figs. 8–10 represent the extents of the bandgaps for a specific volume fraction. A useful characterization of the bandgap is the gap-midgap ratio $\Delta \omega /\omega m,$ where $\Delta \omega $ is the width of the bandgap and $\omega m$ is the frequency in the middle of the gap. The advantage of using this measure is that the ratio is independent of the scale (unit cell size). The first bandgap takes place before the first band, while the second bandgap occurs between the second and third bands. The width of the first TE bandgap is almost the same for the different volume fractions $(\Delta \omega /\omega m\u22482.0)$. On the other hand, the width of the second bandgap increases with the increase of the volume fraction of the conductive phase. The widths of the second bandgap of the P-IPC at 2%, 4.5%, and 10% volume fractions of the conductive phase are $\Delta \omega /\omega m\u22482.26$, ≈2.9, and ≈3.74, respectively. Also, when the volume fraction of the conductive phase is increased, the second TE bandgap starts at lower frequencies. The P-IPC has unique bandgap diagrams when it is compared to the structures available in the literature; the P-IPC forms two bandgaps at low frequencies and low volume fractions of the conductive phase.^{40,52}

Furthermore, it has been found that the P-IPC with the configurations (material properties of the phases, volume fractions, unit cell dimensions) mentioned in Sec. II and III A does not possess any complete TM bandgaps. The formation of TE and TM bandgaps of the P-IPC agrees with the conclusions given in the literature: complete TE bandgaps could be achieved by embedding inclusions with a lower dielectric constant in a matrix with a higher dielectric constant.^{40,53} These prior findings explain the formation of complete TE bandgaps for the P-IPC while the complete TM bandgaps do not exist. The formation and width of bandgaps are functions of the material properties of the phases, phase arrangements (architectures), phase volume fractions, and the unit cell size.

## IV. CONCLUSIONS

The shielding effectiveness of an interpenetrating phase composite based on the Primitive surface is studied. The shielding effectiveness of the P-IPC is compared to two types of composites: with periodic and random spherical inclusions. The shielding effectiveness of the P-IPC is found to be much higher than those of the other two matrix-inclusion composite types. Both phases (conductive phase and polymeric matrix) in the P-IPC are connected in 3D at all volume fractions. This connectivity of the phases induces higher shielding effectiveness even at low volume fractions when compared to the other two composites types. Besides, the TE bandgap diagrams of the P-IPC at different volume fractions are constructed, and it is found that the bandgaps reported in this paper are wider than those of other structures available in the literature, and these bandgaps exist at low volume fractions and low frequencies.

## ACKNOWLEDGMENTS

The authors gratefully acknowledge partial support from the NSF (Grant No. IIP-1362146; I.M.J. and M.O.-S.) and Computational Science and Engineering (CSE) Fellowship (P.K.), and the use of the campus cluster resources provided under the CSE program at the University of Illinois.