The thermoplastic matrix used in this study is epoxy resin (E-44), and solid PAN carbon fiber (T300-1k, shown in Fig. 1) supplied by Nantong Sengyou Carbon Fiber Company, Jiangsu, China, with an electrical resistivity of 1.6×10^-3Ω•cm and the average diameter of 7.8 μm. The content of C_sf in the epoxy resin matrix varied from 0.25wt%-1wt%, and the carbon fibers were cut into three different lengths: 2 mm, 3 mm and 4 mm, respectively. The C_sf was proportionally weighed and dispersed by an ultrasonic bath at room temperature for 30 min in acetone medium. Filled epoxy resin materials were obtained by mixing liquid epoxy resin with a certain amount of C_sf by means of a Haake mixer. After adding hardener( polyamide resin with low molecular weight 650), the mixtures were stirred and then coated by hand. The hybrid mixtures were postcured at 100℃ for 30 min. The thickness of oriented samples was about 2 mm in this study.

Fig. 1

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	Fig. 1 SEM image of carbon fiber

The cross-section morphologies of the composites were observed by scanning electron microscope (SEM, Model SUPRA55, Zeiss, Germany). The dielectric permittivity of the specimens was measured by the rectangle wave-guide method, which was based on the measurements of the reflection and transmission modules, in the fundamental rectangle wave-guide mode TE10 by Agilent E8326B PNA series network analyzer (Palo Alto, CA). Both the real and the imaginary parts of the permittivity calculated with the reflection and transmission coefficients. For a dielectric material ( μ′ = 1, μ′′ = 0) the relative error varies between 1% (pure dielectric) and 10% (highly conductive materials). When the oriented fibers are parallel to the direction of the electric field, axial permittivity can be obtained shown in Fig. 2(a). Otherwise, radial permittivity can be obtained when the oriented fibers are against to the direction of the electric field shown in Fig. 2(b).

Fig. 2

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	Fig. 2 Sketch of the dielectric test

The optical microscope and SEM images of the oriented composites are shown in Fig.3, indicating the short fibers are well oriented along the shear direction (Fig. 3(a)). As can be seen, the short carbon fibers are physically integrated after postcured at 100℃ which is very important for the C_sfto maintain the dielectric properties in the composites. And there are no agglomerations found (Fig. 3(b)). We suggest that the C_sf used in our research are relatively short, and the mass of C_sf used is relatively low. The material properties of the greatest importance to microwave interaction of a dielectric material are the complex permittivity, ε= ε′ + jε′′. The dielectric constant ε′ of a material is a function of its capacitance^{[ 7]}. On the other hand, the ε″ represents the capacity of dielectric loss in the microwave frequency under an applied electric field^{[ 11]}. The complex permittivities of two types of distribution of carbon fibers over the frequency range from 2.6 GHz to 8.2 GHz were characterized. The real and imaginary components of the complex permittivity for both the two materials as functions of concentration and length as well as distributions are presented in Fig. 4 to Fig.6, respectively.

Fig. 3

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	Fig. 3 The images of optical microscope(×50) (a) and SEM (b) of the oriented composites

The effective complex permittivity is significantly anisotropic. The axial permittivity is shown in Fig. 4 and Fig.5. From Fig. 4, when filled with 3 mm carbon fiber, the real part ( ε′) of the permittivity of composites varies from 42-33 to 98-56 and the imaginary part ( ε′′) varies from 2-6 to 33-62, with the content of C_sf in the range from 0.25wt% to 1wt%. When the content of C_sf is 0.5wt%, the real part get maximum values 42 with 2 mm C_sf and 112 with 4 mm C_sf at 2.6 GHz, while the imaginary part increase from 6 to 30 with increasing fiber length from 2 mm to 4 mm at 2.6 GHz (Fig. 5). It is obviously that the complex permittivity increase with increasing C_sf content and length. And at each length, the real parts of the complex permittivity decrease smoothly while the imaginary part increase slightly with increasing frequency. The radial permittivity data illustrated in Fig. 6 indicates a much weaker interaction with microwaves over this frequency range than the previous specimens. As with the former specimens tested, ε′ decreases smoothly with increasing frequency. Maximum ε′ values occur at the low frequency limit (2.6 GHz). These increase with content from 7 at 0.25wt% to 18 at 1wt% in Fig. 6(a). And the imaginary part increase from 1 to 11, respectively. The magnitude of content dependence generally diminishes with increasing frequency.

Fig. 4

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	Fig. 4 The axial permittivity of the composites with different concentration of 3 mm Csf(a) Real part; (b) Imaginary part

Fig. 5

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	Fig. 5 The axial permittivity of the composites with different fiber lengths (The fiber content is 0.5wt%) (a) Real part; (b) Imaginary part

Fig. 6

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	Fig. 6 The radial permittivity of the composites with different fiber content (The fiber length is 3 mm) (a) Real part; (b) Imaginary part

It is well known that the capacitance of a material is a function of its dielectric constant ε′, and ε′ is proportional to the quantity of charge stored on either surface of the sample under an applied electric field, and the ε″ represents the capacity of dielectric loss. When subjected to an alternating electrical field, the free electrons shift with the alternating electrical field. In order to overcome the electrical resistance, the microwave energy is dissipated and converted to thermal energy, which is the combined effect of relaxation polarization loss and electric conductance loss. Therefore, the loss can be expressed as follows:

(1)

Where ε_C′′ is the loss factor due to conductivity, ε_I′′ is another one due to electronic relaxation polarization. Although polarization plays a role in the imaginary part, free electrons have more effects on it, due to the good electrical conductivity of carbon fibers^{[ 12]}. According to the free electric theory, ε′′ could therefore be obtained to be^{[ 11]}

(2)

Eq. (2) shows that σ plays the dominating role in ε′′.

To explain the effects of fiber concentration, length, and distribution on the dielectric properties, the following simple model has been proposed. The anisotropic distribution of C_sf (Fig. 7(a)) is approximated by a network of fibers separated by the insulating EP polymer matrix. Each fiber is assumed to be conductive with an equivalent resistor, and coupled to its neighbor by a capacitor (Fig. 7(b)). Under an applied electric field, only coupling capacitors and resistors in the electric field direction are considered. And we assume that the electric field distribution between the coupled conductors of the microstrip is uniform and parallel or perpendicular to the fiber. In this way, the model comes to be a two layer microstrip transmission line (Fig. 7(d)) formed by the series connection of a resistor ( R_tot) and a capacitor ( C_tot) (Fig. 7(c)). The capacitor is modeled in the two-layer structure of Fig. 7(d) by a corresponding capacity for the up layer.

Fig. 7

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	Fig. 7 The two layered microstrip transmission line model on fiber-filled composites

For the both situations, the complex permittivity increase with the increasing carbon fiber content. The value of C_tot is inversely to the distance between fibers. The distance between inclusions decreases with increasing concentration. For the low concentration, the interaction between coupled fibers is much weaker than high concentration, meaning lager thickness H, otherwise, leading to a higher capacitance and, hence, a larger value of C_tot, characterized by ε′. The ε′ also increase with increasing fiber length. It is observed that ε′=80 is obtained when the sample is added with 0.75wt%, 4 mm fiber, and 79 is obtained when the sample is added with 1wt%, 3 mm fiber. In order to get higher value of ε′, we can use longer fiber instate of increasing fiber concentration, meaning a smaller distance between conductive inclusions and hence, a higher coupling. The two layered model for the fiber is able to predict the conductivity of the composites. By increasing concentration or fiber length, the thickness of conductive layer H_iis enhanced, corresponding, the higher conductivity σ in Eq. (2). It means higher energy loss, characterized by ε′′.

By comparing Fig. 4 and Fig. 6, it is observed that the effective complex permittivity is significantly anisotropic. For example, the real part of the axial permittivity is about five times (filled with 1wt% 3 mm fiber) as high as that of the radial permittivity. In the two layered transmission line model, there are more capacitors in the series connection system, because each resistor is shorted in the direction of each transmission line when the electric field is perpendicular to fibers. As we know, C_tot is calculated by the following law:

(3)

When the capacitors series are connected, the more coupled conductors, the lower C_tot value. On the other hand, the conductivity parallel to the direction of carbon fibers is higher than that perpendicular to the carbon fiber direction, because that the current could not cross the space among fibers and could only flow between the cross sections^{[ 13]}. Those may be the reasons why we can get high axial permittivity.

From the experimental studies, it was found that the distribution of embedded fibers affected the dielectric permittivity of composites at microwave frequencies. Composites with axial distributed (parallel to electric field) show higher permittivity compared to that with radial distributed (perpendicular to electric field) fibers. The results also showed that both the fiber content and length can be used to adjust the dielectric properties of the composites. The two layered transmission line model formed by the series connection of a resistor and a capacitor can explain the phenomena well. Both capacitivity and conducticity were considered in this model.