Microwave radial dielectric properties of carbon fiber bundle: Modeling, validation and application

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ScienceDirect journal homepage: www.elsevier.com/locate/carbon

Microwave radial dielectric properties of carbon fiber bundle: Modeling, validation and application Wen Hong, Peng Xiao *, Zhuan Li, Heng Luo State Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, China

A R T I C L E I N F O

A B S T R A C T

Article history:

The radial dielectric spectroscopy of the carbon fiber bundle (CFB) array/epoxy resin com-

Received 22 April 2014

posites have been performed over the frequency range from 8.2 to 12.4 GHz. The complex

Accepted 7 August 2014

permittivity of the composite increases with the volume fraction, which is attributed to the

Available online 13 August 2014

increase of polarization, and decrease with the CFB diameter along the electric field direction which is ascribed to the increase of depolarization field. A revised Wiener model is proposed for the calculation of the effective permittivity of the composite, and used to extract the radial permittivity of the CFB. The proposed model is validated by the experimental measurements. The proposed model is further applied to derive the radial permittivity of cylinder carbon fiber and analyze the effect of the bundle geometry on the effective permittivity. Ó 2014 Elsevier Ltd. All rights reserved.

1.

Introduction

Though carbon fibers (CFs) are widely used in various fields [1–5], particular in the field of microwave absorbing [6–9], the research of dielectric properties of the CF lags far behind their applications, and this, to some extent, hinders us from making full application of CFs. Until now, the dielectric properties of CFs at microwave frequencies have not been fully investigated yet, and this is one of the main obstacles for the development of microwave CF absorbers. Generally, there are two principle methods of deriving dielectric constant: direct measurement and indirect calculation from the CF composite or carbon fiber bundle (CFB). However, compared with microwave wavelength, either the size of CF or CFB is too small to be discerned by the microwave wave. Therefore, it is unable to detect the slight field change caused by the single CF or CFB, and thus it may be impossible to measure the permittivity of CF or CFB at microwave frequencies directly. Meanwhile, the method of deriving the permittivity

* Corresponding author. E-mail address: [email protected] (P. Xiao). http://dx.doi.org/10.1016/j.carbon.2014.08.012 0008-6223/Ó 2014 Elsevier Ltd. All rights reserved.

of CF from composite or CFB has been hindered by the lack of effective prediction model due to the unsatisfied description of the permittivity variation of both volume fraction and inclusion geometry. Therefore, the key to obtain the permittivity of CF is to establish an effective prediction model. Good sample reproducibility is the basic premise of the model validation. Results show the random mixture permittivity is easily affected by the type of random distribution and inhomogeneous orientation [10–12]. Moreover, as for the non-random distribution (array distributed), the permittivity along the axes is more sensitive than that in the direction perpendicular to the axes due to the relatively larger dielectric constant. In other words, the samples with same volume fraction and preparing process may bring completely different permittivity in random or parallel polarized array distribution. Meanwhile, it was shown in our previous paper [13], the radially polarized parallel distribution is beneficial to attenuate more microwave inside the material due to the excellent impedance match. Hence, here we restrict ourselves

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to investigate the dielectric response of the radially polarized CFB array, establish the effective permittivity model to derive the radial permittivity of the CFB. Our next step is to derive the axial permittivity of the CFB, thus the anisotropy permittivity of CFB is completely obtained. A number of research works have already been carried out to study the effective permittivity of the radially polarized fiber array. For instance, Sareni et al. [14] reported the fiber array mixture under radial electrical field followed the Rayleigh’s formula and Boundary Integral Equation (BIE) method, but the prediction accuracy will decrease with the increase of volume fraction and permittivity contract [15], and the significant departure was confirmed by experiments latterly [16]. Based on the choice of Mori–Tanaka model, Pre´ault et al. [17] indicated the cylindrical Maxwell–Garnett model could predict periodic fiber composite well. However, the Mori– Tanaka model is only suitable for low fiber concentration. Recently, Neo et al. [18] proposed the permittivity of random CF composite obey phenomenon law in both real and imaginary parts. Based on the above researches, here we propose a revised Wiener model, a new phenomenological mixture model, to predict the effective permittivity of the CFB array composite, and the model takes into the effect of bundle geometry on the effective permittivity. In this paper, the radial permittivity of the CFB (or CF array/epoxy resin) is derived from the CFB array/epoxy resin composite by the revised Wiener model, and then validated by experimental measurements. The proposed model is further applied to extract the permittivity of cylinder CF, analyze the effect of bundle geometry on the effective permittivity of the CFB array composite.

2.

Experimental

As shown in Fig. 1, the CFB array composites are prepared as follows: rows of ellipse CFBs are cut from non-woven carbon fiber cloth (12K Toray T700) to stack on the top of one another. The CFBs are closely arranged in the non-woven cloth. The CF preform is obtained by parallel embedding the rows of the CFBs into the rectangular model, in which all CFBs are vertically aligned. Epoxy resin (E44) is heated to 60 °C where its viscosity can be greatly reduced, resin matrix is obtained by mixing 87 parts of the epoxy with 13 parts of the flexibilizer (Di (n-butyl) phthalate) for epoxy resins. The hardener (ethylenediamine) is added at a concentration of 10:100 (by weight) hardener to mixture (epoxy resin and flexibilizer). For preparing

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the composite a method based on vacuum-assisted resin transfer molding is used. After that the composite is heated 24 h under 80 °C for the curing. The bundle volume filling fraction (vol.f.f) is controlled by the row number. Considering the effect of bundle overall profile on the dielectric response, here we prepare two kinds of CFB array samples: the bundle whose major diameter of transversal section is perpendicular to the electrical field direction is called Perpendicular 1 (P1), the bundle whose minor diameter of transversal section is perpendicular to the electrical field direction is called Perpendicular 2 (P2). The average major and minor diameter of the T700 CFB transversal section is 1.5 and 0.2 mm respectively. All samples are ground to the length of 22.86 mm and the width of 10.16 mm for the X-band waveguide measurement. Except the samples prepared for discussing the effect of fiber length (the sample height varies from 3 to 5 mm), all samples have been ground to the height of 3 mm. The dielectric constant of CFB array composites are measured by Agilent N5230A Vector Network Analyzer (VNA) in X-band, the device is carefully calibrated with Short-OpenLoad-Reciprocal thru (SOLR) approach. The electromagnetic parameters are derived from the scattering (S) parameters with standard S parameter retrieval procedure. In order to ensure the measuring validity and accuracy of the samples, the lossless epoxy resin is repeated measured before other samples.

3.

Results and discussion

3.1.

Measurement validity

The real and imaginary parts of the relative complex permittivity of the epoxy resin sample, the CFB array sample P1 and P2 with 58.9 bundle vol.f.f is compared in the Fig. 2. The measured complex permittivity of epoxy resin is 2.9j0.1 (see black solid and open up-triangle lines in Fig. 2), which agrees well with the reported literature. And the permittivity of P1 is larger than that of P2 in both real and imaginary part, which is the result of the relative size of the CFB along the electric field direction. The polarization of the CFB is believed to be proportional to the electrical field inside CFB, and the inside electrical field of arbitrary shape inclusion is: Ein ¼ eM =½eM þ L  ðein  eM Þ  Eout ;

ð1Þ

where Ein and Eout is the electrical field inside and outside the inclusion, ein and eM are the dielectric constant of inclusion and matrix, and the depolarization factor L follows [19]:

Fig. 1 – Schematic illustration of preparing CFB array composite. (A color version of this figure can be viewed online.)

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ein ðLÞ ¼ c  1=½aðL  bÞ;

Fig. 2 – Frequency spectra of the relative complex permittivity of epoxy resin, CFB array sample P1 and P2 with 58.9 bundle vol.f.f. (A color version of this figure can be viewed online.)

L¼

r1 r2 r3 2

Z 0

þ1

du pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðu þ r21 Þ ðu þ r21 Þðu þ r22 Þðu þ r23 Þ

ð2Þ

where r1 is the semi-axes of the inclusion along the electrical field, r2 and r3 denote the radius of the inclusion in other two orthogonal directions. The effective polarization P of arbitrary shape inclusion can be described as follows:

ð6Þ

where uin is the CFB bundle vol.f.f, ein ðLÞ is the CFB permittivity function of depolarization factor L. And a, b and c are fitting parameters, which are mainly dependent on the permittivity of fiber and matrix, inclusion geometry and distribution. Calculated from the Eq. (2), the depolarization factor of CFB in P1 and P2 sample is 0.089 and 0.82 respectively. So the radial permittivity of the CFB ein ð0:089Þ and ein ð0:82Þ can be derived from the composite permittivity of P1 and P2 by the Eq. (5), the radial complex permittivity of the CFB with L of 0.089 and 0.82 is shown in the Fig. 3. The permittivity of the CFB decreases with the depolarization factor L due to the effect of depolarization field. Because of relative small depolarization field, ein ð0:089Þ still decreases with the frequency, and ein ð0:82Þ keeps constant in this band. It can be calculated from Eq. (4), eeff ð0Þ ¼ ein ð0Þ and eeff ð1Þ ¼ 1 þ eM  ein ð1Þ=eM , combined with ein ð0:089Þ and ein ð0:82Þ, the fitting parameters can be obtained. And the permittivity of the CFB ein ð0Þ can be obtained by setting L as zero in the Eq. (6) (shown in Fig. 4). According to the Transmission Line theory, the closer charpffiffiffiffiffiffiffi acteristic impedance of the material (377 l=e) and free space (377) is, the better impedance match could be achieved, which means more electromagnetic wave could enter the material

P ¼ e0 ðeeff  1ÞEout ¼ e0 ðein  1ÞEin ¼ e0 ðein  1ÞeM =½eM þ L  ðein  eM ÞEout ;

ð3Þ

where eeff is the effective permittivity of the arbitrary shape inclusion. So the effective permittivity can be written as: eeff ¼ 1 þ ðein  1ÞeM =½eM þ L  ðein  eM Þ:

ð4Þ

Refer to the definition of the depolarization factor in Eq. (2), the smaller the diameter of inclusion along the electrical field is, the larger the depolarization factor L is. Due to the smaller diameter along the electric field in P2, the depolarization factor LP2 (0.82) is larger than LP1 (0.089), thus the effective mixture permittivity of P1 (see the red solid and open circle lines in Fig. 2) is larger than that of P2 (see the blue solid and open square lines in Fig. 2). Besides, it is found the dielectric constant decreases with the frequency in P1 sample, which is helpful for the impedance match. Due to the effect of depolarization field, the permittivity keeps constant in P2 sample. The bad impedance match will cause more reflection without being attenuated.

3.2.

Fig. 3 – The radial complex permittivity of the CFB with L of 0.089 and 0.82. (A color version of this figure can be viewed online.)

EM modeling of CFB array composite

In order to predict the complex permittivity of the CFB array effectively, here we propose a new phenomenological mixture model – the revised Wiener model. The model borrows the permittivity variation of volume fraction from the Wiener series model, and takes into the effect of CFB geometry on the permittivity. For this purpose, we proposed a revised Wiener model, which takes into account the effect of inclusion geometry on the effective dielectric constant of the composite. The revised Wiener model for CFB array composite is given by: eeff ¼ uin  ein ðLÞ þ ð1  uin Þ  eM ;

ð5Þ

Fig. 4 – The radial permittivity of the CFB in X-band. (A color version of this figure can be viewed online.)

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to be attenuated. When the permeability remains unchanged, the possible way to improve the impedance match is to decrease the permittivity. As it is shown in the Fig. 4, the real part of the CFB permittivity is not only small, but also decreases with the frequency, which means the perpendicular polarized CFB is capable of good impedance match, and the relative large imaginary part of the CFB permittivity will attenuate electromagnetic wave fast inside the material. In order to validate the effectiveness of CFB dielectric constant, we prepared CFB array samples with different bundle vol.f.f in P1 and P2 orientation to test the accuracy of the proposed model. The comparison between measurement and prediction is illustrated in Fig. 5. As shown in the Fig. 5, the derived CFB permittivity could predict the CFB array composites with different bundle vol.f.f and geometry effectively, especially in P2 samples, which means the revised Wiener model is suitable for extracting the permittivity of the CFB. And the hypothesis of neglecting the interaction between fiber bundles behind the Wiener model is reasonable. But the accuracy of the proposed method decreases with the increase of depolarization L and volume fraction, so the difference in P1 samples is larger than that in P2.

3.3.

Model application

The proposed model is applied to derive the radial permittivity of the CF and investigate the effect of bundle geometry

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(like transversal section geometry and bundle length) on the effective permittivity.

3.3.1.

The radial permittivity of the CF

Based on the permittivity of the CFB derived by the proposed Wiener model, the permittivity of cylinder CF is further derived from the dielectric constant of the CFB by the Eq. (5), the radial dielectric spectral of the cylinder CF is shown in Fig. 6. The permittivity of the radial CF permittivity decrease with the frequency, and the imaginary part is significant to attenuate microwave energy quickly. It can be concluded from the radial permittivity of the CF and CFB, the excellent impedance match of radially polarized CF array is stemmed from fiber property itself, not the array structure. The obtained CF permittivity will not only help the relative CF electrical design, but also assist to analyze the dielectric phenomenon of CF composites.

3.3.2.

Bundle transversal section geometry

After ensuring the effectiveness of the permittivity of the CFB, here we show the variation of dielectric constant with the transversal section geometry (depolarization factor). Fig. 7 shows the relationship between permittivity and depolarization factor at 8.2 and 12.4 GHz, and the variation of bundle transversal section with depolarization is also shown in it. It is shown in the Fig. 7, both real and imaginary parts of dielectric constant decrease with depolarization factor and

Fig. 5 – Calculated and measured frequency spectra of the real and imaginary parts of the relative complex permittivity of the CFB array composites: (a) P2 with 43.5 bundle vol.f.f; (b) P2 with 72.5 bundle vol.f.f; (c) P1 with 43.5 bundle vol.f.f; (d) P1 with 72.5 bundle vol.f.f. (A color version of this figure can be viewed online.)

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with polarization factor. The dielectric constant of the CFB decreases with the frequency, and thus the permittivity at 12.4 GHz is smaller than that at 8.2 GHz at every point.

3.3.3.

Fig. 6 – The radial permittivity of the cylinder CF derived from the permittivity of the CFB. (A color version of this figure can be viewed online.)

Bundle length

Theoretical work believed the radial permittivity of fiber is the function of fiber length when the aspect ratio is small [20]. Will the permittivity of CFB change with the bundle length in mm-size? In order to study the effect of bundle length on the permittivity of CFB array composite, we ground the periodic P2 sample with 43.5 bundle vol.f.f from 5 mm to 3 mm, when measuring the dielectric constant, the permittivity of P2 samples with different height is shown in the Fig. 8a. And the bundle length is equal to the sample height in our samples, so the radial permittivity of the CFB with different bundle length can be derived by the Eq. (5), the comparison is shown in the Fig. 8b. The difference between samples with different bundle length is significant, and it can tell from the above figure, the resonant frequency tends to decrease with the increase of fiber length. This may be caused by the half-wave interference, the resonant frequency is the odd integer multiple of the half wavelength in material: f ¼ ð2n þ 1Þ

Fig. 7 – The relationship between dielectric constant and depolarization factor at 8.2 and 12.4 GHz. (A color version of this figure can be viewed online.)

frequency. With the increase of depolarization factor, the depolarize electrical field becomes larger, and then the local field in inclusion material decrease, thus the polarization of inclusions decrease, which bring the decrease of permittivity

c pffiffi ðn ¼ 0; 1; 2 . . .Þ 4d e

ð7Þ

Submitting the dielectric constant of the 3 mm-thick P2 sample into the Eq. (7), it is found the interference of 5 mmthick sample is more likely occurred at 11.8 GHz when n = 1, which agrees well with the experimental result, and the 3 mm-thick and 4 mm-thick sample is likely resonant at 14.75 and 19.67 GHz. Therefore, the interference frequency of 5 mm-thick sample could be well explained with the data in 3 mm-thick sample by the half-wave interference formula, which indicates the radial permittivity of the CFB does not vary with the bundle length, thus the radial permittivity of CF is also independent of length. This is because the aspect ratio of carbon fiber with mm-size length is already large enough, the change of fiber length or bundle length in mmsize does not change the aspect ratio greatly. So the radial permittivity of CFB is independent of bundle length.

Fig. 8 – The radial permittivity variation with bundle length in: (a) P2 samples; (b) CFB. (A color version of this figure can be viewed online.)

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Conclusion

The permittivity of the CFB has been successfully extracted from the radially polarized CFB array composite, which is validated by samples with different bundle vol.f.f. The radial dielectric constant of cylinder CF is further derived from the CFB. The dielectric resonance of different fiber length is well explained by the interference formula, which means the permittivity of CF and CFB is independent of fiber length, the radial permittivity of the CF and CFB indicates radially polarized CF or CFB is a promising candidate for microwave absorber.

Acknowledgments The project was supported by the National Key Basic Research and Development Program under Grant No. 2011CB605804 and the National Natural Science Foundation of China under Grant No. 51205417.

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