Modeling and experimental verification of dielectric constants for three-dimensional woven composites

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1 Available online at Composites Science and Technology 68 (2008) COMPOSITES SCIENCE AND TECHNOLOGY Modeling and experimental verification of dielectric constants for three-dimensional woven composites Lan Yao a,b, Xin Wang a,b, Fei Liang a,b,ruwu c, Bin Hu a,b, Yani Feng a,b, Yiping Qiu a,b, * a Key Laboratory of Textile Science and Technology (Donghua University), Ministry of Education, China b College of Textiles, Donghua University, Shanghai , China c School of Materials Science and Engineering, Beihang University, Beijing , China Received 19 August 2007; received in revised form 23 December 2007; accepted 29 January 2008 Available online 6 February 2008 Abstract In order to determine the dielectric constants of 3D orthogonal woven single fiber type (SFT) and hybrid composites from their component dielectric properties, a theoretical model is proposed based on the rule of binary mixtures. The model shows that with the same fiber volume fraction, a component with a larger cross-sectional area perpendicular to the electric field has a greater contribution to the composite dielectric constant. For experimental verification, SFT basalt/epoxy and aramid (Kevlar 129)/epoxy as well as interply and intraply basalt/aramid/epoxy 3D orthogonal woven hybrid composites were fabricated and their dielectric properties were measured using the waveguide method at a frequency range of 8 12 GHz. At 10 GHz, the experimental results agreed well with the calculated results from the model for the SFT composites, while a positive hybrid effect on the dielectric constant was observed for the two hybrid composites. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: A. Textile composites; A. Hybrid compounds; B. Electrical properties; B. Modeling; A. Aramid fiber 1. Introduction When composites are used as interconnections, printed circuit boards, and airplane skin materials, their dielectric properties become very important to their electrical transparency or microwave reflection. Therefore dielectric properties must be determined before they can be used in these applications. For polymer composites, the dielectric properties are associated with the component volume fractions of the composite [1 3]. Therefore, the component volume fractions are the only independent variables generally considered for estimating the dielectric constant of a composite. The following equation is the most well known the rule of mixtures (ROM) for calculating the dielectric constant * Corresponding author. Address: College of Textiles, Donghua University, Shanghai , China. Tel.: ; fax: address: ypqiu@dhu.edu.cn (Y. Qiu). of a composite initially proposed by Hippel [4] and widely adopted in the literature [5 7] e r ¼ X V i e ri ; ð1þ i where V i and e ri are the volume fraction and the dielectric constant of the ith component. In addition, the fiber orientation in the electric field has also been considered an important factor for the dielectric properties of a composite [8,9]. Seo et al. [9] have investigated the influence of the angle between the fiber orientation and the electric field vector on the dielectric constants of unidirectional E- glass/epoxy composites. Based on their analysis, a model was proposed to include the fiber orientation factor in calculation of dielectric constants for composites. Three dimensionally reinforced composites are introduced to prevent delamination in laminated composites. Among all 3D composites, 3D orthogonal woven composites have the most ordered structures and thus are more ready for embedding electronics and other smart compo /$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi: /j.compscitech

L. Yao et al. / Composites Science and Technology 68 (2008) 1794 1799 1795 nents such as sensors and actuators than other types of 3D composites [10].

2 L. Yao et al. / Composites Science and Technology 68 (2008) nents such as sensors and actuators than other types of 3D composites [10]. In addition, fibers with good dielectric properties may not have good mechanical properties while fibers with superior mechanical properties may not be suitable for embedment of electronics. Therefore, composites with both good electric and adequate mechanical properties are often designed as hybrid composites containing two or more different types of reinforcement fibers [11 13]. Structures of hybrid composites may be classified as interply hybrids, intraply hybrids, intimately mixed (intermingled) hybrids, selective placement and super hybrid composites [14]. Our previous study reported that dielectric properties of aramid/glass/epoxy hybrid composites were dependent on the stacking of the two types of fiber layers [15]. However, no theoretical model other than the simple ROM has been reported in literature to calculate the dielectric constants of a 3D single fiber type (SFT) or hybrid composites. In this study, a theoretical model is proposed for prediction of dielectric constants of 3D SFT and hybrid composites. The dielectric constants of SFT basalt/epoxy and aramid/ epoxy as well as interply and intraply basalt/aramid/epoxy composites were determined experimentally and compared with the model predicted values. The selection of basalt fibers is because it has desirable properties such as high tensile strength, high tensile modulus, and excellent heat resistance. It is produced in a similar way as glass fibers [16] using basalt rock which is an over-ground, effusive volcanic rock with 45 52% SiO 2. For laminated composites, it is relatively easy to use these equations since the laminas in a laminated composite can be represented by parallel capacitors and the overall capacitance is simply the summation of all the individual capacitances. Of course, for multidirectional laminates, the above equations cannot be directly used without a change of basis as proposed in the literature [9]. For 3D orthogonal woven composites, the arrangement of yarns is more complicated and therefore the models for unidirectional composites will have to be modified. In order to do so, a representative volume element (RVE) for a 3D orthogonal woven composite is isolated as shown in Fig. 1 in which, we assume that there are n columns, m rows, and l layers. The electric field is parallel to the Y direction. The cross-section of the composite perpendicular to the electric field direction is composed of n m subareas, namely A ij, i =1,2,...,n and j = 1,2,...,m. The thickness of the layers can be represented by d k, k =1, 2,..., l. Thus, the composite RVE can be divided into n m l subunits, each of which can be represented by a capacitor, C ijk. The equivalent circuit including all capacitors representing all subunits in the composite is shown in Fig. 2 from which a model can be derived. Since all subunits with the same sub-area A ij are serially aligned parallel to the electric field direction, they can be represented as capacitors in series in the equivalent circuit. For each of these series, the equivalent capacitance can be calculated as 2. Theoretical modeling Theoretical models for predicting dielectric properties of a 3D woven composite can be constructed on the basis of the electric circuit model proposed by Chin and Lee [17]. They have proposed a modeling method in which dielectric properties of materials in an alternating electric field are simulated by an electric circuit composed of a parallel resistor R and a capacitor C [17]. The capacitor represents the dielectric constant and the resistor represents the dielectric loss. To estimate the dielectric constant of a composite, an equivalent circuit containing only capacitors can be used. According to Chin and Lee [17], in fiber oriented direction, the dielectric constant of a unidirectional composite, e 0 1,as a function of those of its component fiber, e 0 f, and matrix, e 0 M, can be expressed as e 0 1 ¼ e0 f v f þ e 0 M ð1 v fþ; ð2þ where v f is the fiber volume fraction. This is similar to the Voigt model in micromechanics of composites [18]. In the transverse to the fiber axial direction, as similar to the Reuss model in micromechanics of composites [18], the relation can be expressed as e 0 2 ¼ e0 M þ p pffiffiffiffi v 4 fðe 0 f e 0 M Þ 1 þ p e 0 pffiffiffiffi : ð3þ f 1 ð v 4 e 0 f vf Þ M Fig. 1. Schematic of a RVE for a 3D orthogonal woven composite.

1796 L. Yao et al. / Composites Science and Technology 68 (2008) 1794 1799 C 111 C 121 C nm1 C 112 C 122 C nm2 C 11l C 12l C nml Fig. 2.

3 1796 L. Yao et al. / Composites Science and Technology 68 (2008) C 111 C 121 C nm1 C 112 C 122 C nm2 C 11l C 12l C nml Fig. 2. Equivalent circuit of capacitors representing a 3D orthogonal woven composite. C ij ¼ Xl k¼1 1 C ijk! 1 : ð4þ Since the composite RVE is composed of n m parallel packed such series, the capacitance for the whole RVE can be simply written as! 1 C unit ¼ Xn X m C ij ¼ Xn X m X l 1 : ð5þ j¼1 j¼1 k¼1 C ijk The relation between the capacitance and the corresponding dielectric constant for each subunit is C ijk ¼ e 0 A ij ijk : ð6þ d k And for the whole RVE, C unit ¼ e 0 A unit d ; P m j¼1 A ij and d ¼ P l k¼1 d k. where A ¼ P n Replacing all the capacitances in Eq. (5), we have e 0 unit ¼ Xn ð7þ " X m X l # 1 d k A 1 : ð8þ d A j¼1 k¼1 ij e 0 ijk This is the general equation for calculation of dielectric constant of any 3D orthogonal woven composite. This is a combination of Voigt model and Reuss model using parallel series decomposition method as adopted in micromechanics to predict the effective elastic moduli of composites [18]. For a SFT 3D orthogonal woven composite as shown in Fig. 3, n=m=l=2. The equation can be written as e 0 ¼ e 0 1 A 21 A þ e 0 M e0 2 þ e 0 2 A 12 A A 11 =A e 0 M d 2=d þ e 0 2 d 1=d þ A 22 =A e 0 2 d 2=d þ e 0 M d 1=d ; ð9þ where e 0 1 and e0 2 are the dielectric constant of the composite along the fiber oriented direction and that along the transverse direction to the fibers, respectively, and e 0 M is the dielectric constant of the matrix corresponding to the net resin pocket. Fig. 3. Schematic of a RVE for a single fiber 3D orthogonal woven composite. For 3D hybrid composites, RVEs representing an interply and an intraply hybrid composite are shown in Fig. 4. In both structures, the RVE is composed of subunits, or n=m=l= 4. It would be a very complicated equation if we tried to use an equation similar to Eq. (9). However, with the help of a computer, it is relatively simple to calculate dielectric constants for both hybrid composites using Eq. (8). It is obvious that when the layer thickness ratio, d k /d, decreases, or the area ratio, A ij /A, increases, the contribution of the individual component s dielectric constant, i.e. e 0 1, e0 2,ore0 M, to the overall dielectric constant becomes more significant. In other words, a component with a large area perpendicular to the electric field vector but small thickness will be more significant than a component with a small area but large thickness assuming they both have the same volume fraction. This model takes into account the shape of the reinforcement instead of just the component volume fractions as in the traditional ROM model. 3. Experimental 3.1. Materials Basalt fibers of 6144 dtex and aramid fibers (Kevlar Ò 129) of 3140 dtex were used to weave the 3D performs which were consolidated with Epoxy 618 (q = 1.25 g/cm 3 ) and hardener Iminazole 5510 (q = 1.19 g/cm 3 ) from Shanghai Resin Company, whose chemical and mechanical properties were similar to those of Epon 828. Basalt fiber type 3000 was provided by Shenzhen Research Institute, Harbin Institute of Technology in China. The physical properties of the fibers and the resin are presented in Table Fabrication and consolidations Four reinforcement geometries with six warp and seven weft layers were adopted in making the composites, namely

L. Yao et al. / Composites Science and Technology 68 (2008) 1794 1799 1797 Fig. 4. Schematic of a RVE of the interply hybrid (left) and the intraply hybrid (right) 3D orthogonal woven composites.

4 L. Yao et al. / Composites Science and Technology 68 (2008) Fig. 4. Schematic of a RVE of the interply hybrid (left) and the intraply hybrid (right) 3D orthogonal woven composites. Table 1 Physical properties of the fiber and resin Properties Aramid fiber Basalt fiber Epoxy resin Linear density (tex) Tensile strength (MPa) Modulus (GPa) Elongation at break (%) Density (g/cm 3 ) Dielectric constant (10 GHz) Loss tangent (10 GHz) interply hybrid, intraply hybrid, single basalt type, and single aramid type composites. In the interply hybrids, aramid yarns or basalt yarns were placed in different layers while in the intraply hybrids, the two types of yarns were placed alternately in each layer of warp or weft (see Fig. 4). The fabric counts in warp and weft directions were 5 ends/cm and 5 picks/cm while that of the Z-yarn was also 5 ends/ cm. The thickness of the weft yarns was twice as much as that of the warp yarns and the Z-yarns are all aramid yarns except for the single basalt type composites. Consolidation of the preforms was achieved by the vacuum-assisted resin infusion method, with a 2 h curing at 80 C, followed by half an hour post-curing at 100 C Dielectric property test Measurements of fundamental dielectric properties were carried out on an Agilent 8722ES Vector Network Analyzer (VNA) system using the waveguide method. The waveguide system, short-circuited at one end, was connected with VNA through a coaxial connector as shown in Fig. 5. Five specimens were tested for each type of composites at five frequencies (8, 9, 10, 11, 12 GHz) in the range of X-band. The specimens were cut into the size of mm along the warp direction to fit the size of the waveguide holder. All the specimens were dried in a desiccator for 24 h before testing. 4. Results and discussion A complex dielectric constant is expressed as e r ¼ e 0 r ie00 r. The real part e0 r and the imaginary part e 00 r are known as dielectric constant and dielectric loss, respectively. The dielectric constant and dielectric loss with respect to the frequency for the four types of composites are shown in Figs. 6 and 7. Both the dielectric constant and the dielectric loss of the basalt composite increase as the frequency increases. This could be explained as the phenomenon of reaching an electronic resonance. The other three types of composites show a slow decreasing trend as the frequency increases, which may be in the range of dipolar relaxation and electronic resonances. It is noticed that the dielectric constant of the aramid fiber composite presented a relatively faster decreasing trend than the Source Network analyzer R Receiver Material T Computer Fig. 5. Schematic of the experiment setup for rectangular waveguide method.

5 1798 L. Yao et al. / Composites Science and Technology 68 (2008) Dielectric constant Frequency (GHz) Fig. 6. Dielectric constants versus frequencies in X-band of the four types of composites. Dielectric loss Frequency (GHz) Fig. 7. Dielectric losses versus frequencies in X-band of the four types of composites. hybrid types, showing the unstable dielectric property of aramid fiber as reported previously [15]. The dielectric constants at 10 GHz of the four types of composites are presented in Table 2 along with the estimated results using the current model. The estimated dielectric constants using the proposed model for the two Table 2 Experimental and calculated results of dielectric constant at 10 GHz Composites Dielectric constant Experimental Model predicted e 0 B1 e 0 B2 e 0 A1 e 0 A2 e 0 M RVE Interply Intraply Single basalt Single aramid SFT composites are very close to the experimental results. However, although the model predicted that the hybrid composites dielectric constants would be between the two SFT composites, the experimental results showed the dielectric constants of the two types of hybrid composites to be higher than those of the two SFT composites. Similar results were also reported when the dielectric constants of aramid/glass hybrid composites were investigated in our previous studies of interply glass/aramid/epoxy composites in which the hybrid composites had lower dielectric constants than both SFT composites at low frequencies [15]. This may be called a positive hybrid effect on dielectric properties, which may be partially attributed to the interphase polarization between the fibers and matrix as indicated by Todd and Frank [19]. They found a significant effect of the interphase region on the dielectric constant and therefore developed a unique model to provide physical insights into the complex dielectric properties of the fiber-reinforced composites. It is likely that the interactions between the component fibers and the matrix resulted in a deviation of the dielectric constants from the model prediction. If one looks at Fig. 6, it is obvious that the dielectric constants of the two hybrid composites are initially both above those the SFT composites. As the frequency increases, the dielectric constant of the basalt/epoxy composite also increases and finally reaches a level above all the other composites. Therefore, this deviation seemed to be frequency dependent. Other factors may also affect the dielectric properties of the composites. It has been reported that the void content [20], the moisture content [21], and the size and the shape of filler particles [22] may influence the dielectric constants of the composites. It may be concluded that there must be some kind of interaction among the fibers and the matrix which has not been included in the model and more systematic studies are needed to explore the mechanism behind this phenomenon. In addition, the dielectric constants of all the composites showed a smooth decreasing trend with increasing frequency except for the single basalt fiber composite which presents an increasing trend as shown in Figs. 6 and 7. Moreover, the dielectric loss of the single basalt fiber composite shows a sudden increase at 12 GHz. This should not be a surprise because according to Frasch et al. [23], the imaginary part of the complex permittivity of a basalt rock shows a peak around 12 GHz, which could be explained as dielectric relaxation or a resonance at that frequency. It is likely that the basalt rock used to produce the basalt fibers in our study has a similar composition as that in [23]. 5. Conclusions In this study, a theoretical model was proposed for calculation of dielectric constants of 3D orthogonal woven composites as a function of the fiber volume fraction and the yarn arrangement. The model indicates that a component with a lower thickness and larger area perpendicular to the electric field should have greater contribution to the

6 L. Yao et al. / Composites Science and Technology 68 (2008) overall dielectric constant of the composite. The model predicted dielectric constants for four types of composites, namely interply and intraply basalt/aramid hybrid composites and SFT basalt and aramid composites were compared with the experimental results. The model predicted dielectric constants agreed very well with the experimental results for the SFT composites but not so well with the hybrid composites. The dielectric constants of the two hybrid composites were higher than those of the single basalt and aramid fiber composites, which could be due to the effect of the interaction among the component fibers and the matrix. Acknowledgements This project was jointly sponsored by the National High Technology Research and Development Program of China (No. 2007AA03Z101), the Program for Changjiang Scholars and Innovative Research Team in University (No. IRT0526). References [1] Cheng KC, Lin CM, Wang SF, Lin ST, Yang CF. Dielectric properties of epoxy resin-barium titanate composites at high frequency. Mater Lett 2007;61: [2] Ramajo L, Reboredo M, Castro M. Dielectric response and relaxation phenomena in composites of epoxy resin with BaTiO 3 particles. Composites, Part A 2005;36: [3] Starke TKH, Johnston C, Hill S, Dobson P, Grant PS. The effect of inhomogeneities in particle distribution on the dielectric properties of composite films. J Phys D: Appl Phys 2006;39: [4] Hippel ARV. Dielectrics and waves. New York: Wiley; [5] Barrow DA, Petroff TE, Tandon RP, Sayer M. Characterization of thick lead zirconate titanate films fabricated using a new sol gel based process. J Appl Phys 1997;81: [6] Yoon DH, Zhang JP, Lee BI. Dielectric constant and mixing model of BaTiO 3 composite thick films. Mater Res Bull 2003;38: [7] Yoon SH, Choi GK, Kim DW, Cho SY, Hong KS. Mixture behavior of and microwave dielectric properties of (1 x)cawo 4 xtio 2.J Eur Ceram Soc 2007;27: [8] Kchaou B, Turki C, Salvia M, Fakhfakh Z, Treheux D. Role of fibre matrix interface and fibre direction on dielectric behaviour of epoxy composites. Compos Sci Technol 2004;64: [9] Seo IS, Chin WS, Lee DG. Characterization of electromagnetic properties of polymeric composite materials with free space method. Compos Struct 2004;66: [10] Bogdanovich AE, Wigent DE, Whitney TJ. Fabrication of 3-D woven preforms and composites with integrated fiber optic sensors. Sampe J 2003;39:6 15. [11] Abdullah AlK, Abedin MZ, Beg MDH, Pickering KL, Khan MA. Study on the mechanical properties of jute/glass fiber-reinforced unsaturated polyester hybrid composites: effect of surface modification by ultraviolet radiation. J Reinf Plast Compos 2006;25: [12] Anuar H, Ahmad SH, Rasid R, Daud NSN. Tensile and impact properties of thermoplastic natural rubber reinforced short glass fiber and empty fruit bunch hybrid composites. Polym Plast Technol Eng 2006;45: [13] Xie HQ, Zhang S, Xie D. An efficient way to improve the mechanical properties of polypropylene/short glass fiber composites. J Appl Polym Sci 2005;96: [14] Pegoretti A, Fabbri E, Migliaresi C, Pilati F. Intraply and interply hybrid composites based on E-glass and poly(vinyl alcohol) woven fabrics: tensile and impact properties. Polym Int 2004;53: [15] Yao L, Li WB, Wang N, Li W, Guo X, Qiu YP. Tensile, impact and dielectric properties of three dimensional orthogonal aramid/glass fiber hybrid composites. J Mater Sci 2007;42: [16] Czigány T. Basalt fiber reinforced hybrid polymer composites. Mater Sci Forum 2005; : [17] Chin WS, Lee DG. Binary mixture rule for predicting the dielectric properties of unidirectional E-glass/epoxy composite. Compos Struct 2006;74: [18] Feng XQ, Mai Y-W, Qin QH. A micromechanical model for interpenetrating multiphase composites. Comput Mater Sci 2003;28: [19] Todd MG, Shi FG. Complex permittivity of composite systems: A comprehensive interphase approach. IEEE Trans Dielect Elect Insul 2005;12: [20] Ma J, Liao K, Hing P. Effect of aluminum nitride on the properties of cordierite. J Mater Sci 2000;35: [21] Fraga AN, Frullloni E, Osa ODL, Kenny J M, Vázquez A. Relationship between water absorption and dielectric behavior of glass fiber reinforced unsaturated polyester resin. J Compos Mater 2007;41: [22] Dang ZM, Yu YF, Xu HP, Bai J. Study on microstructure and dielectric property of the BaTiO 3 /epoxy resin composites. Compos Sci Technol 2008;68: [23] Frasch LL, McLean SJ, Olsen RG. Electromagnetic properties of dry and water saturated basalt rock, GHz. IEEE Trans Geosci Remote Sens 1998;36:

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