International ournal of Applied Ceramic Technolog-Sebastian and Jantunen Vol.7,No.4,2010 Licht 0-S2 △-S3 EMt 7S4 Modified Lichteneck -o- S5 10203040506070 BaTiO3 powder(vol%) Variation of relative Permittivity with BaTiO3 poud loading and their particle size (S1=0.151um, S2=0.254um S3=0319μm,S=08321um,S5=0916μum, after Cho et al.3 shape of the filler used in the polymer-ceramic com- posites. A small value of n indicates a near-spherical shape for the filler, while a high value of n shows a largely nonspherically shaped particle. However, the particle size should be small for a better fitting with the theoretical prediction. The EMT model fits well for the polymer-Sm2Si2O7 composites as shown in Fig. 4 However, Teirikangas et al. reported that in som cases, the morphology factor n used in EMT is not in- Volume fraction of Sm2 Si207 dependent of the selection of the polymer. Several quan titative rules of mixture models had been proposed for permitivity(a) polystyrene- Si 07(6) polyethylene-Sm25i207 each componen. s of the dielectric properties of Fig. 4. Comparison of theoretical and experimental relative redictions of the However, the theoretical composites at 8 GHz(after Thomas et al) models do not completely agree with the experimental somewhat sensitive to both the polymer and the ce- that it is difficult to obtain the correct Er of the powders ramic, thus reducing the feasibility of the Lichtnecker Instead, E, of the bulk ceramic is used. The er of the equation for different materials. The relative permit- powder may be different from that of the bulk. In fer- tivity of composites also depends on the distribution of roelectric BaTiO3 powders, Er varies with particle of the filler, shape, and size of the fillers and the interfac between ceramics and polymers. Recently, Rao et al. ace grain sizes. Figure 5 shows the variation of Er with proposed a model(Effective Medium Theory, EMT)to The number of theoretical models available for pre predict the relative permittivity of the composite in listing loss tangent is relatively less, as it is more com which the dielectric property of the composite is treated plicated. 4 The following relations are used to model the as an effective medium whose relative permittivity is dielectric loss tangent of the composites btained by averaging the permittivity values of the constituents. The emt model is a self-consistent model that assumes a random unit cell consisting of each filler an8)2=∑m(tan)2 surrounded by a concentric matrix layer. The model includes a morphology factor"n, "which is determined where tan 8 and tan 8; are the loss tangent of the com- empirically. This correction factor compensates for th osite, the loss tangent of ith material, a is a constant
somewhat sensitive to both the polymer and the ceramic, thus reducing the feasibility of the Lichtnecker equation for different materials.41 The relative permittivity of composites also depends on the distribution of the filler, shape, and size of the fillers and the interface between ceramics and polymers. Recently, Rao et al. 51 proposed a model (Effective Medium Theory, EMT) to predict the relative permittivity of the composite in which the dielectric property of the composite is treated as an effective medium whose relative permittivity is obtained by averaging the permittivity values of the constituents. The EMT model is a self-consistent model that assumes a random unit cell consisting of each filler surrounded by a concentric matrix layer. The model includes a morphology factor ‘‘n,’’ which is determined empirically. This correction factor compensates for the shape of the filler used in the polymer–ceramic composites. A small value of n indicates a near-spherical shape for the filler, while a high value of n shows a largely nonspherically shaped particle. However, the particle size should be small for a better fitting with the theoretical prediction. The EMT model fits well for the polymer–Sm2Si2O7 composites as shown in Fig. 4. However, Teirikangas et al. 41 reported that in some cases, the morphology factor n used in EMT is not independent of the selection of the polymer. Several quantitative rules of mixture models had been proposed for predictions of the basis of the dielectric properties of each component.6,50,52,53 However, the theoretical models do not completely agree with the experimental observations. One of the reasons for the poor fitting is that it is difficult to obtain the correct er of the powders. Instead, er of the bulk ceramic is used. The er of the powder may be different from that of the bulk. In ferroelectric BaTiO3 powders, er varies with particle or grain sizes.53 Figure 5 shows the variation of er with vol% of BaTiO3 powder having different grain sizes. The number of theoretical models available for predicting loss tangent is relatively less, as it is more complicated.54 The following relations are used to model the dielectric loss tangent of the composites. General mixing model55,56 ðtan dcÞ a ¼ Xvfiðtan diÞ a ð6Þ where tan dc and tan di are the loss tangent of the composite, the loss tangent of ith material, a is a constant, Fig. 4. Comparison of theoretical and experimental relative permittivity (a) polystyrene–Sm2Si2O7 (b) polyethylene–Sm2Si2O7 composites at 8 GHz (after Thomas et al.40). Fig. 5. Variation of relative permittivity with BaTiO3 powder loading and their particle size (S1 5 0.151 mm, S2 5 0.254 mm, S3 5 0.319 mm, S4 5 0.832 mm, S5 5 0.916 mm, after Cho et al.53). 420 International Journal of Applied Ceramic Technology—Sebastian and Jantunen Vol. 7, No. 4, 2010
wwceramics. org/ACT Polymer-Ceramic Composites of 0-3 Connectivity 一 Experimental Parallel Serial ewwwM 0.03 Bruggmen “日: Volume Percentage of SCT Fig on of the experimental dielectric loss with the 4 predicted values(afier Subodh et al. 5) 4.5 and vg is the volume fraction of the ith material. The 24.0 value of the constant a determines the mixing rule where a=-1 means serial mixing, a= 1 parallel mix d a=0 gives the logarithmic The Bruggeman model Em)(E+2e/)e Temperature(C) +e-sa)(e+28) (7) polystyrene-Sm i207(b) polyethylene-Sm2i207composites(after Thomas where the e, em,E,e"i, e"m, e" are the real and imag- inary parts of the permittivity of the filler, matrix, and the temperature variation of the relative permittivity the composite, respectively. Figure 6 shows the com- reasonably small and the material is thus useful for parison of the predicted and experimental loss tangent practical applications. Results for other properties im for epoxy/Srg Ce?Ti12O3 composites. The Bruggeman portant for microwave substrates have been reported less model gives a relatively good fit with the experimental frequently. Walpata et al obtained a temperature results for lower filler contents. The dielectric loss de- compensated thermoplastic composite with a high E, by pends on intrinsic and extrinsic factors. The intrinsic using a second filler with contrasting thermal depen are mainly due to the interaction of the ac elec- dence of permittivity(tEr). They prepared a polyp ld with phonons. The extrinsic factors such henylene sulfide/SrTiO3/mica composite. The use of as defects, interfaces, size and shape of the filler, and one ceramic with a positive ter and the other with a micropores also contribute to dielectric loss negative ter allowed to obtain a temperature-compen- is The temperature coefficient of relative permittivity sated compensate figure 8 shows the variation of E, of one of the important properties that control the over- the composite 38/8/54(mica/SrTiO, /PPS)measured at all performance of the substrate materials. Some groups 2 GHz. The same procedure was also followed by Xiang have reported fabrication and calculation methods et al. to modify ter. Table I gives a list of the dielectric perature-compensated composites properties polym Figure 7 shows the temperature dependence of the Among the many reported composites given in Table relative permittivity of polystyrene and polyethylene I, polystyrene-Sr2Ce2TisO1s has a very loss tangent of composites with Sm2Si2O7. In many cases, however, 0.0004 relatively high permittivity of 13.6 at
and vfi is the volume fraction of the ith material. The value of the constant a determines the mixing rule where a 5 1 means serial mixing, a 5 1 parallel mixing, and a 5 0 gives the logarithmic mixing rule.55 The Bruggeman model57 e00 ¼ ðe0 i e0 Þðe0 i þ 2e0 mÞe0 ðe0 i e0 mÞðe0 i þ 2e0 Þe0 m e00 m þ 3ðe0 e0 mÞ ðe0 i e0 mÞðe0 i þ 2e0 Þ e00 i ð7Þ where the e0 i, e0 m, e0 , e00 i, e00 m, e00 are the real and imaginary parts of the permittivity of the filler, matrix, and the composite, respectively. Figure 6 shows the comparison of the predicted and experimental loss tangent for epoxy/Sr9Ce2Ti12O36 composites. The Bruggeman model gives a relatively good fit with the experimental results for lower filler contents. The dielectric loss depends on intrinsic and extrinsic factors. The intrinsic factors are mainly due to the interaction of the ac electric field with phonons. The extrinsic factors such as defects, interfaces, size and shape of the filler, and micropores also contribute to dielectric loss. The temperature coefficient of relative permittivity is one of the important properties that control the overall performance of the substrate materials. Some groups have reported fabrication and calculation methods to enable temperature-compensated composites.38,59 Figure 7 shows the temperature dependence of the relative permittivity of polystyrene and polyethylene composites with Sm2Si2O7. 40 In many cases, however, the temperature variation of the relative permittivity is reasonably small and the material is thus useful for practical applications. Results for other properties important for microwave substrates have been reported less frequently. Walpata et al. 60 obtained a temperaturecompensated thermoplastic composite with a high er by using a second filler with contrasting thermal dependence of permittivity (ter). They prepared a polyphenylene sulfide/SrTiO3/mica composite. The use of one ceramic with a positive ter and the other with a negative ter allowed to obtain a temperature-compensated compensate. Figure 8 shows the variation of er of the composite 38/8/54(mica/SrTiO3/PPS) measured at 2 GHz. The same procedure was also followed by Xiang et al. 59 to modify ter. Table I gives a list of the dielectric properties of several polymer–ceramic composites. Among the many reported composites given in Table I, polystyrene–Sr2Ce2Ti5O15 has a very loss tangent of 0.0004 with a relatively high permittivity of 13.6 at 0 10 20 30 40 0.01 0.02 0.03 0.04 tan δ Volume Percentage of SCT Experimental Parallel Serial Logarithmic Bruggmen Fig. 6. Comparison of the experimental dielectric loss with the predicted values (after Subodh et al.58). Fig. 7. Variation of relative permittivity with temperature (a) polystyrene–Sm2Si2O7 (b) polyethylene–Sm2Si2O7 composites (after Thomas et al.40). www.ceramics.org/ACT Polymer–Ceramic Composites of 0–3 Connectivity 421