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Calculation on the permittivity of the wollastonite ceramic matrix
Solid State Sciences 14 (2012) 1467e1470
Contents lists available at SciVerse ScienceDirect
Solid State Sciences journal homepage: www.elsevier.com/locate/ssscie
Calculation on the permittivity of the wollastonite ceramic matrix Ming He a, b, *, Huaiwu Zhang a, Jing Wan a, c, Hua Su a a
State Key Laboratory of Electronic Thin Film and Integrated Devices, University of Electronic Science and Technology of China, Chengdu, China Communication Engineering Department, Chengdu Technological University, Chengdu, China c College of Chemistry & Environment Protection Environment, Southwest University for Nationalities, Chengdu 610041, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 February 2012 Received in revised form 13 July 2012 Accepted 28 August 2012 Available online 5 September 2012
Wollastonite ceramic matrix composites are multi-materials including the crystalline wollastonite and the residual glass phases. The relative permittivity (simplified as permittivity) of the crystalline wollastonite is used to evaluate the dielectric properties of the wollastonite ceramic matrix. Usually, the permittivity of the crystalline wollastonite is estimated to be 8.6, however, the residual glasses in wollastonite ceramic matrix are often unknown. In this study, the ClausiuseMosotti model is utilized to calculate the permittivity of the crystalline wollastonite from molecular polarizability. Molecular polarizabilities as the crystalline wollastonite polarizability are calculated by the ionic and oxide additivity rules. The permittivity of prepared glasses is used instead of that of the remaining glasses and estimated to 4.65. The composite permittivity of the wollastonite ceramic matrix is calculated by Lichnetecker logarithm rules. Compared with measurements, the calculated permittivity of wollastonite ceramic matrix corresponds well with measurements. The application of this simple calculation technique allows the assessment of the relative permittivity for ceramic matrix. Ó 2012 Elsevier Masson SAS. All rights reserved.
Keywords: Ceramic matrix Permittivity Additivity Glasses
1. Introduction With crystalline wollastonite (b-CaSiO3) the major phase, wollastonite ceramic matrix composites possessing good dielectric properties have recently been widely investigated [1e4]. In actual application, wollastonite ceramic matrix is a multi-phases material including crystalline phases, such as wollastonite, calcium borate, quartz, residual glass phases and pores. So the permittivity research of the single-phase in composites is necessary. The contents of crystalline phases and the residual glass phases in wollastonite ceramic matrix have been studied [5]. And because of the proportions of pores and minor crystalline phases in the wollastonite ceramic matrix materials are relatively low, we only study the permittivity of the crystalline wollastonite and residual glass phases in order to simplify the model in this paper. The relationship between the permittivity and component is totally dependent on the experimental results. The present studies of the composite dielectric materials can only depend on a large number of experimental validation and various component ratios in order to improve the dielectric properties of composite
* Corresponding author. Communication Engineering Department, Chengdu Technological University, Chengdu, China. Tel.: þ86 28 87798391. E-mail addresses: [email protected], [email protected] (M. He). 1293-2558/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.solidstatesciences.2012.08.015
materials. These researches caused the waste of time, material costs, and a large number of personnel. The following discussion about the permittivity model of composite dielectric materials can effectively improve this situation. 2. Experimental and calculation methods 2.1. Experimental 2.1.1. Synthesis of wollastonite ceramic matrix The raw materials are analytical reagent Ca(OH)2, H2SiO3, H3BO3. These materials were placed in a barreled mill with a solvent of deionized water, and milled for 24 h, then dried at 80 C. Prepared powders were calcined at 700 C for 3 h in the air. The 15 wt.% acrylic emulsion was added to the mixture, ground, and pressed at 20 Mpa to make tablets of 23.7 mm in diameter and 10 mm in height. Samples were fired at a heating rate of 5 C min1 from room temperature to 1000 C for 3 h under the normal atmospheric pressure. 2.1.2. Synthesis of glasses The glasses are synthesized by solegel technique and the influential factors of gelation were well controlled in the process. The raw materials to prepare glasses are oxides CaO, SiO2, B2O3. These oxides were dissolved in deionized water, and the solutions
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M. He et al. / Solid State Sciences 14 (2012) 1467e1470
were mixed and stirred at room temperature to form a mix solution. With constant stirring, an appropriate amount of ethanol as dispersant was slowly added into this mixed solution until a transparent sol was obtained. The sol was held until a transparent gel was formed. Then, the transparent gel was poured into a dish and dried at 80 C for 26 h to transform into dried gel. The dried gels were calcined at 500 C for 5 h with a heating rate of 5 C min1 to eliminate organic content and obtain glass powders. 8 wt.% acrylic emulsion was added into and then the powders were pressed into pellets with 23.7 mm in diameter and 10 mm in thickness under a pressure of 20 Mpa. The pellets were subsequently sintered for 30 min in the air at 850 C, which did not contain any crystalline phase confirmed by XRD and SEM results. 2.1.3. Measurements The permittivity was tested by an Agilent 4284A LCR meter with a frequency of 1 MHz utilizing an environmental chamber for the temperature measurements. The density was determined according to Archimedes principle. 2.2. Calculation methods On the basis of permittivity of a large number of compounds [6], it was shown that polarizability of each type of ion is almost identical in different crystals-subject. And the molecular polarizability can be expressed as the summation of the polarizabilities of the constituent ions in one formula unit,
a ¼
X
ai
(1)
P where ai is the sum of the molecular polarizability of all ions in the molecule or chemical formula, and a is the molecular polarizability, which can be calculated as follows,
a ¼
3V K 1 4p K þ 2
(2)
where V is the volume per molecule or chemical formula, which can be calculated by dividing the molar volume by Avogadro’s number; K is the relative permittivity. Equation (2) is rigorous only when applied to the crystals of “diagonal cubic” symmetry, such as NaCl, CaF2, CsCl, but not rigorous to low symmetry crystals. Nevertheless, it does appear to “work” deceptively well when applied to certain titanates of the perovskite-type [7]. In order to determine the best empirical value of K to use for corrected factor L, the value of molecular polarizability employed in equation (2) should be become as equation (3) under asymmetry conditions,
a ¼ Vm
K 1 4p 4p L K L þ 4p 3 3
(3)
L means the deviation of the effected electric field between symmetry and asymmetry structures. For a mixture of n phases, the permittivity is determined by Lichnetecker logarithm rules [8],
logK ¼
X
logKi
(4)
i
X
logVi ¼ 1
i
where Ki and Vi represent the permittivity and the volume fraction of i-phase, respectively.
Table 1 Polarizabilities of ions and oxides.
ai (Ǻ3)
Ion, oxide, wollastonite
Values taken from Lasaga [9] Ca2þ Si4þ O2B3þ CaO SiO2 Wollastonite (aions) Wollastonite (aOxides)
Values taken from Shannon [10]
2.83 0.10 2.37
3.16 0.87 2.01 0.05 5.22 4.82e4.88 10.06 10.04e10.10
5.20 4.84 10.04 10.04
3. Results and discussion 3.1. Permittivity of crystalline wollastonite The polarizability of the crystalline wollastonite, which has been previously avoided by the chemists and physicists due to the complex crystal structure, can now be investigated by utilizing the additivity rule originally proposed by Cygan and Lasaga [9]. Wollastonite is a composite of Ca2þ, Si4þ, O2 ions, therefore the ionic additivity rule can be initially written using ionic components,
aions ¼ aCa þ aSi þ 3aO
(5)
where aions, aCa, aSi and aO are the polarizabilities of wollastonite, Ca2þ, Si4þand O2 ions, respectively. Similarly, in the case of wollastonite, the oxide additivity rule can be used to calculate the polarizabilities by oxide components,
aOxides ¼ aCaO þ aSiO2
(6)
where aOxides, aCaO, and aSiO2 represent the polarizabilities of wollastonite, CaO and SiO2, respectively. Table 1 listed the polarizabilities of the individual ion and oxide obtained from Lasaga and Shannon [9,10]. On the basis of the values, the polarizabilities of wollastonite aions and aOxides could be calculated by the additivity rule from equations (5) and (6). The polarizabilities of wollastonite (aions and aOxides) are 10.04 Ǻ3 and 10.06 Ǻ3 by the ionic additivity rule, but 10.04 Ǻ3 and 10.04e10.10 Ǻ3 by the oxide additivity rule, which are shown in Table 1. The polarizabilities aions using the ionic additivity rule is 10.04 Ǻ3, which is the same as the polarizabilities aOxides by the oxide additivity rule. The values of polarizabilities presented in Table 1 were used in the corrected ClausiuseMosotti equation (equation (3)) to predict the crystalline wollastonite permittivity. For different values of corrected factor L, the permittivities K of crystalline wollastonite are shown in Table 2 and Fig. 1. It can be seen that the calculated value K increases as L increases (L ¼ 0e1.3). While L ¼ 1, the permittivity of the crystalline wollastonite (in Table 3) is almost the same as the measured values of NYCO [11]. Obviously, L ¼ 1 means the polarization effected electric field is the Lorentz effected electric field, and equation (3) is the ClausiuseMosotti equation. 3.2. Permittivity of glasses The permittivity of the remaining glasses in the wollastonite ceramic matrix cannot be measured. So the permittivity of the Table 2 The permittivity K of crystalline wollastonite with different L. 4p L 3 K
0
0.1
0.3
0.5
0.8
1.0
1.2
1.00
1.23
1.82
2.68
5.04
8.60
19.51
1.4 767.36
NYCO [11] 8.60
M. He et al. / Solid State Sciences 14 (2012) 1467e1470
1469
Table 5 The permittivity of oxide.
This work Measured value [10]
KCaO
KSiO2
KB2 O3
10.47 11.95
3.67 4.45
4.86 3e8
6.5
Fig. 1. The permittivity K of crystalline wollastonite vs corrected factor L.
prepared glasses in this paper is used to substitute that of the remaining glasses. Because the statistic distribution of the basic components in macroscopic glass materials is uniform, the permittivity Ks of amorphous glasses can be calculated [10],
Ks ¼
X
Ksi Ni
Ks ¼ KCaO NCaO þ KSiO2 NSiO2 þ KB2 O3 NB2 O3
(8)
where KCaO, KSiO2 , KB2 O3 and NCaO, NSiO2 , NB2 O3 represent the permittivity and the molar fraction of CaO, SiO2, B2O3, respectively. The ratio of oxides CaO, SiO2 and B2O3 and the permittivity of glasses in 1 MHz are shown in Table 4. So the permittivity of the residual glasses is approximately 4.65 according to the average permittivity value of five experimental results. Based on Table 4, linear equations can be obtained,
0:70 0:65 0:60
10 1 1 0 KCaO 0:20 4:59 A @ A @ KSiO2 0:25 ¼ 4:65 A KB2 O3 0:30 4:70
(9)
Equation (9) can be linear fitted by Matlab software, and the permittivity of CaO, SiO2 and B2O3 is shown in Table 5.
Table 3 The permittivity K of wollastonite with corrected factor L ¼ 1. Wollastonite
Lasaga
Rule K
Ionic 6.20
6.1 6 5.9 5.8 5.7 0.34
0.36
0.38 0.4 0.42 0.44 0.46 The volume fraction of crystalline wollastonite
0.48
0.5
Shannon Oxide 6.20
Ionic 6.20
3.3. Permittivity of the composite The measured bulk density of wollastonite ceramic was 2.52e 2.78 g cm3, which is very close to the theoretic results of wollastonite (2.91 g cm3) [11]. At the same time, qualitative analysis results show the total volume fraction of crystalline wollastonite together with the remaining glasses is greater than 90% [5]. These results indicate a small amount of porosity has little effect on the permittivity of wollastonite ceramic matrix. So the permittivity of the wollastonite ceramic matrix is determined by the main phases of the crystalline wollastonite and the remaining glasses and can be calculated by equation (4). The calculated results are shown in Fig. 2. The results show that with the increasing of volume fraction of crystalline wollastonite, the permittivity of composite ceramic matrix of increase, and the permittivity of wollastonite ceramic matrix is about 5.77e6.32, which is higher than that of the measured results (5.12e5.68) at 1 MHz from prepared samples in this paper. The reason could be attributed to the neglect of the minor phases and porosity. Also, the calculated permittivity is consistent with the experimental results from these literature [12,13].
4. Conclusions Oxide 6.22
Table 4 The permittivity Ks of glass.
1 2 3
6.2
Fig. 2. The permittivity of wollastonite ceramic matrix with different volume fraction of crystalline wollastonite.
where Ksi, Ni represent the permittivity and the molar fraction of iphase. For wollastonite ceramic matrix, the compositions in remaining glasses are oxide CaO, SiO2 and B2O3, and equation (7) can be written:
0:10 @ 0:10 0:10
6.3
(7)
i
0
The permittivity of wollastonit ceramics
6.4
NCaO
NSiO2
NB2 O3
Ks
0.05 0.10 0.10
0.70 0.70 0.65
0.25 0.20 0.25
4.31 4.59 4.65
The wollastonite ceramic matrix is considered to be composed of crystalline wollastonite and residual glass phases in order to simplify the permittivity calculated model. The permittivities of crystalline wollastonite and remaining glasses were be estimated by corrected ClausiuseMosotti equation and prepared glasses, respectively. And the composite permittivity of the wollastonite ceramic matrix is calculated using Lichnetecker logarithm rules. The calculated results show that the volume fraction of major crystalline wollastonite has a great effect on the composite permittivity, and the permittivity of wollastonite ceramic matrix is
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M. He et al. / Solid State Sciences 14 (2012) 1467e1470
5.77e6.32 with 35e50% wollastonite. This method could be used to evaluate the permittivity of other ceramic matrix. Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant Nos. 60721001, 51132003 and 61171047. References [1] S.F. Wang, Y.R. Wang, Y.F. Hsu, C.C. Chiang, J. Alloys Compd. 498 (2010) 211e216. [2] A. Mohanram, G.L. Messing, D.J. Green, J. Am. Ceram. Soc. 88 (2005) 2681e 2689.
[3] H.K. Zhu, H.Q. Zhou, M. Liu, P.F. Wei, G. Ning, J. Alloys Compd. 482 (2009) 272e275. [4] C.C. Chiang, S.F. Wang, Ceram. Int. 34 (2008) 599e604. [5] M. He, M.Q. Wu, S.R. Zhang, X.H. Zhou, et al., J. Alloys Compd. 506 (2010) 757e760. [6] S. Roberts, Phys. Rev. 76 (1949) 1215. [7] D.A. Robinson, Soil Sci. Am. J. 68 (2004) 1780e1785. [8] B.R. Li, Inorganic Dielectric, Wuhan: Huazhong University Science and Technology Press, 1995, p. 180. [9] A.C. Lasaga, R.T. Cygan, Am. Mineral 67 (1982) 328e334. [10] R.D. Shannon, J. Appl. Phys. 73 (1992) 348e366. [11] NYCO, IN:299-04-01 Booklet, Premium Quality Wollastonite NYAD M325, NYCO Minerals Inc., Willsboro, NY, 2001. [12] H.P. Wang, S.Q. Xu, S.Q. Lu, S.L. Zhao, B.L. Wang, Ceram. Int. 35 (2009) 2715e 2718. [13] W. Cai, T. Jiang, X.Q. Tan, Q. Wei, Y. Li, Elec. Compd. Mater. 21 (2002) 16e18.
Contents lists available at SciVerse ScienceDirect
Solid State Sciences journal homepage: www.elsevier.com/locate/ssscie
Calculation on the permittivity of the wollastonite ceramic matrix Ming He a, b, *, Huaiwu Zhang a, Jing Wan a, c, Hua Su a a
State Key Laboratory of Electronic Thin Film and Integrated Devices, University of Electronic Science and Technology of China, Chengdu, China Communication Engineering Department, Chengdu Technological University, Chengdu, China c College of Chemistry & Environment Protection Environment, Southwest University for Nationalities, Chengdu 610041, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 February 2012 Received in revised form 13 July 2012 Accepted 28 August 2012 Available online 5 September 2012
Wollastonite ceramic matrix composites are multi-materials including the crystalline wollastonite and the residual glass phases. The relative permittivity (simplified as permittivity) of the crystalline wollastonite is used to evaluate the dielectric properties of the wollastonite ceramic matrix. Usually, the permittivity of the crystalline wollastonite is estimated to be 8.6, however, the residual glasses in wollastonite ceramic matrix are often unknown. In this study, the ClausiuseMosotti model is utilized to calculate the permittivity of the crystalline wollastonite from molecular polarizability. Molecular polarizabilities as the crystalline wollastonite polarizability are calculated by the ionic and oxide additivity rules. The permittivity of prepared glasses is used instead of that of the remaining glasses and estimated to 4.65. The composite permittivity of the wollastonite ceramic matrix is calculated by Lichnetecker logarithm rules. Compared with measurements, the calculated permittivity of wollastonite ceramic matrix corresponds well with measurements. The application of this simple calculation technique allows the assessment of the relative permittivity for ceramic matrix. Ó 2012 Elsevier Masson SAS. All rights reserved.
Keywords: Ceramic matrix Permittivity Additivity Glasses
1. Introduction With crystalline wollastonite (b-CaSiO3) the major phase, wollastonite ceramic matrix composites possessing good dielectric properties have recently been widely investigated [1e4]. In actual application, wollastonite ceramic matrix is a multi-phases material including crystalline phases, such as wollastonite, calcium borate, quartz, residual glass phases and pores. So the permittivity research of the single-phase in composites is necessary. The contents of crystalline phases and the residual glass phases in wollastonite ceramic matrix have been studied [5]. And because of the proportions of pores and minor crystalline phases in the wollastonite ceramic matrix materials are relatively low, we only study the permittivity of the crystalline wollastonite and residual glass phases in order to simplify the model in this paper. The relationship between the permittivity and component is totally dependent on the experimental results. The present studies of the composite dielectric materials can only depend on a large number of experimental validation and various component ratios in order to improve the dielectric properties of composite
* Corresponding author. Communication Engineering Department, Chengdu Technological University, Chengdu, China. Tel.: þ86 28 87798391. E-mail addresses: [email protected], [email protected] (M. He). 1293-2558/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.solidstatesciences.2012.08.015
materials. These researches caused the waste of time, material costs, and a large number of personnel. The following discussion about the permittivity model of composite dielectric materials can effectively improve this situation. 2. Experimental and calculation methods 2.1. Experimental 2.1.1. Synthesis of wollastonite ceramic matrix The raw materials are analytical reagent Ca(OH)2, H2SiO3, H3BO3. These materials were placed in a barreled mill with a solvent of deionized water, and milled for 24 h, then dried at 80 C. Prepared powders were calcined at 700 C for 3 h in the air. The 15 wt.% acrylic emulsion was added to the mixture, ground, and pressed at 20 Mpa to make tablets of 23.7 mm in diameter and 10 mm in height. Samples were fired at a heating rate of 5 C min1 from room temperature to 1000 C for 3 h under the normal atmospheric pressure. 2.1.2. Synthesis of glasses The glasses are synthesized by solegel technique and the influential factors of gelation were well controlled in the process. The raw materials to prepare glasses are oxides CaO, SiO2, B2O3. These oxides were dissolved in deionized water, and the solutions
1468
M. He et al. / Solid State Sciences 14 (2012) 1467e1470
were mixed and stirred at room temperature to form a mix solution. With constant stirring, an appropriate amount of ethanol as dispersant was slowly added into this mixed solution until a transparent sol was obtained. The sol was held until a transparent gel was formed. Then, the transparent gel was poured into a dish and dried at 80 C for 26 h to transform into dried gel. The dried gels were calcined at 500 C for 5 h with a heating rate of 5 C min1 to eliminate organic content and obtain glass powders. 8 wt.% acrylic emulsion was added into and then the powders were pressed into pellets with 23.7 mm in diameter and 10 mm in thickness under a pressure of 20 Mpa. The pellets were subsequently sintered for 30 min in the air at 850 C, which did not contain any crystalline phase confirmed by XRD and SEM results. 2.1.3. Measurements The permittivity was tested by an Agilent 4284A LCR meter with a frequency of 1 MHz utilizing an environmental chamber for the temperature measurements. The density was determined according to Archimedes principle. 2.2. Calculation methods On the basis of permittivity of a large number of compounds [6], it was shown that polarizability of each type of ion is almost identical in different crystals-subject. And the molecular polarizability can be expressed as the summation of the polarizabilities of the constituent ions in one formula unit,
a ¼
X
ai
(1)
P where ai is the sum of the molecular polarizability of all ions in the molecule or chemical formula, and a is the molecular polarizability, which can be calculated as follows,
a ¼
3V K 1 4p K þ 2
(2)
where V is the volume per molecule or chemical formula, which can be calculated by dividing the molar volume by Avogadro’s number; K is the relative permittivity. Equation (2) is rigorous only when applied to the crystals of “diagonal cubic” symmetry, such as NaCl, CaF2, CsCl, but not rigorous to low symmetry crystals. Nevertheless, it does appear to “work” deceptively well when applied to certain titanates of the perovskite-type [7]. In order to determine the best empirical value of K to use for corrected factor L, the value of molecular polarizability employed in equation (2) should be become as equation (3) under asymmetry conditions,
a ¼ Vm
K 1 4p 4p L K L þ 4p 3 3
(3)
L means the deviation of the effected electric field between symmetry and asymmetry structures. For a mixture of n phases, the permittivity is determined by Lichnetecker logarithm rules [8],
logK ¼
X
logKi
(4)
i
X
logVi ¼ 1
i
where Ki and Vi represent the permittivity and the volume fraction of i-phase, respectively.
Table 1 Polarizabilities of ions and oxides.
ai (Ǻ3)
Ion, oxide, wollastonite
Values taken from Lasaga [9] Ca2þ Si4þ O2B3þ CaO SiO2 Wollastonite (aions) Wollastonite (aOxides)
Values taken from Shannon [10]
2.83 0.10 2.37
3.16 0.87 2.01 0.05 5.22 4.82e4.88 10.06 10.04e10.10
5.20 4.84 10.04 10.04
3. Results and discussion 3.1. Permittivity of crystalline wollastonite The polarizability of the crystalline wollastonite, which has been previously avoided by the chemists and physicists due to the complex crystal structure, can now be investigated by utilizing the additivity rule originally proposed by Cygan and Lasaga [9]. Wollastonite is a composite of Ca2þ, Si4þ, O2 ions, therefore the ionic additivity rule can be initially written using ionic components,
aions ¼ aCa þ aSi þ 3aO
(5)
where aions, aCa, aSi and aO are the polarizabilities of wollastonite, Ca2þ, Si4þand O2 ions, respectively. Similarly, in the case of wollastonite, the oxide additivity rule can be used to calculate the polarizabilities by oxide components,
aOxides ¼ aCaO þ aSiO2
(6)
where aOxides, aCaO, and aSiO2 represent the polarizabilities of wollastonite, CaO and SiO2, respectively. Table 1 listed the polarizabilities of the individual ion and oxide obtained from Lasaga and Shannon [9,10]. On the basis of the values, the polarizabilities of wollastonite aions and aOxides could be calculated by the additivity rule from equations (5) and (6). The polarizabilities of wollastonite (aions and aOxides) are 10.04 Ǻ3 and 10.06 Ǻ3 by the ionic additivity rule, but 10.04 Ǻ3 and 10.04e10.10 Ǻ3 by the oxide additivity rule, which are shown in Table 1. The polarizabilities aions using the ionic additivity rule is 10.04 Ǻ3, which is the same as the polarizabilities aOxides by the oxide additivity rule. The values of polarizabilities presented in Table 1 were used in the corrected ClausiuseMosotti equation (equation (3)) to predict the crystalline wollastonite permittivity. For different values of corrected factor L, the permittivities K of crystalline wollastonite are shown in Table 2 and Fig. 1. It can be seen that the calculated value K increases as L increases (L ¼ 0e1.3). While L ¼ 1, the permittivity of the crystalline wollastonite (in Table 3) is almost the same as the measured values of NYCO [11]. Obviously, L ¼ 1 means the polarization effected electric field is the Lorentz effected electric field, and equation (3) is the ClausiuseMosotti equation. 3.2. Permittivity of glasses The permittivity of the remaining glasses in the wollastonite ceramic matrix cannot be measured. So the permittivity of the Table 2 The permittivity K of crystalline wollastonite with different L. 4p L 3 K
0
0.1
0.3
0.5
0.8
1.0
1.2
1.00
1.23
1.82
2.68
5.04
8.60
19.51
1.4 767.36
NYCO [11] 8.60
M. He et al. / Solid State Sciences 14 (2012) 1467e1470
1469
Table 5 The permittivity of oxide.
This work Measured value [10]
KCaO
KSiO2
KB2 O3
10.47 11.95
3.67 4.45
4.86 3e8
6.5
Fig. 1. The permittivity K of crystalline wollastonite vs corrected factor L.
prepared glasses in this paper is used to substitute that of the remaining glasses. Because the statistic distribution of the basic components in macroscopic glass materials is uniform, the permittivity Ks of amorphous glasses can be calculated [10],
Ks ¼
X
Ksi Ni
Ks ¼ KCaO NCaO þ KSiO2 NSiO2 þ KB2 O3 NB2 O3
(8)
where KCaO, KSiO2 , KB2 O3 and NCaO, NSiO2 , NB2 O3 represent the permittivity and the molar fraction of CaO, SiO2, B2O3, respectively. The ratio of oxides CaO, SiO2 and B2O3 and the permittivity of glasses in 1 MHz are shown in Table 4. So the permittivity of the residual glasses is approximately 4.65 according to the average permittivity value of five experimental results. Based on Table 4, linear equations can be obtained,
0:70 0:65 0:60
10 1 1 0 KCaO 0:20 4:59 A @ A @ KSiO2 0:25 ¼ 4:65 A KB2 O3 0:30 4:70
(9)
Equation (9) can be linear fitted by Matlab software, and the permittivity of CaO, SiO2 and B2O3 is shown in Table 5.
Table 3 The permittivity K of wollastonite with corrected factor L ¼ 1. Wollastonite
Lasaga
Rule K
Ionic 6.20
6.1 6 5.9 5.8 5.7 0.34
0.36
0.38 0.4 0.42 0.44 0.46 The volume fraction of crystalline wollastonite
0.48
0.5
Shannon Oxide 6.20
Ionic 6.20
3.3. Permittivity of the composite The measured bulk density of wollastonite ceramic was 2.52e 2.78 g cm3, which is very close to the theoretic results of wollastonite (2.91 g cm3) [11]. At the same time, qualitative analysis results show the total volume fraction of crystalline wollastonite together with the remaining glasses is greater than 90% [5]. These results indicate a small amount of porosity has little effect on the permittivity of wollastonite ceramic matrix. So the permittivity of the wollastonite ceramic matrix is determined by the main phases of the crystalline wollastonite and the remaining glasses and can be calculated by equation (4). The calculated results are shown in Fig. 2. The results show that with the increasing of volume fraction of crystalline wollastonite, the permittivity of composite ceramic matrix of increase, and the permittivity of wollastonite ceramic matrix is about 5.77e6.32, which is higher than that of the measured results (5.12e5.68) at 1 MHz from prepared samples in this paper. The reason could be attributed to the neglect of the minor phases and porosity. Also, the calculated permittivity is consistent with the experimental results from these literature [12,13].
4. Conclusions Oxide 6.22
Table 4 The permittivity Ks of glass.
1 2 3
6.2
Fig. 2. The permittivity of wollastonite ceramic matrix with different volume fraction of crystalline wollastonite.
where Ksi, Ni represent the permittivity and the molar fraction of iphase. For wollastonite ceramic matrix, the compositions in remaining glasses are oxide CaO, SiO2 and B2O3, and equation (7) can be written:
0:10 @ 0:10 0:10
6.3
(7)
i
0
The permittivity of wollastonit ceramics
6.4
NCaO
NSiO2
NB2 O3
Ks
0.05 0.10 0.10
0.70 0.70 0.65
0.25 0.20 0.25
4.31 4.59 4.65
The wollastonite ceramic matrix is considered to be composed of crystalline wollastonite and residual glass phases in order to simplify the permittivity calculated model. The permittivities of crystalline wollastonite and remaining glasses were be estimated by corrected ClausiuseMosotti equation and prepared glasses, respectively. And the composite permittivity of the wollastonite ceramic matrix is calculated using Lichnetecker logarithm rules. The calculated results show that the volume fraction of major crystalline wollastonite has a great effect on the composite permittivity, and the permittivity of wollastonite ceramic matrix is
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5.77e6.32 with 35e50% wollastonite. This method could be used to evaluate the permittivity of other ceramic matrix. Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant Nos. 60721001, 51132003 and 61171047. References [1] S.F. Wang, Y.R. Wang, Y.F. Hsu, C.C. Chiang, J. Alloys Compd. 498 (2010) 211e216. [2] A. Mohanram, G.L. Messing, D.J. Green, J. Am. Ceram. Soc. 88 (2005) 2681e 2689.
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