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A percolation model of metal–insulator composites
Physica B 325 (2003) 195–198
A percolation model of metal–insulator composites Qingzhong Xue* Department of Applied Physics, University of Petroleum, Dongying, Shandong 257062, People’s Republic of China Received 26 March 2002; received in revised form 18 August 2002
Abstract Considering quantum mechanical tunneling effect, we derive a formula for calculating the effective conductivity (electric and thermal) of metal–insulator composites. In order to consider the interaction between the metal particles, a simple self-consistent way is used to improve this expression. Besides, we report the experimental results of effective conductivity (electric and thermal) of Co–ZrO2 and Ni–ZrO2 composites. We find that the theoretical results given by our model are in good agreement with the experimental data. r 2002 Elsevier Science B.V. All rights reserved. PACS: 71.30.Th; 72.10.Bg; 72.15.Cz Keywords: Percolation model; Metal–insulator composite
1. Introduction There has been much work on the percolation properties of two-phase random composite materials [1–12]. The percolation properties of composites are, in general, complicated functions of the percolation properties of the constituents, of the particle shape and size, and of the volume loading and spatial arrangement of the distribution. People have built some model to describe the percolation properties of composites, such as Bruggeman effective medium theory [11], Maxwell–Garnett theory [13–15], and Doyle–Jacobs theory [16]. They express the effective conductivity of a composite as a function of the volume *Department of Materials Science and Engineering, Tsinghua University, Room 183, Building 2#, Beijing 100084, People’s Republic of China. Fax: +86-106-2771160. E-mail address: [email protected] (Q. Xue).
fractions and conductivities of its components. However, due to quantum mechanical tunneling effect, the metallic transition of metal–insulator composite has emerged before the metal particles have actual physical contact. In this paper, we give a corresponding percolation model for metal– insulator composites. In order to test this proposed model, we have reported the experimental results of effective conductivity (electric and thermal) of Co–ZrO2 composites and Ni–ZrO2 composites.
2. The improved percolation model of metal–insulator composites Experimental measurements indicate that when the volume fraction of the metal particles in metal–insulator composite reaches the threshold
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 5 2 3 - 5
196
Q. Xue / Physica B 325 (2003) 195–198
Pc ðPc o1Þ; the metallic transition will occur. In fact, due to quantum mechanical tunneling of electrons through the insulating medium between metal particles [17], the metallic transition of metal–insulator composite has emerged before the metal particles have actual physical contact. That is to say, the quantum mechanical tunneling effect has increased the effective volume of the metal particles. We assume that V0 and V are the real volume and the effective volume of the metal particles, respectively. The relation between V0 and V is V ¼ aV0 ;
ð1Þ
where a ð> 1Þ is the enhancement coefficient. It is related to the nature of the constituents, to the particle shape and size, and to the volume loading and spatial arrangement of the distribution. We assume that seff is the effective electric conductivity of the metal–insulator composite; sc and sm are the electric conductivities of the metal particles and the host insulator medium, respectively. Starting from the average polarization theory, we can obtain the equation for the effective conductivity tensor component of the metal– insulator composite [7] ! X seff sm 1 fk s þ B2;i ðsm seff Þ eff k X seff sc ¼ 0; ð2Þ fk þ s þ Bk;i ðsc seff Þ eff k where Bk;i and B2;i are the depolarization constants of the kth kind of metal particles and the host medium, respectively, fk is the effective volume fraction of the kth metal particles. If the shapes of the metal particles and the host medium particles can be regarded as spherical Bk;i ¼ B2;i ¼ 13 ; from Eq. (2) we obtain the wellknown Bruggeman expression [11] seff sm seff sc ð1 V Þ þV ¼ 0; ð3Þ 2seff þ sm 2seff þ sc where V is the total effective volume fraction of the metal particles. The electric conductivity of the metal–insulator composite is given by Eq. (3) when only dipole interactions are present. For regular arrays this case occurs in the limit of low volume loading.
Higher multipole interactions become important when the metal particles approach contact, so Eq. (3) breaks down in regular arrays at high volume loading. In random or disordered distributions close encounters can occur at any volume loading, so higher multipole corrections are necessary considered in disordered media even at low volume loading [16]. And, the higher multipole interactions intensify rapidly with increasing the volume fraction of the metal particles. In order to consider the higher multipole interactions we give a set of sbi ði ¼ c; mÞ to replace the real electric conductivitysi ði ¼ c; mÞ: The set of sbi ði ¼ c; mÞ have something to do with si ði ¼ m; cÞ and the shape of the particles. If all the particles in the composite are spherical, starting from Maxwell– Garnett theory and the relation between two distinct topological structures (symmetry structure and dissymmetry structure) [13,18,19], they can be expressed as sbc ¼
2V sc ; 3V
sbm ¼
2ð1 V Þ sm : 2þV
ð4Þ
Substituting Eqs. (1) and (4) into Eq. (3) we get the equation for the effective electric conductivity of the metal–insulator composite seff sbm seff sbc ð1 aV0 Þ þ aV ¼ 0: 0 2seff þ sbm 2seff þ sbc
ð5Þ
Using Eq. (5) we can study the percolation properties of metal–insulator composites such as electric conductivity, thermal conductivity, dielectric constant, and so on.
3. Experimental results and discussion In order to test the proposed model we produced a number of specimens of Co–ZrO2 composite and a number of specimens of Ni–ZrO2 composite. The micron-sized powder mixtures with various amounts of Co and ZrO2 (Ni and ZrO2) were milled for 28 h by steel grinding balls. Then the mixtures were compressed into solid slices (F10 mm 10 mm) under a hydrostatic pressure of 15 MPa and at a temperature of 1100 K.
Q. Xue / Physica B 325 (2003) 195–198
The electric conductivity of each specimen was measured at room temperature using a super high resistance meter. As shown in Fig. 1, the electric conductivity thresholds of Co–ZrO2 composite and Ni–ZrO2 composite are 14% and 20%, respectively. According to the thresholds, the enhancement coefficients (a) are fitted as 2.38 and 1.67 for Co–ZrO2 composites and Ni–ZrO2 composites, respectively. The calculated results given by Eq. (5) are in good agreement with the experimental data, whereas the calculated results given by Bruggeman equation are deviated from the experimental data. In order to measure the thermal conductivity each specimen was cut into circular disk with a radius of 10 mm and a thickness of 1.8 mm. The measurement of the thermal conductivity was performed at 1250 K using a kind of laser flash method. Without the quantum mechanical tunneling effect in thermal conduction, we let a ¼ 1 when calculating the thermal conductivity of metal–
197
Fig. 2. Comparisons between the calculated results and the experimental data of effective thermal conductivity (keff ). Symbols (J) and (&) represent the experimental data of the effective thermal conductivity of Co–ZrO2 composites and Ni– ZrO2 composites, respectively. Solid curves 1 and 2 represent the calculated results of the effective thermal conductivity of Co–ZrO2 composites and Ni–ZrO2 composites according to the proposed model, respectively. Dashed curves 3 and 4 represent the calculated results of the effective thermal conductivity of Co–ZrO2 composites and Ni–ZrO2 composites according to Bruggeman equation, respectively.
insulator composites. As shown in Fig. 2, the calculated results given by Eq. (5) are completely consistent with the experimental data, whereas the calculated results given by Bruggeman equation are higher than the experimental data.
4. Conclusions
Fig. 1. Comparisons between the calculated results and the experimental data of the effective electric conductivity (seff ). Symbols (J) and (&) represent the experimental data of the effective electric conductivity of Co–ZrO2 composites and Ni– ZrO2 composites, respectively. Solid curves 1 and 2 represent the calculated results of the effective electric conductivity of Co–ZrO2 composites (a ¼ 2:38) and Ni–ZrO2 composites (a ¼ 1:67) according to the proposed model, respectively. Dashed curves 3 and 4 represent the calculated results of the effective electric conductivity of Co–ZrO2 composites and Ni– ZrO2 composites according to Bruggeman equation, respectively.
In summary, considering quantum mechanical tunneling effect, we derive a formula for calculating the effective conductivity (electric and thermal) of metal–insulator composites. In order to consider the interaction between the metal particles, a simple self-consistent way is used to improve this expression. Besides, we report the experimental results of effective conductivity (electric and thermal) of Co–ZrO2 composites and Ni–ZrO2 composites. We find that the theoretical results given by our model are in good agreement with the experimental data.
198
Q. Xue / Physica B 325 (2003) 195–198
References [1] [2] [3] [4] [5] [6]
S. Kirkpatick, Rev. Mod. Phys. 45 (1973) 574. H. Liu, K.D. Bao, Phys. Rev. B 46 (1992) 9209. K.D. Bao, J. Axell, G. Grimvall, Phys. Rev. B 41 (1990) 4330. R. Hoffman, R.S. Potember, Appl. Opt. 28 (1998) 1417. A. Spanoudaki, R. Pelster R, Phys. Rev. B 64 (2001) 064205. D.A. Robinson, S.P. Friedman, Water Resour. Res. 37 (2001) 33. [7] Q.Z. Xue, J. Mater. Sci. Technol. 16 (2000) 368. [8] D.C. Pham, Acta Materialia 49 (2001) 3333. [9] B. Sareni, L. Krahenbuhl, A. Beroual, J. Appl. Phys. 81 (1997) 2375.
[10] A.N. Goncharenko, V.Z. Lozovski, E.F. Venger, J. Phys.: Condens. Matter 13 (2001) 8217. [11] D.A.G. Bruggeman, Ann. Phys. (Leipzig) 24 (1935) 636. [12] J.J. Wang, Z.Y. Li, Commun. Theor. Phys. 25 (1996) 35. [13] J.C. Maxwell-Garnett, Phil. Trans. R. Soc. A 203 (1904) 385. [14] Z. Hashin, S. Shtrikman, J. Appl. Phys. 33 (1962) 3125. [15] S. Nomura, T.W. Chou, Int. J. Eng. Sci. 24 (1986) 643. [16] T.W. Doyle, I.S. Jacobs, Phys. Rev. B 42 (1990) 9319. [17] G.R. Ruschau, R.E. Newnhaw, J. Compos. Mater. 26 (1992) 2731. [18] Y.Q. Ma, Z.Y. Li, Acta Phys. Sinica 39 (1990) 458. [19] F. Brouers, J. Phys. C 19 (1986) 7183.
A percolation model of metal–insulator composites Qingzhong Xue* Department of Applied Physics, University of Petroleum, Dongying, Shandong 257062, People’s Republic of China Received 26 March 2002; received in revised form 18 August 2002
Abstract Considering quantum mechanical tunneling effect, we derive a formula for calculating the effective conductivity (electric and thermal) of metal–insulator composites. In order to consider the interaction between the metal particles, a simple self-consistent way is used to improve this expression. Besides, we report the experimental results of effective conductivity (electric and thermal) of Co–ZrO2 and Ni–ZrO2 composites. We find that the theoretical results given by our model are in good agreement with the experimental data. r 2002 Elsevier Science B.V. All rights reserved. PACS: 71.30.Th; 72.10.Bg; 72.15.Cz Keywords: Percolation model; Metal–insulator composite
1. Introduction There has been much work on the percolation properties of two-phase random composite materials [1–12]. The percolation properties of composites are, in general, complicated functions of the percolation properties of the constituents, of the particle shape and size, and of the volume loading and spatial arrangement of the distribution. People have built some model to describe the percolation properties of composites, such as Bruggeman effective medium theory [11], Maxwell–Garnett theory [13–15], and Doyle–Jacobs theory [16]. They express the effective conductivity of a composite as a function of the volume *Department of Materials Science and Engineering, Tsinghua University, Room 183, Building 2#, Beijing 100084, People’s Republic of China. Fax: +86-106-2771160. E-mail address: [email protected] (Q. Xue).
fractions and conductivities of its components. However, due to quantum mechanical tunneling effect, the metallic transition of metal–insulator composite has emerged before the metal particles have actual physical contact. In this paper, we give a corresponding percolation model for metal– insulator composites. In order to test this proposed model, we have reported the experimental results of effective conductivity (electric and thermal) of Co–ZrO2 composites and Ni–ZrO2 composites.
2. The improved percolation model of metal–insulator composites Experimental measurements indicate that when the volume fraction of the metal particles in metal–insulator composite reaches the threshold
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 5 2 3 - 5
196
Q. Xue / Physica B 325 (2003) 195–198
Pc ðPc o1Þ; the metallic transition will occur. In fact, due to quantum mechanical tunneling of electrons through the insulating medium between metal particles [17], the metallic transition of metal–insulator composite has emerged before the metal particles have actual physical contact. That is to say, the quantum mechanical tunneling effect has increased the effective volume of the metal particles. We assume that V0 and V are the real volume and the effective volume of the metal particles, respectively. The relation between V0 and V is V ¼ aV0 ;
ð1Þ
where a ð> 1Þ is the enhancement coefficient. It is related to the nature of the constituents, to the particle shape and size, and to the volume loading and spatial arrangement of the distribution. We assume that seff is the effective electric conductivity of the metal–insulator composite; sc and sm are the electric conductivities of the metal particles and the host insulator medium, respectively. Starting from the average polarization theory, we can obtain the equation for the effective conductivity tensor component of the metal– insulator composite [7] ! X seff sm 1 fk s þ B2;i ðsm seff Þ eff k X seff sc ¼ 0; ð2Þ fk þ s þ Bk;i ðsc seff Þ eff k where Bk;i and B2;i are the depolarization constants of the kth kind of metal particles and the host medium, respectively, fk is the effective volume fraction of the kth metal particles. If the shapes of the metal particles and the host medium particles can be regarded as spherical Bk;i ¼ B2;i ¼ 13 ; from Eq. (2) we obtain the wellknown Bruggeman expression [11] seff sm seff sc ð1 V Þ þV ¼ 0; ð3Þ 2seff þ sm 2seff þ sc where V is the total effective volume fraction of the metal particles. The electric conductivity of the metal–insulator composite is given by Eq. (3) when only dipole interactions are present. For regular arrays this case occurs in the limit of low volume loading.
Higher multipole interactions become important when the metal particles approach contact, so Eq. (3) breaks down in regular arrays at high volume loading. In random or disordered distributions close encounters can occur at any volume loading, so higher multipole corrections are necessary considered in disordered media even at low volume loading [16]. And, the higher multipole interactions intensify rapidly with increasing the volume fraction of the metal particles. In order to consider the higher multipole interactions we give a set of sbi ði ¼ c; mÞ to replace the real electric conductivitysi ði ¼ c; mÞ: The set of sbi ði ¼ c; mÞ have something to do with si ði ¼ m; cÞ and the shape of the particles. If all the particles in the composite are spherical, starting from Maxwell– Garnett theory and the relation between two distinct topological structures (symmetry structure and dissymmetry structure) [13,18,19], they can be expressed as sbc ¼
2V sc ; 3V
sbm ¼
2ð1 V Þ sm : 2þV
ð4Þ
Substituting Eqs. (1) and (4) into Eq. (3) we get the equation for the effective electric conductivity of the metal–insulator composite seff sbm seff sbc ð1 aV0 Þ þ aV ¼ 0: 0 2seff þ sbm 2seff þ sbc
ð5Þ
Using Eq. (5) we can study the percolation properties of metal–insulator composites such as electric conductivity, thermal conductivity, dielectric constant, and so on.
3. Experimental results and discussion In order to test the proposed model we produced a number of specimens of Co–ZrO2 composite and a number of specimens of Ni–ZrO2 composite. The micron-sized powder mixtures with various amounts of Co and ZrO2 (Ni and ZrO2) were milled for 28 h by steel grinding balls. Then the mixtures were compressed into solid slices (F10 mm 10 mm) under a hydrostatic pressure of 15 MPa and at a temperature of 1100 K.
Q. Xue / Physica B 325 (2003) 195–198
The electric conductivity of each specimen was measured at room temperature using a super high resistance meter. As shown in Fig. 1, the electric conductivity thresholds of Co–ZrO2 composite and Ni–ZrO2 composite are 14% and 20%, respectively. According to the thresholds, the enhancement coefficients (a) are fitted as 2.38 and 1.67 for Co–ZrO2 composites and Ni–ZrO2 composites, respectively. The calculated results given by Eq. (5) are in good agreement with the experimental data, whereas the calculated results given by Bruggeman equation are deviated from the experimental data. In order to measure the thermal conductivity each specimen was cut into circular disk with a radius of 10 mm and a thickness of 1.8 mm. The measurement of the thermal conductivity was performed at 1250 K using a kind of laser flash method. Without the quantum mechanical tunneling effect in thermal conduction, we let a ¼ 1 when calculating the thermal conductivity of metal–
197
Fig. 2. Comparisons between the calculated results and the experimental data of effective thermal conductivity (keff ). Symbols (J) and (&) represent the experimental data of the effective thermal conductivity of Co–ZrO2 composites and Ni– ZrO2 composites, respectively. Solid curves 1 and 2 represent the calculated results of the effective thermal conductivity of Co–ZrO2 composites and Ni–ZrO2 composites according to the proposed model, respectively. Dashed curves 3 and 4 represent the calculated results of the effective thermal conductivity of Co–ZrO2 composites and Ni–ZrO2 composites according to Bruggeman equation, respectively.
insulator composites. As shown in Fig. 2, the calculated results given by Eq. (5) are completely consistent with the experimental data, whereas the calculated results given by Bruggeman equation are higher than the experimental data.
4. Conclusions
Fig. 1. Comparisons between the calculated results and the experimental data of the effective electric conductivity (seff ). Symbols (J) and (&) represent the experimental data of the effective electric conductivity of Co–ZrO2 composites and Ni– ZrO2 composites, respectively. Solid curves 1 and 2 represent the calculated results of the effective electric conductivity of Co–ZrO2 composites (a ¼ 2:38) and Ni–ZrO2 composites (a ¼ 1:67) according to the proposed model, respectively. Dashed curves 3 and 4 represent the calculated results of the effective electric conductivity of Co–ZrO2 composites and Ni– ZrO2 composites according to Bruggeman equation, respectively.
In summary, considering quantum mechanical tunneling effect, we derive a formula for calculating the effective conductivity (electric and thermal) of metal–insulator composites. In order to consider the interaction between the metal particles, a simple self-consistent way is used to improve this expression. Besides, we report the experimental results of effective conductivity (electric and thermal) of Co–ZrO2 composites and Ni–ZrO2 composites. We find that the theoretical results given by our model are in good agreement with the experimental data.
198
Q. Xue / Physica B 325 (2003) 195–198
References [1] [2] [3] [4] [5] [6]
S. Kirkpatick, Rev. Mod. Phys. 45 (1973) 574. H. Liu, K.D. Bao, Phys. Rev. B 46 (1992) 9209. K.D. Bao, J. Axell, G. Grimvall, Phys. Rev. B 41 (1990) 4330. R. Hoffman, R.S. Potember, Appl. Opt. 28 (1998) 1417. A. Spanoudaki, R. Pelster R, Phys. Rev. B 64 (2001) 064205. D.A. Robinson, S.P. Friedman, Water Resour. Res. 37 (2001) 33. [7] Q.Z. Xue, J. Mater. Sci. Technol. 16 (2000) 368. [8] D.C. Pham, Acta Materialia 49 (2001) 3333. [9] B. Sareni, L. Krahenbuhl, A. Beroual, J. Appl. Phys. 81 (1997) 2375.
[10] A.N. Goncharenko, V.Z. Lozovski, E.F. Venger, J. Phys.: Condens. Matter 13 (2001) 8217. [11] D.A.G. Bruggeman, Ann. Phys. (Leipzig) 24 (1935) 636. [12] J.J. Wang, Z.Y. Li, Commun. Theor. Phys. 25 (1996) 35. [13] J.C. Maxwell-Garnett, Phil. Trans. R. Soc. A 203 (1904) 385. [14] Z. Hashin, S. Shtrikman, J. Appl. Phys. 33 (1962) 3125. [15] S. Nomura, T.W. Chou, Int. J. Eng. Sci. 24 (1986) 643. [16] T.W. Doyle, I.S. Jacobs, Phys. Rev. B 42 (1990) 9319. [17] G.R. Ruschau, R.E. Newnhaw, J. Compos. Mater. 26 (1992) 2731. [18] Y.Q. Ma, Z.Y. Li, Acta Phys. Sinica 39 (1990) 458. [19] F. Brouers, J. Phys. C 19 (1986) 7183.