Cu/Cu_2O金属陶瓷逾渗与分形分析
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摘要
本文通过对Cu/Cu_2O金属陶瓷微观组织表征和导电性的测量,系统研究了导通相Cu含量、形貌、尺寸对金属陶瓷逾渗行为的影响规律,并在此基础上,分析了逾渗临界指数的非普适性,提出以骨架密度表征有效导通相含量,为在更深层次上认识导体/绝缘体材料的逾渗机理做出了一定的贡献。
研究发现,球形Cu粉制备的Cu/Cu_2O金属陶瓷导通门槛值最高,大约在22%左右。随着球形导通相尺寸的减小,导通门槛值会略有减小,但是减小幅度不明显。由枝状Cu粉制备的Cu/Cu_2O金属陶瓷导通门槛值最低,大约在10%左右。而由还原法制备得到的Cu/Cu_2O金属陶瓷导通门槛值基本上和球形Cu粉制备的材料相同,大约在22%左右。通过对晶格逾渗体系分析,在导通相体积相同的条件下,导通相尺寸的减小和长径比的增加都会降低逾渗门槛值。通过对配位数变化的计算得知,同比改变导通相的长径比比减小导通相尺寸使导通门槛值降低更快。对导通相体积分数相同、结构及分布不同的Cu/Cu_2O金属陶瓷电导率进行比较发现,在相同体积分数的前提下,不同导通相结构及分布会导致完全不同的电导率。枝状导通相制备的材料电导率最高,还原法制备的其次,球形导通相的最低。
在还原法制备的Cu/Cu_2O金属陶瓷的中,首次得到了小于逾渗临界指数普适值的导体/绝缘体真实材料体系,t = 0.87±0.1。并用骨架密度的概念,假设逾渗体系中的导通相服从泊松分布,给出了逾渗临界指数的预测模型。说明了逾渗临界指数是逾渗体系中骨架密度的函数。随着体系中骨架密度的增加,逾渗临界指数会降低。在该模型中,逾渗临界指数还受结构因子κ影响。导通相结构越复杂,κ值越小;导通相结构越简单,κ值越大。在门槛值附近,逾渗临界指数会出现一个先上升后下降的趋势,说明随着导通相体积分数的增加,体系中的骨架密度在逾渗门槛值附近不是随着导通相体积分数的增加而线性增加的。不同导通相结构和分布会使得直流电导率在相同导通相体积分数增加时获得不同的增量。当导通相体积分数与逾渗门槛值差值p - p_c从0.01增加到0.1时,枝状导通相制备的金属陶瓷增量最大、球形的基本相同、还原法制备的增量最小。
随着导通相Cu含量体积分数的增加,所有Cu/Cu_2O金属陶瓷的分形维数都呈现增加趋势。由球形Cu粉和枝状Cu粉制备的Cu/Cu_2O金属陶瓷,其分形维数在门槛值附近都有一个先下降后上升的趋势。说明在门槛值附近,其导通相的结构复杂程度会出现一个波动,也就是说,其骨架密度会出现波动,并不是随着导通相体积分数的增加而单调增加的。在相同导通相体积分数的条件下,不同的导通相的结构和分布,会导致Cu/Cu_2O金属陶瓷具有不同的分形维数。含枝状导通相Cu/Cu_2O金属陶瓷的分形维数最低,球形导通相和还原法制备得到的Cu/Cu_2O金属陶瓷分形维数接近。并且,球形导通相的分形维数会随着导通相尺寸减小而减小。随着p-p_c的增加,Cu/Cu_2O金属陶瓷导通相骨架密度的增量会导致其分形维数继续保持增长。对于不同的导通相结构,其增长速度不同。在本论文的材料体系中,枝状导通相的增长速度比球形导通相的增长速度快,而还原法制备的增长速度基本为零。
通过对二维四方晶格逾渗体系的定量计算可知,在逾渗体系中,无限网络总量和骨架总量都是随着导通相体积分数的增加而增加的。但是骨架总量和死端总量的比值,也就是骨架密度,在逾渗门槛值附近会出现波动,意味着骨架密度在逾渗门槛值附近并不随着导通相体积分数的增加而增加。在二维四方晶格逾渗体系中,通过对逾渗临界指数的计算,也得到了逾渗临界指数先上升后下降的现象。
Cu/Cu_2O cermet materials were prepared with hot pressing and hot pressing re-duction methods. The Cu/Cu_2O cermets prepared with hot pressing technology wereproduced by hot pressing the spherical and Cu_2O powders or branched Cu and Cu_2Opowders. The hot pressing reduction method was hot pressing C and Cu_2O powders.Using C to reduce Cu_2O matrix into Cu as the conductor. The dc electrical conductiv-ity of Cu/Cu_2O cermets, with different volume content, structure, and distribution, weretested. The percolation and fractal behavior were analyzed, and the relationship betweenmicrostructure and properties has been found.
The percolation thresholds with different conductor structure and distribution wereobtained by the dc electrical conductivity test. The highest percolation threshold was theone prepared with 75μm spherical Cu. The percolation thresholds of Cu/Cu_2O cermetsprepared with different sizes of spherical Cu were very close, they were all about 0.22.The percolation threshold of Cu/Cu_2O cermet prepared with branched Cu was the lowest,it was about 10%. The percolation threshold of Cu/Cu_2O cermets prepared with hotpressing reduction method was close to the one prepared with spherical Cu, it was about22% as well. By the lattice analysis, it can be concluded that the percolation thresholddecreases with increasing of aspect ratio and decreasing of size of conductors, and thevariation rate of the percolation threshold based on aspect ratio dependence is faster thanthe one based on size dependence. By comparing the Cu/Cu_2O cermets prepared withsame volume content of Cu, it was found that the Cu/Cu_2O cermets with branched Cuhas the highest dc electrical conductivity, and the Cu/Cu_2O cermets with spherical Cuhas the lowest dc electrical conductivity.
By the analyzing of the Cu/Cu_2O cermets prepared by the hot pressing reductionmethod, the electrical conductivity percolation critical exponent lower than the univer-sal value, t = 0.87±0.1, was firstly obtained in real conductor/insulator composites.The definition of the backbone density was introduced into the percolation system, andpresuming the conductors were obeyed the Poisson distribution. A model predicting thepercolation critical exponent were given and demonstrated that the percolation thresh-old is proportional to the backbone density. A“structure factor”was introduced into themodel. The more complex the conductor structure, the lower the structure factor. Nearthe percolation threshold, the critical exponent was not a constant but has a ?uctuation, which means the backbone density was not linearly increasing with increasing of volumecontent of conductors. It was also found that the increasing rate of the dc electrical con-ductivity, with same increasing rate of volume content of conductor, was different withdifferent structure and distribution of conductors. The Cu/Cu_2O cermets prepared withbranched Cu has the highest increasing rate, and the one prepared with reduction methodhas the lowest increasing rate.
The fractal dimensions of all Cu/Cu_2O cermets increase with increasing volumecontent of the conductors. The fractal dimension of those prepared with spherical andbranched Cu have a ?uctuation near the percolation threshold. This means the complexityof the conductor’s structure varied near the percolation threshold. Cu/Cu_2O cermets withsame conductor volume content but different structure and distribution leads to differentfractal dimensions. Those prepared with branched Cu has lower fractal dimension, andthose prepared with spherical Cu and reduction method have higher fractal dimensions.The fractal dimension of those prepared with spherical Cu decreases with decreasing sizeof conductors. The increasing rates of the fractal dimension were different with differentstructure and distribution of conductors. Those prepared with branched Cu was higherthan those prepared with spherical Cu, and the increasing rate of those prepared withreduction method is close to zero.
By the quantitative calculation of the 2d square lattice percolation system, it wasfound that the quantities of infinite cluster and backbone increase with increasing vol-ume content of conductors, but the backbone density has a ?uctuation near the perco-lation threshold. by the quantitative calculation of the percolation critical exponent, italso found that near the percolation threshold, the percolation critical exponent has a?uctuation as well.
研究发现,球形Cu粉制备的Cu/Cu_2O金属陶瓷导通门槛值最高,大约在22%左右。随着球形导通相尺寸的减小,导通门槛值会略有减小,但是减小幅度不明显。由枝状Cu粉制备的Cu/Cu_2O金属陶瓷导通门槛值最低,大约在10%左右。而由还原法制备得到的Cu/Cu_2O金属陶瓷导通门槛值基本上和球形Cu粉制备的材料相同,大约在22%左右。通过对晶格逾渗体系分析,在导通相体积相同的条件下,导通相尺寸的减小和长径比的增加都会降低逾渗门槛值。通过对配位数变化的计算得知,同比改变导通相的长径比比减小导通相尺寸使导通门槛值降低更快。对导通相体积分数相同、结构及分布不同的Cu/Cu_2O金属陶瓷电导率进行比较发现,在相同体积分数的前提下,不同导通相结构及分布会导致完全不同的电导率。枝状导通相制备的材料电导率最高,还原法制备的其次,球形导通相的最低。
在还原法制备的Cu/Cu_2O金属陶瓷的中,首次得到了小于逾渗临界指数普适值的导体/绝缘体真实材料体系,t = 0.87±0.1。并用骨架密度的概念,假设逾渗体系中的导通相服从泊松分布,给出了逾渗临界指数的预测模型。说明了逾渗临界指数是逾渗体系中骨架密度的函数。随着体系中骨架密度的增加,逾渗临界指数会降低。在该模型中,逾渗临界指数还受结构因子κ影响。导通相结构越复杂,κ值越小;导通相结构越简单,κ值越大。在门槛值附近,逾渗临界指数会出现一个先上升后下降的趋势,说明随着导通相体积分数的增加,体系中的骨架密度在逾渗门槛值附近不是随着导通相体积分数的增加而线性增加的。不同导通相结构和分布会使得直流电导率在相同导通相体积分数增加时获得不同的增量。当导通相体积分数与逾渗门槛值差值p - p_c从0.01增加到0.1时,枝状导通相制备的金属陶瓷增量最大、球形的基本相同、还原法制备的增量最小。
随着导通相Cu含量体积分数的增加,所有Cu/Cu_2O金属陶瓷的分形维数都呈现增加趋势。由球形Cu粉和枝状Cu粉制备的Cu/Cu_2O金属陶瓷,其分形维数在门槛值附近都有一个先下降后上升的趋势。说明在门槛值附近,其导通相的结构复杂程度会出现一个波动,也就是说,其骨架密度会出现波动,并不是随着导通相体积分数的增加而单调增加的。在相同导通相体积分数的条件下,不同的导通相的结构和分布,会导致Cu/Cu_2O金属陶瓷具有不同的分形维数。含枝状导通相Cu/Cu_2O金属陶瓷的分形维数最低,球形导通相和还原法制备得到的Cu/Cu_2O金属陶瓷分形维数接近。并且,球形导通相的分形维数会随着导通相尺寸减小而减小。随着p-p_c的增加,Cu/Cu_2O金属陶瓷导通相骨架密度的增量会导致其分形维数继续保持增长。对于不同的导通相结构,其增长速度不同。在本论文的材料体系中,枝状导通相的增长速度比球形导通相的增长速度快,而还原法制备的增长速度基本为零。
通过对二维四方晶格逾渗体系的定量计算可知,在逾渗体系中,无限网络总量和骨架总量都是随着导通相体积分数的增加而增加的。但是骨架总量和死端总量的比值,也就是骨架密度,在逾渗门槛值附近会出现波动,意味着骨架密度在逾渗门槛值附近并不随着导通相体积分数的增加而增加。在二维四方晶格逾渗体系中,通过对逾渗临界指数的计算,也得到了逾渗临界指数先上升后下降的现象。
Cu/Cu_2O cermet materials were prepared with hot pressing and hot pressing re-duction methods. The Cu/Cu_2O cermets prepared with hot pressing technology wereproduced by hot pressing the spherical and Cu_2O powders or branched Cu and Cu_2Opowders. The hot pressing reduction method was hot pressing C and Cu_2O powders.Using C to reduce Cu_2O matrix into Cu as the conductor. The dc electrical conductiv-ity of Cu/Cu_2O cermets, with different volume content, structure, and distribution, weretested. The percolation and fractal behavior were analyzed, and the relationship betweenmicrostructure and properties has been found.
The percolation thresholds with different conductor structure and distribution wereobtained by the dc electrical conductivity test. The highest percolation threshold was theone prepared with 75μm spherical Cu. The percolation thresholds of Cu/Cu_2O cermetsprepared with different sizes of spherical Cu were very close, they were all about 0.22.The percolation threshold of Cu/Cu_2O cermet prepared with branched Cu was the lowest,it was about 10%. The percolation threshold of Cu/Cu_2O cermets prepared with hotpressing reduction method was close to the one prepared with spherical Cu, it was about22% as well. By the lattice analysis, it can be concluded that the percolation thresholddecreases with increasing of aspect ratio and decreasing of size of conductors, and thevariation rate of the percolation threshold based on aspect ratio dependence is faster thanthe one based on size dependence. By comparing the Cu/Cu_2O cermets prepared withsame volume content of Cu, it was found that the Cu/Cu_2O cermets with branched Cuhas the highest dc electrical conductivity, and the Cu/Cu_2O cermets with spherical Cuhas the lowest dc electrical conductivity.
By the analyzing of the Cu/Cu_2O cermets prepared by the hot pressing reductionmethod, the electrical conductivity percolation critical exponent lower than the univer-sal value, t = 0.87±0.1, was firstly obtained in real conductor/insulator composites.The definition of the backbone density was introduced into the percolation system, andpresuming the conductors were obeyed the Poisson distribution. A model predicting thepercolation critical exponent were given and demonstrated that the percolation thresh-old is proportional to the backbone density. A“structure factor”was introduced into themodel. The more complex the conductor structure, the lower the structure factor. Nearthe percolation threshold, the critical exponent was not a constant but has a ?uctuation, which means the backbone density was not linearly increasing with increasing of volumecontent of conductors. It was also found that the increasing rate of the dc electrical con-ductivity, with same increasing rate of volume content of conductor, was different withdifferent structure and distribution of conductors. The Cu/Cu_2O cermets prepared withbranched Cu has the highest increasing rate, and the one prepared with reduction methodhas the lowest increasing rate.
The fractal dimensions of all Cu/Cu_2O cermets increase with increasing volumecontent of the conductors. The fractal dimension of those prepared with spherical andbranched Cu have a ?uctuation near the percolation threshold. This means the complexityof the conductor’s structure varied near the percolation threshold. Cu/Cu_2O cermets withsame conductor volume content but different structure and distribution leads to differentfractal dimensions. Those prepared with branched Cu has lower fractal dimension, andthose prepared with spherical Cu and reduction method have higher fractal dimensions.The fractal dimension of those prepared with spherical Cu decreases with decreasing sizeof conductors. The increasing rates of the fractal dimension were different with differentstructure and distribution of conductors. Those prepared with branched Cu was higherthan those prepared with spherical Cu, and the increasing rate of those prepared withreduction method is close to zero.
By the quantitative calculation of the 2d square lattice percolation system, it wasfound that the quantities of infinite cluster and backbone increase with increasing vol-ume content of conductors, but the backbone density has a ?uctuation near the perco-lation threshold. by the quantitative calculation of the percolation critical exponent, italso found that near the percolation threshold, the percolation critical exponent has a?uctuation as well.
引文
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54 C. Thomsen. Critical Exponent and Percolation in Two-dimensional Systems witha Finite Interplane Coupling. Physical Review E. 2002, 65:065104
55 A. V. Goncharenko, E. F. Venger. Percolation Threshold for Bruggeman Compos-ites. Physical Review E. 2004, 70:057102
56 C. Domb, M. F. Sykes. Cluster Size in Random Mixtures and Percolation Pro-cesses. Physical Review. 1961, 122:77–78
57 S. I. Lee, Y. Song, T. W. Noh, et al. Experimental Observation of NonuniversalBehavior of the Conductivity Exponent for Three Dimensional Continuum Perco-lation System. Physical Review B. 1986, 34(10):6719–6724
58 D.Toker, D.Azulay, N.Shimoni, et al. Tunneling and Percolation in Metal-insulatorComposite Materials. Physical Review B. 2003, 68:041403
59 M. B. Heaney. Measurement and Interpretation of Nonuniversal Critical Expo-nents in Disordered Conductor-insulator Composites. Physical Review B. 1995,52(17):12477–12480
60 G.E.Pike. Electrical Transport and Optical Properties of Inhomogeneous Media.AIP Conf. Proc. No.40. New York, 1978:366
61 J. Wu, D.S.McLachlan. Percolation Exponents and Threshold Obtained from theNearly Ideal Continuum Percolation System Graphite-boron Nitride. Physical Re-view B. 1997, 56(3):1236–1248
62 P.Keblinski, F.Cleri. Contact Resistance in Percolating Networks. Physical ReviewB. 2004, 69:184201
63 J. C.Grunlan, A. R. Mehrabi, M. V. Bannon, et al. Water Based Single Walled Nan-otube Filled Polymer Composite with an Exceptionally Low Percolation Thresh-old. Advanced Materials. 2004, 16(2):150–153
64 P. M. Ajayan, L. S. Schadler, C. Giannaris, et al. Single Walled Carbon Nan-otube Polymer Composite: Strength and Weakness. Advanced Materials. 2000,12(10):750–753
65 K. Nozaki, T. Itami. The Determination of the Full Set of Characteristic Val-ues of Percolation, Percolation Threshold and Crtical Exponents for the ArtificalComposite with Ionic Conduction Ag4 Rbi5-β-agi. Journal of Physics CondensedMatter. 1997, 56(3):1236–1248
66 C. R. Scullard, R. M. Ziff. Critical Surfaces for General Bond Percolation Prob-lems. Physical Review Letters. 2008, 100:185701
67 B. J. Last, D. J. Thouless. Percolation Theory and Electrical Conductivity. PhysicalReview Letters. 1971, 27:1719
68 Y. Deng, H. W.J.Blote. Surface Critical Phenomena in Three-dimension Percola-tion. Physical Review E. 2005, 71:016117
69 Y. Shen, Z. Yue, M. Li, et al. Enhanced Initial Permeability and Dielectric Constantin a Double Percolating Ni0.3zn0.7fe1.95o4-ni-polymer Composite. AdvancedFunctional Materials. 2005, 15:1100–1103
70 J. C. Wierman. Percolation Threshold Is Not a Decreasing Function of the AverageCoordination Number. Physical Review E. 2002, 66:046125
71 L.J.Adriaanse, J.A.Reedijk, P.A.A.Teunissen, et al. High-dilution Carbon-black/polymer Composites: Hierarchical Percolating Network Derived from Hzto Thz Ac Conductivity. Physical Review Letters. 1997, 78(9):1755
72 S.Rul, F.Lefevre-schlick, E.Capria, et al. Percolation of Single Walled CarbonNanotubes in Ceramic Matrix Nanocomposites. Acta Materialia. 2004, 52:1061–1067
73 C. Zhang, C.-A. Ma, P. Wang, et al. Temperature Demendence of Electrical Re-sistivity for Carbon Black Filled Ultra-high Molecular Weight Polyethylene Com-posites Prepared by Hot Compaction. Carbon. 2005, 43:2544–2553
74 M. Hindermann-Bischoff, F. Ehrbruger-Dolle. Electrical Conductivity of CarbonBlack-polyethylene Composites Experimental Evidence of the Change of ClusterConnectivity in the Ptc Effect. Carbon. 2001, 39:375–382
75 S.Shekhar, V.Prasad, S.V.Subramanyam. Transport Properties of Conducting Am-porphous Carbon-poly(vinyl Chloride) Composite. Carbon. 2006, 44:334–340
76 F. O. Pfeiffer, H. Rieger. Critical Properties of Loop Percolation Models withOptimization Constraints. Physical Review E. 2003, 67:056113
77 H. E. Stanley, A. Coniglio. Flow in Porous Media: The“backbone”Fractal at thePercolation Threshold. Physical Review B. 1984, 29(1):522–524
78 A.R.Day, R.R.Tremblay, A.M.S.Tremblay. Rigi Backbone: A New Geometry forPercolation. Physical Review Letters. 1986, 56(23):2501–2504
79 C. Moukarzel, P.M.Duxbruy. Stressed Backbone and Elasticity of Random CentralForce Systems. Physical Review Letters. 1995, 75(22):4055–4058
80 G. Paul, H. Stanley. Beyond Blobs in Percolation Cluster Structure: The Dis-tribution of 3 Blocks at the Percolation Threshold. Physical Review E. 2002,65:056126
81 G. Paul, S. V. Buldyrev, P. R. Y. L. Nikolay V.Dokholyan, Shlomo Havlin, et al.Dependence of Conductance on Percolation Backbone Mass. Physical Review E.2000, 61(4):3435–3439
82 D. Y. Ki, K. Y. Woo, S. B. Lee. Static and Dynamic Properties of the BackboneNetwork for the Irreversible Kinetic Gelation Model. Physical Review E. 2000,62:821–827
83 H.J.Herrmann, D.C.Hong, H.E.Stanley. Backbone and Elastic Backbone of Perco-lation Clusters Obtained by the New Method of“burning”. J.Phys.A: Math. Gen.1984, 17:L261–L266
84 S.Roux, A.Hansen. A New Algorithm to Extract the Backbone in a Random Re-sistor Network. J.Phys.A: Math. Gen. 1987, 20:L1281–L1285
85 B.B.Mandelbrot. Fractal Geometry of Nature. New york edn. Freeman, 1983
86 K. Falconer. Fractal Geometry-mathematical Foundations and Applications. 2ndedn. John Wiley and Sons Ltd., 2003
87 S.S.Manna, D.Dhar. Fractal Dimension of Backbone of Eden Trees. PhysicalReview E. 1996, 54:R3063–R3066
88 S.S.Albuquerque, F. Moura, M.L.Lyra. Fractality of Largest Clusters and thePercolation Transition in Power-law Diluted Chains. Physical Review E. 2005,72:016116
89 G. Paul, H. Stanley. Fractal Dimension of 3-blocks in Four-, Five, and Six-dimensional Percolatoin Systems. Physical Review E. 2003, 67:026103
90 J. Fournier, G. Boiteus, G. Seytre. Fractal Analysis of the Percolation Network inEpoxy-polypyrrlole Composites. Physical Review B. 1997, 56:5207–5212
91 K.K.Bardhan, R.K.Chakrabarty. Identical Scalling Behavior of Dc and Ac Re-sponse Near the Percolation Threshold in Conductor-insulator Mixtures. PhyscalReview Letters. 1994, 72(7):1068–1071
92 H. G. Yoon, K. W. Kwon, K. Nagata, et al. Changing the Percolation Threshold of aCarbon Black/polymer Composite by a Coupling Treatment of the Black. Carbon.2004, 42:1877–1879
93 P. Potschke, A. R.Bhattacharyya, A. Janke. Carbon Nanotube-filled PolycarbonateComposites Produced by Melt Mixing and Their Use in Blends with Polyethylene.Carbon. 2004, 42:965–969
94 I. H. Campbell, T. W. Hagler, D. L. Smith. Direct Measurement of ConjugatedPolymer Electronic Excitation Energies Using Metaly Polymery Metal Structures.Physical Review Letters. 1996, 76(11):1900–1903
95 M. Wildan, H.J.Edrees, A. Hendry. Ceramic Matrix Composites of Zirconia Rein-forced with Metal Particles. Materials Chemistry and Physics. 2002, 75:276–283
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51 J. J. Wu, D. S. McLachlan. Percolation Exponents and Threshold in Two NearlyIdeal Aniostropic Continuum Systems. Physica A. 1997, 241:360–366
52 V. Cornette, A. J. R. Pastor, F. Nieto. Dependence of the Percolation Threshold onthe Size of the Percolating Species. Physica A. 2003, 327:71–75
53 Y. B. Yi, A. M. Sastry. Analytical Approoximation of the Two-dimensional Per-colation Threshold for Fields of Overlapping Ellipses. Physical Review E. 2002,66:066130
54 C. Thomsen. Critical Exponent and Percolation in Two-dimensional Systems witha Finite Interplane Coupling. Physical Review E. 2002, 65:065104
55 A. V. Goncharenko, E. F. Venger. Percolation Threshold for Bruggeman Compos-ites. Physical Review E. 2004, 70:057102
56 C. Domb, M. F. Sykes. Cluster Size in Random Mixtures and Percolation Pro-cesses. Physical Review. 1961, 122:77–78
57 S. I. Lee, Y. Song, T. W. Noh, et al. Experimental Observation of NonuniversalBehavior of the Conductivity Exponent for Three Dimensional Continuum Perco-lation System. Physical Review B. 1986, 34(10):6719–6724
58 D.Toker, D.Azulay, N.Shimoni, et al. Tunneling and Percolation in Metal-insulatorComposite Materials. Physical Review B. 2003, 68:041403
59 M. B. Heaney. Measurement and Interpretation of Nonuniversal Critical Expo-nents in Disordered Conductor-insulator Composites. Physical Review B. 1995,52(17):12477–12480
60 G.E.Pike. Electrical Transport and Optical Properties of Inhomogeneous Media.AIP Conf. Proc. No.40. New York, 1978:366
61 J. Wu, D.S.McLachlan. Percolation Exponents and Threshold Obtained from theNearly Ideal Continuum Percolation System Graphite-boron Nitride. Physical Re-view B. 1997, 56(3):1236–1248
62 P.Keblinski, F.Cleri. Contact Resistance in Percolating Networks. Physical ReviewB. 2004, 69:184201
63 J. C.Grunlan, A. R. Mehrabi, M. V. Bannon, et al. Water Based Single Walled Nan-otube Filled Polymer Composite with an Exceptionally Low Percolation Thresh-old. Advanced Materials. 2004, 16(2):150–153
64 P. M. Ajayan, L. S. Schadler, C. Giannaris, et al. Single Walled Carbon Nan-otube Polymer Composite: Strength and Weakness. Advanced Materials. 2000,12(10):750–753
65 K. Nozaki, T. Itami. The Determination of the Full Set of Characteristic Val-ues of Percolation, Percolation Threshold and Crtical Exponents for the ArtificalComposite with Ionic Conduction Ag4 Rbi5-β-agi. Journal of Physics CondensedMatter. 1997, 56(3):1236–1248
66 C. R. Scullard, R. M. Ziff. Critical Surfaces for General Bond Percolation Prob-lems. Physical Review Letters. 2008, 100:185701
67 B. J. Last, D. J. Thouless. Percolation Theory and Electrical Conductivity. PhysicalReview Letters. 1971, 27:1719
68 Y. Deng, H. W.J.Blote. Surface Critical Phenomena in Three-dimension Percola-tion. Physical Review E. 2005, 71:016117
69 Y. Shen, Z. Yue, M. Li, et al. Enhanced Initial Permeability and Dielectric Constantin a Double Percolating Ni0.3zn0.7fe1.95o4-ni-polymer Composite. AdvancedFunctional Materials. 2005, 15:1100–1103
70 J. C. Wierman. Percolation Threshold Is Not a Decreasing Function of the AverageCoordination Number. Physical Review E. 2002, 66:046125
71 L.J.Adriaanse, J.A.Reedijk, P.A.A.Teunissen, et al. High-dilution Carbon-black/polymer Composites: Hierarchical Percolating Network Derived from Hzto Thz Ac Conductivity. Physical Review Letters. 1997, 78(9):1755
72 S.Rul, F.Lefevre-schlick, E.Capria, et al. Percolation of Single Walled CarbonNanotubes in Ceramic Matrix Nanocomposites. Acta Materialia. 2004, 52:1061–1067
73 C. Zhang, C.-A. Ma, P. Wang, et al. Temperature Demendence of Electrical Re-sistivity for Carbon Black Filled Ultra-high Molecular Weight Polyethylene Com-posites Prepared by Hot Compaction. Carbon. 2005, 43:2544–2553
74 M. Hindermann-Bischoff, F. Ehrbruger-Dolle. Electrical Conductivity of CarbonBlack-polyethylene Composites Experimental Evidence of the Change of ClusterConnectivity in the Ptc Effect. Carbon. 2001, 39:375–382
75 S.Shekhar, V.Prasad, S.V.Subramanyam. Transport Properties of Conducting Am-porphous Carbon-poly(vinyl Chloride) Composite. Carbon. 2006, 44:334–340
76 F. O. Pfeiffer, H. Rieger. Critical Properties of Loop Percolation Models withOptimization Constraints. Physical Review E. 2003, 67:056113
77 H. E. Stanley, A. Coniglio. Flow in Porous Media: The“backbone”Fractal at thePercolation Threshold. Physical Review B. 1984, 29(1):522–524
78 A.R.Day, R.R.Tremblay, A.M.S.Tremblay. Rigi Backbone: A New Geometry forPercolation. Physical Review Letters. 1986, 56(23):2501–2504
79 C. Moukarzel, P.M.Duxbruy. Stressed Backbone and Elasticity of Random CentralForce Systems. Physical Review Letters. 1995, 75(22):4055–4058
80 G. Paul, H. Stanley. Beyond Blobs in Percolation Cluster Structure: The Dis-tribution of 3 Blocks at the Percolation Threshold. Physical Review E. 2002,65:056126
81 G. Paul, S. V. Buldyrev, P. R. Y. L. Nikolay V.Dokholyan, Shlomo Havlin, et al.Dependence of Conductance on Percolation Backbone Mass. Physical Review E.2000, 61(4):3435–3439
82 D. Y. Ki, K. Y. Woo, S. B. Lee. Static and Dynamic Properties of the BackboneNetwork for the Irreversible Kinetic Gelation Model. Physical Review E. 2000,62:821–827
83 H.J.Herrmann, D.C.Hong, H.E.Stanley. Backbone and Elastic Backbone of Perco-lation Clusters Obtained by the New Method of“burning”. J.Phys.A: Math. Gen.1984, 17:L261–L266
84 S.Roux, A.Hansen. A New Algorithm to Extract the Backbone in a Random Re-sistor Network. J.Phys.A: Math. Gen. 1987, 20:L1281–L1285
85 B.B.Mandelbrot. Fractal Geometry of Nature. New york edn. Freeman, 1983
86 K. Falconer. Fractal Geometry-mathematical Foundations and Applications. 2ndedn. John Wiley and Sons Ltd., 2003
87 S.S.Manna, D.Dhar. Fractal Dimension of Backbone of Eden Trees. PhysicalReview E. 1996, 54:R3063–R3066
88 S.S.Albuquerque, F. Moura, M.L.Lyra. Fractality of Largest Clusters and thePercolation Transition in Power-law Diluted Chains. Physical Review E. 2005,72:016116
89 G. Paul, H. Stanley. Fractal Dimension of 3-blocks in Four-, Five, and Six-dimensional Percolatoin Systems. Physical Review E. 2003, 67:026103
90 J. Fournier, G. Boiteus, G. Seytre. Fractal Analysis of the Percolation Network inEpoxy-polypyrrlole Composites. Physical Review B. 1997, 56:5207–5212
91 K.K.Bardhan, R.K.Chakrabarty. Identical Scalling Behavior of Dc and Ac Re-sponse Near the Percolation Threshold in Conductor-insulator Mixtures. PhyscalReview Letters. 1994, 72(7):1068–1071
92 H. G. Yoon, K. W. Kwon, K. Nagata, et al. Changing the Percolation Threshold of aCarbon Black/polymer Composite by a Coupling Treatment of the Black. Carbon.2004, 42:1877–1879
93 P. Potschke, A. R.Bhattacharyya, A. Janke. Carbon Nanotube-filled PolycarbonateComposites Produced by Melt Mixing and Their Use in Blends with Polyethylene.Carbon. 2004, 42:965–969
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